From 2d1ec7cd8fddaedafede5aa146b719d5d7a970a2 Mon Sep 17 00:00:00 2001 From: Dylan Roy Date: Thu, 1 Oct 2026 14:25:20 +0000 Subject: [PATCH 1/6] C8 Theorem L': the radial ledger on the torus with a vanishing remainder Math-#215 (Theorem L) bounds A_r/A_0^ref - 1 by a(R) r^2 + eps_d with an r-independent eps_d, and lists a vanishing-remainder rate as not given (carried among the open C8 items in Math-#193). This packet compares A_r with the contact integrand of the same kernel and proves, for d = 2, 3, every torus side L >= 10, every frame and the Gaussian kernel, on the band b in [0, 1], k in [1/2, 2], 0 < r <= R <= 1/64: -(a_dn r^2 + beta_dn r) <= A_r/A_0 - 1 <= a_up r^2 + beta_up r, with a(R) = #215's Theorem L constants (never below them; equal for R <= 1/2048), beta_2 <= 1.036e-9, beta_3 <= 1.914e-7 on the torus, and beta = 0 exactly for the Gaussian kernel. Corollary L'1 expands ell^(1/3) nu_cand(ell) about the torus's own c_{B,K} with remainder c1 ell^(1/3) + c2 ell^(2/3) (c1/c <= 1.53e-10 and 3.99e-8). Method: interval-coefficient Taylor models with remainder proportional to r^9; the torus part split coefficientwise (|tau_p| <= T0 e^p from the Cauchy estimates of #206); increments against r = 0 for the pin density and the main normalizer term (derivative bound in d = 2, gradient coupling with |grad det^2 1{B>0}| <= ||B||_F^3 and a Sylvester square-root perturbation in d = 3); far set by Markov at order 64. theorem_v.py is new; theorem_b.py, engine_e2.py and engine_e3.py are byte-identical pinned copies from #215 (d77cabf). --check passes under -B -S and -B -O -S; five mutants exit 1 in both modes. Scientific effect NONE. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .../c8-torus-vanishing-remainder.yml | 76 + .../NOTE.md | 276 + .../README.md | 19 + .../RESULTS.json | 18129 ++++++++++++++++ .../SOURCE_FILES.json | 53 + .../SOURCE_MAP.json | 87 + .../engine_e2.py | 1244 ++ .../engine_e3.py | 1709 ++ .../theorem_b.py | 1162 + .../theorem_v.py | 849 + 10 files changed, 23604 insertions(+) create mode 100644 .github/workflows/c8-torus-vanishing-remainder.yml create mode 100644 frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md create mode 100644 frontiers/c8_torus_vanishing_remainder_20261001/README.md create mode 100644 frontiers/c8_torus_vanishing_remainder_20261001/RESULTS.json create mode 100644 frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json create mode 100644 frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_MAP.json create mode 100644 frontiers/c8_torus_vanishing_remainder_20261001/engine_e2.py create mode 100644 frontiers/c8_torus_vanishing_remainder_20261001/engine_e3.py create mode 100644 frontiers/c8_torus_vanishing_remainder_20261001/theorem_b.py create mode 100644 frontiers/c8_torus_vanishing_remainder_20261001/theorem_v.py diff --git a/.github/workflows/c8-torus-vanishing-remainder.yml b/.github/workflows/c8-torus-vanishing-remainder.yml new file mode 100644 index 000000000..db74dc54a --- /dev/null +++ b/.github/workflows/c8-torus-vanishing-remainder.yml @@ -0,0 +1,76 @@ +name: C8 torus vanishing remainder (certified replay) + +'on': + pull_request: + paths: + - 'frontiers/c8_torus_vanishing_remainder_20261001/**' + - '.github/workflows/c8-torus-vanishing-remainder.yml' + - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md' + workflow_dispatch: + +permissions: + contents: read + +jobs: + certified-replay: + runs-on: ubuntu-latest + timeout-minutes: 120 + strategy: + fail-fast: false + matrix: + flags: ['-B -S', '-B -O -S'] + name: certified-replay (${{ matrix.flags }}) + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + persist-credentials: false + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - name: Verify manifest, main-resident pin and packet copies; replay the certificate; reject the mutants + env: + PYTHONDONTWRITEBYTECODE: '1' + FLAGS: ${{ matrix.flags }} + run: | + set -euo pipefail + python -B -S - <<'PY' + import hashlib, json, os, pathlib, subprocess, sys + root = pathlib.Path('frontiers/c8_torus_vanishing_remainder_20261001').resolve() + manifest = json.loads((root / 'SOURCE_FILES.json').read_text())['files'] + present = sorted(p.name for p in root.iterdir()) + if present != sorted([e['path'] for e in manifest] + ['SOURCE_FILES.json']): + raise ValueError('packet tree differs from manifest: ' + repr(present)) + def blob(data): + return hashlib.sha1(b'blob %d\0' % len(data) + data).hexdigest() + for entry in manifest: + path = root / entry['path'] + if pathlib.PurePosixPath(entry['path']).name != entry['path'] or path.is_symlink(): + raise ValueError('flat regular source required: ' + entry['path']) + data = path.read_bytes() + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or blob(data) != entry['git_blob']: + raise ValueError('source identity mismatch: ' + entry['path']) + pins = json.loads((root / 'SOURCE_MAP.json').read_text()) + for entry in pins['sources']: + if entry.get('current_required'): + data = pathlib.Path(entry['path']).read_bytes() + elif entry.get('packet_copy'): + data = (root / entry['packet_copy']).read_bytes() + else: + continue + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or blob(data) != entry['git_blob']: + raise ValueError('pinned source identity mismatch: ' + entry['path']) + flags = os.environ['FLAGS'].split() + script = str(root / 'theorem_v.py') + out = subprocess.run([sys.executable, *flags, script, '--check'], cwd=root, capture_output=True, timeout=3600) + if out.returncode != 0: + raise RuntimeError('--check failed: ' + out.stdout.decode()[-400:] + out.stderr.decode()[-400:]) + print(flags, out.stdout.decode().strip()) + muts = ('torus-const-only', 'exp-radius', 'no-lin', 'no-om-term', 'coupling-half') + procs = [subprocess.Popen([sys.executable, *flags, script, '--mutant', m], cwd=root, + stdout=subprocess.DEVNULL, stderr=subprocess.DEVNULL) for m in muts] + for m, p in zip(muts, procs): + if p.wait(timeout=3600) != 1: + raise RuntimeError('mutant ' + m + ' not rejected') + print(json.dumps({'scientific_effect': 'NONE', 'mutants_rejected': len(muts)})) + PY + git diff --exit-code diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md b/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md new file mode 100644 index 000000000..138521e82 --- /dev/null +++ b/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md @@ -0,0 +1,276 @@ +# C8: the radial ledger on the torus with a vanishing remainder (Theorem L′) + +**Object:** CL-C8-TORUS-VANISHING-REMAINDER-20261001-v1. +**Author:** Anthropic Claude (Claude Code session `session_015wNj8LPTKXsaT68G3DgPPh`), 1 October 2026. +**Disposition:** author-side theorem with certified constants; no review yet. It extends Math-#215 (Theorem L) and closes +the item that #215 lists as not given ("a vanishing-remainder rate"), which Math-#193 (comment 5932553031) records among the +open C8 items as "a vanishing torus remainder". +**Scientific effect:** NONE. No `STATUS`, `PROOF_INDEX`, `GRAPH`, claim, catalog, prize or source-body change. Same GitHub +account as every lane; zero organizational-independence credit. Nothing in Math-#215, #206 or #212 is touched. +**Delivered under:** Dylan Roy's explicit instructions in this session, given after the 2026-09-27 owner stop (quoted in +`SOURCE_MAP.json`, `delivered_under`). `OWNER_STOP.md` addresses Cursor agents and automations and is untouched. + +## 0. Statement + +Notation as in Math-#215 ([B]) and [LP]: `A_r(b, k, u) = 12 pi_r(R; v_r) Z_r / r^2` is the radial ledger of [LP] (10.2) for +the field with kernel `K`, the band is `B x K = [0, 1] x [1/2, 2]`, and `A_0 = lim_{r -> 0} A_r` is the contact integrand +**of the same kernel**. For the Gaussian kernel `A_0 = A_0^ref` (closed form, [B] §1); for the torus kernel `K_L` it is +its own `A_0^(L)`, which [B] only locates within `eps_d` of `A_0^ref`. + +**Theorem L′.** For `d = 2, 3`, every torus side `L >= 10`, every frame, every `b in B`, `k in K`, every `R` in the +table and every `0 < r <= R`, + + -(a_dn(R) r^2 + beta_dn r) <= A_r(b, k, u) / A_0(b, k, u) - 1 <= a_up(R) r^2 + beta_up r. + +The constants are exact rationals in `RESULTS.json` (key `rate_torus`), shown rounded up: + +| `R` | `a_up`, `d = 2` | `a_dn`, `d = 2` | `a_up`, `d = 3` | `a_dn`, `d = 3` | +|---|---|---|---|---| +| `1/4096` | `13.822900` | `3.637234` | `25.666574` | `25.516405` | +| `1/2048` | `13.825546` | `3.643346` | `26.256538` | `26.108063` | +| `1/1024` | `13.830904` | `3.655573` | `27.446809` | `27.301701` | +| `1/512` | `13.841883` | `3.680046` | `29.869058` | `29.730582` | +| `1/256` | `13.864900` | `3.729062` | `34.883096` | `34.757367` | +| `1/128` | `13.915213` | `3.827402` | `45.611669` | `45.508415` | +| `1/64` | `14.033374` | `4.025510` | `70.060722` | `69.981331` | + +with first-order coefficients + + beta_2 <= 1.036 x 10^-9, beta_3 <= 1.914 x 10^-7 (every R in the table). + +**For the Gaussian kernel the first-order coefficients vanish identically** (`rate_reference_kernel`: `beta = 0` exactly), +and the `a` columns are the same: `|A_r / A_0^ref - 1| <= a(R) r^2`, with no additive remainder. + +What changes against [B]: +- **The remainder vanishes.** [B] proves `|A_r / A_0^ref - 1| <= a r^2 + eps_d`, with `eps_2 <= 3.96 x 10^-10` and + `eps_3 <= 1.79 x 10^-8` independent of `r`. Theorem L′ compares `A_r` with the limit of the same kernel, and every + remainder carries a factor `r` or `r^2`. In particular `A_r^(L) = A_0^(L) (1 + O(r))` on the torus with explicit + constants, and `A_r = A_0^ref (1 + O(r^2))` for the Gaussian kernel with no remainder at all. +- **The second-order coefficients are [B]'s.** + - `a(R)` is never below [B]'s Theorem L value. + - It equals [B]'s to the displayed digits for `R <= 1/2048`. + - It exceeds [B]'s by at most `1.1 x 10^-5` for `R <= 1/512` and by at most `7.4 x 10^-4` at `R = 1/64`. The excess comes + from the cross terms and the `r^9` remainders, §§2, 5. +- **The first-order coefficient is a torus artefact.** `beta` is a bound on the torus part of the `r^1` coefficients + (§2, Lemma V2). It is not sharp: the actual torus corrections are of order `e^-50`, while `beta` comes from Cauchy + estimates at `M_1(10) < 4.46 x 10^-15`. + +**Corollary L′1 (the densities with a vanishing remainder).** Let `nu_cand`, `nu_eld` be the compact-window densities of [LP] +Theorem B in the versions (11.2), for the torus kernel or the Gaussian kernel, and `c_{B,K} = |S^(d-1)| int A_0/(3 k^(2/3))` +the leading constant **of the same kernel**. For `0 < ell < R^3 / 2`: + + -(c2_dn ell^(2/3) + c1_dn ell^(1/3)) <= ell^(1/3) nu_cand(ell) - c_{B,K} <= c2_up ell^(2/3) + c1_up ell^(1/3), + ell^(1/3) nu_cand(ell) - C_{B,K}(R) ell <= ell^(1/3) nu_eld(ell) <= ell^(1/3) nu_cand(ell), + +with `C_{B,K}(R)` of [B] Corollary L1, and (`corollary_torus`; ratios to `c_{B,K}`, rounded up): + +| `R` | `c2_up/c`, `d = 2` | `c2_dn/c`, `d = 2` | `c2_up/c`, `d = 3` | `c2_dn/c`, `d = 3` | +|---|---|---|---|---| +| `1/4096` | `6.282558` | `4.464593` | `23.153357` | `22.427357` | +| `1/512` | `6.322008` | `4.505526` | `27.377121` | `26.661848` | +| `1/64` | `6.643378` | `4.836304` | `67.060479` | `66.396981` | + +and `c1/c <= 1.531 x 10^-10` (`d = 2`), `<= 3.985 x 10^-8` (`d = 3`). For the Gaussian kernel `c1 = 0` +(`corollary_reference_kernel`). + +So, on the torus, both densities are `c_{B,K} ell^(-1/3) + O(1) + O(ell^(1/3))` about the **torus** constant, with explicit +constants and a vanishing remainder. [B] gave the same expansion only about `c^ref_{B,K}`, up to `eps'_d ell^(-1/3)`. + +## 1. Where [B]'s remainder came from + +[B]'s `eps_d` collects five things, each `r`-independent: +1. the Taylor-model remainder `e` (rounding, truncated powers `r^j`, `j > 8`, the covariance tail beyond `g_x = 60`); +2. the torus part of each covariance, as one constant `M_1`-bound; +3. the kernel normalization `Theta` in the pin density (`n theta_1 / 2`); +4. in `d = 3`, the Loewner coupling's truncation at radius 12 (`out_D2`, `tail_c`), present even for the Gaussian kernel; +5. the far set `G_0^c`, bounded at its worst radius `R`. + +Theorem L′ removes all five: +1. Every rounding goes into a coefficient radius and every truncation into a remainder proportional to `r^9` (§2). +2. The torus part is split coefficientwise (Lemma V2). Its constant term cancels against `A_0^(L)`, and its higher terms + carry powers of `r`. +3. `Theta` multiplies only increments (Lemma V3). +4. The main normalizer term is compared at `r` and at `0` directly, without truncation (Lemmas V4, V5). +5. The far set is bounded by Markov's inequality at order 64, which gives `O(r^2)` (Lemma V6). + +## 2. Interval-coefficient Taylor models (Lemmas V1, V2) + +**Lemma V1 (the class `TV`).** A `TV` on `[0, R_0]`, `R_0 = 1/64`, is a list of centers `C_j`, radii `D_j` (`j <= 8`) and a +remainder coefficient `E`. It encloses a function `f` if there are **fixed** numbers `c_j in [C_j - D_j, C_j + D_j]` and a +function `rho` with `|rho(r)| <= E r^9` such that `f(r) = sum c_j r^j + rho(r)` for every `r in [0, R_0]`. Then `f(0) = c_0`. +Sums, products, scalar multiples and reciprocals of enclosed functions are enclosed by the operations of `theorem_v.py`: +- **Products.** The coefficient `sum_{i+j=k} a_i b_j` lies in the interval product. For `k <= 8` its center is rounded + with the error added to the radius. For `k > 8` the term is at most `(|S_k| + Rd_k) R_0^(k-9) r^9`. The cross terms + `rho_a g_b`, `rho_b g_a`, `rho_a rho_b` are at most `(E_a |g_b| + E_b |g_a| + E_a E_b R_0^9) r^9`, where `|g| <= sum (|C_j| + D_j) R_0^j`. +- **Reciprocals.** Write `f = c_0 + h`, with `h(0) = 0` and `u = 1/c_0` in an interval. The interval is centered at + `1/C_0`, so that centers stay the reference values. Then `g = h u` has `|g(r)| <= G_1 r`, and + `1/f = u sum_{n <= 8} (-g)^n + u theta (G_1 r)^9/(1 - G_1 R_0)`. The certificate requires `G_1 R_0 < 1/2` and `g(0) = 0`. + +Nothing is assumed about the coefficients beyond their intervals. In particular the same unknown number appears +consistently wherever an enclosed function is reused, because each operation is sound for every admissible choice. + +**Readers.** For `0 < r <= R`, with `H = sum_{j >= 3} (|C_j| + D_j) R^(j-2) + E R^7`: + + f(r) - f(0) in [c_1^lo r + (c_2^lo - H) r^2, c_1^hi r + (c_2^hi + H) r^2]. + +**Lemma V2 (the torus part, coefficientwise).** For the covariance of two preconditioned functionals with weight `w`: +- [E] §2(c) writes the torus part as `sum_{g_x >= w} partial^gamma rho(0) pi_gamma r^(g_x - w)`, with + `|partial^gamma rho(0)| <= g_x! g_y! M_1` (Cauchy estimates on the unit polydisc). +- `|pi_gamma| <= sum |c c'| |d|^m/m!`, with `m = g_x - A`, `|d| <= 1`. +- `g_x!/(g_x - A)! <= w^A e^(g_x - w)`, because `A log x <= w(x - 1)` for `x = g_x/w >= 1`. + +So the coefficient of `r^p` in the torus part is at most `T_0 e^p`, with `T_0 = sum |c c'| g_y! max(w, 1)^A M_1`, the +quantity [E] uses at `p`-sum level. The certificate adds: +- radii `T_0 e^p` to the coefficients `p = 0, ..., 8`; +- `T_0 e^9 / (1 - e R_0) r^9` to the remainder. + +The reference series gives the centers exactly. The truncation tail beyond `g_x = 60` is at most +`2 |c c'| |h(g_y)| 61^A r^(61 - w)` for `r <= 1/64` ([E] §2(b)), so it goes into the `r^9` remainder (`61 - w >= 9` is +checked). `M_1(10) < 4.46 x 10^-15` (`d = 2`) and `< 1.94 x 10^-13` (`d = 3`) decrease in `L`, so the models hold for every +`L >= 10` and every frame; with `M_1 = 0` they are the Gaussian kernel's. + +The six pins, the targets of [B] §2 and Gaussian elimination without pivoting give `TV`s for `det S_r`, the energies `e_bb`, +`e_bk`, `e_kk`, the regression coefficients `alpha`, `beta` of the transverse Hessian and of `Omega'`, their covariances, and in +`d = 3` the regression of `Omega'` on `B = -A`. The certificate checks that the centers are [B]'s reference laws. In +particular the law of the transverse Hessian is flat in `r` ([B] Lemma 1): every center of `alpha_A + 1`, `beta_A` and +`Cov(A) - Cov(GOE)` vanishes to `10^-60`, so on the torus only radii survive. + +## 3. The pin density (Lemma V3) + +With `S_r` the unnormalized pin covariance and `Theta in [1, 1 + theta_1]` the normalization, `pi_r ∝ det(S_r/Theta)^(-1/2) +exp(-Theta E_r/2)`. So + + log(pi_r / pi_0) = -(1/2) log(det S_r / det S_0) - Theta (E_r - E_0)/2. + +`Theta^(n/2)` cancels and `Theta` multiplies only the increment `E_r - E_0`. With `x = (det S_r - det S_0)/det S_0` and +`|x| <= x_m < 1/2`: +- `-(1/2) log(1 + x) <= max(0, -x)/(2(1 - x_m))`; +- `(1/2) log(1 + x) <= max(0, x)/2`. + +The increments of `det S` and of the energies are read off the `TV`s as above. On a box with `b, k >= 0` the energy weights +`b^2`, `2bk`, `k^2` are nonnegative and the bounds are taken coefficientwise. There is no constant term: +`log(pi_r/pi_0) in [-(dn_a r^2 + dn_b r), up_a r^2 + up_b r]`. + +## 4. The normalizer (Lemmas V4–V6) + +As in [B] §4 (exact expansion of `W_r = r^2 P1 P2` on the typed set, the three first-order cancellations, Hölder for the +second-order terms): + + Z_r / (36 k^2 r^2) = m_r + (first-order term) + (second-order terms), + +where `m_r = E_r[lambda_+^2]` (`d = 2`) or `E_r[det(B)^2 1{B > 0}]` (`d = 3`), under the law at `r`, and `m_0` is the same +under the law at `0`. Then `A_r / A_0 = (pi_r/pi_0) (Z_r / (36 k^2 r^2 m_0))`, and every term below is divided by `m_0`. + +**Lemma V4 (`d = 2`, main term).** `lambda ~ N(mu, sigma^2)` with `mu = -(b alpha_w + k beta_w)`, `sigma^2 = C_ww / Theta`, and +`F(mu, sigma) = E[(mu + sigma z)_+^2]`. Then `dF/dmu = 2 E[lambda_+]` and `dF/d(sigma^2) = Phi(mu/sigma) <= 1`. Along the +segment from `(mu_0, sigma_0^2)` to `(mu_r, sigma_r^2)`, which stays in the band's range, + + |m_r - m_0| <= 2 F_1,max |mu_r - mu_0| + |C_ww(r) - C_ww(0)|, m_0 >= F_min. + +Both increments are `TV` increments. For the Gaussian kernel they vanish ([B] Lemma 1). + +**Lemma V5 (`d = 3`, main term).** In natural coordinates `(B_11, B_22, sqrt2 B_12)` the Frobenius norm is Euclidean. Couple +`B_r = mu_r + C_r^(1/2) Z` and `B_0 = mu_0 + C_0^(1/2) Z`. +- `g(B) = det(B)^2 1{B > 0}` is continuous, since `det B = 0` on the boundary of the cone. +- `g` is piecewise smooth along segments, with `|grad g| = 2 |det B| ||adj B||_F <= ||B||_F^3` (`2 x 2`: + `|det B| <= ||B||_F^2/2`, `||adj B||_F = ||B||_F`). + +Hence + + |m_r - m_0| <= E[max(||B_r||, ||B_0||)^3 ||B_r - B_0||] <= sqrt2 ||B||_8^3 (||mu_r - mu_0|| + ||C_r^(1/2) - C_0^(1/2)||_F). + +Square roots satisfy `||C_r^(1/2) - C_0^(1/2)||_F <= ||C_r - C_0||_F / (2 sqrt(lambda_min))`. Proof: +- `X = C_r^(1/2) - C_0^(1/2)` solves `C_r^(1/2) X + X C_0^(1/2) = C_r - C_0`. +- In the eigenbases, `|X_ij| (s_i + t_j) = |(C_r - C_0)_ij|`. +- Here `lambda_min` is the Gershgorin floor over the band. + +`||B||_8` is the band's Frobenius moment bound, and `m_0 >= m_3(b) rho_lo`, with `rho_lo` from [B]'s Loewner coupling over +`[0, R]` (which contains `r = 0`). For the Gaussian kernel the increments vanish. + +**First-order terms.** For `d = 2`, the term is `-36 k^2 r E[lambda_+ om]`. For `d = 3`, it is `-36 k^2 r E[D tr(A* Omega') 1]`. +- [B] writes each through the regression, with coefficients `mu_om`, `bw_om` (`d = 2`) and `beta(r)`, `mu_Omega`, `DG` + (`d = 3`). +- Each `TV` coefficient satisfies `|t(r)| <= c_0 + s r`. +- The explicit factor `r` turns `c_0` (a torus radius; `0` for the Gaussian kernel) into a contribution to `beta`. It turns + `s` into a contribution to `a`. +- In `d = 3`, `r beta(r) - r^2/2 = c_0 r + (c_1 - 1/2) r^2 + ...`, with `c_1` centered at `1/2`. + +**Second-order terms.** These are [B]'s, with its interval band laws, divided by `m_0` instead of the reference moment: +- in `d = 2`: the signed `kappa` term, `T_4`, `T_5`, `T_6` and the thin set; +- in `d = 3`: `U_2`, ..., `U_10` and the thin set. + +**Lemma V6 (the far set).** `G_0^c = {|T| > 3k/(2r)} u {|tau| > 3k/(2r^2)}` enters through +`E[W_r 1_(G_0^c)]/r^2 <= P1n P2n P(G_0^c)^(1/2)` ([B] §5). By Markov's inequality at order `p = 64`: + + P(G_0^c) <= (2 ||T||_64/(3 k_0))^64 R^60 r^4 + (2 ||tau||_64/(3 k_0))^64 R^124 r^4, + +so this term is `O(r^2)`. + +## 5. Assembly + +`(1 + p)(1 + z) - 1 <= p + z + p z(R)` and `1 - (1 - p)(1 - z) <= p + z`, with `exp(x) - 1 <= x exp(x(R))` for the pin factor. +This gives Theorem L′ on each of the 96 boxes `[i/8, (i+1)/8] x [1/2 + j/8, 1/2 + (j+1)/8]` (as in [B]) and each `R`. + +**Corollary L′1.** As in [B] Corollary L1: +- `ell^(1/3) nu_cand(ell) = |S| int A_r / (3 k^(2/3))` with `r = (ell/k)^(1/3) <= R`. +- Integrate Theorem L′ box by box: `a r^2` gives the weight `int A_0 / (3 k^(4/3))`, and `beta r` gives `int A_0 / (3 k)`. +- The weights of the torus `A_0^(L)` are at most `(1 + eps_0)` times the reference weights. Here `eps_0` is [B]'s + `|A_0/A_0^ref - 1|` bound, recomputed by the pinned `theorem_b.py` at `R = 1/4096`: + `eps_0 <= 3.95 x 10^-10` (`d = 2`), `1.78 x 10^-8` (`d = 3`). +- The elder line is [B]'s. + +## 6. Verification + +- `python3 -B -S theorem_v.py --check` recomputes everything and compares it with `RESULTS.json` exactly: + - the `TV` laws of both kernels; + - the 2 x 7 x 96 box bounds per kernel; + - the reference integrals and the `r^1` weights; + - `eps_0`; + - the corollaries. + + It takes about 3.5 minutes on one core and passes under `-B -S` and `-B -O -S`. +- Five mutants, each of which must exit 1: + - `torus-const-only`: the torus part only in the constant coefficient; + - `exp-radius`: `e^p` replaced by `1` in the torus radii; + - `no-lin`: drop the `r^1` coefficients; + - `no-om-term`: drop the first-order terms; + - `coupling-half`: halve Lemma V5's constant. +- The workflow `.github/workflows/c8-torus-vanishing-remainder.yml` checks: + - the manifest; + - the main-resident pin [LP]; + - the packet copies [B], [E], [E3], byte for byte; + - both modes; + - the mutants; + - a clean tree. + +**Consistency with [B].** With `beta` set aside, `a(R)` reproduces [B]'s Theorem L table, as in §0. The `c2/c` ratios +reproduce [B]'s Corollary L1 to within `5 x 10^-4`. Both come from an independent assembly, using increments rather than +deviations from `A_0^ref`. + +## 7. What this does not do + +- **Still the compact window `B x K`, `L >= 10`, `d = 2, 3`, `R <= 1/64`.** These are [B]'s ranges. +- **`beta` is not the true first-order coefficient.** It bounds the torus part of the `r^1` coefficients through + `M_1`. Whether the torus `A_r` has a nonzero `r^1` term at all (of order `e^-50`) is not decided. +- **Consumed, not reviewed:** + - [LP]'s sections 8–12, as in [B]; + - [B]'s §§2–5 algebra (exact expansions, first-order cancellations, Hölder bounds, Loewner coupling for the moment + bounds); + - the interval band laws and transcendentals of [E], [E3]. + + [B] is author-side with an OpenAI amendment ACCEPT (5378815703). [E] and [E3] are author-side and unmerged. +- No `C`, `r_*`, `z_*` beyond [B], #206, #212 and #195. No new count, moment or elder statement beyond Corollary L′1. +- Same GitHub account as every lane; zero organizational-independence credit. **I will not merge.** + +## 8. Provenance + +- **Sources** (`SOURCE_MAP.json`): + - [LP] `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` (on `main`, verified by the workflow); + - [B] Math-#215 `theorem_b.py` at `d77cabf` (packet copy `theorem_b.py`) and its `NOTE.md` and `RESULTS.json` (provenance); + - [E] Math-#206 `theorem_a.py` (packet copy `engine_e2.py`); + - [E3] Math-#212 `theorem_a3.py` (packet copy `engine_e3.py`). +- **Files:** + - `NOTE.md`, `README.md`; + - `theorem_v.py` (new); + - the copies `theorem_b.py`, `engine_e2.py`, `engine_e3.py`; + - `RESULTS.json`, `SOURCE_MAP.json`, `SOURCE_FILES.json`. diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/README.md b/frontiers/c8_torus_vanishing_remainder_20261001/README.md new file mode 100644 index 000000000..c7aed0099 --- /dev/null +++ b/frontiers/c8_torus_vanishing_remainder_20261001/README.md @@ -0,0 +1,19 @@ +# C8: the radial ledger on the torus with a vanishing remainder + +Theorem L′ (`NOTE.md`): for the torus kernel (every side `L >= 10`, every frame) and the Gaussian kernel, `d = 2, 3`, on the +band `b in [0, 1]`, `k in [1/2, 2]`, + + -(a_dn r^2 + beta_dn r) <= A_r / A_0 - 1 <= a_up r^2 + beta_up r (0 < r <= R <= 1/64), + +with `A_0` the contact integrand of the same kernel. The `a(R)` are Math-#215's Theorem L constants (13.82 / 3.64 in the +plane, 25.67 / 25.52 in `d = 3` at `R = 1/4096`); `beta_2 <= 1.04e-9`, `beta_3 <= 1.92e-7` on the torus and `beta = 0` for +the Gaussian kernel. This removes the `r`-independent `eps_d` of Math-#215. Corollary L′1 gives `ell^(1/3) nu(ell)` about +the torus's own `c_{B,K}` with remainder `c1 ell^(1/3) + c2 ell^(2/3)`. + +Replay (standard library, about 3.5 minutes): + + python3 -B -S theorem_v.py --check + python3 -B -S theorem_v.py --mutant torus-const-only # must exit 1 (also exp-radius, no-lin, no-om-term, coupling-half) + +`theorem_b.py`, `engine_e2.py` and `engine_e3.py` are byte-identical copies of Math-#215's files (and of the engines of +Math-#206, #212), pinned in `SOURCE_MAP.json`. 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"AUTHOR_SIDE_THEOREM_WITH_CERTIFIED_CONSTANTS", + "source_repository": "d6g8k5htny-coder/Math-", + "main_at_creation": "c2f1270d28192b3c06dd31f728ebf04d03d4246e", + "delivered_under": "explicit owner instructions of Dylan Roy in Claude Code session session_015wNj8LPTKXsaT68G3DgPPh, given after OWNER_STOP (2026-09-27): 2026-09-28 \"you have my full permisions and approval to make the proper changes and uploads\"; 2026-09-30 \"Work on closing as many mathematical items as possible while working with and supporting the other models in the GitHub. Also advance the mathematics in every place that's justified based on what we currently know. Ultra tokens, ultra depth @GitHub @Superpowers\" (repeated 2026-09-30 as \"Continue to Work on closing ...\"). Pull requests only, never merged by their author; no register, GRAPH, STATUS, lemma_closed, prize or premise edits. OWNER_STOP.md addresses Cursor agents and automations and is untouched.", + "note": "current_required sources ([LP]) are verified on the tree by the workflow; packet_copy sources ([B], [E], [E3]) are verified against the packet copies byte for byte; [B-NOTE] and [B-RESULTS] are read-only provenance pinned by identity", + "sources": [ + { + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "id": "LP", + "commit": "c2f1270d28192b3c06dd31f728ebf04d03d4246e", + "current_required": true, + "on_main": true, + "role": "sections 1, 3, 8-12: the model, the preconditioned pins, the radial ledger A_r = 12 pi_r Z_r / r^2 (10.2) and its limit A_0 (10.3), the lifetime pushforward (11.1)-(11.3); section 11 records that A_r -> A_0 was not assigned a numerical rate. 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The proof bounds +Q^W(G_r^c) = E_Q[W_r 1{G_r^c}] / Z_r from above by an exact Gaussian regression near r = 0: + + * covariances of the preconditioned pin and target functionals are exact power series in r, obtained from the Taylor + coefficients of the kernel at 0 (every negative power cancels identically, checked); the torus remainder is bounded + uniformly in r and in the frame by Cauchy estimates on a polydisc; + * the failure event is reduced to {lambda <= (4/(3k)) r M_3^2} u {r M_4 > 3k/10}; M_3 is bounded through exact averaging + identities of the pinned jets and M_4 through the Taylor series at the midpoint; + * the main term is integrated in lambda = -f_yy(M) against its Gaussian density after an exact regression on w = f_yy(M); + the remaining expectations are closed-form incomplete Gaussian moments plus Hoelder-controlled perturbations; + * the normalizer Z_r is bounded below in closed form; the rare branches are Gaussian tails. +All arithmetic is exact rational interval arithmetic (outward rounding to 2^-160); standard library only. +""" +import argparse +import json +import math +import multiprocessing +import os +import sys +from fractions import Fraction as Fr + +HERE = os.path.dirname(os.path.abspath(__file__)) +MUTANTS = ('no-delta', 'cap-half', 'no-zfloor-correction', 'no-om-term') +MUT = None +PBITS = 160 +L_MIN = 10 +NX = 60 # Taylor order in x of the covariance series (exact); the rest is a bounded tail +NB = 24 # truncation order of the Taylor tails of the 4-jets at the midpoint (the rest is bounded) +X0 = Fr(1, 64) # default threshold for the quadratic near-branch analysis (records use COMBOS) +THETA = Fr(1, 32) # default splitting parameter in the max-moment lemma (records use COMBOS) +THETA_E = Fr(1, 16) # splitting parameter for the (r nu + v)^2 factor +QS = (2, 4, 8, 16) # Hoelder exponents of the rare branches: E[W 1_E] <= ||W||_q P(E)^(1 - 1/q), smallest bound used +R_STARS = (Fr(1, 4096), Fr(1, 2048), Fr(1, 1024), Fr(1, 512), Fr(1, 256), Fr(1, 128), Fr(1, 64)) +BAND_B = (Fr(0), Fr(1)) +BAND_K = (Fr(1, 2), Fr(2)) + + +def require(cond, msg): + if not cond: + print(json.dumps({'check': 'FAIL', 'reason': msg, 'mutant': MUT})) + sys.exit(1) + + +# ----------------------------------------------------------------------------- rational intervals (from Math-#195 / #203) +def rdown(x): + return Fr(x.numerator * (1 << PBITS) // x.denominator, 1 << PBITS) + + +def rup(x): + return Fr(-((-x.numerator) * (1 << PBITS) // x.denominator), 1 << PBITS) + + +class IV: + __slots__ = ('lo', 'hi') + + def __init__(self, lo, hi=None): + lo = Fr(lo) + hi = lo if hi is None else Fr(hi) + if hi < lo: + raise ValueError('empty interval') + self.lo, self.hi = rdown(lo), rup(hi) + + def __repr__(self): + return 'IV(%s, %s)' % (float(self.lo), float(self.hi)) + + def __add__(self, o): + o = o if isinstance(o, IV) else IV(o) + return IV(self.lo + o.lo, self.hi + o.hi) + __radd__ = __add__ + + def __neg__(self): + return IV(-self.hi, -self.lo) + + def __sub__(self, o): + return self + (-(o if isinstance(o, IV) else IV(o))) + + def __rsub__(self, o): + return IV(o) - self + + def __mul__(self, o): + o = o if isinstance(o, IV) else IV(o) + c = (self.lo * o.lo, self.lo * o.hi, self.hi * o.lo, self.hi * o.hi) + return IV(min(c), max(c)) + __rmul__ = __mul__ + + def inv(self): + if self.lo <= 0 <= self.hi: + raise ZeroDivisionError('interval contains zero') + return IV(1 / self.hi, 1 / self.lo) + + def __truediv__(self, o): + o = o if isinstance(o, IV) else IV(o) + return self * o.inv() + + def __rtruediv__(self, o): + return IV(o) * self.inv() + + def sq(self): + if self.lo >= 0: + return IV(self.lo * self.lo, self.hi * self.hi) + if self.hi <= 0: + return IV(self.hi * self.hi, self.lo * self.lo) + return IV(0, max(self.lo * self.lo, self.hi * self.hi)) + + def meet(self, o): + lo, hi = max(self.lo, o.lo), min(self.hi, o.hi) + if hi < lo: + raise ValueError('inconsistent enclosures') + return IV(lo, hi) + + def width(self): + return self.hi - self.lo + + def mid(self): + return (self.lo + self.hi) / 2 + + def contains(self, x): + return self.lo <= x <= self.hi + + +def isqrt_iv(x): + """Enclosure of sqrt(x) for an interval x with x.lo > 0.""" + if x.lo <= 0: + raise ValueError('sqrt of interval reaching zero: %r' % x) + + def lo_root(v): + n = v.numerator * (1 << (2 * PBITS)) // v.denominator + s = math.isqrt(n) + return Fr(s, 1 << PBITS) + + def hi_root(v): + n = -((-v.numerator * (1 << (2 * PBITS))) // v.denominator) + s = math.isqrt(n) + 1 + return Fr(s, 1 << PBITS) + return IV(lo_root(x.lo), hi_root(x.hi)) + + +def exp_neg_point(t): + """Enclosure of exp(-t) for a rational t >= 0: halve t to t <= 1/2, evaluate the alternating series in fixed point + (ulp = 2^-(PBITS+40)) with an integer bound on the accumulated rounding error, then square back.""" + t = Fr(t) + m = 0 + while t > Fr(1, 2): + t /= 2 + m += 1 + P2 = PBITS + 40 + ONE = 1 << P2 + X = t.numerator * ONE // t.denominator # X / ONE <= t < (X + 1) / ONE + # series at x_lo = X / ONE: exp(-x_lo) = sum (-1)^n x_lo^n / n! (alternating, terms decreasing for x_lo <= 1/2) + S = ONE + T = ONE + E = 0 # error bound of T in ulps + Etot = 0 + n = 0 + while T > 0: + n += 1 + T = (T * X) // (ONE * n) + E = (E * X) // (ONE * n) + 2 + S += (-1) ** n * T + Etot += E + Etot += E + 1 # first omitted exact term is below (T_n + E) <= E + 1 ulps + # exp(-t) in [exp(-(X + 1)/ONE), exp(-X/ONE)] and exp(-(X + 1)/ONE) >= exp(-X/ONE) - 1/ONE (Lipschitz 1) + v = IV(Fr(S - Etot - 1, ONE), Fr(S + Etot, ONE)) + for _ in range(m): + v = v.sq() + return v + + +def exp_neg(t): + """exp(-t) on an interval t with t.lo >= 0 (monotone decreasing).""" + return IV(exp_neg_point(t.hi).lo, exp_neg_point(t.lo).hi) + + +def _atan_inv(n): + """Enclosure of arctan(1/n), n >= 2, alternating series.""" + x = Fr(1, n) + s = Fr(0) + term = x + k = 0 + while True: + s += term / (2 * k + 1) + k += 1 + term = term * (-x * x) + if abs(term) / (2 * k + 1) < Fr(1, 1 << (PBITS + 8)): + break + return IV(s - abs(term), s + abs(term)) + + +PI = 4 * (4 * _atan_inv(5) - _atan_inv(239)) +SQRT2 = isqrt_iv(IV(2)) +SQRT2PI = isqrt_iv(2 * PI) + + +def phi_point(x): + """Enclosure of Phi(x) = P(N(0,1) <= x) for a rational x with |x| <= 12, from the series + Phi(x) = 1/2 + (1/sqrt(2 pi)) sum_n (-1)^n x^(2n+1) / (2^n n! (2n+1)), evaluated in fixed point (ulp = 2^-(PBITS+40)) + with an explicit integer bound E on the accumulated rounding error (in ulps) and the alternating-tail bound.""" + x = Fr(x) + if x < 0: + p = phi_point(-x) + return IV(1 - p.hi, 1 - p.lo) + if x > 12: + raise ValueError('phi_point range') + P2 = PBITS + 40 + ONE = 1 << P2 + X = x.numerator * ONE // x.denominator # x_lo = X / ONE <= x < (X + 1) / ONE ; |Phi(x) - Phi(x_lo)| <= 1 ulp + if X == 0: # 0 <= x < 1 ulp: Phi(x) in [1/2, 1/2 + ulp] + return IV(Fr(1, 2), Fr(1, 2) + Fr(1, ONE)) + X2u = (X + 1) * (X + 1) # upper bound of x^2 ONE^2 + S = 0 + T = X # T_n ~ x^(2n+1) / (2^n n!) in ulps + E = 1 # error bound of T_n in ulps + Etot = 1 # error of S in ulps (starts with the x rounding) + n = 0 + while T > 0: + S += (-1) ** n * (T // (2 * n + 1)) + Etot += E + 1 + n += 1 + T = (T * X * X) // (ONE * ONE * 2 * n) + E = (E * X2u) // (ONE * ONE * 2 * n) + 2 + # once T == 0 in fixed point the remaining terms are each below 1 ulp; if x^2 >= 2 n they are still growing in + # exact arithmetic, so continue counting until they decrease: bound the tail by the first omitted exact term + # <= (T_n + E) ulps <= (E + 1) ulps when T_n == 0 ... but growth may resume; require the decreasing regime. + if x * x > 2 * n: + raise ValueError('series terminated before the decreasing regime') + Etot += E + 1 + core = IV(Fr(S - Etot, ONE), Fr(S + Etot, ONE)) / SQRT2PI + return IV(Fr(1, 2) + core.lo, Fr(1, 2) + core.hi) + + +def Phi(x): + """Phi on an interval (monotone); arguments clipped to [-12, 12] (clipping only widens).""" + lo = min(max(x.lo, Fr(-12)), Fr(12)) + hi = max(min(x.hi, Fr(12)), Fr(-12)) + if x.lo < -12: + plo = Fr(0) + else: + plo = phi_point(lo).lo + if x.hi > 12: + phi_ = Fr(1) + else: + phi_ = phi_point(hi).hi + return IV(plo, phi_) + + + + +def phi_pt(x): + """phi(x) for rational x (interval).""" + x = Fr(x) + return exp_neg_point(x * x / 2) / SQRT2PI + + +def Phibar_pt(x): + """1 - Phi(x) for rational x (interval).""" + p = phi_point(Fr(x)) if abs(Fr(x)) <= 12 else None + if p is None: + if x > 0: + # Mills: Phibar(x) <= phi(x)/x + return IV(0, (phi_pt(x) / x).hi) + return IV(1 - (phi_pt(-x) / (-x)).hi, 1) + return IV(1 - p.hi, 1 - p.lo) + + +def upper_moments(x, J): + """[M_0(x), ..., M_J(x)], M_j(x) = integral_x^inf t^j phi(t) dt, for rational x >= 0 (intervals).""" + x = Fr(x) + require(x >= 0, 'upper moment at negative point') + ph = phi_pt(x) + M = [Phibar_pt(x), ph] + for j in range(2, J + 1): + M.append(ph * x ** (j - 1) + M[j - 2] * (j - 1)) + return M + + +def dfact(n): + """(n)!! for n >= -1.""" + return math.prod(range(n, 0, -2)) if n > 0 else 1 + + +def rup_rel(x, bits=200): + """Round a rational x >= 0 up to a dyadic with `bits` significant bits (upper bounds stay upper bounds).""" + x = Fr(x) + if x <= 0: + return x + e = x.numerator.bit_length() - x.denominator.bit_length() + sh = bits - e + if sh >= 0: + return Fr(-((-x.numerator << sh) // x.denominator), 1 << sh) + return Fr(-((-x.numerator) // (x.denominator << -sh)) << -sh) + + +def sqrt_up(x): + """Rational upper bound of sqrt(x), x >= 0 rational, relative error below 2^-200 (no absolute rounding floor).""" + x = Fr(x) + if x == 0: + return Fr(0) + n, d = x.numerator, x.denominator + nd = n * d + m = max(0, (420 - nd.bit_length()) // 2 + 1) + return rup_rel(Fr(math.isqrt(nd << (2 * m)) + 1, d << m)) + + +def iroot(x, p): + """Upper bound (rational) of x^(1/p) for rational x >= 0 and p a power of two.""" + v = Fr(x) + while p > 1: + v = sqrt_up(v) + p //= 2 + return v + + +def pow2(p): + q = 1 + while q < p: + q *= 2 + return q + + +ZN = {} + + +def znorm(p): + """Upper bound of ||N(0,1)||_p for p a power of two (p = 1 uses sqrt(2/pi) < 4/5).""" + if p == 1: + return Fr(4, 5) + if p not in ZN: + ZN[p] = iroot(Fr(dfact(p - 1)), p) + return ZN[p] + + +def habs_moment(j): + """E|zeta|^j (interval).""" + if j % 2 == 0: + return IV(dfact(j - 1)) + return SQRT2 / isqrt_iv(PI) * (2 ** ((j - 1) // 2) * math.factorial((j - 1) // 2)) + + +# ----------------------------------------------------------------------------- functionals and exact covariance series +HALF = Fr(1, 2) +XM, XS, X0PT = -HALF, HALF, Fr(0) + + +def term(c, p, a, b, x0): + """c r^p d_x^a d_y^b f(x0 r, 0).""" + return (Fr(c), p, a, b, Fr(x0)) + + +def functionals(): + pins = { + 'p1': [term(HALF, 0, 0, 0, XM), term(HALF, 0, 0, 0, XS)], + 'p2': [term(-1, -1, 0, 0, XM), term(1, -1, 0, 0, XS)], + 'p3p': [term(6, -2, 1, 0, XM), term(6, -2, 1, 0, XS), term(12, -3, 0, 0, XM), term(-12, -3, 0, 0, XS)], + 'p4': [term(-1, -1, 1, 0, XM), term(1, -1, 1, 0, XS)], + 'q1': [term(HALF, 0, 0, 1, XM), term(HALF, 0, 0, 1, XS)], + 'q2': [term(-1, -1, 0, 1, XM), term(1, -1, 0, 1, XS)], + } + targets = { + 'w': [term(1, 0, 0, 2, XM)], + 'v': [term(1, -1, 1, 1, XM), term(1, -2, 0, 1, XM), term(-1, -2, 0, 1, XS)], + 'om': [term(1, -1, 0, 2, XS), term(-1, -1, 0, 2, XM)], + 'Y': [term(1, 0, 0, 3, X0PT)], + 'T': [term(1, -2, 2, 0, XM), term(4, -3, 1, 0, XM), term(2, -3, 1, 0, XS), term(6, -4, 0, 0, XM), term(-6, -4, 0, 0, XS)], + 'tau': [term(1, -3, 2, 0, XS), term(-1, -3, 2, 0, XM), term(-6, -4, 1, 0, XM), term(-6, -4, 1, 0, XS), + term(-12, -5, 0, 0, XM), term(12, -5, 0, 0, XS)], + 'nu': [term(1, -2, 1, 1, XM), term(1, -2, 1, 1, XS), term(2, -3, 0, 1, XM), term(-2, -3, 0, 1, XS)], + 'X40': [term(1, 0, 4, 0, X0PT)], 'X31': [term(1, 0, 3, 1, X0PT)], 'X22': [term(1, 0, 2, 2, X0PT)], + 'X13': [term(1, 0, 1, 3, X0PT)], 'X04': [term(1, 0, 0, 4, X0PT)], + } + return pins, targets + + +PIN_NAMES = ('p1', 'p2', 'p3p', 'p4', 'q1', 'q2') +TARGET_NAMES = ('w', 'v', 'om', 'Y', 'T', 'tau', 'nu', 'X40', 'X31', 'X22', 'X13', 'X04') +JET4 = ('X40', 'X31', 'X22', 'X13', 'X04') +JET4_IDX = {'X40': (4, 0), 'X31': (3, 1), 'X22': (2, 2), 'X13': (1, 3), 'X04': (0, 4)} + + +def weight(F): + """Scaling weight a - (power of r), common to all terms of a preconditioned functional.""" + ws = {t[2] - t[1] for t in F} + require(len(ws) == 1, 'functional not homogeneous') + return ws.pop() + + +def hcoef(n): + """d^n/dx^n exp(-x^2/2) at 0.""" + if n % 2: + return 0 + return (-1) ** (n // 2) * dfact(n - 1) + + +PAIR_CACHE = {} + + +def pair_series(F, G, key): + """Exact data of Cov_K(F, G) = sum_gamma kappa_gamma pi_gamma r^(gx - wF - wG): + returns (reference series {power: rational}, weight w, list of (|c c'|, a+a', b+b') for the remainder bounds).""" + if key in PAIR_CACHE: + return PAIR_CACHE[key] + wF, wG = weight(F), weight(G) + w = wF + wG + pi = {} + absterms = [] + for (cf, pf, af, bf, xf) in F: + for (cg, pg, ag, bg, xg) in G: + gy = bf + bg + sgn = (-1) ** (ag + bg) + d = xf - xg + absterms.append((abs(cf * cg), af + ag, gy)) + for gx in range(af + ag, NX + 1): + m = gx - af - ag + val = cf * cg * sgn * d ** m / math.factorial(m) + if val != 0: + pi[(gx, gy)] = pi.get((gx, gy), Fr(0)) + val + for (gx, gy), v in pi.items(): + require(v == 0 or gx - w >= 0, 'negative power of r in a covariance (regularity fails)') + ser = {} + for (gx, gy), v in pi.items(): + kap = hcoef(gx) * hcoef(gy) + if kap and v: + ser[gx - w] = ser.get(gx - w, Fr(0)) + kap * v + out = ({p: c for p, c in ser.items() if c != 0}, w, absterms) + PAIR_CACHE[key] = out + return out + + +def torus_M1(L): + """Bound of |sum_{n != 0} G(z + p_n)| on the polydisc |z_1|, |z_2| <= 1 (every frame, |p_n| = |n| L): + exp(1) * sum_{j >= 1} 8 j exp(-(j L - 3/2)^2 / 2).""" + L = Fr(L) + tot = Fr(0) + for j in range(1, 6): + tot += 8 * j * exp_neg_point((j * L - Fr(3, 2)) ** 2 / 2).hi + tail = 8 * 7 * exp_neg_point((6 * L - Fr(3, 2)) ** 2 / 2).hi * 4 # j >= 6, very crude + return Fr(27183, 10000) * (tot + tail) + + +def theta_minus_one(L): + L = Fr(L) + tot = Fr(0) + for j in range(1, 6): + tot += 8 * j * exp_neg_point((j * L) ** 2 / 2).hi + return tot + 8 * 7 * exp_neg_point((6 * L) ** 2 / 2).hi * 4 + + +def cov_band(F, G, key, r0, r1, M1): + """Enclosure of the covariance (unnormalized torus kernel sum_n G(z + p_n)) of F and G, for r in [r0, r1].""" + ser, w, absterms = pair_series(F, G, key) + val = IV(0) + for p, c in ser.items(): + val = val + IV(r0 ** p, r1 ** p) * c + A = max((t[1] for t in absterms), default=0) + # Taylor truncation tail of the reference kernel (orders gx > NX) + tail = Fr(0) + for (acc, aa, gy) in absterms: + tail += acc * abs(hcoef(gy)) + tail *= 2 * Fr(NX + 1) ** A * r1 ** (NX + 1 - w) + # torus remainder: Cauchy estimates |rho_gamma| <= gx! gy! M1 on the unit polydisc + tor = Fr(0) + if True: + for (acc, aa, gy) in absterms: + tor += acc * math.factorial(gy) * Fr(max(w, 1)) ** aa + tor = tor * M1 / (1 - Fr(27183, 10000) * r1) + err = tail + tor + return val + IV(-err, err) + + +# ----------------------------------------------------------------------------- interval linear algebra +def solve_iv(A, B): + """Interval Gaussian elimination with partial pivoting on midpoints: X[c] solves A X[c] = B[c].""" + n = len(A) + M = [[A[i][j] for j in range(n)] + [B[c][i] for c in range(len(B))] for i in range(n)] + for col in range(n): + p = max(range(col, n), key=lambda i: abs(M[i][col].mid())) + M[col], M[p] = M[p], M[col] + piv = M[col][col] + for i in range(col + 1, n): + f = M[i][col] / piv + for j in range(col, len(M[i])): + M[i][j] = M[i][j] - f * M[col][j] + X = [] + for c in range(len(B)): + x = [None] * n + for i in reversed(range(n)): + s = M[i][n + c] + for j in range(i + 1, n): + s = s - M[i][j] * x[j] + x[i] = s / M[i][i] + X.append(x) + return X + + +def cholesky3(C): + """Interval Cholesky of a 3 x 3 SPD interval matrix.""" + L = [[IV(0)] * 3 for _ in range(3)] + for i in range(3): + for j in range(i + 1): + s = C[i][j] - sum((L[i][l] * L[j][l] for l in range(j)), IV(0)) + if i == j: + require(s.lo > 0, 'Cholesky pivot not positive') + L[i][j] = isqrt_iv(s) + else: + L[i][j] = s / L[j][j] + return L + + +def absup(x): + return max(abs(x.lo), abs(x.hi)) + + +# ----------------------------------------------------------------------------- the conditional law on an r-band +class Law: + pass + + +def band_law(r0, r1, L=L_MIN): + """Interval enclosures, uniform over r in [r0, r1], every frame and every torus side >= L, of the Q-law of the + targets: mean = b alpha + k beta, covariance C (normalized kernel), and the pin energy coefficients.""" + pins, targets = functionals() + P = [pins[n] for n in PIN_NAMES] + T = [targets[n] for n in TARGET_NAMES] + M1 = torus_M1(L) + th1 = theta_minus_one(L) + nt = len(T) + Spp = [[cov_band(P[i], P[j], ('p', i, j), r0, r1, M1) for j in range(6)] for i in range(6)] + Stp = [[cov_band(T[i], P[j], ('tp', i, j), r0, r1, M1) for j in range(6)] for i in range(nt)] + Stt = [[cov_band(T[i], T[j], ('tt', i, j), r0, r1, M1) for j in range(nt)] for i in range(nt)] + r = IV(r0, r1) + kvec = [-(r * r * r) * HALF, -(r * r), IV(12), IV(0), IV(0), IV(0)] + e1 = [IV(1), IV(0), IV(0), IV(0), IV(0), IV(0)] + W = solve_iv(Spp, Stp) + xs = solve_iv(Spp, [e1, kvec]) + law = Law() + law.r0, law.r1, law.L, law.M1, law.th1 = r0, r1, L, M1, th1 + law.alpha = {n: W[i][0] for i, n in enumerate(TARGET_NAMES)} + law.beta = {n: sum((W[i][j] * kvec[j] for j in range(6)), IV(0)) for i, n in enumerate(TARGET_NAMES)} + scale = IV(1 / (1 + th1), 1) # K_L = (unnormalized) / Theta, Theta in [1, 1 + th1] + C = {} + for i, a in enumerate(TARGET_NAMES): + for j, c in enumerate(TARGET_NAMES): + if j < i: + continue + v = Stt[i][j] - sum((W[i][l] * Stp[j][l] for l in range(6)), IV(0)) + if i != j: + v2 = Stt[j][i] - sum((W[j][l] * Stp[i][l] for l in range(6)), IV(0)) + v = v.meet(v2) + C[(a, c)] = C[(c, a)] = v * scale + law.C = C + escale = IV(1, 1 + th1) + law.e_bb = xs[0][0] * escale + law.e_bk = sum((kvec[j] * xs[0][j] for j in range(6)), IV(0)) * escale + law.e_kk = sum((kvec[j] * xs[1][j] for j in range(6)), IV(0)) * escale + # regression on w and the g-parts + sw2 = C[('w', 'w')] + require(sw2.lo > 0, 'Var w not positive') + law.sw2 = sw2 + law.sw = isqrt_iv(sw2) + law.bw = {n: C[(n, 'w')] / sw2 for n in TARGET_NAMES if n != 'w'} + Cg = {} + for a in TARGET_NAMES: + for c in TARGET_NAMES: + if a != 'w' and c != 'w': + Cg[(a, c)] = C[(a, c)] - C[(a, 'w')] * C[(c, 'w')] / sw2 + law.Cg = Cg + law.L3 = cholesky3([[Cg[(a, c)] for c in ('v', 'om', 'Y')] for a in ('v', 'om', 'Y')]) + law.sdg = {n: isqrt_iv(IV(max(Cg[(n, n)].lo, Fr(0)), Cg[(n, n)].hi)) if Cg[(n, n)].lo > 0 else IV(0, isqrt_iv(IV(Cg[(n, n)].hi)).hi) + for n in TARGET_NAMES if n != 'w'} + law.sdQ = {n: isqrt_iv(C[(n, n)]) for n in TARGET_NAMES} + # crude higher jets: Taylor-tail coefficients S_alpha(r1) = sum_{beta != 0} sd0_{alpha+beta} (2 r1)^{|beta|} / beta! + law.S = {n: taylor_tail_coefficient(JET4_IDX[n], r1, M1) for n in JET4} + return law + + +def sd0_bound(gx, gy, M1): + """Upper bound of sqrt(Var d^gamma f(0)) = sqrt(|d^{2 gamma} K_L(0)|) (normalized, every frame, L >= L_MIN).""" + v = dfact(2 * gx - 1) * dfact(2 * gy - 1) + math.factorial(2 * gx) * math.factorial(2 * gy) * M1 + return isqrt_iv(IV(v)).hi + + +def taylor_tail_coefficient(alpha, r1, M1): + ax, ay = alpha + tot = Fr(0) + for n in range(1, NB + 1): + for bx in range(n + 1): + by = n - bx + tot += sd0_bound(ax + bx, ay + by, M1) * (2 * r1) ** n / (math.factorial(bx) * math.factorial(by)) + # tail |beta| > NB: sd0_gamma <= 2^|gamma| gx! gy! sqrt(1 + M1) (binom(2m, m) <= 4^m), so a term is at most + # 2^|alpha| (4 + n)^4 (4 r1)^n 2 per multi-index; n + 1 multi-indices of order n + q = 4 * r1 + require(q <= Fr(1, 16), 'Taylor tail ratio too large') + tail = Fr(0) + base = 2 ** (ax + ay) * 2 + n = NB + 1 + # sum_{n > NB} (n + 1)(n + 4)^8 q^n <= (NB + 2)(NB + 5)^8 q^(NB+1) / (1 - 2 q) (consecutive ratio <= 2q for these n) + tail = base * (NB + 2) * Fr(NB + 5) ** 8 * q ** (NB + 1) / (1 - 2 * q) + return tot + tail + + +# ----------------------------------------------------------------------------- moment calculus (normalized by K = 12 k) +def binom(n, k): + return math.comb(n, k) + + +def gauss_norm(mu_abs, sd, p): + """Upper bound of ||X||_p for X Gaussian with |mean| <= mu_abs and sd <= sd.""" + return rup_rel(mu_abs + sd * znorm(pow2(p))) + + +def pos_part_moments(s, c, nmax, extra_max): + """P[n][e] = upper bound of E[u^e (u^n - c^n)_+], u = s |zeta|, for 1 <= n <= nmax, 0 <= e <= extra_max.""" + x = c / s + M = upper_moments(x, nmax + extra_max) + out = {} + for n in range(1, nmax + 1): + for e in range(extra_max + 1): + out[(n, e)] = rup_rel(((M[n + e] * s ** (n + e) - M[e] * c ** n * s ** e) * 2).hi) + return out + + +def normalized_moments(q, nmax, theta): + """Component moments, normalized by K = 12 k (K >= Klo), for n <= nmax: + E1[n], E1o[n], E1v[n] : upper bounds of E[y1^n], E[o0 y1^n], E[v0^2 y1^n], y1 = 1 + cmu1/K + (r1/K) Delta1; + Tn[n], To[n], Tv[n] : upper bounds of sum_{i=2,3,4} E[(y_i^n - c^n)_+] (and with the factors o0, v0^2), + y_i <= u_i + delta_i (u_i = s_i |zeta| independent half-normals), threshold c = q['c']. + o0 = s3 |z2| and v0^2 = s2^2 z1^2 / 4 are unnormalized.""" + th = theta + cp = q['c'] * (1 - th) # <= c (1 + th)^{-(n-1)/n} for every n (Bernoulli) + PP = {i: pos_part_moments(q['s'][i], cp, nmax, 2) for i in (2, 3, 4)} + Eu3 = (habs_moment(1) * q['s3']).hi + Eu2sq = (habs_moment(2) * q['s2'] * q['s2']).hi + n_o0_2 = q['s3'] * znorm(2) + n_u2sq_2 = q['s2'] * q['s2'] * znorm(4) ** 2 + r3 = q['s3'] / q['s'][3] + r2sq = (q['s2'] / q['s'][2]) ** 2 + E1, E1o, E1v, Tn, To, Tv = {}, {}, {}, {}, {}, {} + base = 1 + q['cmt1'] + for n in range(0, nmax + 1): + ey = base ** n + eo = base ** n * Eu3 + ev = base ** n * Eu2sq + for j in range(1, n + 1): + coef = binom(n, j) * base ** (n - j) * q['rK'] ** j + ey += coef * q['nD1'](pow2(j)) ** j + eo += coef * n_o0_2 * q['nD1'](pow2(2 * j)) ** j + ev += coef * n_u2sq_2 * q['nD1'](pow2(2 * j)) ** j + E1[n], E1o[n], E1v[n] = rup_rel(ey), rup_rel(eo), rup_rel(ev / 4) + if n == 0: + continue + f1 = (1 + th) ** (n - 1) + f2 = (1 + 1 / th) ** (n - 1) + tn = to = tv = Fr(0) + for i in (2, 3, 4): + P = PP[i] + dn = q['nd'](i, pow2(n)) ** n + dn2 = q['nd'](i, pow2(2 * n)) ** n + tn += f1 * P[(n, 0)] + f2 * dn + to += (f1 * r3 * P[(n, 1)] if i == 3 else f1 * Eu3 * P[(n, 0)]) + f2 * n_o0_2 * dn2 + tv += (f1 * r2sq * P[(n, 2)] if i == 2 else f1 * Eu2sq * P[(n, 0)]) + f2 * n_u2sq_2 * dn2 + Tn[n], To[n], Tv[n] = rup_rel(tn), rup_rel(to), rup_rel(tv / 4) + return E1, E1o, E1v, Tn, To, Tv + + +def elam_pos(m, s): + """E[(s z + m)_+] at a point (rational m, s > 0).""" + x = Fr(m) / s + return m * Phi(IV(x)) + phi_pt(x) * s + + +def elam2_pos(m, s): + """E[(s z + m)_+^2] at a point.""" + x = Fr(m) / s + return (m * m + s * s) * Phi(IV(x)) + phi_pt(x) * (m * s) + + +def box_bound(law, bb, kb, r0, r1, last_band=False, x0=None, theta=None): + """Certified upper bound of sup_{r in [r0, r1]} Q^W(G_r^c) / r^3 over b in bb, k in kb, with its pieces, for the near-branch + threshold x0 (X_0 of NOTE section 4) and the split parameter theta (section 5); every choice gives a valid bound.""" + x0 = X0 if x0 is None else Fr(x0) + theta = THETA if theta is None else Fr(theta) + b = IV(bb[0], bb[1]) + k = IV(kb[0], kb[1]) + klo, khi = kb[0], kb[1] + Klo = 12 * klo + mu = {n: b * law.alpha[n] + k * law.beta[n] for n in TARGET_NAMES} + mabs = {n: absup(mu[n]) for n in TARGET_NAMES} + E = b * b * law.e_bb + b * k * law.e_bk * 2 + k * k * law.e_kk + require(E.hi > 0, 'energy') + sqE = isqrt_iv(IV(E.hi)).hi + sw_lo, sw_hi = law.sw.lo, law.sw.hi + muw_abs = mabs['w'] + bw = {n: absup(law.bw[n]) for n in law.bw} + t = {n: law.S[n] / sw_lo for n in JET4} + a_fac = Fr(1, 2) if MUT == 'cap-half' else Fr(1) # the cap constant 4/(3k) (mutant: halved) + B = lambda n: bw[n] + t[n] + bbars = {1: r1 * (Fr(5, 2) * B('X40') + 2 * B('X31')), + 2: 2 * bw['v'] + r1 * (Fr(5, 2) * B('X31') + 2 * B('X22')), + 3: bw['om'] + r1 * (Fr(5, 2) * B('X22') + 2 * B('X13')), + 4: bw['Y'] + r1 * (2 * B('X13') + 2 * B('X04'))} + bbars = {i: rup_rel(v) for i, v in bbars.items()} + bbar = max(bbars.values()) + L3 = law.L3 + s2, s3, s4 = 2 * L3[0][0].hi, L3[1][1].hi, L3[2][2].hi + l21, l31, l32 = absup(L3[1][0]), absup(L3[2][0]), absup(L3[2][1]) + + memo = {} + + def nA(n, p): # ||A_alpha^g||_p, A_alpha^g = |X_alpha,g| + T_alpha^g + key = ('A', n, p) + if key not in memo: + memo[key] = rup_rel(gauss_norm(mabs[n], law.sdg[n].hi, p) + law.S[n] * (sqE + znorm(pow2(p)))) + return memo[key] + + def nDelta(i, p): + key = ('D', i, p) + if key not in memo: + memo[key] = rup_rel(nDelta0(i, p)) + return memo[key] + + def nDelta0(i, p): + if MUT == 'no-delta': + return Fr(0) + if i == 1: + return Fr(5, 2) * nA('X40', p) + 2 * nA('X31', p) + if i == 2: + return Fr(5, 2) * nA('X31', p) + 2 * nA('X22', p) + if i == 3: + return Fr(5, 2) * nA('X22', p) + 2 * nA('X13', p) + return 2 * nA('X13', p) + 2 * nA('X04', p) + + def nd(i, p): + key = ('d', i, p) + if key not in memo: + memo[key] = rup_rel(nd0(i, p)) + return memo[key] + + def nd0(i, p): # ||delta_i||_p / Klo, delta_i including beta_i |mu_w| + zp = znorm(pow2(p)) + cm = bbars[i] * muw_abs + if i == 2: + v = 2 * mabs['v'] + cm + r1 * nDelta(2, p) + elif i == 3: + v = mabs['om'] + l21 * zp + cm + r1 * nDelta(3, p) + else: + v = mabs['Y'] + (l31 + l32) * zp + cm + r1 * nDelta(4, p) + return v / Klo + + kappa = 3 / (1 - 3 * x0) + Fk = 1 + kappa * x0 # (1 + kappa x_i) <= Fk on the event {x_i <= x0} + cth = 1 - kappa * x0 / 2 # <= Fk^{-1/2} + NMAX = 24 + q = {'cmt1': bbars[1] * muw_abs / Klo, 'rK': r1 / Klo, 'nD1': lambda p: nDelta(1, p), + 's': {2: s2 / Klo, 3: s3 / Klo, 4: s4 / Klo}, 'nd': nd, 's3': s3, 's2': s2, 'c': cth} + E1, E1o, E1v, Tn, To, Tv = normalized_moments(q, NMAX, theta) + et1 = rup_rel(16 * kappa * r1 * bbars[1] * a_fac) # eta_1 K, k-free + aK2_hi = 192 * khi * a_fac # a K^2 = 192 k + + # moments of Ut = U / (a K^2) <= max(Z1, Fk y_i^2), Z1 = y1^2 (1 + et1 y1) + def EUt(p): # upper bound of E[Ut^p] + key = ('U', p) + if key not in memo: + memo[key] = rup_rel(sum(binom(p, l) * et1 ** l * E1[2 * p + l] for l in range(p + 1)) + Fk ** p * Tn[2 * p]) + return memo[key] + + def EoUt2(): + return sum(binom(2, l) * et1 ** l * E1o[4 + l] for l in range(3)) + Fk ** 2 * To[4] + + def EvUt2(): + return sum(binom(2, l) * et1 ** l * E1v[4 + l] for l in range(3)) + Fk ** 2 * Tv[4] + + def sq(x): + return isqrt_iv(IV(x)).hi + + U3_2 = sq(EUt(6)) # ||Ut^3||_2 + U2_2 = sq(EUt(4)) # ||Ut^2||_2 + + def nU(p): # ||U||_p <= a K^2 Fk ||max_i y_i||_2p^2 (on the event x_i <= X0) + require(2 * p <= NMAX and pow2(p) == p, 'norm order') + key = ('nU', p) + if key not in memo: + memo[key] = rup_rel(aK2_hi * Fk * iroot(E1[2 * p] + Tn[2 * p], p)) + return memo[key] + + def nbar(n, p): # ||F_bar||_p, F_bar = |F_g| + |beta_F| (|mu_w| + r1 U) + return gauss_norm(mabs[n], law.sdg[n].hi, p) + bw[n] * (muw_abs + r1 * nU(p)) + + six_klo = 6 * klo + nT2, nT4, nt2, nt4 = nbar('T', 2), nbar('T', 4), nbar('tau', 2), nbar('tau', 4) + epsc2 = (2 * r1 * nT2 + r1 * r1 * nt2) / six_klo + (r1 * nT4) * (r1 * nT4 + r1 * r1 * nt4) / six_klo ** 2 + nT8, nt8 = nbar('T', 8), nbar('tau', 8) + epsc4 = (2 * r1 * nT4 + r1 * r1 * nt4) / six_klo + (r1 * nT8) * (r1 * nT8 + r1 * r1 * nt8) / six_klo ** 2 + # Term A / (36 k^2) + A_n = EUt(3) + epsc2 * U3_2 + termA = aK2_hi ** 3 / 3 * A_n + # Term B / (36 k^2) + ndo = lambda p: l21 * znorm(pow2(p)) + mabs['om'] + bw['om'] * (muw_abs + r1 * nU(p)) + no4 = s3 * znorm(4) + ndo(4) + B_n = EoUt2() + (ndo(2) + epsc4 * no4) * U2_2 + termB = aK2_hi ** 2 / 2 * B_n + if MUT == 'no-om-term': + termB = Fr(0) + # Term C / (36 k^2) = (a K^2)^2 / (12 k) * E[(1 + r T/(6k)) e Ut^2] + th = THETA_E + dv = lambda p: mabs['v'] + bw['v'] * (muw_abs + r1 * nU(p)) + ebar4 = (r1 * nbar('nu', 8) + L3[0][0].hi * znorm(8) + dv(8)) ** 2 + C_n = ((1 + th) ** 2 * EvUt2() + + ((1 + th) * (1 + 1 / th) * dv(4) ** 2 + (1 + 1 / th) * r1 * r1 * nbar('nu', 4) ** 2) * U2_2 + + r1 * nT4 / six_klo * ebar4 * U2_2) + termC = aK2_hi ** 2 / (12 * klo) * C_n + # one more power of U for the density refinement: the same three terms with lambda inserted, divided by r + # (int_0^{rU} lambda W dlambda <= r^6 |c1| [|c2| (U^4/4 + om U^3/3) + e U^3/3]) + U3_2_ = U3_2 + termAp = aK2_hi ** 4 / 4 * (EUt(4) + epsc2 * sq(EUt(8))) + EoUt3 = sum(binom(3, l) * et1 ** l * E1o[6 + l] for l in range(4)) + Fk ** 3 * To[6] + EvUt3 = sum(binom(3, l) * et1 ** l * E1v[6 + l] for l in range(4)) + Fk ** 3 * Tv[6] + termBp = aK2_hi ** 3 / 3 * (EoUt3 + (ndo(2) + epsc4 * no4) * U3_2_) + if MUT == 'no-om-term': + termBp = Fr(0) + termCp = aK2_hi ** 3 / (18 * klo) * ((1 + th) ** 2 * EvUt3 + + ((1 + th) * (1 + 1 / th) * dv(4) ** 2 + (1 + 1 / th) * r1 * r1 * nbar('nu', 4) ** 2) * U3_2_ + + r1 * nT4 / six_klo * ebar4 * U3_2_) + # density of lambda = -w: pbar = its supremum on [0, inf); on the failure window [0, rU] also p(lambda) <= p0 + lambda Lp + # with p0 = sup over the box of the density at 0 and Lp = sup |p'| <= phi(1)/sigma^2, phi(1) < 0.24198 + mlo = max(Fr(0), mu['w'].lo) + pbar = (phi_pt(mlo / sw_hi) / sw_lo).hi + m0 = Fr(0) if mu['w'].lo <= 0 <= mu['w'].hi else min(abs(mu['w'].lo), abs(mu['w'].hi)) + p0 = (phi_pt(m0 / sw_hi) / sw_lo).hi + Lp = Fr(24198, 100000) / (sw_lo * sw_lo) + # ---- normalizer floor Z_r / (36 k^2 r^2) >= El2 - corr + m_lam = -mu['w'] + El2 = elam2_pos(m_lam.lo, sw_lo).lo + Elp = elam_pos(m_lam.hi, sw_hi).hi + lam_n = lambda p: muw_abs + sw_hi * znorm(pow2(p)) + sdT, sdtau = law.sdQ['T'].hi, law.sdQ['tau'].hi + q_bad = gauss_tail2(mabs['T'], sdT, 3 * klo / (2 * r1)) + gauss_tail2(mabs['tau'], sdtau, 3 * klo / (2 * r1 * r1)) + Eom_g = mabs['om'] + law.sdg['om'].hi * Fr(4, 5) + Evg2 = mabs['v'] ** 2 + law.sdg['v'].hi ** 2 + E_om_lam = Eom_g * Elp + bw['om'] * (muw_abs * lam_n(1) + lam_n(2) ** 2) + E_v2_lam = 2 * Evg2 * Elp + 2 * bw['v'] ** 2 * (sw_hi * znorm(4)) ** 2 * lam_n(2) + corr = (lam_n(4) ** 2 * iroot(q_bad, 2) if q_bad > 0 else Fr(0)) \ + + (2 * r1 * gauss_norm(mabs['T'], sdT, 2) + r1 * r1 * gauss_norm(mabs['tau'], sdtau, 2)) * lam_n(4) ** 2 / six_klo \ + + r1 * (E_v2_lam / (3 * klo) + E_om_lam) + if MUT == 'no-zfloor-correction': + corr = Fr(0) + zt = El2 - corr + require(zt > 0, 'normalizer floor not positive') + main = min(pbar * (termA + termB + termC), p0 * (termA + termB + termC) + r1 * Lp * (termAp + termBp + termCp)) / zt + # ---- rare branches at the band top: E[W 1_E] <= ||W 1_typed||_q P(E)^(1 - 1/q) (Hoelder), ||W 1_typed||_q / r^2 <= g1 g2, + # for each q in QS; the smallest resulting bound is used + six_khi = 6 * khi + + def wnorm(q): + p4 = 4 * q + nTQ = gauss_norm(mabs['T'], sdT, p4) + ntauQ = gauss_norm(mabs['tau'], sdtau, p4) + g1 = (six_khi + r1 * nTQ) * lam_n(p4) + g2 = (six_khi + r1 * nTQ + r1 * r1 * ntauQ) * (lam_n(p4) + r1 * gauss_norm(mabs['om'], law.sdQ['om'].hi, p4)) \ + + r1 * (r1 * gauss_norm(mabs['nu'], law.sdQ['nu'].hi, p4) + gauss_norm(mabs['v'], law.sdQ['v'].hi, p4)) ** 2 + return g1 * g2 + thr = 3 * klo / (10 * r1) + qA2 = Fr(0) + args = [] + for n in JET4: + args.append((thr * Fr(7, 8) - mabs[n]) / law.sdQ[n].hi) + qA2 += gauss_tail2(mabs[n], law.sdQ[n].hi, thr * Fr(7, 8)) + qA2 += (law.S[n] * (sqE + znorm(32)) / (thr / 8)) ** 32 + a_hi = Fr(4, 3) / klo * a_fac + qfar = Fr(0) + qx = Fr(0) + if bbar > 0: + lam_far = 1 / (4 * a_hi * r1 * bbar * bbar) + args.append((lam_far - muw_abs) / sw_hi) + qfar = gauss_tail1(muw_abs, sw_hi, lam_far) + for i in (1, 2, 3, 4): # x_i > X0 <=> y_i' > X0 / (a r1 beta_i) + if bbars[i] == 0: + continue + tx = x0 / (a_hi * r1 * bbars[i]) + if i == 1: + cb = 12 * khi + bbars[1] * muw_abs + qx += Fr(1) if tx <= cb else (r1 * nDelta(1, 32) / (tx - cb)) ** 32 + else: + s_i = {2: s2, 3: s3, 4: s4}[i] + args.append(tx / (2 * s_i)) + qx += gauss_tail2(0, s_i, tx / 2) + (nd(i, 32) * Klo / (tx / 2)) ** 32 + qs = [min(Fr(1), x) for x in (qA2, qfar, qx)] + tail_num = min(wnorm(q) * sum(pow_tail(x, q) for x in qs) for q in QS) + if last_band: + # (0, R] is the union of [R 2^-(j+1), R 2^-j]; every Gaussian tail argument scales at least like 1/r and is >= 3 at R, + # so each Gaussian piece drops by more than 64 when r halves; every Markov piece is a power r^m with m >= 32; after the + # power 1 - 1/q (q >= 2) each piece still drops by more than 8, so for the q chosen at R the supremum of tail / r^3 + # over (0, R] is attained on [R/2, R] + require(min(args) >= 3, 'last-band tail arguments') + # no piece is capped at 1, so the factor-64 decrease of every piece per halving carries over to each sum + require(max(qA2, qfar, qx) < 1, 'last-band tail pieces') + rdiv = r1 / 2 + else: + rdiv = r0 + tail = tail_num / (36 * klo * klo * zt * rdiv ** 3) + total = main + tail + return {'total': total, 'main': main, 'tail': tail, 'zt': zt, 'pbar': pbar, 'termA': termA, 'termB': termB, + 'termC': termC, 'qA2': qA2, 'qfar': qfar, 'qx': qx, 'El2': El2, 'corr': corr, 'min_arg': min(args)} + + +def pow_tail(x, q): + """Upper bound of x^(1 - 1/q) for 0 <= x <= 1 (q a power of two).""" + if x <= 0: + return Fr(0) + return iroot(rup_rel(x, 64) ** (q - 1), q) + + +def phibar_upper(x): + """Rational upper bound of 1 - Phi(x) for rational x >= 0: the series value for x <= 12, and for x > 12 the Mills bound + phi(x)/x with exp(-y) <= m!/y^m (m = 40), which stays exact far below the 2^-160 rounding floor.""" + x = Fr(x) + if x <= 12: + return Phibar_pt(x).hi + y = x * x / 2 + m = 40 + return Fr(math.factorial(m)) / y ** m / (x * Fr(5, 2)) # sqrt(2 pi) > 5/2 + + +def gauss_tail2(mu_abs, sd, t): + """Upper bound of P(|X| > t) for X Gaussian with |mean| <= mu_abs, sd <= sd.""" + if t <= mu_abs: + return Fr(1) + return min(Fr(1), 2 * phibar_upper((t - mu_abs) / sd)) + + +def gauss_tail1(mu_abs, sd, t): + if t <= mu_abs: + return Fr(1) + return min(Fr(1), phibar_upper((t - mu_abs) / sd)) + + +# ----------------------------------------------------------------------------- bands, boxes, runs +B_BOXES = ((Fr(0), Fr(1, 64)), (Fr(1, 64), Fr(1, 16)), (Fr(1, 16), Fr(1, 4)), (Fr(1, 4), Fr(1))) +K_BOXES = ((Fr(1, 2), Fr(1)), (Fr(1), Fr(3, 2)), (Fr(3, 2), Fr(7, 4)), (Fr(7, 4), Fr(15, 8)), (Fr(15, 8), Fr(31, 16)), + (Fr(31, 16), Fr(63, 32)), (Fr(63, 32), Fr(2))) +HIGH_OCTAVES = 3 # octaves [2^-(j+1), 2^-j], j = 6..8 (r up to 1/64), split into 8 sub-bands +TOP_OCTAVES = 4 # octaves j = 9..12, split into 16 sub-bands +LOW_OCTAVES = 16 # octaves j = 13..28, split into 4 sub-bands +R_LAST = Fr(1, 2 ** 29) # last band [0, 2^-29] + + +def bands(): + """Sub-bands [r0, r1] covering (0, 1/64]; each octave split into equal sub-bands; the last band is [0, R_LAST].""" + out = [] + for j in range(6, 9 + TOP_OCTAVES + LOW_OCTAVES): + hi = Fr(1, 2 ** j) + lo = hi / 2 + m = 8 if j < 9 else (16 if j < 9 + TOP_OCTAVES else 4) + for i in range(m): + out.append((lo + (hi - lo) * i / m, lo + (hi - lo) * (i + 1) / m)) + out.append((Fr(0), R_LAST)) + require(min(b[0] for b in out) == 0 and max(b[1] for b in out) == Fr(1, 64), 'band cover') + return sorted(out) + + +def up(x, digits=6): + """Decimal string of a rational rounded UP to `digits` significant digits (as a string float).""" + x = Fr(x) + if x <= 0: + return '0' + e = len(str(x.numerator // x.denominator)) if x >= 1 else -len(str(x.denominator // x.numerator)) + scale = Fr(10) ** (digits - e) + n = -((-x.numerator * scale.numerator) // (x.denominator * scale.denominator)) + return str(Fr(n) / scale) if scale.denominator == 1 else '%s' % (Fr(n) / scale) + + +def dec_down(x, places=4): + x = Fr(x) + n = (x.numerator * 10 ** places) // x.denominator + s = str(abs(n)).rjust(places + 1, '0') + return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:] + + +def dec_up(x, places=4): + x = Fr(x) + n = -((-x.numerator * 10 ** places) // x.denominator) + s = str(abs(n)).rjust(places + 1, '0') + return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:] + + +BOXES = tuple((bb, kb) for bb in B_BOXES for kb in K_BOXES) # the order of the per-box lists in a band record +# (x0, theta) choices; each gives a valid bound and the record keeps the smaller one +COMBOS = ((Fr(1, 64), Fr(1, 32)), (Fr(1, 1024), Fr(1, 256))) +# the fixed-axis SIDE24 point of [CAP] section 1 (b = 6/5, kappa = 1/6), a single (b, k) point outside the band +SIDE24_B, SIDE24_K = (Fr(6, 5), Fr(6, 5)), (Fr(1, 6), Fr(1, 6)) +CG_02_LO, CG_SIDE24_LO = Fr('5324360.4426'), Fr('35736.7352') # certified lower ends of c_G at (0, 2), (6/5, 1/6) ([G]) + + +def best_box(law, bb, kb, r0, r1, last): + best = None + for x0, theta in COMBOS: + res = box_bound(law, bb, kb, r0, r1, last_band=last, x0=x0, theta=theta) + if best is None or res['total'] < best['total']: + best = res + return best + + +def band_record(args): + """Per band: the certified bound of sup Q^W(G_r^c)/r^3 (rounded up, 4 places) and the floor of Z_r/r^2 (rounded down, + 6 places) for each of the 28 boxes in BOXES order, the largest tail part (rounded up, 12 places), and the same two + numbers at the SIDE24 point.""" + r0, r1 = args + law = band_law(r0, r1) + last = (r0 == 0) + bound, zfloor, tails = [], [], [] + for bb, kb in BOXES: + res = best_box(law, bb, kb, r0, r1, last) + bound.append(dec_up(res['total'])) + zfloor.append(dec_down(36 * kb[0] * kb[0] * res['zt'], 6)) + tails.append(res['tail']) + s24 = best_box(law, SIDE24_B, SIDE24_K, r0, r1, last) + return {'r': [str(r0), str(r1)], 'bound': bound, 'zfloor': zfloor, 'tail_max': dec_up(max(tails), 12), + 'side24': [dec_up(s24['total']), dec_down(36 * SIDE24_K[0] ** 2 * s24['zt'], 6)]} + + +def full_run(procs): + bl = bands() + with multiprocessing.Pool(procs) as pool: + recs = pool.map(band_record, bl) + return assemble(recs) + + +def assemble(recs): + table = {} + for rs in R_STARS: + mx, arg, zmin = Fr(0), None, None + for rec in recs: + if Fr(rec['r'][1]) <= rs: + for (bb, kb), bs, zs in zip(BOXES, rec['bound'], rec['zfloor']): + v = Fr(bs) + if v > mx: + mx, arg = v, {'r': rec['r'], 'b': [str(bb[0]), str(bb[1])], 'k': [str(kb[0]), str(kb[1])]} + z = Fr(zs) + zmin = z if zmin is None or z < zmin else zmin + table[str(rs)] = {'C': str(mx), 'argmax': arg, 'C_over_cG02': dec_up(mx / CG_02_LO, 6), + 'C_rstar3': dec_up(mx * rs ** 3, 10), 'z_star': str(zmin)} + side = {} + for rs in R_STARS: + rows = [rec for rec in recs if Fr(rec['r'][1]) <= rs] + mx = max(Fr(rec['side24'][0]) for rec in rows) + arg = [rec['r'] for rec in rows if Fr(rec['side24'][0]) == mx][0] + zmin = min(Fr(rec['side24'][1]) for rec in rows) + side[str(rs)] = {'C': str(mx), 'argmax_r': arg, 'C_over_cG': dec_up(mx / CG_SIDE24_LO, 6), + 'C_rstar3': dec_up(mx * rs ** 3, 10), 'z_star': str(zmin)} + return {'schema': 1, 'object': 'CL-C8-THEOREM-A-CONSTANTS-PLANAR-20260930-v1', 'scientific_effect': 'NONE', + 'certified': True, 'mutant': MUT, 'L_min': L_MIN, + 'band': {'b': [str(BAND_B[0]), str(BAND_B[1])], 'k': [str(BAND_K[0]), str(BAND_K[1])]}, + 'side24_point': {'b': str(SIDE24_B[0]), 'k': str(SIDE24_K[0])}, + 'parameters': params(), 'C_table': table, 'C_table_side24': side, 'bands': recs} + + +def params(): + return {'PBITS': PBITS, 'NX': NX, 'NB': NB, 'COMBOS_X0_THETA': [[str(a), str(b)] for a, b in COMBOS], 'THETA_E': str(THETA_E), 'QS': list(QS), + 'R_STARS': [str(x) for x in R_STARS], 'HIGH_OCTAVES': HIGH_OCTAVES, 'TOP_OCTAVES': TOP_OCTAVES, + 'LOW_OCTAVES': LOW_OCTAVES, 'R_LAST': str(R_LAST), + 'B_BOXES': [[str(a), str(b)] for a, b in B_BOXES], 'K_BOXES': [[str(a), str(b)] for a, b in K_BOXES]} + + +def results_text(res): + """RESULTS.json layout: the summary indented, then one line per band record.""" + head = json.dumps({key: val for key, val in res.items() if key != 'bands'}, indent=1, sort_keys=True) + lines = ',\n'.join(' ' + json.dumps(rec, sort_keys=True) for rec in res['bands']) + return head[:-2] + ',\n "bands": [\n' + lines + '\n ]\n}\n' + + +def write_results(res): + with open(os.path.join(HERE, 'RESULTS.json'), 'w') as fh: + fh.write(results_text(res)) + + +def load_results(): + with open(os.path.join(HERE, 'RESULTS.json')) as fh: + return json.load(fh) + + +def replay_sample(): + """Band indices replayed by --check: the band ending at each r_*, the bands ending at 2^-14, 2^-20, 2^-26, the last band.""" + bl = bands() + idx = [] + for rs in R_STARS: + j = max(i for i, b in enumerate(bl) if b[1] == rs) + idx.append(j) + idx += [i for i, b in enumerate(bl) if b[1] in (Fr(1, 2 ** 14), Fr(1, 2 ** 20), Fr(1, 2 ** 26))] + idx.append(0) + return sorted(set(idx)) + + +def check_run(procs, full=False): + ref = load_results() + with open(os.path.join(HERE, 'RESULTS.json')) as fh: + require(fh.read() == results_text(ref), 'RESULTS.json is not in the canonical layout') + require(ref['parameters'] == params(), 'parameters differ from RESULTS.json') + require(ref['mutant'] is None, 'RESULTS.json produced under a mutant') + require(abs(PI.mid() - Fr('3.14159265358979323846')) < Fr(1, 10 ** 19) and PI.width() < Fr(1, 10 ** 30), 'pi enclosure') + require(abs(Phi(IV(1)).mid() - Fr('0.841344746068542948585')) < Fr(1, 10 ** 19), 'Phi(1) enclosure') + bl = bands() + require([rec['r'] for rec in ref['bands']] == [[str(a), str(b)] for a, b in bl], 'band list differs') + idx = list(range(len(bl))) if full else replay_sample() + with multiprocessing.Pool(procs) as pool: + recs = pool.map(band_record, [bl[i] for i in idx]) + for i, rec in zip(idx, recs): + require(rec == ref['bands'][i], 'band %d (%s) differs from RESULTS.json' % (i, rec['r'])) + # the table is the maximum over the stored records + asm = assemble(ref['bands']) + require(asm['C_table'] == ref['C_table'] and asm['C_table_side24'] == ref['C_table_side24'], + 'C tables are not the maxima of the stored bounds') + # sanity: every C exceeds the sharp reference-kernel cap coefficient c_G(0, 2) of Math-#203 (a lower bound for every + # cap-route constant valid for all L >= 10, since c_G^{L,R} -> c_G^{ref} as L -> infinity) + for rs, row in ref['C_table'].items(): + require(Fr(row['C']) > CG_02_LO, 'C below the sharp cap coefficient') + for rs, row in ref['C_table_side24'].items(): + require(Fr(row['C']) > CG_SIDE24_LO, 'SIDE24 C below the sharp cap coefficient') + print(json.dumps({'check': 'ok', 'bands_replayed': len(idx), 'C_table': {rs: row['C'] for rs, row in ref['C_table'].items()}, + 'C_table_side24': {rs: row['C'] for rs, row in ref['C_table_side24'].items()}, 'mutant': MUT})) + + +def main(): + global MUT + ap = argparse.ArgumentParser() + ap.add_argument('--check', action='store_true') + ap.add_argument('--check-full', action='store_true') + ap.add_argument('--mutant', choices=MUTANTS) + ap.add_argument('--procs', type=int, default=4) + ap.add_argument('--mc', nargs=5, metavar=('R', 'B', 'K', 'N', 'SEED'), help='floating end-to-end control') + args = ap.parse_args() + MUT = args.mutant + if args.mc: + rr, bb, kk, nn, sd = args.mc + print(json.dumps(mc_event(float(Fr(rr)), float(Fr(bb)), float(Fr(kk)), int(nn), int(sd)))) + return + if args.check or args.check_full: + check_run(args.procs, full=args.check_full) + return + res = full_run(args.procs) + write_results(res) + print(json.dumps({'band': {rs: row['C'] for rs, row in res['C_table'].items()}, + 'side24': {rs: row['C'] for rs, row in res['C_table_side24'].items()}})) + + + +# ----------------------------------------------------------------------------- floating control (not part of the certificate) +def mc_event(r, b, k, n, seed, deg=8, grid=9): + """End-to-end Monte Carlo of Q^W(G_r^c) / r^3 and Z_r / r^2 for the reference kernel (floating point): the Taylor jet of + order deg at the midpoint is sampled under Q, the field near D is rebuilt from it, W_r and G_r (M_3, M_4 on a grid of D) + are evaluated directly from the definitions.""" + import random + pins, _ = functionals() + P = [pins[nm] for nm in PIN_NAMES] + jets = [(i, m - i) for m in range(deg + 1) for i in range(m, -1, -1)] + J = [[term(1, 0, a, bb, X0PT)] for (a, bb) in jets] + + def ev(F, G, key): + ser, w, _ = pair_series(F, G, key) + return sum(float(c) * r ** p for p, c in ser.items()) + Spp = [[ev(P[i], P[j], ('p', i, j)) for j in range(6)] for i in range(6)] + Sjp = [[ev(J[i], P[j], ('jp', deg, i, j)) for j in range(6)] for i in range(len(J))] + Sjj = [[ev(J[i], J[j], ('jj', deg, i, j)) for j in range(len(J))] for i in range(len(J))] + + def solve(A, bv): + m = len(A) + M = [A[i][:] + [bv[i]] for i in range(m)] + for c in range(m): + p = max(range(c, m), key=lambda i: abs(M[i][c])) + M[c], M[p] = M[p], M[c] + for i in range(c + 1, m): + f = M[i][c] / M[c][c] + for j in range(c, m + 1): + M[i][j] -= f * M[c][j] + x = [0.0] * m + for i in reversed(range(m)): + x[i] = (M[i][m] - sum(M[i][j] * x[j] for j in range(i + 1, m))) / M[i][i] + return x + vr = [b - k * r ** 3 / 2, -k * r * r, 12 * k, 0, 0, 0] + Wt = [solve(Spp, Sjp[i]) for i in range(len(J))] + mu = [sum(Wt[i][j] * vr[j] for j in range(6)) for i in range(len(J))] + C = [[Sjj[i][j] - sum(Wt[i][l] * Sjp[j][l] for l in range(6)) for j in range(len(J))] for i in range(len(J))] + # symmetric square root (Jacobi) + m = len(J) + A = [row[:] for row in C] + V = [[float(i == j) for j in range(m)] for i in range(m)] + for _ in range(100): + off = sum(A[p][q] ** 2 for p in range(m) for q in range(p + 1, m)) + if off < 1e-26: + break + for p in range(m - 1): + for q in range(p + 1, m): + if abs(A[p][q]) < 1e-300: + continue + th = 0.5 * math.atan2(2 * A[p][q], A[q][q] - A[p][p]) + c, s = math.cos(th), math.sin(th) + for kk in range(m): + akp, akq = A[kk][p], A[kk][q] + A[kk][p], A[kk][q] = c * akp - s * akq, s * akp + c * akq + for kk in range(m): + apk, aqk = A[p][kk], A[q][kk] + A[p][kk], A[q][kk] = c * apk - s * aqk, s * apk + c * aqk + for kk in range(m): + vkp, vkq = V[kk][p], V[kk][q] + V[kk][p], V[kk][q] = c * vkp - s * vkq, s * vkp + c * vkq + lam = [max(A[i][i], 0.0) for i in range(m)] + S = [[sum(V[i][kk] * math.sqrt(lam[kk]) * V[j][kk] for kk in range(m)) for j in range(m)] for i in range(m)] + fact = [math.factorial(i) for i in range(deg + 2)] + pts = [(-2 * r + 4 * r * i / (grid - 1), -2 * r + 4 * r * j / (grid - 1)) for i in range(grid) for j in range(grid)] + + def deriv(cf, a0, b0, x, y): + tot = 0.0 + for idx, (a, bb) in enumerate(jets): + if a >= a0 and bb >= b0: + tot += cf[idx] * x ** (a - a0) * y ** (bb - b0) / (fact[a - a0] * fact[bb - b0]) + return tot + rng = random.Random(seed) + num = den = 0.0 + acap = 4 / (3 * k) + for _ in range(n): + z = [rng.gauss(0, 1) for _ in jets] + cf = [mu[i] + sum(S[i][j] * z[j] for j in range(m)) for i in range(m)] + hM = [deriv(cf, 2, 0, -r / 2, 0.0), deriv(cf, 1, 1, -r / 2, 0.0), deriv(cf, 0, 2, -r / 2, 0.0)] + hS = [deriv(cf, 2, 0, r / 2, 0.0), deriv(cf, 1, 1, r / 2, 0.0), deriv(cf, 0, 2, r / 2, 0.0)] + dM = hM[0] * hM[2] - hM[1] ** 2 + dS = hS[0] * hS[2] - hS[1] ** 2 + if not (hM[0] < 0 and dM > 0 and dS < 0): + continue + Wv = abs(dM * dS) + den += Wv + lamv = -hM[2] + M3 = max(abs(deriv(cf, a, 3 - a, x, y)) for (x, y) in pts for a in range(4)) + M4 = max(abs(deriv(cf, a, 4 - a, x, y)) for (x, y) in pts for a in range(5)) + if not (lamv > acap * r * M3 * M3 and r * M4 <= 3 * k / 10): + num += Wv + return {'r': r, 'b': b, 'k': k, 'n': n, 'seed': seed, 'QW_Gc_over_r3': num / den / r ** 3, 'Z_over_r2': den / n / r ** 2} + + +if __name__ == '__main__': + main() diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/engine_e3.py b/frontiers/c8_torus_vanishing_remainder_20261001/engine_e3.py new file mode 100644 index 000000000..ac809dbd3 --- /dev/null +++ b/frontiers/c8_torus_vanishing_remainder_20261001/engine_e3.py @@ -0,0 +1,1709 @@ +#!/usr/bin/env python3 +"""Explicit cap-route constants for Theorem A of [LP] in dimension d = 3, for the torus kernel K_L with L >= 10 in every +frame: for 0 < r <= r_*, b in [0, 1], k in [1/2, 2], + + 1 - p_r(b, k, R) <= Q^W(G_r^c) <= C3(r_*) r^3, Z_r / r^2 >= z3(r_*), + +with G_r the good event of [LP] section 7 / [CAP] (1) (partial-block operator norms on the cylinder +D = [-2r, 2r] x ball(0, 2r)) and C3(r_*), z3(r_*) certified here (NOTE.md Theorem E3). The proof bounds +Q^W(G_r^c) = E_Q[W_r 1{G_r^c}] / Z_r from above by an exact Gaussian regression near r = 0: + + * covariances of the eight preconditioned pins and the 31 targets are exact power series in r, obtained from the Taylor + coefficients of the kernel at 0 (every negative power cancels identically, checked); the torus remainder is bounded + uniformly in r and in the frame by Cauchy estimates on a polydisc in C^3; + * the failure event is reduced to {lambda_1 <= (4/(3k)) r M_3^2} u {r M_4 > 3k/10}; M_3 is bounded blockwise through exact + averaging identities of the pinned jets and M_4 through the Taylor series at the midpoint (Frobenius norms); + * the main term is integrated over the transverse Hessian B = -D_y^2 f(M) in eigenvalue coordinates after an exact + regression on its three entries, against an angle-free Gaussian majorant of its density; the remaining expectations are + closed-form incomplete Gaussian and chi moments plus Hoelder-controlled perturbations; + * the normalizer Z_r is bounded below through a Loewner-order coupling with the contact law, whose moments + E[det(B)^a tr(B)^c 1{B > 0}] are in closed form; the rare branches are Gaussian and chi tails. +All arithmetic is exact rational interval arithmetic (outward rounding to 2^-160); standard library only. +""" +import argparse +import json +import math +import multiprocessing +import os +import sys +from fractions import Fraction as Fr + +HERE = os.path.dirname(os.path.abspath(__file__)) +MUTANTS = ('no-delta', 'cap-half', 'no-zfloor-correction', 'no-om-term', 'no-lambda2') +MUT = None +PBITS = 160 +L_MIN = 10 +NX = 60 # Taylor order in x of the covariance series (exact); the rest is a bounded tail +NB = 24 # truncation order of the Taylor tails of the 4-jets at the midpoint (the rest is bounded) +X0 = Fr(1, 64) # default threshold for the quadratic near-branch analysis (records use COMBOS) +THETA = Fr(1, 32) # default splitting parameter in the max-moment lemma (records use COMBOS) +THETA_E = Fr(1, 16) # splitting parameter for the ||r nu - v||^2 factor +QS = (2, 4, 8, 16) # Hoelder exponents of the rare branches: E[W 1_E] <= ||W||_q P(E)^(1 - 1/q), smallest bound used +R0_COUPLING = 12 # radius of the Loewner coupling of the transverse Hessian with the contact law (floor, section 6) +R_STARS = (Fr(1, 4096), Fr(1, 2048), Fr(1, 1024), Fr(1, 512), Fr(1, 256), Fr(1, 128), Fr(1, 64)) +BAND_B = (Fr(0), Fr(1)) +BAND_K = (Fr(1, 2), Fr(2)) + + +def require(cond, msg): + if not cond: + print(json.dumps({'check': 'FAIL', 'reason': msg, 'mutant': MUT})) + sys.exit(1) + + +# ----------------------------------------------------------------------------- rational intervals (from Math-#195 / #203 / #206) +def rdown(x): + return Fr(x.numerator * (1 << PBITS) // x.denominator, 1 << PBITS) + + +def rup(x): + return Fr(-((-x.numerator) * (1 << PBITS) // x.denominator), 1 << PBITS) + + +class IV: + __slots__ = ('lo', 'hi') + + def __init__(self, lo, hi=None): + lo = Fr(lo) + hi = lo if hi is None else Fr(hi) + if hi < lo: + raise ValueError('empty interval') + self.lo, self.hi = rdown(lo), rup(hi) + + def __repr__(self): + return 'IV(%s, %s)' % (float(self.lo), float(self.hi)) + + def __add__(self, o): + o = o if isinstance(o, IV) else IV(o) + return IV(self.lo + o.lo, self.hi + o.hi) + __radd__ = __add__ + + def __neg__(self): + return IV(-self.hi, -self.lo) + + def __sub__(self, o): + return self + (-(o if isinstance(o, IV) else IV(o))) + + def __rsub__(self, o): + return IV(o) - self + + def __mul__(self, o): + o = o if isinstance(o, IV) else IV(o) + c = (self.lo * o.lo, self.lo * o.hi, self.hi * o.lo, self.hi * o.hi) + return IV(min(c), max(c)) + __rmul__ = __mul__ + + def inv(self): + if self.lo <= 0 <= self.hi: + raise ZeroDivisionError('interval contains zero') + return IV(1 / self.hi, 1 / self.lo) + + def __truediv__(self, o): + o = o if isinstance(o, IV) else IV(o) + return self * o.inv() + + def __rtruediv__(self, o): + return IV(o) * self.inv() + + def sq(self): + if self.lo >= 0: + return IV(self.lo * self.lo, self.hi * self.hi) + if self.hi <= 0: + return IV(self.hi * self.hi, self.lo * self.lo) + return IV(0, max(self.lo * self.lo, self.hi * self.hi)) + + def meet(self, o): + lo, hi = max(self.lo, o.lo), min(self.hi, o.hi) + if hi < lo: + raise ValueError('inconsistent enclosures') + return IV(lo, hi) + + def width(self): + return self.hi - self.lo + + def mid(self): + return (self.lo + self.hi) / 2 + + def contains(self, x): + return self.lo <= x <= self.hi + + +def isqrt_iv(x): + """Enclosure of sqrt(x) for an interval x with x.lo > 0.""" + if x.lo <= 0: + raise ValueError('sqrt of interval reaching zero: %r' % x) + + def lo_root(v): + n = v.numerator * (1 << (2 * PBITS)) // v.denominator + s = math.isqrt(n) + return Fr(s, 1 << PBITS) + + def hi_root(v): + n = -((-v.numerator * (1 << (2 * PBITS))) // v.denominator) + s = math.isqrt(n) + 1 + return Fr(s, 1 << PBITS) + return IV(lo_root(x.lo), hi_root(x.hi)) + + +def exp_neg_point(t): + """Enclosure of exp(-t) for a rational t >= 0: halve t to t <= 1/2, evaluate the alternating series in fixed point + (ulp = 2^-(PBITS+40)) with an integer bound on the accumulated rounding error, then square back.""" + t = Fr(t) + m = 0 + while t > Fr(1, 2): + t /= 2 + m += 1 + P2 = PBITS + 40 + ONE = 1 << P2 + X = t.numerator * ONE // t.denominator # X / ONE <= t < (X + 1) / ONE + # series at x_lo = X / ONE: exp(-x_lo) = sum (-1)^n x_lo^n / n! (alternating, terms decreasing for x_lo <= 1/2) + S = ONE + T = ONE + E = 0 # error bound of T in ulps + Etot = 0 + n = 0 + while T > 0: + n += 1 + T = (T * X) // (ONE * n) + E = (E * X) // (ONE * n) + 2 + S += (-1) ** n * T + Etot += E + Etot += E + 1 # first omitted exact term is below (T_n + E) <= E + 1 ulps + # exp(-t) in [exp(-(X + 1)/ONE), exp(-X/ONE)] and exp(-(X + 1)/ONE) >= exp(-X/ONE) - 1/ONE (Lipschitz 1) + v = IV(Fr(S - Etot - 1, ONE), Fr(S + Etot, ONE)) + for _ in range(m): + v = v.sq() + return v + + +def exp_neg(t): + """exp(-t) on an interval t with t.lo >= 0 (monotone decreasing).""" + return IV(exp_neg_point(t.hi).lo, exp_neg_point(t.lo).hi) + + +def _atan_inv(n): + """Enclosure of arctan(1/n), n >= 2, alternating series.""" + x = Fr(1, n) + s = Fr(0) + term = x + k = 0 + while True: + s += term / (2 * k + 1) + k += 1 + term = term * (-x * x) + if abs(term) / (2 * k + 1) < Fr(1, 1 << (PBITS + 8)): + break + return IV(s - abs(term), s + abs(term)) + + +PI = 4 * (4 * _atan_inv(5) - _atan_inv(239)) +SQRT2 = isqrt_iv(IV(2)) +SQRT2PI = isqrt_iv(2 * PI) + + +def phi_point(x): + """Enclosure of Phi(x) = P(N(0,1) <= x) for a rational x with |x| <= 12, from the series + Phi(x) = 1/2 + (1/sqrt(2 pi)) sum_n (-1)^n x^(2n+1) / (2^n n! (2n+1)), evaluated in fixed point (ulp = 2^-(PBITS+40)) + with an explicit integer bound E on the accumulated rounding error (in ulps) and the alternating-tail bound.""" + x = Fr(x) + if x < 0: + p = phi_point(-x) + return IV(1 - p.hi, 1 - p.lo) + if x > 12: + raise ValueError('phi_point range') + P2 = PBITS + 40 + ONE = 1 << P2 + X = x.numerator * ONE // x.denominator # x_lo = X / ONE <= x < (X + 1) / ONE ; |Phi(x) - Phi(x_lo)| <= 1 ulp + if X == 0: # 0 <= x < 1 ulp: Phi(x) in [1/2, 1/2 + ulp] + return IV(Fr(1, 2), Fr(1, 2) + Fr(1, ONE)) + X2u = (X + 1) * (X + 1) # upper bound of x^2 ONE^2 + S = 0 + T = X # T_n ~ x^(2n+1) / (2^n n!) in ulps + E = 1 # error bound of T_n in ulps + Etot = 1 # error of S in ulps (starts with the x rounding) + n = 0 + while T > 0: + S += (-1) ** n * (T // (2 * n + 1)) + Etot += E + 1 + n += 1 + T = (T * X * X) // (ONE * ONE * 2 * n) + E = (E * X2u) // (ONE * ONE * 2 * n) + 2 + # once T == 0 in fixed point the remaining terms are each below 1 ulp; if x^2 >= 2 n they are still growing in + # exact arithmetic, so continue counting until they decrease: bound the tail by the first omitted exact term + # <= (T_n + E) ulps <= (E + 1) ulps when T_n == 0 ... but growth may resume; require the decreasing regime. + if x * x > 2 * n: + raise ValueError('series terminated before the decreasing regime') + Etot += E + 1 + core = IV(Fr(S - Etot, ONE), Fr(S + Etot, ONE)) / SQRT2PI + return IV(Fr(1, 2) + core.lo, Fr(1, 2) + core.hi) + + +def Phi(x): + """Phi on an interval (monotone); arguments clipped to [-12, 12] (clipping only widens).""" + lo = min(max(x.lo, Fr(-12)), Fr(12)) + hi = max(min(x.hi, Fr(12)), Fr(-12)) + if x.lo < -12: + plo = Fr(0) + else: + plo = phi_point(lo).lo + if x.hi > 12: + phi_ = Fr(1) + else: + phi_ = phi_point(hi).hi + return IV(plo, phi_) + + +def phi_pt(x): + """phi(x) for rational x (interval).""" + x = Fr(x) + return exp_neg_point(x * x / 2) / SQRT2PI + + +def Phibar_pt(x): + """1 - Phi(x) for rational x (interval).""" + x = Fr(x) + if abs(x) <= 12: + p = phi_point(x) + return IV(1 - p.hi, 1 - p.lo) + if x > 0: + return IV(0, (phi_pt(x) / x).hi) # Mills: Phibar(x) <= phi(x)/x + return IV(1 - (phi_pt(-x) / (-x)).hi, 1) + + +def dfact(n): + """(n)!! for n >= -1.""" + return math.prod(range(n, 0, -2)) if n > 0 else 1 + + +def gauss_moment(j): + """E[Z^j] for Z ~ N(0, 1).""" + return 0 if j % 2 else dfact(j - 1) + + +def upper_moments(x, J): + """[M_0(x), ..., M_J(x)], M_j(x) = integral_x^inf t^j phi(t) dt, for rational x (intervals). For x < 0, + M_j(x) = E[Z^j] - (-1)^j M_j(-x).""" + x = Fr(x) + if x < 0: + M = upper_moments(-x, J) + return [IV(gauss_moment(j)) - M[j] * (-1) ** j for j in range(J + 1)] + ph = phi_pt(x) + M = [Phibar_pt(x), ph] + for j in range(2, J + 1): + M.append(ph * x ** (j - 1) + M[j - 2] * (j - 1)) + return M[:J + 1] + + +def rup_rel(x, bits=200): + """Round a rational x >= 0 up to a dyadic with `bits` significant bits (upper bounds stay upper bounds).""" + x = Fr(x) + if x <= 0: + return x + e = x.numerator.bit_length() - x.denominator.bit_length() + sh = bits - e + if sh >= 0: + return Fr(-((-x.numerator << sh) // x.denominator), 1 << sh) + return Fr(-((-x.numerator) // (x.denominator << -sh)) << -sh) + + +def rdown_rel(x, bits=200): + """Round a rational x >= 0 down to a dyadic with `bits` significant bits.""" + x = Fr(x) + if x <= 0: + return x + e = x.numerator.bit_length() - x.denominator.bit_length() + sh = bits - e + if sh >= 0: + return Fr((x.numerator << sh) // x.denominator, 1 << sh) + return Fr((x.numerator // (x.denominator << -sh)) << -sh) + + +def sqrt_up(x): + """Rational upper bound of sqrt(x), x >= 0 rational, relative error below 2^-200 (no absolute rounding floor).""" + x = Fr(x) + if x == 0: + return Fr(0) + n, d = x.numerator, x.denominator + nd = n * d + m = max(0, (420 - nd.bit_length()) // 2 + 1) + return rup_rel(Fr(math.isqrt(nd << (2 * m)) + 1, d << m)) + + +def sqrt_down(x): + """Rational lower bound of sqrt(x), x >= 0 rational.""" + x = Fr(x) + if x <= 0: + return Fr(0) + n, d = x.numerator, x.denominator + nd = n * d + m = max(0, (420 - nd.bit_length()) // 2 + 1) + return rdown_rel(Fr(math.isqrt(nd << (2 * m)), d << m)) + + +def iroot(x, p): + """Upper bound (rational) of x^(1/p) for rational x >= 0 and p a power of two.""" + v = Fr(x) + while p > 1: + v = sqrt_up(v) + p //= 2 + return v + + +def pow2(p): + q = 1 + while q < p: + q *= 2 + return q + + +ZN = {} + + +def znorm(p): + """Upper bound of ||N(0,1)||_p for p a power of two (p = 1 uses sqrt(2/pi) < 4/5).""" + if p == 1: + return Fr(4, 5) + if p not in ZN: + ZN[p] = iroot(Fr(dfact(p - 1)), p) + return ZN[p] + + +def znorm1(p): + """max(1, ||N(0,1)||_p): the factor in ||(|mu| + ||Sigma^(1/2) z||)||_p <= |mu| + sqrt(tr Sigma) znorm1(p).""" + return Fr(1) if p <= 2 else znorm(p) + + +def frob_norm(mu_abs, tr_hi, p): + """Upper bound of || ||X|| ||_p for a Gaussian vector X with ||E X|| <= mu_abs and tr Cov X <= tr_hi (Minkowski in + L^(p/2) on the eigen-decomposition for p >= 2; Lyapunov for p <= 2).""" + return rup_rel(mu_abs + sqrt_up(tr_hi) * znorm1(pow2(p))) + + +def gamma_half(n2): + """Gamma(n2 / 2) for an integer n2 >= 1, as an interval (sqrt(pi) for odd n2).""" + if n2 % 2 == 0: + return IV(math.factorial(n2 // 2 - 1)) + n = (n2 - 1) // 2 # Gamma(n + 1/2) = (2n - 1)!! sqrt(pi) / 2^n + return isqrt_iv(PI) * Fr(dfact(2 * n - 1), 2 ** n) + + +def chi_moment(nu, j): + """E[chi_nu^j] = 2^(j/2) Gamma((nu + j)/2) / Gamma(nu/2) (interval), integer j >= 0.""" + if j % 2 == 0: + return IV(math.prod(nu + 2 * i for i in range(j // 2))) + out = gamma_half(nu + j) / gamma_half(nu) + for _ in range(j): + out = out * SQRT2 + return out + + +CN = {} + + +def chi_norm(nu, p): + """Upper bound of ||chi_nu||_p for p a power of two.""" + if p == 1: + return sqrt_up(Fr(nu)) + key = (nu, p) + if key not in CN: + CN[key] = iroot(Fr(math.prod(nu + 2 * i for i in range(p // 2))), p) + return CN[key] + + +CC = {} + + +def chi_const(nu): + """C_nu sqrt(2 pi) with C_nu = 1 / (2^(nu/2 - 1) Gamma(nu/2)): E[chi_nu^j 1{chi > x}] = C_nu sqrt(2 pi) M_(j+nu-1)(x).""" + if nu not in CC: + p2 = IV(Fr(2) ** ((nu - 2) // 2)) if nu % 2 == 0 else IV(Fr(2) ** ((nu - 3) // 2)) * SQRT2 + CC[nu] = SQRT2PI / (p2 * gamma_half(nu)) + return CC[nu] + + +def chi_upper_moments(nu, x, J): + """[E[chi_nu^j 1{chi_nu > x}] for j = 0..J] (intervals), rational x >= 0.""" + M = upper_moments(x, J + nu - 1) + c = chi_const(nu) + return [M[j + nu - 1] * c for j in range(J + 1)] + + +def chi_tail(nu, x): + """Rational upper bound of P(chi_nu > x) for rational x >= 0 (Mills-type bound beyond the series range).""" + x = Fr(x) + if x <= 0: + return Fr(1) + if x <= 12: + return min(Fr(1), chi_upper_moments(nu, x, 0)[0].hi) + # x > 12: P(chi_nu > x) = c_nu sqrt(2pi) M_(nu-1)(x), M_j(x) <= phi(x) x^(j-1) (1 + 2 (j-1)/x^2) for x^2 >= 2 j, and + # phi(x) <= m!/(x^2/2)^m / sqrt(2 pi) (m = 40) + y = x * x / 2 + ph = Fr(math.factorial(40)) / y ** 40 * Fr(2, 5) # 1/sqrt(2 pi) < 2/5 + j = nu - 1 + Mj = ph * (x ** (j - 1) if j >= 1 else 1 / x) * (1 + Fr(2 * max(j - 1, 1)) / (x * x)) + return min(Fr(1), Mj * Fr(51, 20)) # c_nu sqrt(2 pi) <= sqrt(2 pi) < 51/20 + + +def phibar_upper(x): + """Rational upper bound of 1 - Phi(x) for rational x >= 0: the series value for x <= 12, and for x > 12 the Mills bound + phi(x)/x with exp(-y) <= m!/y^m (m = 40), which stays exact far below the 2^-160 rounding floor.""" + x = Fr(x) + if x <= 12: + return Phibar_pt(x).hi + y = x * x / 2 + m = 40 + return Fr(math.factorial(m)) / y ** m / (x * Fr(5, 2)) # sqrt(2 pi) > 5/2 + + +def gauss_tail2(mu_abs, sd, t): + """Upper bound of P(|X| > t) for X Gaussian with |mean| <= mu_abs, sd <= sd.""" + if t <= mu_abs: + return Fr(1) + return min(Fr(1), 2 * phibar_upper((t - mu_abs) / sd)) + + +def gauss_tail1(mu_abs, sd, t): + if t <= mu_abs: + return Fr(1) + return min(Fr(1), phibar_upper((t - mu_abs) / sd)) + + +def gauss_norm(mu_abs, sd, p): + """Upper bound of ||X||_p for X Gaussian with |mean| <= mu_abs and sd <= sd.""" + return rup_rel(mu_abs + sd * znorm(pow2(p))) + + +def absup(x): + return max(abs(x.lo), abs(x.hi)) + + +def pow_tail(x, q): + """Upper bound of x^(1 - 1/q) for 0 <= x <= 1 (q a power of two).""" + if x <= 0: + return Fr(0) + return iroot(rup_rel(x, 64) ** (q - 1), q) + + +# ----------------------------------------------------------------------------- functionals and exact covariance series (d = 3) +HALF = Fr(1, 2) +XM, XS, X0PT = -HALF, HALF, Fr(0) +JET4_IDX = {'X%d%d%d' % (a, b1, 4 - a - b1): (a, b1, 4 - a - b1) for a in range(5) for b1 in range(5 - a)} +JET4 = tuple(sorted(JET4_IDX, key=lambda n: (-JET4_IDX[n][0], -JET4_IDX[n][1]))) +BLOCKS = ((4, 0), (3, 1), (2, 2), (1, 3), (0, 4)) # partial blocks (a, c) of the fourth derivatives, a + c = 4 + + +def block_members(a, c): + """[(name, Frobenius multiplicity c!/beta!)] of the 4-jet block d_x^a D_y^c f(0).""" + out = [] + for b1 in range(c, -1, -1): + b2 = c - b1 + out.append(('X%d%d%d' % (a, b1, b2), math.factorial(c) // (math.factorial(b1) * math.factorial(b2)))) + return out + + +def term(c, p, a, b1, b2, x0): + """c r^p d_x^a d_y1^b1 d_y2^b2 f(x0 r, 0, 0).""" + return (Fr(c), p, a, b1, b2, Fr(x0)) + + +def functionals(): + pins = { + 'p1': [term(HALF, 0, 0, 0, 0, XM), term(HALF, 0, 0, 0, 0, XS)], + 'p2': [term(-1, -1, 0, 0, 0, XM), term(1, -1, 0, 0, 0, XS)], + 'p3p': [term(6, -2, 1, 0, 0, XM), term(6, -2, 1, 0, 0, XS), term(12, -3, 0, 0, 0, XM), term(-12, -3, 0, 0, 0, XS)], + 'p4': [term(-1, -1, 1, 0, 0, XM), term(1, -1, 1, 0, 0, XS)], + } + for j, (e1, e2) in ((1, (1, 0)), (2, (0, 1))): + pins['q1%d' % j] = [term(HALF, 0, 0, e1, e2, XM), term(HALF, 0, 0, e1, e2, XS)] + pins['q2%d' % j] = [term(-1, -1, 0, e1, e2, XM), term(1, -1, 0, e1, e2, XS)] + targets = { + 'a11': [term(1, 0, 0, 2, 0, XM)], 'a22': [term(1, 0, 0, 0, 2, XM)], 'a12': [term(1, 0, 0, 1, 1, XM)], + 'o11': [term(1, -1, 0, 2, 0, XS), term(-1, -1, 0, 2, 0, XM)], + 'o22': [term(1, -1, 0, 0, 2, XS), term(-1, -1, 0, 0, 2, XM)], + 'o12': [term(1, -1, 0, 1, 1, XS), term(-1, -1, 0, 1, 1, XM)], + 'T': [term(1, -2, 2, 0, 0, XM), term(4, -3, 1, 0, 0, XM), term(2, -3, 1, 0, 0, XS), term(6, -4, 0, 0, 0, XM), + term(-6, -4, 0, 0, 0, XS)], + 'tau': [term(1, -3, 2, 0, 0, XS), term(-1, -3, 2, 0, 0, XM), term(-6, -4, 1, 0, 0, XM), term(-6, -4, 1, 0, 0, XS), + term(-12, -5, 0, 0, 0, XM), term(12, -5, 0, 0, 0, XS)], + } + for j, (e1, e2) in ((1, (1, 0)), (2, (0, 1))): + targets['v%d' % j] = [term(1, -1, 1, e1, e2, XM), term(1, -2, 0, e1, e2, XM), term(-1, -2, 0, e1, e2, XS)] + targets['nu%d' % j] = [term(1, -2, 1, e1, e2, XM), term(1, -2, 1, e1, e2, XS), term(2, -3, 0, e1, e2, XM), + term(-2, -3, 0, e1, e2, XS)] + for b1 in range(3, -1, -1): + targets['y%s' % ('1' * b1 + '2' * (3 - b1))] = [term(1, 0, 0, b1, 3 - b1, X0PT)] + for n, (a, b1, b2) in JET4_IDX.items(): + targets[n] = [term(1, 0, a, b1, b2, X0PT)] + return pins, targets + + +PIN_NAMES = ('p1', 'p2', 'p3p', 'p4', 'q11', 'q21', 'q12', 'q22') +BNAMES = ('a11', 'a22', 'a12') # the transverse Hessian A = D_y^2 f(M) = -B +VNAMES = ('v1', 'v2') +ONAMES = ('o11', 'o22', 'o12') +YNAMES = ('y111', 'y112', 'y122', 'y222') +FREE = VNAMES + ONAMES + YNAMES # the free third-derivative blocks, in Cholesky order +FREE_W = (1, 1, 1, 1, 2, 1, 3, 3, 1) # Frobenius multiplicities (Euclidean norms in these weights) +TARGET_NAMES = BNAMES + FREE + ('T', 'tau', 'nu1', 'nu2') + JET4 +NP = len(PIN_NAMES) + + +def weight(F): + """Scaling weight a - (power of r), common to all terms of a preconditioned functional.""" + ws = {t[2] - t[1] for t in F} + require(len(ws) == 1, 'functional not homogeneous') + return ws.pop() + + +def hcoef(n): + """d^n/dx^n exp(-x^2/2) at 0.""" + if n % 2: + return 0 + return (-1) ** (n // 2) * dfact(n - 1) + + +PAIR_CACHE = {} + + +def pair_series(F, G, key): + """Exact data of Cov_K(F, G) = sum_gamma kappa_gamma pi_gamma r^(gx - wF - wG), gamma = (gx, gy1, gy2): + returns (reference series {power: rational}, weight w, list of (|c c'|, a+a', gy1, gy2) for the remainder bounds).""" + if key in PAIR_CACHE: + return PAIR_CACHE[key] + wF, wG = weight(F), weight(G) + w = wF + wG + pi = {} + absterms = [] + for (cf, pf, af, bf1, bf2, xf) in F: + for (cg, pg, ag, bg1, bg2, xg) in G: + g1, g2 = bf1 + bg1, bf2 + bg2 + sgn = (-1) ** (ag + bg1 + bg2) + d = xf - xg + absterms.append((abs(cf * cg), af + ag, g1, g2)) + for gx in range(af + ag, NX + 1): + m = gx - af - ag + val = cf * cg * sgn * d ** m / math.factorial(m) + if val != 0: + pi[(gx, g1, g2)] = pi.get((gx, g1, g2), Fr(0)) + val + for (gx, g1, g2), v in pi.items(): + require(v == 0 or gx - w >= 0, 'negative power of r in a covariance (regularity fails)') + ser = {} + for (gx, g1, g2), v in pi.items(): + kap = hcoef(gx) * hcoef(g1) * hcoef(g2) + if kap and v: + ser[gx - w] = ser.get(gx - w, Fr(0)) + kap * v + out = ({p: c for p, c in ser.items() if c != 0}, w, absterms) + PAIR_CACHE[key] = out + return out + + +E_UP = Fr(27183, 10000) # > e +SQRT3_UP = Fr(7, 4) # > sqrt(3) + + +def torus_M1(L): + """Bound of |sum_{n != 0} G(z + p_n)| on the polydisc |z_1|, |z_2|, |z_3| <= 1 (every frame, |p_n| = |n| L): + exp(3/2) sum_{j >= 1} (24 j^2 + 2) exp(-(j L - sqrt 3)^2 / 2); 24 j^2 + 2 lattice points have sup-norm j.""" + L = Fr(L) + e32 = exp_neg_point(Fr(3, 2)).inv().hi + tot = Fr(0) + for j in range(1, 6): + tot += (24 * j * j + 2) * exp_neg_point((j * L - SQRT3_UP) ** 2 / 2).hi + # j >= 6: (24 j^2 + 2) exp(-(jL - 7/4)^2/2) decreases by more than 1/2 per step for L >= 10 + tail = 2 * (24 * 36 + 2) * exp_neg_point((6 * L - SQRT3_UP) ** 2 / 2).hi + return e32 * (tot + tail) + + +def theta_minus_one(L): + L = Fr(L) + tot = Fr(0) + for j in range(1, 6): + tot += (24 * j * j + 2) * exp_neg_point((j * L) ** 2 / 2).hi + return tot + 2 * (24 * 36 + 2) * exp_neg_point((6 * L) ** 2 / 2).hi + + +def cov_band(F, G, key, r0, r1, M1): + """Enclosure of the covariance (unnormalized torus kernel sum_n G(z + p_n)) of F and G, for r in [r0, r1].""" + ser, w, absterms = pair_series(F, G, key) + val = IV(0) + for p, c in ser.items(): + val = val + IV(r0 ** p, r1 ** p) * c + A = max((t[1] for t in absterms), default=0) + # Taylor truncation tail of the reference kernel (orders gx > NX) + tail = Fr(0) + for (acc, aa, g1, g2) in absterms: + tail += acc * abs(hcoef(g1) * hcoef(g2)) + tail *= 2 * Fr(NX + 1) ** A * r1 ** (NX + 1 - w) + # torus remainder: Cauchy estimates |rho_gamma| <= gx! gy1! gy2! M1 on the unit polydisc + tor = Fr(0) + for (acc, aa, g1, g2) in absterms: + tor += acc * math.factorial(g1) * math.factorial(g2) * Fr(max(w, 1)) ** aa + tor = tor * M1 / (1 - E_UP * r1) + err = tail + tor + return val + IV(-err, err) + + +# ----------------------------------------------------------------------------- interval linear algebra +def solve_iv(A, B): + """Interval Gaussian elimination with partial pivoting on midpoints: X[c] solves A X[c] = B[c].""" + n = len(A) + M = [[A[i][j] for j in range(n)] + [B[c][i] for c in range(len(B))] for i in range(n)] + for col in range(n): + p = max(range(col, n), key=lambda i: abs(M[i][col].mid())) + M[col], M[p] = M[p], M[col] + piv = M[col][col] + for i in range(col + 1, n): + f = M[i][col] / piv + for j in range(col, len(M[i])): + M[i][j] = M[i][j] - f * M[col][j] + X = [] + for c in range(len(B)): + x = [None] * n + for i in reversed(range(n)): + s = M[i][n + c] + for j in range(i + 1, n): + s = s - M[i][j] * x[j] + x[i] = s / M[i][i] + X.append(x) + return X + + +def cholesky(C): + """Interval Cholesky of an n x n SPD interval matrix (pivots must be positive).""" + n = len(C) + L = [[IV(0)] * n for _ in range(n)] + for i in range(n): + for j in range(i + 1): + s = C[i][j] - sum((L[i][l] * L[j][l] for l in range(j)), IV(0)) + if i == j: + require(s.lo > 0, 'Cholesky pivot not positive') + L[i][j] = isqrt_iv(s) + else: + L[i][j] = s / L[j][j] + return L + + +def opnorm_lower_tri(L, rows, cols): + """Upper bound of the operator norm of the block L[rows][cols] of an interval lower-triangular matrix: + max |diagonal| + Frobenius norm of the rest (square diagonal blocks), Frobenius norm otherwise.""" + if rows == cols: + dg = max(absup(L[i][i]) for i in rows) + off = sum(absup(L[i][j]) ** 2 for i in rows for j in cols if j < i) + return rup_rel(dg + sqrt_up(off)) + return sqrt_up(sum(absup(L[i][j]) ** 2 for i in rows for j in cols)) + + +def sym3_eig_bounds(C): + """(lower bound of lambda_min, upper bound of lambda_max) of a symmetric 3 x 3 interval matrix (Gershgorin).""" + lo = min(C[i][i].lo - sum(absup(C[i][j]) for j in range(3) if j != i) for i in range(3)) + hi = max(C[i][i].hi + sum(absup(C[i][j]) for j in range(3) if j != i) for i in range(3)) + return lo, hi + + +def det3(C): + a, b, c = C[0] + d, e, f = C[1] + g, h, i = C[2] + return a * (e * i - f * h) - b * (d * i - f * g) + c * (d * h - e * g) + + +# ----------------------------------------------------------------------------- the conditional law on an r-band +class Law: + pass + + +def band_law(r0, r1, L=L_MIN): + """Interval enclosures, uniform over r in [r0, r1], every frame and every torus side >= L, of the Q-law of the + targets: mean = b alpha + k beta, covariance C (normalized kernel), and the pin energy coefficients; then the + regression on the transverse Hessian (three entries) and the g-law of the remaining targets.""" + pins, targets = functionals() + P = [pins[n] for n in PIN_NAMES] + T = [targets[n] for n in TARGET_NAMES] + M1 = torus_M1(L) + th1 = theta_minus_one(L) + nt = len(T) + Spp = [[None] * NP for _ in range(NP)] + for i in range(NP): + for j in range(i, NP): + Spp[i][j] = Spp[j][i] = cov_band(P[i], P[j], ('p', i, j), r0, r1, M1) + Stp = [[cov_band(T[i], P[j], ('tp', i, j), r0, r1, M1) for j in range(NP)] for i in range(nt)] + Stt = [[None] * nt for _ in range(nt)] + for i in range(nt): + for j in range(i, nt): + Stt[i][j] = Stt[j][i] = cov_band(T[i], T[j], ('tt', i, j), r0, r1, M1) + r = IV(r0, r1) + kvec = [-(r * r * r) * HALF, -(r * r), IV(12)] + [IV(0)] * (NP - 3) + e1 = [IV(1)] + [IV(0)] * (NP - 1) + W = solve_iv(Spp, Stp) + xs = solve_iv(Spp, [e1, kvec]) + law = Law() + law.r0, law.r1, law.L, law.M1, law.th1 = r0, r1, L, M1, th1 + law.alpha = {n: W[i][0] for i, n in enumerate(TARGET_NAMES)} + law.beta = {n: sum((W[i][j] * kvec[j] for j in range(NP)), IV(0)) for i, n in enumerate(TARGET_NAMES)} + scale = IV(1 / (1 + th1), 1) # K_L = (unnormalized) / Theta, Theta in [1, 1 + th1] + C = {} + for i, a in enumerate(TARGET_NAMES): + for j in range(i, nt): + c = TARGET_NAMES[j] + v = Stt[i][j] - sum((W[i][l] * Stp[j][l] for l in range(NP)), IV(0)) + C[(a, c)] = C[(c, a)] = v * scale + law.C = C + escale = IV(1, 1 + th1) + law.e_bb = xs[0][0] * escale + law.e_bk = sum((kvec[j] * xs[0][j] for j in range(NP)), IV(0)) * escale + law.e_kk = sum((kvec[j] * xs[1][j] for j in range(NP)), IV(0)) * escale + regress_on_B(law) + law.S = {blk: taylor_tail_coefficient(blk, r1, M1) for blk in BLOCKS} + law.net_var = {blk: net_max_variance(law.C, blk) for blk in BLOCKS if blk[1] >= 2} + return law + + +NET_M = 128 # rational unit vectors ((1 - t^2), 2 t)/(1 + t^2), t = j/NET_M, and their rotations + + +def circle_net(): + """Rational unit vectors u_j covering angles [0, pi) with consecutive angular gaps <= 2/NET_M (theta = 2 arctan t has + d theta/dt <= 2), together with the rotated copies (-u2, u1).""" + pts = [] + for j in range(NET_M + 1): + t = Fr(j, NET_M) + pts.append(((1 - t * t) / (1 + t * t), 2 * t / (1 + t * t))) + return pts + [(-u2, u1) for (u1, u2) in pts] + + +NET = circle_net() + + +def block_form_weights(blk, u): + """w_beta = (c!/beta!) u1^beta1 u2^beta2: d_x^a d_u^c f(0) = sum_beta w_beta X_(a,beta).""" + a, c = blk + return [(n, m * u[0] ** int(n[2]) * u[1] ** int(n[3])) for n, m in block_members(a, c)] + + +def net_max_variance(C, blk): + """Upper bound of max_j Var_Q(d_x^a d_(u_j)^c f(0)) over the net.""" + best = Fr(0) + for u in NET: + w = block_form_weights(blk, u) + v = sum(abs(wi) * abs(wj) * absup(C[(ni, nj)]) if ni != nj else wi * wi * C[(ni, ni)].hi + for ni, wi in w for nj, wj in w) + best = max(best, v) + return best + + +FROB_B = (IV(1), IV(1), SQRT2) # (B11, B22, B12) -> Frobenius-isometric coordinates (B11, B22, sqrt2 B12) + + +def regress_on_B(law): + """Regression of every target on the three entries of the transverse Hessian, in the Frobenius-isometric coordinates + b~ = (B11, B22, sqrt2 B12) of B = -A: F = F_g + beta~_F . (b~ - mu~), with F_g independent of B.""" + C = law.C + # covariance of b~ (B = -A: signs cancel in the covariance) + Cb = [[C[(BNAMES[i], BNAMES[j])] * FROB_B[i] * FROB_B[j] for j in range(3)] for i in range(3)] + law.Cb = Cb + law.detCb = det3(Cb) + require(law.detCb.lo > 0, 'transverse Hessian covariance not positive') + lo, hi = sym3_eig_bounds(Cb) + require(lo > 0, 'transverse Hessian covariance eigenvalue bound') + law.sB_lo2, law.sB_hi2 = lo, hi + law.sB_lo = sqrt_down(lo) + # regression coefficients of F on b~: beta~_F = Cov(F, b~) Cb^-1, Cov(F, b~_i) = -Cov(F, A_i) FROB_B[i] + others = [n for n in TARGET_NAMES if n not in BNAMES] + rhs = [[-C[(n, BNAMES[i])] * FROB_B[i] for i in range(3)] for n in others] + sol = solve_iv(Cb, rhs) + law.bt = {n: sol[i] for i, n in enumerate(others)} + law.bnorm = {n: sqrt_up(sum(absup(x) ** 2 for x in law.bt[n])) for n in others} + Cg = {} + for a in others: + for c in others: + Cg[(a, c)] = C[(a, c)] - sum((law.bt[a][i] * (-C[(c, BNAMES[i])] * FROB_B[i]) for i in range(3)), IV(0)) + law.Cg = Cg + # the free blocks in Frobenius coordinates: v, omega = (o11, o22, sqrt2 o12), Y~ = (y111, sqrt3 y112, sqrt3 y122, y222) + fw = [isqrt_iv(IV(w)) if w != 1 else IV(1) for w in FREE_W] + law.Lfree = cholesky([[Cg[(FREE[i], FREE[j])] * fw[i] * fw[j] for j in range(9)] for i in range(9)]) + law.fw = fw + law.sdg = {n: sqrt_up(max(Cg[(n, n)].hi, Fr(0))) for n in others} + law.sdQ = {n: sqrt_up(C[(n, n)].hi) for n in TARGET_NAMES} + + +def sd0_sq(g, M1): + """Upper bound of Var d^g f(0) = |d^(2g) K_L(0)| (normalized kernel, every frame, L >= L_MIN), g = (gx, gy1, gy2): + the reference value prod (2 g_i - 1)!! plus the Cauchy bound (2g)! M1 of the torus remainder.""" + return math.prod(dfact(2 * x - 1) for x in g) + math.prod(math.factorial(2 * x) for x in g) * M1 + + +def taylor_tail_coefficient(blk, r1, M1): + """S_(a,c)(r1) = sum_{gamma != 0} sigma_(a,c),gamma (2 r1)^|gamma| / gamma!, sigma^2 = sum_beta (c!/beta!) s^2_(a+gx, beta+gy): + the Frobenius norm of the Taylor tail of the block d_x^a D_y^c f at the midpoint over D.""" + a, c = blk + tot = Fr(0) + for n in range(1, NB + 1): + for gx in range(n + 1): + for g1 in range(n - gx + 1): + g2 = n - gx - g1 + s2 = Fr(0) + for b1 in range(c + 1): + b2 = c - b1 + mult = math.factorial(c) // (math.factorial(b1) * math.factorial(b2)) + s2 += mult * sd0_sq((a + gx, b1 + g1, b2 + g2), M1) + tot += sqrt_up(s2) * (2 * r1) ** n / (math.factorial(gx) * math.factorial(g1) * math.factorial(g2)) + # tail |gamma| = n > NB: s_delta <= 2^|delta| delta! sqrt(1 + M1) and (alpha + gamma)!/gamma! <= (n + 4)^4, so + # sigma (2 r1)^n / gamma! <= 2^(c/2) 2^(4 + n) (n + 4)^4 (2 r1)^n sqrt(1 + M1); (n+1)(n+2)/2 multi-indices of order n; + # the ratio of consecutive bounds is below 2 q for these n (q = 4 r1 <= 1/16) + q = 4 * r1 + require(q <= Fr(1, 16), 'Taylor tail ratio too large') + n = NB + 1 + base = 2 * 2 ** 4 * sqrt_up(Fr(2) ** c * (1 + M1)) + tail = base * Fr((n + 1) * (n + 2), 2) * Fr(n + 4) ** 4 * q ** n / (1 - 2 * q) + return rup_rel(tot + tail) + + +# ----------------------------------------------------------------------------- closed forms for the contact transverse Hessian +# B = m I + GOE_2 (diagonal N(0, 2), off-diagonal N(0, 1)): its eigenvalues are s -+ rho with s = m + (G11 + G22)/2 ~ N(m, 1) +# and rho = sqrt(((G11 - G22)/2)^2 + G12^2) Rayleigh, independent; det B = s^2 - rho^2, tr B = 2 s, B > 0 iff s > rho. +def upper_moments_iv(x, J): + """[M_0(x), ..., M_J(x)] over an interval x whose interior does not contain 0 (each M_j is monotone there).""" + if x.lo == x.hi: + return upper_moments(x.lo, J) + require(not (x.lo < 0 < x.hi), 'incomplete moment over an interval containing 0') + A, B = upper_moments(x.lo, J), upper_moments(x.hi, J) + return [IV(min(a.lo, b.lo), max(a.hi, b.hi)) for a, b in zip(A, B)] + + +def es_pos(m, n): + """[E[s^j 1{s > 0}] for j = 0..n], s ~ N(m, 1), rational m (intervals).""" + M = upper_moments(-Fr(m), n) + return [sum((M[i] * (math.comb(j, i) * Fr(m) ** (j - i)) for i in range(j + 1)), IV(0)) for j in range(n + 1)] + + +def isqrt2_pow(i): + """2^(-i/2) (interval).""" + return IV(Fr(1, 2 ** (i // 2))) if i % 2 == 0 else IV(Fr(1, 2 ** (i // 2))) / SQRT2 + + +def es_pos_exp(m, n): + """[E[s^j exp(-s^2/2) 1{s > 0}] for j = 0..n], s ~ N(m, 1), rational m: + = exp(-m^2/4)/sqrt2 . E[(m/2 + U/sqrt2)^j 1{U > -m/sqrt2}], U ~ N(0, 1).""" + m = Fr(m) + M = upper_moments_iv(-(IV(m) / SQRT2), n) + pre = exp_neg_point(m * m / 4) / SQRT2 + out = [] + for j in range(n + 1): + tot = IV(0) + for i in range(j + 1): + tot = tot + M[i] * (math.comb(j, i) * (m / 2) ** (j - i)) * isqrt2_pow(i) + out.append(pre * tot) + return out + + +def inner_rho_polys(a): + """integral_0^s (s^2 - rho^2)^a rho exp(-rho^2/2) d rho = P_a(s) - exp(-s^2/2) Q_a(s) (exact polynomial coefficient + dicts {power: Fraction}); with u = rho^2/2 it is integral_0^(s^2/2) (s^2 - 2u)^a e^(-u) du.""" + P, Q = {}, {} + for i in range(a + 1): + c = Fr(math.comb(a, i) * (-2) ** i * math.factorial(i)) + P[2 * (a - i)] = P.get(2 * (a - i), Fr(0)) + c + for l in range(i + 1): + pw = 2 * (a - i) + 2 * l + Q[pw] = Q.get(pw, Fr(0)) + c / (2 ** l * math.factorial(l)) + return P, Q + + +def contact_moment(a, c, m): + """E[det(B)^a tr(B)^c 1{B > 0}] for B = m I + GOE_2, rational m (interval).""" + P, Q = inner_rho_polys(a) + n = 2 * a + c + E1 = es_pos(m, n) + E2 = es_pos_exp(m, n) + tot = IV(0) + for pw, cf in P.items(): + tot = tot + E1[pw + c] * (cf * 2 ** c) + for pw, cf in Q.items(): + tot = tot - E2[pw + c] * (cf * 2 ** c) + return tot + + +def m3(m): + """m_(3,m) = E[det(B)^2 1{B > 0}] = E[(s^4 - 4 s^2 + 8) 1{s > 0}] - 4 sqrt2 exp(-m^2/4) Phi(m/sqrt2).""" + return contact_moment(2, 0, m) + + +def gauss_half_moment(j, m, sig): + """J_j(m, sig) = integral_0^inf lambda^j exp(-(lambda - m)^2/(2 sig^2)) d lambda for rational m, sig > 0 (interval): + sig sqrt(2 pi) E[(m + sig Z)^j 1{m + sig Z > 0}].""" + m, sig = Fr(m), Fr(sig) + M = upper_moments(-m / sig, j) + tot = sum((M[i] * (math.comb(j, i) * m ** (j - i) * sig ** i) for i in range(j + 1)), IV(0)) + return tot * SQRT2PI * sig + + +# ----------------------------------------------------------------------------- moment calculus (normalized by K = 12 k) +def binom(n, k): + return math.comb(n, k) + + +def pos_part_moments_chi(s, c, nu, nmax, extra_max): + """P[(n, e)] = upper bound of E[u^e (u^n - c^n)_+], u = s chi_nu, for 1 <= n <= nmax, 0 <= e <= extra_max.""" + x = c / s + M = chi_upper_moments(nu, x, nmax + extra_max) + out = {} + for n in range(1, nmax + 1): + for e in range(extra_max + 1): + out[(n, e)] = rup_rel((M[n + e] * s ** (n + e) - M[e] * c ** n * s ** e).hi) + return out + + +NU = {2: 2, 3: 3, 4: 4} # chi degrees of freedom of the free blocks v (2), omega (3), Y~ (4) + + +def normalized_moments(q, nmax, theta): + """Component moments, normalized by K = 12 k (K >= Klo), for n <= nmax, under the product of the g-law and nu: + E1[n], E1o[n], E1v[n] : upper bounds of E[y1^n], E[o0 y1^n], E[v0^2 y1^n], y1 = 1 + cmt1 + (r1/Klo) D1; + Tn[n], To[n], Tv[n] : upper bounds of sum_{i=2,3,4} E[(y_i^n - c^n)_+] (and with the factors o0, v0^2), + y_i <= u_i + delta_i (u_i = s_i chi_nu_i independent), threshold c = q['c']. + o0 = Klo u_3 and v0^2 = Klo^2 u_2^2 / 4 are unnormalized.""" + th = theta + cp = q['c'] * (1 - th) # <= c (1 + th)^{-(n-1)/n} for every n (Bernoulli) + PP = {i: pos_part_moments_chi(q['s'][i], cp, NU[i], nmax, 2) for i in (2, 3, 4)} + Eo0 = (chi_moment(3, 1) * q['s3']).hi + Ev0sq = (chi_moment(2, 2) * q['s2'] * q['s2']).hi / 4 + n_o0_2 = q['s3'] * chi_norm(3, 2) + n_v0sq_2 = q['s2'] * q['s2'] * chi_norm(2, 4) ** 2 / 4 + Klo = q['Klo'] + E1, E1o, E1v, Tn, To, Tv = {}, {}, {}, {}, {}, {} + base = rup_rel(1 + q['cmt1']) + bp = [Fr(1)] + for _ in range(nmax): + bp.append(rup_rel(bp[-1] * base)) + # (r/K) D1 terms: ||(rK D1)^j||_1 <= (rK ||D1||_pow2(j))^j and ||(rK D1)^j||_2 <= (rK ||D1||_pow2(2j))^j + d1 = [Fr(1)] + [rup_rel((q['rK'] * q['nD1'](pow2(j))) ** j) for j in range(1, nmax + 1)] + d2 = [Fr(1)] + [rup_rel((q['rK'] * q['nD1'](pow2(2 * j))) ** j) for j in range(1, nmax + 1)] + for n in range(0, nmax + 1): + ey = bp[n] + eo = bp[n] * Eo0 + ev = bp[n] * Ev0sq + for j in range(1, n + 1): + coef = binom(n, j) * bp[n - j] + ey += coef * d1[j] + eo += coef * n_o0_2 * d2[j] + ev += coef * n_v0sq_2 * d2[j] + E1[n], E1o[n], E1v[n] = rup_rel(ey), rup_rel(eo), rup_rel(ev) + if n == 0: + continue + f1 = (1 + th) ** (n - 1) + f2 = (1 + 1 / th) ** (n - 1) + tn = to = tv = Fr(0) + for i in (2, 3, 4): + P = PP[i] + dn = rup_rel(q['nd'](i, pow2(n)) ** n) + dn2 = rup_rel(q['nd'](i, pow2(2 * n)) ** n) + tn += f1 * P[(n, 0)] + f2 * dn + to += (f1 * Klo * P[(n, 1)] if i == 3 else f1 * Eo0 * P[(n, 0)]) + f2 * n_o0_2 * dn2 + tv += (f1 * Klo * Klo / 4 * P[(n, 2)] if i == 2 else f1 * Ev0sq * P[(n, 0)]) + f2 * n_v0sq_2 * dn2 + Tn[n], To[n], Tv[n] = rup_rel(tn), rup_rel(to), rup_rel(tv) + return E1, E1o, E1v, Tn, To, Tv + + +def half_gauss_moments(m, sig, J): + """[J_j(m, sig) for j = 0..J], J_j = integral_0^inf l^j exp(-(l - m)^2/(2 sig^2)) dl, rational m >= 0, sig > 0 (intervals): + J_0 = sig sqrt(2 pi) Phi(m/sig), J_1 = m J_0 + sig^2 exp(-m^2/(2 sig^2)), J_(j+1) = m J_j + j sig^2 J_(j-1) + (integration by parts; every term is positive for m >= 0).""" + m, sig = Fr(m), Fr(sig) + require(m >= 0 and sig > 0, 'half-line Gaussian moments') + s2 = sig * sig + out = [SQRT2PI * sig * (1 - Phibar_pt(m / sig))] + out.append(out[0] * m + exp_neg_point(m * m / (2 * s2)) * s2) + for j in range(1, J): + nxt = out[j] * m + out[j - 1] * (j * s2) + out.append(IV(rdown_rel(nxt.lo), rup_rel(nxt.hi))) + return out + + +def lam2_moments(b_lo, b_hi, eps, sig): + """Bounds on the lambda_2-integrals of the density majorant g(l) = exp(-(l - b)^2/(2 sig^2) + eps |l - b|/sig^2), + uniformly over b in [b_lo, b_hi] (b >= 0): I_hi[j] >= integral_0^inf l^j g(l) dl (j = 0..JM), I3_lo <= integral l^3 g. + For l >= 0, eps |l - b| <= eps (l + b), so g(l) <= exp((4 b eps + eps^2)/(2 sig^2)) exp(-(l - b - eps)^2/(2 sig^2)); + each J_j(m, sig) increases with m; and g(l) >= exp(-(l - b)^2/(2 sig^2)).""" + JM = 3 + 64 + shift = (4 * b_hi * eps + eps * eps) / (2 * sig * sig) + fac = exp_neg_point(shift).inv().hi + Jh = half_gauss_moments(b_hi + eps, sig, JM) + I_hi = [rup_rel(x.hi * fac) for x in Jh] + I3_lo = rdown_rel(half_gauss_moments(b_lo, sig, 3)[3].lo) + return I_hi, I3_lo + + +DETN = {} + + +def det_norm(p, bpp, muB, sig): + """Upper bound of ||det(B) 1{B > 0}||_p (p a power of two) for B = mu_B + Gaussian with Loewner coupling + B <= (b'' + eta ||z||) I + GOE (b'' rounded up to a 1/1024 grid): the contact moment E[det^p 1{B > 0}] at b'' on + {||z|| <= R0}, plus E[(||B||_F^2/2)^p 1{||z|| > R0}] <= 2^-p E[(||mu_B|| + sig chi_3)^(2p) 1{chi_3 > R0}].""" + bg = Fr(math.ceil(bpp * 1024), 1024) + key = (p, bg) + if key not in DETN: + DETN[key] = contact_moment(p, 0, bg).hi + cm = DETN[key] + R0 = Fr(R0_COUPLING) + U = chi_upper_moments(3, R0, 2 * p) + out = sum((U[i] * (binom(2 * p, i) * muB ** (2 * p - i) * sig ** i) for i in range(2 * p + 1)), IV(0)).hi / 2 ** p + return iroot(rup_rel(cm + out), p) + + +def box_bound(law, bb, kb, r0, r1, last_band=False, x0=None, theta=None): + """Certified upper bound of sup_{r in [r0, r1]} Q^W(G_r^c) / r^3 over b in bb, k in kb, with its pieces, for the near-branch + threshold x0 (X_0 of NOTE section 4) and the split parameter theta (section 5); every choice gives a valid bound.""" + x0 = X0 if x0 is None else Fr(x0) + theta = THETA if theta is None else Fr(theta) + b = IV(bb[0], bb[1]) + k = IV(kb[0], kb[1]) + blo, bhi = bb + klo, khi = kb[0], kb[1] + require(blo >= 0, 'b >= 0 assumed by the density majorant') + Klo = 12 * klo + mu = {n: b * law.alpha[n] + k * law.beta[n] for n in TARGET_NAMES} + mabs = {n: absup(mu[n]) for n in TARGET_NAMES} + E = b * b * law.e_bb + b * k * law.e_bk * 2 + k * k * law.e_kk + require(E.hi > 0, 'energy') + sqE = sqrt_up(E.hi) + nl2 = MUT != 'no-lambda2' + # ---- the transverse Hessian B = -A: mean in Frobenius coordinates, its distance to b I, the density majorant + dmu = [b * (-law.alpha['a11'] - 1) + k * (-law.beta['a11']), b * (-law.alpha['a22'] - 1) + k * (-law.beta['a22']), + (b * law.alpha['a12'] + k * law.beta['a12']) * SQRT2] + eps = sqrt_up(sum(absup(x) ** 2 for x in dmu)) # sup over the box of ||mu_B - b I||_F + muB = rup_rel(bhi * sqrt_up(Fr(2)) + eps) # sup ||mu_B||_F + sig = sqrt_up(law.sB_hi2) # rational, >= sqrt(lambda_max(Cov b~)) + sB_lo = law.sB_lo # <= sqrt(lambda_min(Cov b~)) + KB = rup_rel((PI * SQRT2 / (2 * PI * SQRT2PI) / isqrt_iv(law.detCb)).hi) # pi sqrt2 (2 pi)^(-3/2) det^(-1/2) + s2g = sig * sig + ex0 = blo * blo / (2 * s2g) - eps * bhi / s2g # g(0; b) <= exp(-ex0) over the box + g0 = exp_neg_point(ex0).hi if ex0 >= 0 else exp_neg_point(-ex0).inv().hi + gbar = exp_neg_point(eps * eps / (2 * s2g)).inv().hi + # slope of the linear density bound g(l) <= g0 + l Lg on l >= 0: the global |g'| <= exp(-1/2) gbar / sig, or the chord + # of min(gbar, g0 exp(c l)) with c = (b_hi + eps)/sig^2 (g(l; b) <= g(0; b) exp((b + eps) l / sig^2) for l >= 0): + # g0 c (x - 1)/log x with log x = log(gbar/g0) = y (rational), (x - 1)/log x increasing in x + Lg = rup_rel(exp_neg_point(Fr(1, 2)).hi * gbar / sig) + y = (eps * eps + blo * blo - 2 * eps * bhi) / (2 * s2g) + if y > 0: + cc = (bhi + eps) / s2g + Lg = min(Lg, rup_rel(g0 * cc * (exp_neg_point(y).inv().hi - 1) / y)) + else: + Lg = Fr(0) # gbar <= g0: the bound g <= gbar alone is the smaller one + I_hi, I3_lo = lam2_moments(blo, bhi, eps, sig) + require(I3_lo > 0, 'lambda_2 moment') + + def l2norm(p): # ||lambda_2||_p under nu (and under nu'), p a power of two + if not nl2: + return Fr(0) + return iroot(rup_rel(I_hi[3 + p] / I3_lo), p) + # ---- regression on B and the coefficients bar-beta_i (NOTE section 4) + bn = law.bnorm + bv = sqrt_up(bn['v1'] ** 2 + bn['v2'] ** 2) + bo = sqrt_up(bn['o11'] ** 2 + bn['o22'] ** 2 + 2 * bn['o12'] ** 2) + by = sqrt_up(bn['y111'] ** 2 + 3 * bn['y112'] ** 2 + 3 * bn['y122'] ** 2 + bn['y222'] ** 2) + bnu = sqrt_up(bn['nu1'] ** 2 + bn['nu2'] ** 2) + a_fac = Fr(1, 2) if MUT == 'cap-half' else Fr(1) # the cap constant 4/(3k) (mutant: halved) + + def Bblk(blk): + return rup_rel(sqrt_up(sum(m * bn[n] ** 2 for n, m in block_members(*blk))) + law.S[blk] / sB_lo) + Bb = {blk: Bblk(blk) for blk in BLOCKS} + bb1 = rup_rel(Fr(5, 2) * Bb[(4, 0)] + 2 * Bb[(3, 1)]) # bar-beta_1 / r + bbars = {1: rup_rel(r1 * bb1), + 2: rup_rel(2 * bv + r1 * (Fr(5, 2) * Bb[(3, 1)] + 2 * Bb[(2, 2)])), + 3: rup_rel(bo + r1 * (Fr(5, 2) * Bb[(2, 2)] + 2 * Bb[(1, 3)])), + 4: rup_rel(by + r1 * (2 * Bb[(1, 3)] + 2 * Bb[(0, 4)]))} + bbar = max(bbars.values()) + # ---- free blocks: Cholesky factors in Frobenius coordinates (v | omega | Y~) + Lf = law.Lfree + Iv, Io, Iy = (0, 1), (2, 3, 4), (5, 6, 7, 8) + Lvv, Low, Loo = opnorm_lower_tri(Lf, Iv, Iv), opnorm_lower_tri(Lf, Io, Iv), opnorm_lower_tri(Lf, Io, Io) + Lyv, Lyo, Lyy = opnorm_lower_tri(Lf, Iy, Iv), opnorm_lower_tri(Lf, Iy, Io), opnorm_lower_tri(Lf, Iy, Iy) + mu_v = sqrt_up(mabs['v1'] ** 2 + mabs['v2'] ** 2) + mu_o = sqrt_up(mabs['o11'] ** 2 + mabs['o22'] ** 2 + 2 * mabs['o12'] ** 2) + mu_y = sqrt_up(mabs['y111'] ** 2 + 3 * mabs['y112'] ** 2 + 3 * mabs['y122'] ** 2 + mabs['y222'] ** 2) + mu_nu = sqrt_up(mabs['nu1'] ** 2 + mabs['nu2'] ** 2) + s2, s3, s4 = 2 * Lvv, Loo, Lyy + + memo = {} + + def blk_g_norm(blk, p): # || ||X_blk,g||_F ||_p + mem = block_members(*blk) + m_ = sqrt_up(sum(m * mabs[n] ** 2 for n, m in mem)) + tr = sum(m * max(law.Cg[(n, n)].hi, Fr(0)) for n, m in mem) + return frob_norm(m_, tr, p) + + def nA(blk, p): # ||A^g_blk||_p, A^g = ||X_blk,g||_F + T^g_blk + key = ('A', blk, p) + if key not in memo: + memo[key] = rup_rel(blk_g_norm(blk, p) + law.S[blk] * (sqE + znorm1(pow2(p)))) + return memo[key] + + def nDelta(i, p): + key = ('D', i, p) + if key not in memo: + memo[key] = rup_rel(nDelta0(i, p)) + return memo[key] + + def nDelta0(i, p): + if MUT == 'no-delta': + return Fr(0) + if i == 1: + return Fr(5, 2) * nA((4, 0), p) + 2 * nA((3, 1), p) + if i == 2: + return Fr(5, 2) * nA((3, 1), p) + 2 * nA((2, 2), p) + if i == 3: + return Fr(5, 2) * nA((2, 2), p) + 2 * nA((1, 3), p) + return 2 * nA((1, 3), p) + 2 * nA((0, 4), p) + + def nd(i, p, lam=True): + key = ('d', i, p, lam) + if key not in memo: + memo[key] = rup_rel(nd0(i, p, lam)) + return memo[key] + + def nd0(i, p, lam): # ||delta_i||_p / Klo, delta_i including bar-beta_i (||mu_B|| + lambda_2) + cm = bbars[i] * (muB + (l2norm(p) if lam else 0)) + if i == 2: + v = 2 * mu_v + cm + r1 * nDelta(2, p) + elif i == 3: + v = mu_o + Low * chi_norm(2, pow2(p)) + cm + r1 * nDelta(3, p) + else: + v = mu_y + Lyv * chi_norm(2, pow2(p)) + Lyo * chi_norm(3, pow2(p)) + cm + r1 * nDelta(4, p) + return v / Klo + + kappa = 3 / (1 - 3 * x0) + Fk = 1 + kappa * x0 # (1 + kappa x_i) <= Fk on the event {x_i <= x0} + cth = 1 - kappa * x0 / 2 # <= Fk^{-1/2} + NMAX = 24 + q = {'cmt1': bbars[1] * muB / Klo, 'rK': r1 / Klo, + 'nD1': lambda p: rup_rel(nDelta(1, p) + bb1 * l2norm(pow2(p))), + 's': {2: s2 / Klo, 3: s3 / Klo, 4: s4 / Klo}, 'nd': nd, 's3': s3, 's2': s2, 'c': cth, 'Klo': Klo} + E1, E1o, E1v, Tn, To, Tv = normalized_moments(q, NMAX, theta) + et1 = rup_rel(16 * kappa * r1 * bbars[1] * a_fac) # eta_1 K, k-free + aK2_hi = 192 * khi * a_fac # a K^2 = 192 k + + def EUt(p): # upper bound of E[Ut^p], Ut = U/(a K^2) <= max(Z1, Fk y_i^2) + key = ('U', p) + if key not in memo: + memo[key] = rup_rel(sum(binom(p, l) * et1 ** l * E1[2 * p + l] for l in range(p + 1)) + Fk ** p * Tn[2 * p]) + return memo[key] + + def EoUt(p): + return sum(binom(p, l) * et1 ** l * E1o[2 * p + l] for l in range(p + 1)) + Fk ** p * To[2 * p] + + def EvUt(p): + return sum(binom(p, l) * et1 ** l * E1v[2 * p + l] for l in range(p + 1)) + Fk ** p * Tv[2 * p] + + def sq(x): + return sqrt_up(x) + + def nUt(m, p): # ||Ut^m||_p <= Fk^m ||max_i y_i||_2mp^2m (on the event x_i <= X0) + require(2 * m * p <= NMAX and pow2(p) == p, 'norm order') + key = ('nUt', m, p) + if key not in memo: + memo[key] = rup_rel(Fk ** m * iroot(E1[2 * m * p] + Tn[2 * m * p], p)) + return memo[key] + + def nU(p): + return rup_rel(aK2_hi * nUt(1, p)) + + def ndel(p): # || ||B - mu_B||_F ||_p <= ||mu_B|| + lambda_1 + lambda_2 <= ||mu_B|| + 2 lambda_2 + return rup_rel(muB + 2 * l2norm(p)) + + def nbar(n, p): # ||F_bar||_p, F_bar = |F_g| + ||beta_F|| ||B - mu_B||_F + return rup_rel(gauss_norm(mabs[n], law.sdg[n], p) + bn[n] * ndel(p)) + + six_klo = 6 * klo + nT2, nT4, nt2, nt4 = nbar('T', 2), nbar('T', 4), nbar('tau', 2), nbar('tau', 4) + epsc2 = (2 * r1 * nT2 + r1 * r1 * nt2) / six_klo + (r1 * nT4) * (r1 * nT4 + r1 * r1 * nt4) / six_klo ** 2 + nT8, nt8 = nbar('T', 8), nbar('tau', 8) + epsc4 = (2 * r1 * nT4 + r1 * r1 * nt4) / six_klo + (r1 * nT8) * (r1 * nT8 + r1 * r1 * nt8) / six_klo ** 2 + U2_2, U3_2, U4_2 = sq(EUt(4)), sq(EUt(6)), sq(EUt(8)) + U2_4, U3_4 = nUt(2, 4), nUt(3, 4) + # the Omega majorant: Omega_bar = o0 + d_o, o0 = s3 chi_3 (shares u_3), d_o = rest + ndo = lambda p: rup_rel(mu_o + Low * chi_norm(2, pow2(p)) + bo * ndel(p)) + no4 = rup_rel(s3 * chi_norm(3, 4) + ndo(4)) + # the ||r nu - v||^2 majorant (Term C) + th = THETA_E + dv = lambda p: rup_rel(mu_v + bv * ndel(p)) + nnu = lambda p: rup_rel(frob_norm(mu_nu, law.Cg[('nu1', 'nu1')].hi + law.Cg[('nu2', 'nu2')].hi, p) + + bnu * ndel(p)) + ebar4 = rup_rel((r1 * nnu(8) + Lvv * chi_norm(2, 8) + dv(8)) ** 2) # ||e_bar||_4 + ebar8 = rup_rel((r1 * nnu(16) + Lvv * chi_norm(2, 16) + dv(16)) ** 2) # ||e_bar||_8 + cC = (1 + th) * (1 + 1 / th) * dv(4) ** 2 + (1 + 1 / th) * r1 * r1 * nnu(4) ** 2 + # ---- Term A, B, C (one power of U per lambda_1 integration), normalized by 36 k^2 r^5 and by the lambda_2 weight + termA = aK2_hi ** 3 / 3 * (EUt(3) + epsc2 * U3_2) + termB = aK2_hi ** 2 / 2 * (EoUt(2) + (ndo(2) + epsc4 * no4) * U2_2) + termC = aK2_hi ** 2 / (12 * klo) * ((1 + th) ** 2 * EvUt(2) + cC * U2_2 + r1 * nT4 / six_klo * ebar4 * U2_2) + termAp = aK2_hi ** 4 / 4 * (EUt(4) + epsc2 * U4_2) + termBp = aK2_hi ** 3 / 3 * (EoUt(3) + (ndo(2) + epsc4 * no4) * U3_2) + termCp = aK2_hi ** 3 / (18 * klo) * ((1 + th) ** 2 * EvUt(3) + cC * U3_2 + r1 * nT4 / six_klo * ebar4 * U3_2) + if MUT == 'no-om-term': + termB = termBp = Fr(0) + # ---- the Jacobian-correction terms: the part r Omega_bar of (lambda_2 + r Omega_bar), integrated against nu' + # D = E[Omega_bar (A + B + C integrands)], D' = E[Omega_bar (A' + B' + C' integrands)] + nT16 = nbar('T', 16) + termD = (aK2_hi ** 3 / 3 * (EoUt(3) + (ndo(2) + epsc4 * no4) * U3_2) + + aK2_hi ** 2 / 2 * no4 * no4 * (U2_2 + epsc4 * U2_4) + + aK2_hi ** 2 / (12 * klo) * no4 * (1 + r1 * nT16 / six_klo) * ebar8 * U2_4) + termDp = (aK2_hi ** 4 / 4 * (EoUt(4) + (ndo(2) + epsc4 * no4) * U4_2) + + aK2_hi ** 3 / 3 * no4 * no4 * (U3_2 + epsc4 * U3_4) + + aK2_hi ** 3 / (18 * klo) * no4 * (1 + r1 * nT16 / six_klo) * ebar8 * U3_4) + if MUT == 'no-om-term': + termD = termDp = Fr(0) + I2, I3 = I_hi[2], I_hi[3] + lead = I3 * (termA + termB + termC) + r1 * I2 * termD + leadp = I3 * (termAp + termBp + termCp) + r1 * I2 * termDp + # ---- normalizer floor Z_r / (36 k^2 r^2) >= m3(b'_lo) - corrections (NOTE section 6) + Lb = cholesky([[IV(x.lo, x.hi) for x in row] for row in law.Cb]) + eta = sqrt_up(sum(absup(Lb[i][j] - (SQRT2 if i == j else 0)) ** 2 for i in range(3) for j in range(i + 1))) + R0 = Fr(R0_COUPLING) + bp_lo = blo - eps - eta * R0 + bp_hi = bhi + eps + eta * R0 + bp_lo = Fr(math.floor(bp_lo * 2 ** 40), 2 ** 40) + bp_hi = Fr(math.ceil(bp_hi * 2 ** 40), 2 ** 40) + pR0 = chi_tail(3, R0) + m3lo = m3(bp_lo).lo + tail_c = sqrt_up(contact_moment(4, 0, bp_lo).hi) * sqrt_up(pR0) + # upper bounds of E[h(B)] for Loewner-monotone h through b'' = bp_hi, plus the outside-ball part + nBF16 = rup_rel(muB + sig * chi_norm(3, 16)) # || ||B||_F ||_16 + out_ball = sqrt_up(pR0) + + def up_moment(a, c): # E[det^a tr^c 1{B > 0}] <= contact value at b'' + outside-ball part + crude = rup_rel(nBF16 ** (2 * a + c)) # det <= ||B||^2, tr <= sqrt2 ||B|| <= 2 ||B||: crude, times P^(1/2) + return rup_rel(contact_moment(a, c, bp_hi).hi + 2 ** c * crude * out_ball) + F2 = up_moment(4, 0) # E[(det B)^4 1{B > 0}] + DT, TDT, TTDT = up_moment(1, 1), up_moment(1, 2), up_moment(1, 3) + sdT, sdtau = law.sdQ['T'], law.sdQ['tau'] + q_bad = gauss_tail2(mabs['T'], sdT, 3 * klo / (2 * r1)) + gauss_tail2(mabs['tau'], sdtau, 3 * klo / (2 * r1 * r1)) + nF = sqrt_up(F2) + corr_G0 = nF * sqrt_up(q_bad) if q_bad > 0 else Fr(0) + corr_eps = (2 * r1 * gauss_norm(mabs['T'], sdT, 2) + r1 * r1 * gauss_norm(mabs['tau'], sdtau, 2)) / six_klo * nF + trO = sum(m * max(law.Cg[(n, n)].hi, Fr(0)) for n, m in zip(ONAMES, (1, 1, 2))) + trV = law.Cg[('v1', 'v1')].hi + law.Cg[('v2', 'v2')].hi + Eog = rup_rel(mu_o + sqrt_up(trO)) + corr_om = Eog * DT + bo * (TDT + muB * DT) + Evg2 = mu_v ** 2 + trV + corr_v = (Fr(17, 16) * Evg2 * DT + 17 * bv ** 2 * (TTDT + 2 * muB * TDT + muB * muB * DT)) / (4 * klo) + corr = tail_c + corr_G0 + corr_eps + r1 * (corr_om + corr_v) + if MUT == 'no-zfloor-correction': + corr = Fr(0) + zt = m3lo - corr + require(zt > 0, 'normalizer floor not positive') + main = KB * min(gbar * lead, g0 * lead + r1 * Lg * leadp) / zt + # ---- rare branches at the band top: E[W 1_E] <= ||W 1_typed||_q P(E)^(1 - 1/q), ||W 1_typed||_q / r^2 <= g1 g2 + six_khi = 6 * khi + trB = sum(x.hi for x in (law.Cb[0][0], law.Cb[1][1], law.Cb[2][2])) + trOQ = sum(m * law.C[(n, n)].hi for n, m in zip(ONAMES, (1, 1, 2))) + trVQ = law.C[('v1', 'v1')].hi + law.C[('v2', 'v2')].hi + trNQ = law.C[('nu1', 'nu1')].hi + law.C[('nu2', 'nu2')].hi + + # W_r / r^2 <= |c1| D [|c2| (D + r Om tr + r^2 Om^2) + r ||e||^2 (tr + r Om)] on the typed support, D = det B 1{B > 0}, + # tr = tr B, Om = ||Omega'||_F (NOTE section 7) + sd_tr = sqrt_up(law.C[('a11', 'a11')].hi + law.C[('a22', 'a22')].hi + 2 * absup(law.C[('a11', 'a22')])) + m_tr = absup(mu['a11'] + mu['a22']) + + def wnorm(qq): + p4, p8 = 4 * qq, 8 * qq + nD = lambda p: det_norm(p, bp_hi, muB, sig) + ntr = lambda p: gauss_norm(m_tr, sd_tr, p) + no = lambda p: frob_norm(mu_o, trOQ, p) + nT = lambda p: gauss_norm(mabs['T'], sdT, p) + nc2 = lambda p: six_khi + r1 * nT(p) + r1 * r1 * gauss_norm(mabs['tau'], sdtau, p) + ne = lambda p: r1 * frob_norm(mu_nu, trNQ, p) + frob_norm(mu_v, trVQ, p) + g1 = (six_khi + r1 * nT(p4)) * nD(p4) + g2 = (nc2(p4) * (nD(p4) + r1 * no(p8) * ntr(p8) + r1 * r1 * no(p8) ** 2) + + r1 * ne(p8) ** 2 * (ntr(p4) + r1 * no(p4))) + return g1 * g2 + thr = 3 * klo / (10 * r1) + qA2 = Fr(0) + args = [] + for blk in BLOCKS: + a_, c_ = blk + mem = block_members(*blk) + m_ = sqrt_up(sum(m * mabs[n] ** 2 for n, m in mem)) # |mean of d_x^a d_u^c f(0)| <= ||mean||_F for unit u + best = None + for frac in (Fr(7, 8), Fr(15, 16)): # split t = frac t + (1 - frac) t between jet and Taylor tail + if c_ == 0: # scalar block + x_ = (thr * frac - m_) / law.sdQ[mem[0][0]] + pj = gauss_tail2(m_, law.sdQ[mem[0][0]], thr * frac) + elif c_ == 1: # vector block: ||X - mean|| <= sqrt(lambda_max) chi_2 + lmax = max(sum(absup(law.C[(ni, nj)]) for nj, mj in mem) for ni, mi in mem) + x_ = (thr * frac - m_) / sqrt_up(lmax) + pj = chi_tail(2, x_) if x_ > 0 else Fr(1) + else: # ||X||_op = max_theta |Q(theta)|, Q a trigonometric + rho = 1 - Fr(c_, NET_M) # polynomial of degree c: Bernstein |Q'| <= c max|Q| and + sdn = sqrt_up(law.net_var[blk]) # net gaps <= 2/NET_M give max|Q| <= max_j |Q_j| / rho + x_ = (thr * frac * rho - m_) / sdn + pj = min(Fr(1), len(NET) * gauss_tail2(m_, sdn, thr * frac * rho)) + pj += (law.S[blk] * (sqE + znorm(32)) / (thr * (1 - frac))) ** 32 + if best is None or pj < best[0]: + best = (pj, x_) + args.append(best[1]) + qA2 += best[0] + a_hi = Fr(4, 3) / klo * a_fac + qfar = Fr(0) + qx = Fr(0) + sdB11 = sqrt_up(law.C[('a11', 'a11')].hi) + if bbar > 0: + lam_far = 1 / (4 * a_hi * r1 * bbar * bbar) + args.append((lam_far - absup(mu['a11'])) / sdB11) + qfar = gauss_tail1(absup(mu['a11']), sdB11, lam_far) + + def lam2_tail(s): # P(lambda_2 > s) <= P(sig chi_3 > s - ||mu_B||) + x_ = (s - muB) / sig + args.append(x_) + return chi_tail(3, x_) if x_ > 0 else Fr(1) + for i in (1, 2, 3, 4): # x_i > X0 <=> y_i' + bar-beta_i lambda_2 > X0 / (a r1 bar-beta_i) + if bbars[i] == 0: + continue + tx = x0 / (a_hi * r1 * bbars[i]) + if i == 1: + cb = 12 * khi + bbars[1] * muB + if tx <= cb: + qx += Fr(1) + else: + qx += (r1 * nDelta(1, 32) / ((tx - cb) / 2)) ** 32 + qx += lam2_tail((tx - cb) / (2 * bbars[1])) + else: + s_i = {2: s2, 3: s3, 4: s4}[i] + x_ = tx / (3 * s_i) + args.append(x_) + qx += chi_tail(NU[i], x_) + (nd(i, 32, lam=False) * Klo / (tx / 3)) ** 32 + lam2_tail(tx / (3 * bbars[i])) + qs = [min(Fr(1), x) for x in (qA2, qfar, qx)] + tail_num = min(wnorm(qq) * sum(pow_tail(x, qq) for x in qs) for qq in QS) + if last_band: + # (0, R] is the union of [R 2^-(j+1), R 2^-j]; every Gaussian or chi tail argument scales at least like 1/r and is + # >= 3 at R, so each such piece drops by more than 64 when r halves; every Markov piece is a power r^m with m >= 32; + # after the power 1 - 1/q (q >= 2) each piece still drops by more than 8, so for the q chosen at R the supremum of + # tail / r^3 over (0, R] is attained on [R/2, R] + require(min(args) >= 3, 'last-band tail arguments') + require(max(qA2, qfar, qx) < 1, 'last-band tail pieces') + rdiv = r1 / 2 + else: + rdiv = r0 + tail = tail_num / (36 * klo * klo * zt * rdiv ** 3) + total = main + tail + return {'total': total, 'main': main, 'tail': tail, 'zt': zt, 'tail_num': tail_num, 'KB': KB, 'g0': g0, 'gbar': gbar, 'I3': I3, + 'termA': termA, 'termB': termB, 'termC': termC, 'termD': termD, 'qA2': qA2, 'qfar': qfar, 'qx': qx, + 'm3lo': m3lo, 'corr': corr, 'eps': eps, 'eta': eta, 'min_arg': min(args)} +# ----------------------------------------------------------------------------- the (k, b) boxes of the sweep +F_ = Fr +GRID = ( + ((F_(1, 2), F_(3, 4)), (F_(0), F_(1, 2), F_(1))), + ((F_(3, 4), F_(1)), (F_(0), F_(1, 2), F_(1))), + ((F_(1), F_(5, 4)), (F_(0), F_(1, 2), F_(1))), + ((F_(5, 4), F_(3, 2)), (F_(0), F_(1, 2), F_(1))), + ((F_(3, 2), F_(13, 8)), (F_(0), F_(243, 512), F_(1))), + ((F_(13, 8), F_(7, 4)), (F_(0), F_(309, 1024), F_(899, 1024), F_(1))), + ((F_(7, 4), F_(15, 8)), (F_(0), F_(37, 256), F_(105, 256), F_(933, 1024), F_(1))), + ((F_(15, 8), F_(31, 16)), (F_(0), F_(37, 512), F_(207, 1024), F_(225, 512), F_(113, 128), F_(1))), + ((F_(31, 16), F_(63, 32)), (F_(0), F_(37, 1024), F_(103, 1024), F_(111, 512), F_(439, 1024), F_(421, 512), F_(1))), + ((F_(63, 32), F_(2)), (F_(0), F_(1, 512), F_(5, 1024), F_(5, 512), F_(19, 1024), F_(35, 1024), F_(1, 16), F_(29, 256), + F_(209, 1024), F_(377, 1024), F_(685, 1024), F_(1))), +) + + +# ----------------------------------------------------------------------------- bands, boxes, runs +TOP_OCTAVES = 7 # octaves [2^-(j+1), 2^-j], j = 6..12 (r up to 1/64), split into 8 sub-bands +LOW_OCTAVES = 16 # octaves j = 13..28, split into 4 sub-bands +R_LAST = Fr(1, 2 ** 29) # last band [0, 2^-29] +COMBO = (Fr(1, 64), Fr(1, 32)) # (X_0, theta) of NOTE sections 4-5 + + +def bands(): + """Sub-bands [r0, r1] covering (0, 1/64]; each octave split into equal sub-bands; the last band is [0, R_LAST].""" + out = [] + for j in range(6, 6 + TOP_OCTAVES + LOW_OCTAVES): + hi = Fr(1, 2 ** j) + lo = hi / 2 + m = 8 if j < 6 + TOP_OCTAVES else 4 + for i in range(m): + out.append((lo + (hi - lo) * i / m, lo + (hi - lo) * (i + 1) / m)) + out.append((Fr(0), R_LAST)) + require(min(b[0] for b in out) == 0 and max(b[1] for b in out) == Fr(1, 64), 'band cover') + out = sorted(out) + require(all(out[i][1] == out[i + 1][0] for i in range(len(out) - 1)), 'bands contiguous') + return out + + +def dec_down(x, places=4): + x = Fr(x) + n = (x.numerator * 10 ** places) // x.denominator + s = str(abs(n)).rjust(places + 1, '0') + return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:] + + +def dec_up(x, places=4): + x = Fr(x) + n = -((-x.numerator * 10 ** places) // x.denominator) + s = str(abs(n)).rjust(places + 1, '0') + return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:] + + +def grid_boxes(): + out = [] + for kb, bg in GRID: + require(bg[0] == 0 and bg[-1] == 1 and all(bg[i] < bg[i + 1] for i in range(len(bg) - 1)), 'b grid') + for i in range(len(bg) - 1): + out.append(((Fr(bg[i]), Fr(bg[i + 1])), (Fr(kb[0]), Fr(kb[1])))) + ks = [kb for kb, _ in GRID] + require(ks[0][0] == BAND_K[0] and ks[-1][1] == BAND_K[1] and all(ks[i][1] == ks[i + 1][0] for i in range(len(ks) - 1)), + 'k boxes cover K') + return tuple(out) + + +CG3_02_LO = Fr('22424320.655069') # certified lower end of c_G^(3)(0, 2) (Math-#208) + + +def band_record(args): + """Per band: the certified bound of sup Q^W(G_r^c)/r^3 (rounded up, 4 places) and the floor of Z_r/r^2 (rounded down, + 6 places) for each box in BOXES order, and the largest tail part (rounded up, 12 places).""" + r0, r1 = args + law = band_law(r0, r1) + last = (r0 == 0) + bound, zfloor, tails = [], [], [] + for bb, kb in BOXES: + res = box_bound(law, bb, kb, r0, r1, last_band=last, x0=COMBO[0], theta=COMBO[1]) + bound.append(dec_up(res['total'])) + zfloor.append(dec_down(36 * kb[0] * kb[0] * res['zt'], 6)) + tails.append(res['tail']) + return {'r': [str(r0), str(r1)], 'bound': bound, 'zfloor': zfloor, 'tail_max': dec_up(max(tails), 12)} + + +def full_run(procs): + bl = bands() + with multiprocessing.Pool(procs) as pool: + recs = pool.map(band_record, bl) + return assemble(recs) + + +def assemble(recs): + table = {} + for rs in R_STARS: + mx, arg, zmin = Fr(0), None, None + for rec in recs: + if Fr(rec['r'][1]) <= rs: + for (bb, kb), bs, zs in zip(BOXES, rec['bound'], rec['zfloor']): + v = Fr(bs) + if v > mx: + mx, arg = v, {'r': rec['r'], 'b': [str(bb[0]), str(bb[1])], 'k': [str(kb[0]), str(kb[1])]} + z = Fr(zs) + zmin = z if zmin is None or z < zmin else zmin + table[str(rs)] = {'C': str(mx), 'argmax': arg, 'C_over_cG3_02': dec_up(mx / CG3_02_LO, 6), + 'C_rstar3': dec_up(mx * rs ** 3, 10), 'z_star': str(zmin)} + return {'schema': 1, 'object': 'CL-C8-THEOREM-A-CONSTANTS-D3-20261001-v1', 'scientific_effect': 'NONE', + 'certified': True, 'mutant': MUT, 'L_min': L_MIN, 'dimension': 3, + 'band': {'b': [str(BAND_B[0]), str(BAND_B[1])], 'k': [str(BAND_K[0]), str(BAND_K[1])]}, + 'parameters': params(), 'C_table': table, 'bands': recs} + + +def params(): + return {'PBITS': PBITS, 'NX': NX, 'NB': NB, 'COMBO_X0_THETA': [str(COMBO[0]), str(COMBO[1])], 'THETA_E': str(THETA_E), + 'QS': list(QS), 'R0_COUPLING': R0_COUPLING, 'NET_M': NET_M, 'R_STARS': [str(x) for x in R_STARS], + 'TOP_OCTAVES': TOP_OCTAVES, 'LOW_OCTAVES': LOW_OCTAVES, 'R_LAST': str(R_LAST), + 'GRID': [[[str(kb[0]), str(kb[1])], [str(x) for x in bg]] for kb, bg in GRID]} + + +def results_text(res): + """RESULTS.json layout: the summary indented, then one line per band record.""" + head = json.dumps({key: val for key, val in res.items() if key != 'bands'}, indent=1, sort_keys=True) + lines = ',\n'.join(' ' + json.dumps(rec, sort_keys=True) for rec in res['bands']) + return head[:-2] + ',\n "bands": [\n' + lines + '\n ]\n}\n' + + +def write_results(res): + with open(os.path.join(HERE, 'RESULTS.json'), 'w') as fh: + fh.write(results_text(res)) + + +def load_results(): + with open(os.path.join(HERE, 'RESULTS.json')) as fh: + return json.load(fh) + + +def replay_sample(): + """Band indices replayed by --check: the band ending at each r_*, the bands ending at 2^-14, 2^-20, 2^-26, the last band.""" + bl = bands() + idx = [] + for rs in R_STARS: + j = max(i for i, b in enumerate(bl) if b[1] == rs) + idx.append(j) + idx += [i for i, b in enumerate(bl) if b[1] in (Fr(1, 2 ** 14), Fr(1, 2 ** 20), Fr(1, 2 ** 26))] + idx.append(0) + return sorted(set(idx)) + + +def check_run(procs, full=False): + ref = load_results() + with open(os.path.join(HERE, 'RESULTS.json')) as fh: + require(fh.read() == results_text(ref), 'RESULTS.json is not in the canonical layout') + require(ref['parameters'] == params(), 'parameters differ from RESULTS.json') + require(ref['mutant'] is None, 'RESULTS.json produced under a mutant') + require(abs(PI.mid() - Fr('3.14159265358979323846')) < Fr(1, 10 ** 19) and PI.width() < Fr(1, 10 ** 30), 'pi enclosure') + require(abs(Phi(IV(1)).mid() - Fr('0.841344746068542948585')) < Fr(1, 10 ** 19), 'Phi(1) enclosure') + require(abs(m3(0).mid() - (Fr(7, 2) - 2 * SQRT2.mid())) < Fr(1, 10 ** 40), 'm_(3,0) closed form') + bl = bands() + require([rec['r'] for rec in ref['bands']] == [[str(a), str(b)] for a, b in bl], 'band list differs') + idx = list(range(len(bl))) if full else replay_sample() + with multiprocessing.Pool(procs) as pool: # in order; the first differing band exits (and stops the pool) + for i, rec in zip(idx, pool.imap(band_record, [bl[i] for i in idx])): + require(rec == ref['bands'][i], 'band %d (%s) differs from RESULTS.json' % (i, rec['r'])) + # the table is the maximum over the stored records + asm = assemble(ref['bands']) + require(asm['C_table'] == ref['C_table'], 'C table is not the maximum of the stored bounds') + # sanity: every C exceeds the sharp reference-kernel cap coefficient c_G^(3)(0, 2) of Math-#208 (a lower bound for every + # d = 3 cap-route constant valid at (0, 2) for the reference kernel, which this certificate covers) + for rs, row in ref['C_table'].items(): + require(Fr(row['C']) > CG3_02_LO, 'C below the sharp cap coefficient') + print(json.dumps({'check': 'ok', 'bands_replayed': len(idx), 'C_table': {rs: row['C'] for rs, row in ref['C_table'].items()}, + 'z_star': {rs: row['z_star'] for rs, row in ref['C_table'].items()}, 'mutant': MUT})) + + +def main(): + global MUT + ap = argparse.ArgumentParser() + ap.add_argument('--check', action='store_true') + ap.add_argument('--check-full', action='store_true') + ap.add_argument('--mutant', choices=MUTANTS) + ap.add_argument('--procs', type=int, default=4) + ap.add_argument('--mc', nargs=5, metavar=('R', 'B', 'K', 'N', 'SEED'), help='floating end-to-end control') + args = ap.parse_args() + MUT = args.mutant + if args.mc: + rr, bb, kk, nn, sd = args.mc + print(json.dumps(mc_event(float(Fr(rr)), float(Fr(bb)), float(Fr(kk)), int(nn), int(sd)))) + return + if args.check or args.check_full: + check_run(args.procs, full=args.check_full) + return + res = full_run(args.procs) + write_results(res) + print(json.dumps({'C': {rs: row['C'] for rs, row in res['C_table'].items()}, + 'z': {rs: row['z_star'] for rs, row in res['C_table'].items()}})) +BOXES = grid_boxes() + + +# ----------------------------------------------------------------------------- floating control (not part of the certificate) +def mc_event(r, b, k, n, seed, deg=7, nang=24): + """End-to-end Monte Carlo of Q^W(G_r^c) / r^3 and Z_r / r^2 for the reference kernel in d = 3 (floating point): the + Taylor jet of order deg at the midpoint is sampled under Q (exact pin series evaluated in floating point), W_r is + evaluated from the Hessians at M and S, and M_3, M_4 are the partial-block operator norms ([CAP] (1)) maximized over a + grid of the cylinder D (binary forms maximized over nang angles; floating, so slightly below the true suprema).""" + import random + pins, _ = functionals() + P = [pins[nm] for nm in PIN_NAMES] + jets = [(a, b1, m - a - b1) for m in range(deg + 1) for a in range(m, -1, -1) for b1 in range(m - a, -1, -1)] + J = [[term(1, 0, a, b1, b2, X0PT)] for (a, b1, b2) in jets] + nj = len(J) + + def ev(F, G, key): + ser, w, _ = pair_series(F, G, key) + return sum(float(c) * r ** p for p, c in ser.items()) + Spp = [[ev(P[i], P[j], ('p', i, j)) for j in range(NP)] for i in range(NP)] + Sjp = [[ev(J[i], P[j], ('jp3', deg, i, j)) for j in range(NP)] for i in range(nj)] + Sjj = [[ev(J[i], J[j], ('jj3', deg, i, j)) for j in range(nj)] for i in range(nj)] + + def solve(A, bv): + m = len(A) + M = [A[i][:] + [bv[i]] for i in range(m)] + for c in range(m): + p = max(range(c, m), key=lambda i: abs(M[i][c])) + M[c], M[p] = M[p], M[c] + for i in range(c + 1, m): + f = M[i][c] / M[c][c] + for j in range(c, m + 1): + M[i][j] -= f * M[c][j] + x = [0.0] * m + for i in reversed(range(m)): + x[i] = (M[i][m] - sum(M[i][j] * x[j] for j in range(i + 1, m))) / M[i][i] + return x + vr = [b - k * r ** 3 / 2, -k * r * r, 12 * k] + [0.0] * (NP - 3) + Wt = [solve(Spp, Sjp[i]) for i in range(nj)] + mu = [sum(Wt[i][j] * vr[j] for j in range(NP)) for i in range(nj)] + C = [[Sjj[i][j] - sum(Wt[i][l] * Sjp[j][l] for l in range(NP)) for j in range(nj)] for i in range(nj)] + # Cholesky with clamped pivots (the conditional covariance is positive semidefinite up to rounding) + L = [[0.0] * nj for _ in range(nj)] + for i in range(nj): + for j in range(i + 1): + s = C[i][j] - sum(L[i][l] * L[j][l] for l in range(j)) + if i == j: + L[i][i] = math.sqrt(s) if s > 1e-300 else 0.0 + else: + L[i][j] = s / L[j][j] if L[j][j] > 0 else 0.0 + index = {al: i for i, al in enumerate(jets)} + fact = [math.factorial(i) for i in range(deg + 2)] + + def deriv(cf, be, p): + """d^be f(p) from the jet (Taylor at 0).""" + tot = 0.0 + for al, i in index.items(): + if al[0] >= be[0] and al[1] >= be[1] and al[2] >= be[2]: + e = (al[0] - be[0], al[1] - be[1], al[2] - be[2]) + tot += cf[i] * (p[0] ** e[0] * p[1] ** e[1] * p[2] ** e[2]) / (fact[e[0]] * fact[e[1]] * fact[e[2]]) + return tot + angs = [(math.cos(math.pi * t / nang), math.sin(math.pi * t / nang)) for t in range(nang)] + pts = [(x, 0.0, 0.0) for x in (-2 * r, -r, 0.0, r, 2 * r)] + pts += [(x, 2 * r * math.cos(2 * math.pi * t / 6), 2 * r * math.sin(2 * math.pi * t / 6)) for x in (-2 * r, 0.0, 2 * r) + for t in range(6)] + + def block_norm(vals, c): + """Operator norm of the symmetric c-form with components vals[(b1, b2)] (b1 + b2 = c) over unit vectors.""" + if c == 0: + return abs(vals[(0, 0)]) + best = 0.0 + for (u1, u2) in angs: + q = sum(math.comb(c, b1) * vals[(b1, c - b1)] * u1 ** b1 * u2 ** (c - b1) for b1 in range(c + 1)) + best = max(best, abs(q)) + return best + + def Mj(cf, j): + best = 0.0 + for p in pts: + for a in range(j + 1): + c = j - a + vals = {(b1, c - b1): deriv(cf, (a, b1, c - b1), p) for b1 in range(c + 1)} + best = max(best, block_norm(vals, c)) + return best + + def hess(cf, p): + return [[deriv(cf, tuple(int(x == i) + int(x == j) for x in range(3)), p) for j in range(3)] for i in range(3)] + + def det3f(H): + return (H[0][0] * (H[1][1] * H[2][2] - H[1][2] * H[2][1]) - H[0][1] * (H[1][0] * H[2][2] - H[1][2] * H[2][0]) + + H[0][2] * (H[1][0] * H[2][1] - H[1][1] * H[2][0])) + + def neg_def(H): + return H[0][0] < 0 and H[0][0] * H[1][1] - H[0][1] ** 2 > 0 and det3f(H) < 0 + + def index2(H): + """Number of negative eigenvalues of a symmetric 3 x 3 matrix is 2: det > 0 and not positive definite.""" + d = det3f(H) + pd = H[0][0] > 0 and H[0][0] * H[1][1] - H[0][1] ** 2 > 0 and d > 0 + return d > 0 and not pd + rng = random.Random(seed) + num = den = 0.0 + acap = 4 / (3 * k) + for _ in range(n): + z = [rng.gauss(0, 1) for _ in range(nj)] + cf = [mu[i] + sum(L[i][j] * z[j] for j in range(i + 1)) for i in range(nj)] + HM, HS = hess(cf, (-r / 2, 0.0, 0.0)), hess(cf, (r / 2, 0.0, 0.0)) + if not (neg_def(HM) and index2(HS)): + continue + Wv = abs(det3f(HM) * det3f(HS)) + den += Wv + a11, a22, a12 = -HM[1][1], -HM[2][2], -HM[1][2] + lam1 = (a11 + a22) / 2 - math.sqrt(((a11 - a22) / 2) ** 2 + a12 * a12) + M3 = Mj(cf, 3) + if lam1 <= acap * r * M3 * M3: + num += Wv + continue + if r * Mj(cf, 4) > 3 * k / 10: + num += Wv + return {'r': r, 'b': b, 'k': k, 'n': n, 'seed': seed, 'QW_Gc_over_r3': num / den / r ** 3 if den else None, + 'Z_over_r2': den / n / r ** 2} + + +if __name__ == '__main__': + main() diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/theorem_b.py b/frontiers/c8_torus_vanishing_remainder_20261001/theorem_b.py new file mode 100644 index 000000000..f0e788cb0 --- /dev/null +++ b/frontiers/c8_torus_vanishing_remainder_20261001/theorem_b.py @@ -0,0 +1,1162 @@ +#!/usr/bin/env python3 +"""Explicit constants for [LP] Theorem B on the C8 band, d = 2 and d = 3 (NOTE.md Theorem L and its corollaries). + +For the radial ledger A_r = 12 pi_r(R; v_r) Z_r / r^2 of [LP] (10.2) the certificate proves, for 0 < r <= R, every frame, +every torus side L >= 10 and the reference kernel, + + -(a_dn r^2 + eps) <= A_r(b, k, u) / A_0^ref(b, k) - 1 <= a_up r^2 + eps (b in [0, 1], k in [1/2, 2]), + +with A_0^ref the closed-form contact integrand of the Gaussian kernel, and assembles from it two-sided second-order bounds for +nu_cand and nu_eld, the difference constant C_{B,K} of [LP] (1.2), and the constants of (12.1) and (12.3). + +Standard library only; exact rational interval arithmetic throughout. +- Taylor models in r (fixed-point integer coefficients, every rounding carried in the remainder) for the pin covariance, its + determinant, the pin energies and the law of the transverse Hessian and of Omega' (section 2 of NOTE.md). +- The interval band laws and the box-bound engines of Math-#206 (d = 2) and Math-#212 (d = 3), as byte-identical pinned + copies engine_e2.py and engine_e3.py (section 2; SOURCE_MAP.json). + + python3 -B -S theorem_b.py --procs N regenerate RESULTS.json + python3 -B -S theorem_b.py --check --procs N replay: Taylor models, rate sweep, reference integrals, a sample of + cap bands, and the assembled tables (exact comparison) + python3 -B -S theorem_b.py --check-full --procs N replay every cap band as well + python3 -B -S theorem_b.py --mutant NAME --check a semantic mutation; the replay must then fail (exit 1) + python3 -B -S theorem_b.py --controls floating controls (not part of the certificate) +""" +import argparse +import importlib.util +import json +import math +import multiprocessing +import os +import sys +from fractions import Fraction as Fr + +HERE = os.path.dirname(os.path.abspath(__file__)) +MUTANTS = ('no-om-term', 'no-bad-set', 'no-det', 'no-torus', 'cap-half') +MUT = None + +R_STARS = (Fr(1, 4096), Fr(1, 2048), Fr(1, 1024), Fr(1, 512), Fr(1, 256), Fr(1, 128), Fr(1, 64)) +TM_S = 6 # Taylor models on [0, 2^-TM_S] = [0, 1/64] +TM_N = 8 # Taylor-model degree +TM_P = 288 # fixed-point bits of the Taylor-model coefficients +R0C = Fr(12) # Loewner coupling radius (d = 3) +TINY = Fr(1, 10 ** 60) # polynomial coefficients below this are rounding residue of exact zeros +SUB_B = tuple((Fr(i, 8), Fr(i + 1, 8)) for i in range(8)) +SUB_K = tuple((Fr(1, 2) + Fr(j, 8), Fr(1, 2) + Fr(j + 1, 8)) for j in range(12)) +SUBBOXES = tuple((bb, kb) for kb in SUB_K for bb in SUB_B) +NH = 4096 # b-cells of the reference integrals +BLOCK = 16 # cells per second-derivative block +CAP_B = tuple((Fr(i, 4), Fr(i + 1, 4)) for i in range(4)) +CAP_K = tuple((Fr(1, 2) + Fr(i, 32), Fr(1, 2) + Fr(i + 1, 32)) for i in range(8)) \ + + tuple((Fr(3, 4) + Fr(i, 16), Fr(3, 4) + Fr(i + 1, 16)) for i in range(4)) + ((Fr(1), Fr(3, 2)), (Fr(3, 2), Fr(2))) +CAP_BOXES = tuple((bb, kb) for kb in CAP_K for bb in CAP_B) +C_REF_197 = Fr('0.0023869380211529483091019117') # Math-#197, reference c_{B,K} in d = 2 (cross-check only) +C_REF_205 = Fr('0.0009743823660242504624710565') # Math-#205, reference c_{B,K} in d = 3 (cross-check only) + + +def require(cond, msg): + if not cond: + raise SystemExit('FAIL: ' + msg) + + +def load_engine(name, fn): + spec = importlib.util.spec_from_file_location(name, os.path.join(HERE, fn)) + mod = importlib.util.module_from_spec(spec) + spec.loader.exec_module(mod) + return mod + + +E2 = load_engine('engine_e2', 'engine_e2.py') +E3 = load_engine('engine_e3', 'engine_e3.py') +ENG = {2: E2, 3: E3} + + +def exp_minus(M, t): + """Enclosure of exp(-t) for rational t of either sign.""" + t = Fr(t) + return M.exp_neg_point(t) if t >= 0 else M.exp_neg_point(-t).inv() + + +def grid_up(x, g=2 ** 40): + return Fr(math.ceil(Fr(x) * g), g) + + +def grid_dn(x, g=2 ** 40): + return Fr(math.floor(Fr(x) * g), g) + + +def dec_up(x, places): + x = Fr(x) + n = -((-x.numerator * 10 ** places) // x.denominator) + s = str(abs(n)).rjust(places + 1, '0') + return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:] + + +def dec_down(x, places): + x = Fr(x) + n = (x.numerator * 10 ** places) // x.denominator + s = str(abs(n)).rjust(places + 1, '0') + return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:] + + +# ============================================================================= 1. Taylor models in r +def cdiv(a, b): + return -((-a) // b) + + +class TM: + """f(r) = sum_{j <= N} (C_j / 2^P) r^j + theta (E / 2^P), |theta| <= 1, for every r in [0, 2^-s].""" + __slots__ = ('C', 'E', 's', 'N') + + def __init__(self, C, E, s, N): + self.C, self.E, self.s, self.N = C, E, s, N + + @classmethod + def from_fracs(cls, coeffs, e, s, N): + items = coeffs.items() if isinstance(coeffs, dict) else enumerate(coeffs) + C = [0] * (N + 1) + E = Fr(e) + for p, x in items: + x = Fr(x) + if x == 0: + continue + if p <= N: + v = round(x * (1 << TM_P)) + C[p] += v + E += abs(x - Fr(v, 1 << TM_P)) / Fr(2) ** (s * p) + else: + E += abs(x) / Fr(2) ** (s * p) + return cls(C, cdiv(E.numerator << TM_P, E.denominator), s, N) + + @classmethod + def const(cls, x, s, N): + return cls.from_fracs({0: Fr(x)}, 0, s, N) + + def polybound(self): + return sum(cdiv(abs(c), 1 << (self.s * j)) for j, c in enumerate(self.C) if c) + + def bound(self): + return self.polybound() + self.E + + def frac_coeffs(self): + return [Fr(c, 1 << TM_P) for c in self.C] + + def frac_e(self): + return Fr(self.E, 1 << TM_P) + + def __add__(self, o): + if not isinstance(o, TM): + o = TM.const(o, self.s, self.N) + return TM([a + b for a, b in zip(self.C, o.C)], self.E + o.E, self.s, self.N) + __radd__ = __add__ + + def __neg__(self): + return TM([-a for a in self.C], self.E, self.s, self.N) + + def __sub__(self, o): + if not isinstance(o, TM): + o = TM.const(o, self.s, self.N) + return self + (-o) + + def scale(self, x): + x = Fr(x) + C = [] + E = Fr(self.E, 1 << TM_P) * abs(x) + for j, c in enumerate(self.C): + v = Fr(c, 1 << TM_P) * x + w = round(v * (1 << TM_P)) + C.append(w) + E += abs(v - Fr(w, 1 << TM_P)) / Fr(2) ** (self.s * j) + return TM(C, cdiv(E.numerator << TM_P, E.denominator), self.s, self.N) + + def __mul__(self, o): + if not isinstance(o, TM): + return self.scale(o) + N, s = self.N, self.s + S = [0] * (2 * N + 1) + for i, a in enumerate(self.C): + if a: + for j, b in enumerate(o.C): + if b: + S[i + j] += a * b + C, E = [], 0 + for k in range(N + 1): + v = S[k] >> TM_P # floor: error below one unit, times r^k <= 1 + C.append(v) + if S[k] - (v << TM_P): + E += 1 + for k in range(N + 1, 2 * N + 1): + if S[k]: + E += cdiv(abs(S[k]), 1 << (TM_P + s * k)) + E += cdiv(self.E * o.polybound() + o.E * self.polybound() + self.E * o.E, 1 << TM_P) + return TM(C, E, s, N) + __rmul__ = __mul__ + + def inv(self): + c0 = Fr(self.C[0], 1 << TM_P) + require(c0 != 0, 'Taylor-model inverse at a zero constant term') + h = (self - TM([self.C[0]] + [0] * self.N, 0, self.s, self.N)).scale(1 / c0) + H = Fr(h.bound(), 1 << TM_P) + require(H < Fr(1, 2), 'Taylor-model inverse: |h| too large') + acc = TM.const(1, self.s, self.N) + p = TM.const(1, self.s, self.N) + mh = -h + for _ in range(self.N): + p = p * mh + acc = acc + p + tail = H ** (self.N + 1) / (1 - H) + acc = TM(acc.C, acc.E + cdiv(tail.numerator << TM_P, tail.denominator), self.s, self.N) + return acc.scale(1 / c0) + + def enclose(self, r0, r1): + lo = hi = Fr(0) + for j, c in enumerate(self.C): + if c: + x = Fr(c, 1 << TM_P) + a, b = Fr(r0) ** j, Fr(r1) ** j + lo += x * (a if x >= 0 else b) + hi += x * (b if x >= 0 else a) + e = self.frac_e() + return lo - e, hi + e + + +def cov_tm(M, F, G, key, s, N, M1): + """Taylor model on [0, 2^-s] of the covariance of F, G under the unnormalized torus kernel: the engine's exact series, + its truncation tail and its torus remainder (as in the engines' cov_band, at r1 = 2^-s).""" + R = Fr(1, 2 ** s) + ser, w, absterms = M.pair_series(F, G, key) + A = max((t[1] for t in absterms), default=0) + tail = tor = Fr(0) + for t in absterms: + acc, aa, gys = t[0], t[1], t[2:] + hprod = fprod = 1 + for g in gys: + hprod *= M.hcoef(g) + fprod *= math.factorial(g) + tail += acc * abs(hprod) + tor += acc * fprod * Fr(max(w, 1)) ** aa + tail *= 2 * Fr(M.NX + 1) ** A * R ** (M.NX + 1 - w) + tor = tor * M1 / (1 - Fr(27183, 10000) * R) + return TM.from_fracs(ser, tail + tor, s, N) + + +def solve_tm(A, B): + """Gaussian elimination without pivoting (A(0) is positive definite): X[c] solves A X[c] = B[c]; also det A.""" + n = len(A) + Mx = [[A[i][j] for j in range(n)] + [B[c][i] for c in range(len(B))] for i in range(n)] + det, invs = None, [] + for col in range(n): + piv = Mx[col][col] + det = piv if det is None else det * piv + ip = piv.inv() + invs.append(ip) + for i in range(col + 1, n): + f = Mx[i][col] * ip + for j in range(col, len(Mx[i])): + Mx[i][j] = Mx[i][j] - f * Mx[col][j] + X = [] + for c in range(len(B)): + x = [None] * n + for i in reversed(range(n)): + acc = Mx[i][n + c] + for j in range(i + 1, n): + acc = acc - Mx[i][j] * x[j] + x[i] = acc * invs[i] + X.append(x) + return X, det + + +class TMLaw: + pass + + +TM_NAMES = {2: ('w', 'om'), 3: ('a11', 'a22', 'a12', 'o11', 'o22', 'o12')} + + +def tm_law(d, s=TM_S, N=TM_N, L=10): + """Taylor-model law over r in [0, 2^-s] of the targets TM_NAMES[d] (every frame, every torus side >= L, the reference + kernel): alpha, beta (mean = b alpha + k beta), the unnormalized covariance block, the unnormalized pin energies, det.""" + M = ENG[d] + names = TM_NAMES[d] + pins, targets = M.functionals() + P = [pins[nm] for nm in M.PIN_NAMES] + idx = [M.TARGET_NAMES.index(nm) for nm in names] + T = [targets[nm] for nm in names] + M1 = Fr(0) if MUT == 'no-torus' else M.torus_M1(L) + n, nt = len(P), len(T) + Spp = [[None] * n for _ in range(n)] + for i in range(n): + for j in range(i, n): + Spp[i][j] = Spp[j][i] = cov_tm(M, P[i], P[j], ('p', i, j), s, N, M1) + Stp = [[cov_tm(M, T[a], P[j], ('tp', idx[a], j), s, N, M1) for j in range(n)] for a in range(nt)] + Stt = [[None] * nt for _ in range(nt)] + for a in range(nt): + for b in range(a, nt): + i, j = idx[a], idx[b] + if i <= j: + Stt[a][b] = Stt[b][a] = cov_tm(M, T[a], T[b], ('tt', i, j), s, N, M1) + else: + Stt[a][b] = Stt[b][a] = cov_tm(M, T[b], T[a], ('tt', j, i), s, N, M1) + zero = TM.const(0, s, N) + kvec = [TM.from_fracs({3: Fr(-1, 2)}, 0, s, N), TM.from_fracs({2: Fr(-1)}, 0, s, N), TM.const(12, s, N)] + [zero] * (n - 3) + e1 = [TM.const(1, s, N)] + [zero] * (n - 1) + X, det = solve_tm(Spp, Stp + [e1, kvec]) + W, xs0, xs1 = X[:nt], X[nt], X[nt + 1] + law = TMLaw() + law.d, law.M1, law.th1, law.s, law.N = d, M1, (Fr(0) if MUT == 'no-torus' else M.theta_minus_one(L)), s, N + law.alpha = {nm: W[a][0] for a, nm in enumerate(names)} + law.beta = {nm: sum((W[a][j] * kvec[j] for j in range(n)), zero) for a, nm in enumerate(names)} + law.Cu = {} + for a, na in enumerate(names): + for b, nb in enumerate(names): + law.Cu[(na, nb)] = Stt[a][b] - sum((W[a][l] * Stp[b][l] for l in range(n)), zero) + law.e_bb = xs0[0] + law.e_bk = sum((kvec[j] * xs0[j] for j in range(n)), zero) + law.e_kk = sum((kvec[j] * xs1[j] for j in range(n)), zero) + law.det = det + if d == 3: + # E[O - mu_O | B] = sum_l G[O][l] (B_l - mu_B_l) in natural coordinates (B11, B22, B12), B = -A: + # G = -Cu(O, A) Cu(A, A)^-1 + an = ('a11', 'a22', 'a12') + Xg, _ = solve_tm([[law.Cu[(x, y)] for y in an] for x in an], [[law.Cu[(o, y)] for y in an] for o in ('o11', 'o22', 'o12')]) + law.G = {o: [-Xg[c][l] for l in range(3)] for c, o in enumerate(('o11', 'o22', 'o12'))} + return law + + +def tm_summary(law): + """Exact low-order coefficients (decimal, 30 places) and remainders of the Taylor models, for the record.""" + def row(t): + c = t.frac_coeffs() + return {'c': [dec_down(x, 30) if x >= 0 else dec_up(x, 30) for x in c[:5]], 'rem': dec_up(t.frac_e(), 30)} + out = {'det': row(law.det), 'e_bb': row(law.e_bb), 'e_bk': row(law.e_bk), 'e_kk': row(law.e_kk)} + for nm in law.alpha: + out['alpha_' + nm] = row(law.alpha[nm]) + out['beta_' + nm] = row(law.beta[nm]) + names = TM_NAMES[law.d] + for i, a in enumerate(names): + for b in names[i:]: + out['C_%s_%s' % (a, b)] = row(law.Cu[(a, b)]) + return out + + +# ============================================================================= 2. (a, eps) forms and Taylor-model readers +def ae_add(*xs): + return (sum(x[0] for x in xs), sum(x[1] for x in xs)) + + +def ae_mul(x, c): + return (x[0] * c, x[1] * c) + + +def ae_at(x, R): + return x[0] * R * R + x[1] + + +def exp_up(M, x, R): + """exp(a0 r^2 + e0) - 1 <= (a0 r^2 + e0) exp(a0 R^2 + e0) on (0, R].""" + f = exp_minus(M, -ae_at(x, R)).hi + return (x[0] * f, x[1] * f) + + +def tm_dev(t, ref, R): + """(a, eps): |t(r) - ref| <= a r^2 + eps for 0 <= r <= R.""" + c = t.frac_coeffs() + eps = abs(c[0] - ref) + abs(c[1]) * R + t.frac_e() + a, Rp = Fr(0), Fr(1) + for x in c[2:]: + a += abs(x) * Rp + Rp *= R + return (a, eps) + + +def tm_lin(t, R): + """(s, eps): |t(r)| <= s r + eps for 0 <= r <= R.""" + c = t.frac_coeffs() + s, Rp = Fr(0), Fr(1) + for x in c[1:]: + s += abs(x) * Rp + Rp *= R + return (s, abs(c[0]) + t.frac_e()) + + +def tm_shift(t, ref, R): + """(lo, hi, eps): t(r) - ref in [lo r^2 - eps, hi r^2 + eps] for 0 <= r <= R.""" + c = t.frac_coeffs() + eps = abs(c[0] - ref) + abs(c[1]) * R + t.frac_e() + lo = hi = c[2] + Rp = R + for x in c[3:]: + v = x * Rp + lo += min(v, 0) + hi += max(v, 0) + Rp *= R + return (lo, hi, eps) + + +def assert_flat(t, ref, what): + c = t.frac_coeffs() + require(abs(c[0] - ref) <= TINY and all(abs(x) <= TINY for x in c[1:]), + 'the reference law of %s is not the contact law for every r' % what) + + +def gersh_min(C): + n = len(C) + return min(C[i][i].lo - sum(max(abs(C[i][j].lo), abs(C[i][j].hi)) for j in range(n) if j != i) for i in range(n)) + + +def wq_bound(M, Cg, rows, vecs): + """Upper bound of ||V' Cg^-1 V|| (rows of V given as magnitude lists): Cg = D^(1/2)(I + E)D^(1/2), Cg^-1 <= D^-1/(1 - ||E||).""" + n = len(rows) + D = [Cg[(rows[i], rows[i])].lo for i in range(n)] + Ef2 = Fr(0) + for i in range(n): + for j in range(n): + if i != j: + c = max(abs(Cg[(rows[i], rows[j])].lo), abs(Cg[(rows[i], rows[j])].hi)) + Ef2 += c * c / (D[i] * D[j]) + Ef = M.sqrt_up(Ef2) + require(Ef < Fr(1, 2), 'diagonal scaling of the conditional covariance') + return sum(sum(x * x for x in vecs[i]) / D[i] for i in range(n)) / (1 - Ef) + + +def half_moments(M, c): + """H_j = int_0^inf u^j exp(-u^2/(2c)) du, j = 0..3: sqrt(pi c/2), c, c sqrt(pi c/2), 2 c^2 (upper bounds).""" + s = M.isqrt_iv(M.PI * c / 2).hi + return [s, Fr(c), c * s, 2 * Fr(c) ** 2] + + +# ============================================================================= 3. the ratio A_r / A_0^ref on a (b, k) box +def pi_ratio(M, tml, n, Q, R): + """log(pi_r / pi_0^ref) in [-dn, up], with dn, up in (a, eps) form; pi_0^ref has det 12 and energy 3b^2/2 + 24k^2.""" + b0, b1, k0, k1 = Q + th1 = tml.th1 + dlo, dhi, de = tm_shift(tml.det, 12, R) + if MUT == 'no-det': + dlo = dhi = de = Fr(0) + xm = (max(abs(dlo), abs(dhi)) * R * R + de) / 12 + require(xm < Fr(1, 2), 'pin determinant deviation') + half_log_up = (max(Fr(0), -dlo) / (24 * (1 - xm)), de / (24 * (1 - xm))) + half_log_dn = (max(Fr(0), dhi) / 24, de / 24) + parts = ((tml.e_bb, Fr(3, 2), (b0 * b0, b1 * b1)), (tml.e_bk, Fr(0), (2 * b0 * k0, 2 * b1 * k1)), + (tml.e_kk, Fr(24), (k0 * k0, k1 * k1))) + Elo = Ehi = Ee = Eu = Fr(0) + for t, ref, (wlo, whi) in parts: + lo, hi, e = tm_shift(t, ref, R) + Elo += min(wlo * lo, whi * lo) + Ehi += max(wlo * hi, whi * hi) + Ee += whi * e + Eu += whi * (abs(ref) + max(abs(lo), abs(hi)) * R * R + e) + up = ae_add(half_log_up, (max(Fr(0), -Elo / 2), Ee / 2 + n * th1 / 2)) + dn = ae_add(half_log_dn, (max(Fr(0), Ehi / 2), (Ee + th1 * Eu) / 2)) + return up, dn + + +def combine(M, p_up, p_dn, z_up, z_dn, R): + pu = exp_up(M, p_up, R) + zR = ae_at(z_up, R) + up = (pu[0] + z_up[0] + pu[0] * zR, pu[1] + z_up[1] + pu[1] * zR) + return up, ae_add(p_dn, z_dn) + + +def bounds_d2(Q, R, tml, il): + """(up, dn): A_r/A_0^ref - 1 <= up(r), 1 - A_r/A_0^ref <= dn(r) on the box, for 0 < r <= R (NOTE sections 3-5).""" + M = E2 + b0, b1, k0, k1 = Q + th1 = tml.th1 + assert_flat(tml.alpha['w'], -1, 'alpha_w') + assert_flat(tml.beta['w'], 0, 'beta_w') + assert_flat(tml.Cu[('w', 'w')], 2, 'Var w') + # the law of lambda = -w: mean b + D, variance 2 + S (remainder only) + Dae = ae_add(ae_mul(tm_dev(tml.alpha['w'], -1, R), b1), ae_mul(tm_dev(tml.beta['w'], 0, R), k1)) + aC = tm_dev(tml.Cu[('w', 'w')], 2, R) + Sae = (aC[0], aC[1] + (2 + ae_at(aC, R)) * th1) + Dm, Sm = ae_at(Dae, R), ae_at(Sae, R) + r2 = M.isqrt_iv(M.IV(2)) + s_lo, s_hi = M.isqrt_iv(M.IV(2 - Sm)).lo, M.isqrt_iv(M.IV(2 + Sm)).hi + sa, sb = min(s_lo, r2.lo), max(s_hi, r2.hi) + m_lo, m_hi = b0 - Dm, b1 + Dm + mmax = max(abs(m_lo), abs(m_hi)) + Fmin = M.elam2_pos(m_lo, sa).lo + Fmax = M.elam2_pos(m_hi, sb).hi + F1max = M.elam_pos(m_hi, sb).hi + Fae = ae_add(ae_mul(Dae, 2 * F1max / Fmin), ae_mul(Sae, 2 * sb / Fmin * Fr(3, 4))) + Fref = M.elam2_pos(b0, r2.lo).lo + # secondary law: the engine's interval law over [0, R] + b, k = M.IV(b0, b1), M.IV(k0, k1) + ma = {nm: M.absup(b * il.alpha[nm] + k * il.beta[nm]) for nm in M.TARGET_NAMES} + gn = lambda nm, p: M.gauss_norm(ma[nm], il.sdQ[nm].hi, p) + lam = lambda p: M.gauss_norm(mmax, s_hi, p) + T, tau, om, v, nu = (lambda p, x=x: gn(x, p) for x in ('T', 'tau', 'om', 'v', 'nu')) + kap = lambda p: 6 * k1 * tau(p) + T(2 * p) ** 2 + R * T(2 * p) * tau(2 * p) + norm36 = 36 * k0 * k0 * Fref + # first order: r E[lambda_+ om] = r [mu_om F_1 - bw_om (F - m F_1)], with mu_om, bw_om = O(r) (Taylor models) + sa_, ea_ = tm_lin(tml.alpha['om'], R) + sb_, eb_ = tm_lin(tml.beta['om'], R) + sc_, ec_ = tm_lin(tml.Cu[('om', 'w')], R) + cwlo = 2 - ae_at(aC, R) + T2 = (((b1 * sa_ + k1 * sb_) * F1max + sc_ / cwlo * (Fmax + mmax * F1max)) / Fref, + R * ((b1 * ea_ + k1 * eb_) * F1max + ec_ / cwlo * (Fmax + mmax * F1max)) / Fref) + if MUT == 'no-om-term': + T2 = (Fr(0), Fr(0)) + # second order (explicit r^2). kappa = 6k tau - T^2 - r T tau with E[tau | w] = mu_tau + bw_tau (w - mu_w): + # 6k mu_tau F + 6k bw_tau E[(w - mu_w) lambda_+^2] - E[T^2 lambda_+^2] - r E[T tau lambda_+^2] (signed in the leading part) + mt = M.IV(6 * k0, 6 * k1) * (b * il.alpha['tau'] + k * il.beta['tau']) * M.IV(Fmin, Fmax) + side = 6 * k1 * M.absup(il.bw['tau']) * s_hi * lam(4) ** 2 + R * T(4) * tau(4) * lam(4) ** 2 + T3_up = max(Fr(0), mt.hi + side) / norm36 + T3_dn = max(Fr(0), -mt.lo + side + T(4) ** 2 * lam(4) ** 2) / norm36 + if MUT == 'no-om-term': + T3_up = T3_dn = Fr(0) + T4 = R * kap(4) * lam(4) * om(2) + xi2 = 6 * k1 * (R * nu(4) ** 2 + 2 * nu(4) * v(4)) + T(4) * (R * R * nu(8) ** 2 + 2 * R * nu(8) * v(8) + 2 * v(8) ** 2) \ + + R * tau(4) * v(8) ** 2 + T5 = lam(2) * xi2 + T6 = v(8) ** 2 * (6 * k1 + R * T(4) + R * R * tau(4)) * om(4) + v(4) ** 2 * (R * nu(4) + v(4)) ** 2 + # the thin set {0 < lambda <= r h} on G_0 (conditional density of lambda given (v, om, nu)) + Yn = ('v', 'om', 'nu') + prec = 1 / (2 - Sm) + wq_bound(M, il.Cg, Yn, [[M.absup(il.bw[nm])] for nm in Yn]) + pbar = Fr(2, 5) * M.sqrt_up(prec) + h = lambda p: Fr(2) / (9 * k0) * v(2 * p) ** 2 + om(p) + bad = R * pbar * h(2) * (Fr(15, 2) * k1 * h(4) + v(8) ** 2) * (9 * k1 * (h(4) + om(4)) + (R * nu(8) + v(8)) ** 2) + if MUT == 'no-bad-set': + bad = Fr(0) + # G_0^c (not r^2-scaled; astronomically small) + q = M.gauss_tail2(ma['T'], il.sdQ['T'].hi, 3 * k0 / (2 * R)) + M.gauss_tail2(ma['tau'], il.sdQ['tau'].hi, 3 * k0 / (2 * R * R)) + P1n = (6 * k1 + R * T(8)) * lam(8) + R * v(8) ** 2 + P2n = (6 * k1 + R * T(8) + R * R * tau(8)) * (lam(8) + R * om(8)) + R * (R * nu(8) + v(8)) ** 2 + g0c = P1n * P2n * M.sqrt_up(q) if q > 0 else Fr(0) + X = ae_add(T2, ((T4 + T5 + T6 + bad) / norm36, g0c / norm36)) + p_up, p_dn = pi_ratio(M, tml, 6, Q, R) + return combine(M, p_up, p_dn, ae_add(exp_up(M, Fae, R), X, (T3_up, Fr(0))), ae_add(Fae, X, (T3_dn, Fr(0))), R) + + +def bounds_d3(Q, R, tml, il): + """(up, dn) in d = 3 (NOTE sections 3-5).""" + M = E3 + b0, b1, k0, k1 = Q + th1 = tml.th1 + names = ('a11', 'a22', 'a12') + for nm, ref in (('a11', -1), ('a22', -1), ('a12', 0)): + assert_flat(tml.alpha[nm], ref, 'alpha_' + nm) + assert_flat(tml.beta[nm], 0, 'beta_' + nm) + for (x, y), ref in ((('a11', 'a11'), 2), (('a22', 'a22'), 2), (('a12', 'a12'), 1), (('a11', 'a22'), 0), + (('a11', 'a12'), 0), (('a22', 'a12'), 0)): + assert_flat(tml.Cu[(x, y)], ref, 'Cov(%s, %s)' % (x, y)) + # Loewner coupling with the contact law at the same b: eps = sup ||mu_B - b I||_F, eta = ||chol(Cb) - sqrt2 I|| + comps = [] + for nm, ref, f in (('a11', -1, 1), ('a22', -1, 1), ('a12', 0, 2)): + comps.append(ae_at(ae_add(ae_mul(tm_dev(tml.alpha[nm], ref, R), b1), ae_mul(tm_dev(tml.beta[nm], 0, R), k1)), R) * f) + eps = M.sqrt_up(sum(x * x for x in comps)) + Fw = (Fr(1), Fr(1), M.SQRT2) + Cb = [] + for i in range(3): + row = [] + for j in range(3): + lo, hi = tml.Cu[(names[i], names[j])].enclose(0, R) + row.append(M.IV(lo, hi) * Fw[i] * Fw[j] * M.IV(1 / (1 + th1), 1)) + Cb.append(row) + Lb = M.cholesky(Cb) + eta = M.sqrt_up(sum(M.absup(Lb[i][j] - (M.SQRT2 if i == j else 0)) ** 2 for i in range(3) for j in range(i + 1))) + Dl = M.rup_rel(eps + eta * R0C) + cm = M.contact_moment + bA, bB = grid_dn(b0 - Dl), grid_up(b1 + Dl) + m3b0 = cm(2, 0, b0).lo + L3up = 2 * cm(1, 1, bB).hi / m3b0 + L3dn = 2 * cm(1, 1, b1).hi / cm(2, 0, bA).lo + pR0 = M.chi_tail(3, R0C) + muB = M.rup_rel(b1 * M.sqrt_up(Fr(2)) + eps) + sig = M.sqrt_up(max(x.hi for x in (Cb[0][0], Cb[1][1], Cb[2][2])) + + 2 * max(M.absup(Cb[i][j]) for i in range(3) for j in range(3) if i != j)) + U = M.chi_upper_moments(3, R0C, 4) + out_D2 = sum((U[i] * (M.binom(4, i) * muB ** (4 - i) * sig ** i) for i in range(5)), M.IV(0)).hi / 4 + tail_c = M.sqrt_up(cm(4, 0, b1).hi) * M.sqrt_up(pR0) + nBF16 = M.rup_rel(muB + sig * M.chi_norm(3, 16)) + out_ball = M.sqrt_up(pR0) + + def up_moment(a, c): + return M.rup_rel(cm(a, c, bB).hi + 2 ** c * M.rup_rel(nBF16 ** (2 * a + c)) * out_ball) + ED2, ED4, ED2T2, ET4, DT = up_moment(2, 0), up_moment(4, 0), up_moment(2, 2), up_moment(0, 4), up_moment(1, 1) + trCb = sum(x.hi for x in (Cb[0][0], Cb[1][1], Cb[2][2])) + BF = lambda p: M.frob_norm(muB, trCb, p) + # secondary law (engine interval law over [0, R]) + b, k = M.IV(b0, b1), M.IV(k0, k1) + ma = {nm: M.absup(b * il.alpha[nm] + k * il.beta[nm]) for nm in M.TARGET_NAMES} + T = lambda p: M.gauss_norm(ma['T'], il.sdQ['T'], p) + tau = lambda p: M.gauss_norm(ma['tau'], il.sdQ['tau'], p) + mu_o = M.sqrt_up(ma['o11'] ** 2 + ma['o22'] ** 2 + 2 * ma['o12'] ** 2) + trO = sum(m * il.C[(nm, nm)].hi for nm, m in zip(M.ONAMES, (1, 1, 2))) + Om = lambda p: M.frob_norm(mu_o, trO, p) + mu_v = M.sqrt_up(ma['v1'] ** 2 + ma['v2'] ** 2) + v = lambda p: M.frob_norm(mu_v, il.C[('v1', 'v1')].hi + il.C[('v2', 'v2')].hi, p) + mu_nu = M.sqrt_up(ma['nu1'] ** 2 + ma['nu2'] ** 2) + nu = lambda p: M.frob_norm(mu_nu, il.C[('nu1', 'nu1')].hi + il.C[('nu2', 'nu2')].hi, p) + e = lambda p: R * nu(p) + v(p) + kap = lambda p: 6 * k1 * tau(p) + T(2 * p) ** 2 + R * T(2 * p) * tau(2 * p) + c1 = lambda p: 6 * k1 + R * T(p) + c2 = lambda p: 6 * k1 + R * T(p) + R * R * tau(p) + norm36 = 36 * k0 * k0 * m3b0 + # the mean of Omega': ||mu_Omega'||_F <= s_mu r + e_mu (Taylor models; beta_O = r^2 on the diagonal) + s_mu2 = e_mu2 = Fr(0) + for nm, mult in zip(M.ONAMES, (1, 1, 2)): + sa_, ea_ = tm_lin(tml.alpha[nm], R) + sb_, eb_ = tm_lin(tml.beta[nm], R) + s_mu2 += mult * (b1 * sa_ + k1 * sb_) ** 2 + e_mu2 += mult * (b1 * ea_ + k1 * eb_) ** 2 + muO = M.sqrt_up(s_mu2) * R + M.sqrt_up(e_mu2) # sup over (0, R] (it multiplies r^2 or more) + # the regression of Omega' on B: E[Omega' - mu | B] = beta (B - mu_B) + DY, beta = (r - r^3/4)/2 + ..., DY remainder-only + G = tml.G + beta_t = G['o11'][0] + bc = beta_t.frac_coeffs() + require(abs(bc[0]) <= TINY, 'regression of Omega on B at r = 0') + for (o, l) in (('o22', 1), ('o12', 2)): + assert_flat(G[o][l] - beta_t, 0, 'regression of %s on B' % o) + for o, ls in (('o11', (1, 2)), ('o22', (0, 2)), ('o12', (0, 1))): + for l in ls: + assert_flat(G[o][l], 0, 'off-diagonal regression of %s on B' % o) + dG2 = Fr(0) + for o, mult in zip(('o11', 'o22', 'o12'), (1, 1, 2)): + for l in range(3): + t = G[o][l] - beta_t if (o, l) in (('o11', 0), ('o22', 1), ('o12', 2)) else G[o][l] + dG2 += mult * Fr(t.bound(), 1 << TM_P) ** 2 + dGn = M.sqrt_up(dG2) + b1c = bc[1] # = 1/2 + require(b1c > 0, 'sign of the first-order regression coefficient of Omega on B') + sb_beta = sum(abs(x) * R ** (j - 2) for j, x in enumerate(bc) if j >= 2) + e_beta = beta_t.frac_e() + # E[D tr(A* Omega') 1] = beta (2 E[D^2 1] - E[D tr(A* mu_B) 1]) + R1, tr(A* mu_B) = b tr B + tr(A* dmu), ||dmu||_F <= eps + DTlo = cm(1, 1, bA).lo - M.sqrt_up(cm(2, 2, b1).hi) * M.sqrt_up(pR0) + m3up = cm(2, 0, b1).hi + rho_lo = exp_minus(M, Dl * L3dn).lo - tail_c / m3b0 # m3_r / m3(b) + rho_hi = exp_minus(M, -(Dl * L3up)).hi + out_D2 / m3b0 + tau_lo = b0 * max(DTlo, Fr(0)) / m3up # b E[D tr B 1] / m3(b) + tau_hi = b1 * DT / m3b0 + X3lo = 2 * rho_lo - tau_hi - eps * DT / m3b0 + X3hi = 2 * rho_hi - tau_lo + eps * DT / m3b0 + X3m = max(abs(X3lo), abs(X3hi)) + # |R1|/m3(b) <= (s_mu r + e_mu) DT/m3 + ||DG|| sqrt(E[D^2 tr^2 1]) sqrt(tr Cb)/m3, so r |R1|/m3 <= r^2 s_mu DT/m3 + R (...) + R1a = M.sqrt_up(s_mu2) * DT / m3b0 + R1e = (M.sqrt_up(e_mu2) * DT + dGn * M.sqrt_up(ED2T2) * M.sqrt_up(trCb)) / m3b0 + # -r E[D tr(A* Omega') 1]/m3(b) = -(b1c r^2 + (r beta - b1c r^2)) X3 - r R1/m3(b), |r beta - b1c r^2| <= sb_beta R r^2 + e_beta r + U3_up = (max(Fr(0), -b1c * X3lo) + sb_beta * R * X3m + R1a, R * (e_beta * X3m + R1e)) + U3_dn = (max(Fr(0), b1c * X3hi) + sb_beta * R * X3m + R1a, R * (e_beta * X3m + R1e)) + if MUT == 'no-om-term': + U3_up = U3_dn = (Fr(0), Fr(0)) + # explained parts of Omega' by B (for U4) and by zeta = (B, v, nu) (for U6, U7): traces of C_Oz C_zz^-1 C_zO + zB = ('a11', 'a22', 'a12') + zA = zB + tuple(M.VNAMES) + ('nu1', 'nu2') + def cov_Oz(o, z): + if z in zB: + s_, e_ = tm_lin(tml.Cu[(o, z)], R) + return s_ * R + e_ + return M.absup(il.C[(o, z)]) + def explained(zs): + rows = [] + for o, mult in zip(M.ONAMES, (1, 1, 2)): + rows.append([cov_Oz(o, z) for z in zs]) + D = [il.C[(z, z)].lo for z in zs] + Ef2 = Fr(0) + for i in range(len(zs)): + for j in range(len(zs)): + if i != j: + c = M.absup(il.C[(zs[i], zs[j])]) + Ef2 += c * c / (D[i] * D[j]) + Ef = M.sqrt_up(Ef2) + require(Ef < Fr(1, 2), 'diagonal scaling (explained part of Omega)') + return sum(mult * sum(x * x / Dz for x, Dz in zip(row, D)) for row, mult in zip(rows, (1, 1, 2))) / (1 - Ef) + tB, tK = explained(zB), explained(zA) + YB4, YK4 = M.frob_norm(0, tB, 4), M.frob_norm(0, tK, 4) + detX = M.absup(il.Cg[('o11', 'o22')] - il.Cg[('o12', 'o12')]) # |E[det(Omega' - E[Omega' | B])]| + E1 = up_moment(1, 0) + # second order (explicit r^2) + # signed kappa term: 6k mu_tau E[D^2 1] + 6k E[(tau - mu_tau) D^2 1] - E[T^2 D^2 1] - r E[T tau D^2 1] + ED2lo = max(Fr(0), cm(2, 0, bA).lo - tail_c) + mt = M.IV(6 * k0, 6 * k1) * (b * il.alpha['tau'] + k * il.beta['tau']) * M.IV(ED2lo, ED2) + side = 6 * k1 * il.bnorm['tau'] * M.sqrt_up(trCb) * M.sqrt_up(ED4) + R * T(4) * tau(4) * M.sqrt_up(ED4) + U2_up = max(Fr(0), mt.hi + side) / norm36 + U2_dn = max(Fr(0), -mt.lo + side + T(4) ** 2 * M.sqrt_up(ED4)) / norm36 + if MUT == 'no-om-term': + U2_up = U2_dn = Fr(0) + U2 = Fr(0) + U3r = R * kap(4) * M.sqrt_up(ED2T2) * Om(4) + # U4 = r^2 E[c1 c2 D det Omega' 1]: E[det Omega' | B] = det(mu + Y_B) + E[det X'] (36 k^2 cancels; relative) + U4rel = (detX * E1 + M.sqrt_up(ED2) * (muO + YB4) ** 2 / 2) / m3b0 + U4 = R * R * kap(4) * M.sqrt_up(ED2) * Om(8) ** 2 / 2 + U5 = M.sqrt_up(ED2T2) * ((2 * T(4) + R * tau(4)) * v(8) ** 2 + 2 * c1(4) * nu(8) * v(8) + R * c1(4) * nu(8) ** 2) + # U6, U7: the 6k parts through E[Omega' | B, v, nu] = mu + Y_K (Omega' - E[Omega' | zeta] has conditional mean 0) + U67rel = (M.sqrt_up(ED2) * (muO * e(4) ** 2 + e(8) ** 2 * YK4) + M.sqrt_up(ET4) * (muO * v(4) ** 2 + v(8) ** 2 * YK4)) / (6 * k0 * m3b0) + U6 = R * T(8) * M.sqrt_up(ED2) * Om(8) * e(8) ** 2 + U7 = R * (T(8) + R * tau(8)) * M.sqrt_up(ET4) * Om(8) * v(8) ** 2 + U8 = R * c2(8) * Om(16) ** 2 / 2 * M.iroot(ET4, 4) * v(16) ** 2 + U9 = M.sqrt_up(ET4) * v(8) ** 2 * e(8) ** 2 + U10 = R * M.iroot(ET4, 4) * v(8) ** 2 * Om(4) * e(8) ** 2 + # the thin set {0 < lambda_1 <= r h} on G_0: conditional law of B given Y = (v, Omega', nu), eigenvalue coordinates + Yn = tuple(M.VNAMES) + tuple(M.ONAMES) + ('nu1', 'nu2') + lmB = gersh_min(Cb) + rows = [] + for nm in Yn: + if nm in M.ONAMES: + cov = [(tm_lin(tml.Cu[(nm, an)], R)[0] * R + tm_lin(tml.Cu[(nm, an)], R)[1]) * f for an, f in zip(names, (1, 1, 2))] + rows.append([M.sqrt_up(sum(x * x for x in cov)) / lmB]) + else: + rows.append([M.absup(x) for x in il.bt[nm]]) + Wt = wq_bound(M, il.Cg, Yn, rows) + c_hi = sig * sig + cl = 1 / (1 / lmB + Wt) + KBp = M.rup_rel((M.PI * M.SQRT2 / (2 * M.PI * M.SQRT2PI)).hi / (cl * M.sqrt_down(cl))) + m1 = muB + c_hi * M.sqrt_up(Wt) * M.chi_norm(7, 8) + Hs = half_moments(M, c_hi) + Ja = m1 ** 4 / 4 + sum(M.binom(3, i) * m1 ** (3 - i) * Hs[i] for i in range(4)) + Jb = m1 ** 3 / 3 + sum(M.binom(2, i) * m1 ** (2 - i) * Hs[i] for i in range(3)) + h = lambda p: Fr(2) / (9 * k0) * v(2 * p) ** 2 + Om(p) + nhg = h(4) * (Fr(15, 2) * k1 * h(8) + v(16) ** 2) * (9 * k1 * (h(8) + Om(8)) + e(16) ** 2) + nhgO = h(8) * (Fr(15, 2) * k1 * h(16) + v(32) ** 2) * (9 * k1 * (h(16) + Om(16)) + e(32) ** 2) * Om(8) + bad = R * KBp * (Ja * nhg + R * Jb * nhgO) + if MUT == 'no-bad-set': + bad = Fr(0) + q = M.gauss_tail2(ma['T'], il.sdQ['T'], 3 * k0 / (2 * R)) + M.gauss_tail2(ma['tau'], il.sdQ['tau'], 3 * k0 / (2 * R * R)) + P1n = c1(8) * BF(16) ** 2 + R * BF(8) * v(16) ** 2 + P2n = c2(8) * (BF(16) + R * Om(16)) ** 2 + R * (BF(8) + R * Om(8)) * e(16) ** 2 + g0c = P1n * P2n * M.sqrt_up(q) if q > 0 else Fr(0) + X = ((U2 + U3r + U4 + U5 + U6 + U7 + U8 + U9 + U10 + bad) / norm36 + U4rel + U67rel, g0c / norm36) + z_up = ae_add(exp_up(M, (Fr(0), Dl * L3up), R), (Fr(0), out_D2 / m3b0), U3_up, X, (U2_up, Fr(0))) + z_dn = ae_add((Fr(0), Dl * L3dn), (Fr(0), tail_c / m3b0), U3_dn, X, (U2_dn, Fr(0))) + p_up, p_dn = pi_ratio(M, tml, 8, Q, R) + return combine(M, p_up, p_dn, z_up, z_dn, R) + + +def rate_sweep(d, tml): + """For every R in R_STARS and every sub-box: (a_up, eps_up, a_dn, eps_dn) as decimal strings (rounded up).""" + M = ENG[d] + fn = bounds_d2 if d == 2 else bounds_d3 + out = {} + for R in R_STARS: + il = M.band_law(Fr(0), R) + rows = [] + for bb, kb in SUBBOXES: + up, dn = fn((bb[0], bb[1], kb[0], kb[1]), R, tml, il) + rows.append([dec_up(up[0], 6), dec_up(up[1], 20), dec_up(dn[0], 6), dec_up(dn[1], 20)]) + out[str(R)] = {'boxes': rows, + 'a_up': max((r[0] for r in rows), key=Fr), 'eps_up': max((r[1] for r in rows), key=Fr), + 'a_dn': max((r[2] for r in rows), key=Fr), 'eps_dn': max((r[3] for r in rows), key=Fr)} + return out + + +# ============================================================================= 4. certified reference integrals +PREC = 240 + + +def iroot_int(N, p): + """Largest y >= 0 with y^p <= N.""" + if N < 2: + return N + y = 1 << ((N.bit_length() + p - 1) // p) + while True: + z = ((p - 1) * y + N // y ** (p - 1)) // p + if z >= y: + break + y = z + while y ** p > N: + y -= 1 + while (y + 1) ** p <= N: + y += 1 + return y + + +def pow_frac(M, x, num, den): + """Enclosure of x^(num/den), rational x > 0.""" + x = Fr(x) ** num + S = 1 << (den * PREC) + lo_n = (x.numerator * S) // x.denominator + hi_n = -((-x.numerator * S) // x.denominator) + ylo, yhi = iroot_int(lo_n, den), iroot_int(hi_n, den) + if yhi ** den < hi_n: + yhi += 1 + return M.IV(Fr(ylo, 1 << PREC), Fr(yhi, 1 << PREC)) + + +def lower_gamma(M, an, ad, x): + """gamma(a, x) = x^a e^-x sum_n x^n / (a (a+1) ... (a+n)), a = an/ad > 0, rational x > 0 (series with a geometric tail).""" + a, x = Fr(an, ad), Fr(x) + term = s = 1 / a + n = 0 + while True: + n += 1 + term = term * x / (a + n) + s += term + if a + n + 1 > 2 * x and term < Fr(1, 10 ** 70) * s: + break + rho = x / (a + n + 1) + return pow_frac(M, x, an, ad) * M.exp_neg_point(x) * M.IV(s, s + term * rho / (1 - rho)) + + +def k_integral(M, qn, qd, k0, k1): + """int_{k0}^{k1} k^q exp(-12 k^2) dk = (1/2) 12^(-a) [gamma(a, 12 k1^2) - gamma(a, 12 k0^2)], a = (q + 1)/2.""" + an, ad = qn + qd, 2 * qd + g = math.gcd(an, ad) + an, ad = an // g, ad // g + pre = pow_frac(M, Fr(1, 12), an, ad) * Fr(1, 2) + return pre * (lower_gamma(M, an, ad, 12 * Fr(k1) ** 2) - lower_gamma(M, an, ad, 12 * Fr(k0) ** 2)) + + +def m_ref(d, b): + """The reference contact moment: F(b, sqrt2) = E[(b + sqrt2 z)_+^2] (d = 2), m_3(b) = E[det(B)^2 1{B > 0}] (d = 3).""" + if d == 2: + s = E2.isqrt_iv(E2.IV(2)) + return E2.IV(E2.elam2_pos(Fr(b), s.lo).lo, E2.elam2_pos(Fr(b), s.hi).hi) + return E3.contact_moment(2, 0, Fr(b)) + + +def m_ref_derivs(d, b): + """Upper bounds of m, m', m'' at b (each increasing in b).""" + if d == 2: + s = E2.isqrt_iv(E2.IV(2)).hi + return E2.elam2_pos(Fr(b), s).hi, 2 * E2.elam_pos(Fr(b), s).hi, Fr(2) + cm = E3.contact_moment + return cm(2, 0, Fr(b)).hi, 2 * cm(1, 1, Fr(b)).hi, 2 * cm(0, 2, Fr(b)).hi + 4 * cm(1, 0, Fr(b)).hi + + +def b_cells(d): + """Enclosures of int over [j h, (j+1) h] of exp(-3b^2/4) m(b) db, h = 1/NH (midpoint rule, |g''| bounded per block).""" + M = ENG[d] + h = Fr(1, NH) + out = [] + for blk in range(NH // BLOCK): + x0, x1 = blk * BLOCK * h, (blk + 1) * BLOCK * h + m, m1, m2 = m_ref_derivs(d, x1) + g2 = M.exp_neg_point(3 * x0 * x0 / 4).hi * ((Fr(9, 4) * x1 * x1 + Fr(3, 2)) * m + 3 * x1 * m1 + m2) + err = h ** 3 / 24 * g2 + for j in range(BLOCK): + c = x0 + (j + Fr(1, 2)) * h + out.append(M.exp_neg_point(3 * c * c / 4) * m_ref(d, c) * h + M.IV(-err, err)) + return out + + +def b_integral(M, cells, b0, b1): + i0, i1 = Fr(b0) * NH, Fr(b1) * NH + require(i0.denominator == 1 and i1.denominator == 1, 'b-box off the cell grid') + return sum((cells[j] for j in range(int(i0), int(i1))), M.IV(0)) + + +def K_const(d): + """|S^(d-1)| 144 (2 pi)^(-(d+1)) / sqrt12: A_0^ref/(3 k^p) = K k^(2-p) e^(-12 k^2) e^(-3 b^2/4) m(b).""" + M = ENG[d] + tp = 2 * M.PI + pw = M.IV(1) + for _ in range(d + 1): + pw = pw * tp + return (tp if d == 2 else 2 * tp) * 144 / pw / M.isqrt_iv(M.IV(12)) + + +def reference(d): + M = ENG[d] + cells = b_cells(d) + K = K_const(d) + Ib = b_integral(M, cells, 0, 1) + c_ref = K * Ib * k_integral(M, 4, 3, Fr(1, 2), Fr(2)) + J23, J43, J13 = {}, {}, {} + for kb in SUB_K: + J23[kb] = k_integral(M, 2, 3, kb[0], kb[1]) + J43[kb] = k_integral(M, 4, 3, kb[0], kb[1]) + for kb in CAP_K: + J13[kb] = k_integral(M, 1, 3, kb[0], kb[1]) + Ibs = {bb: b_integral(M, cells, bb[0], bb[1]) for bb in set(SUB_B) | set(CAP_B)} + return {'c_ref': [dec_down(c_ref.lo, 24), dec_up(c_ref.hi, 24)], + 'w_c': [dec_up((K * Ibs[bb] * J43[kb]).hi, 24) for bb, kb in SUBBOXES], # int A_0/(3 k^(2/3)) + 'w_2': [dec_up((K * Ibs[bb] * J23[kb]).hi, 24) for bb, kb in SUBBOXES], # int A_0/(3 k^(4/3)) + 'w_cap': [dec_up((K * Ibs[bb] * J13[kb]).hi, 24) for bb, kb in CAP_BOXES]} # int A_0/(3 k^(5/3)) + + +# ============================================================================= 5. the refined cap sweep (engines of #206, #212) +CAPM = {} + + +def cap_init(d): + CAPM['M'] = ENG[d] + CAPM['d'] = d + if MUT == 'cap-half': + CAPM['M'].MUT = 'cap-half' + + +def cap_band(args): + """Certified bound of sup over the band of Q^W(G_r^c)/r^3 on every refined box (rounded up, 4 places).""" + d, r0, r1 = args + M = ENG[d] + if MUT == 'cap-half': + M.MUT = 'cap-half' + law = M.band_law(r0, r1) + x0, th = (M.COMBOS[0] if d == 2 else M.COMBO) + return {'r': [str(r0), str(r1)], + 'bound': [M.dec_up(M.box_bound(law, bb, kb, r0, r1, last_band=(r0 == 0), x0=x0, theta=th)['total']) + for bb, kb in CAP_BOXES]} + + +def cap_sweep(d, procs, which=None): + bl = ENG[d].bands() + idx = range(len(bl)) if which is None else which + jobs = [(d, bl[i][0], bl[i][1]) for i in idx] + if procs <= 1: + return [cap_band(j) for j in jobs] + with multiprocessing.Pool(procs, initializer=cap_init, initargs=(d,)) as pool: + return pool.map(cap_band, jobs, chunksize=1) + + +def cap_sample(d): + """Band indices replayed by --check: the band ending at each r_*, the band ending at 2^-20 and the last band.""" + bl = ENG[d].bands() + ends = set(R_STARS) | {Fr(1, 2 ** 20)} + return sorted({i for i, (r0, r1) in enumerate(bl) if r1 in ends} | {0}) + + +# ============================================================================= 6. assembly +def assemble(d, rate, ref, cap): + """The C_{B,K}(r_*) table, the second-order density constants, and the (12.1), (12.3) constants.""" + c_lo, c_hi = Fr(ref['c_ref'][0]), Fr(ref['c_ref'][1]) + wc = [Fr(x) for x in ref['w_c']] + w2 = [Fr(x) for x in ref['w_2']] + wcap = [Fr(x) for x in ref['w_cap']] + out = {} + for R in R_STARS: + rr = rate[str(R)] + a_up, e_up, a_dn, e_dn = Fr(rr['a_up']), Fr(rr['eps_up']), Fr(rr['a_dn']), Fr(rr['eps_dn']) + c2_up = sum(Fr(row[0]) * w for row, w in zip(rr['boxes'], w2)) + c2_dn = sum(Fr(row[2]) * w for row, w in zip(rr['boxes'], w2)) + e_c_up = sum(Fr(row[1]) * w for row, w in zip(rr['boxes'], wc)) + e_c_dn = sum(Fr(row[3]) * w for row, w in zip(rr['boxes'], wc)) + Cbox = [Fr(0)] * len(CAP_BOXES) + for rec in cap: + if Fr(rec['r'][1]) <= R: + for i, x in enumerate(rec['bound']): + Cbox[i] = max(Cbox[i], Fr(x)) + S = sum(c * w for c, w in zip(Cbox, wcap)) + CBK = (1 + a_up * R * R + e_up) * S + out[str(R)] = {'C_BK': dec_up(CBK, 4), 'C_121': dec_up(Fr(3, 5) * CBK, 4), + 'c2_up': dec_up(c2_up, 12), 'c2_dn': dec_up(c2_dn, 12), + 'epsc_up': dec_up(e_c_up, 30), 'epsc_dn': dec_up(e_c_dn, 30), + 'rel2_up': dec_up(c2_up / c_lo, 6), 'rel2_dn': dec_up(c2_dn / c_lo, 6), + 'ell_max': '%s' % (R ** 3 / 2)} + return out + + +def params(): + return {'R_STARS': [str(x) for x in R_STARS], 'TM': {'s': TM_S, 'N': TM_N, 'P': TM_P}, 'R0_COUPLING': str(R0C), + 'SUB_B': [[str(a), str(b)] for a, b in SUB_B], 'SUB_K': [[str(a), str(b)] for a, b in SUB_K], + 'NH': NH, 'BLOCK': BLOCK, 'CAP_B': [[str(a), str(b)] for a, b in CAP_B], 'CAP_K': [[str(a), str(b)] for a, b in CAP_K], + 'band': {'b': ['0', '1'], 'k': ['1/2', '2']}, 'L_min': 10} + + +def compute_core(d): + tml = tm_law(d) + return tm_summary(tml), rate_sweep(d, tml), reference(d) + + +def full_run(procs): + res = {'schema': 1, 'object': 'CL-C8-LIFETIME-DIFFERENCE-CONSTANT-20261001-v1', 'scientific_effect': 'NONE', + 'certified': True, 'mutant': None, 'parameters': params(), 'dims': {}} + for d in (2, 3): + summ, rate, ref = compute_core(d) + cap = cap_sweep(d, procs) + res['dims'][str(d)] = {'tm_law': summ, 'rate': rate, 'reference': ref, 'table': assemble(d, rate, ref, cap), + 'cap_boxes': [[str(bb[0]), str(bb[1]), str(kb[0]), str(kb[1])] for bb, kb in CAP_BOXES], + 'cap_bands': cap} + return res + + +def results_text(res): + """Canonical layout: everything indented except the cap-band records, one line each.""" + caps = {d: res['dims'][d].pop('cap_bands') for d in res['dims']} + head = json.dumps(res, indent=1, sort_keys=True) + for d in res['dims']: + res['dims'][d]['cap_bands'] = caps[d] + tail = ',\n "cap_bands": {\n' + ',\n'.join( + ' "%s": [\n' % d + ',\n'.join(' ' + json.dumps(rec, sort_keys=True) for rec in caps[d]) + '\n ]' + for d in sorted(caps)) + '\n }\n}\n' + return head[:-2] + tail + + +def load_results(): + with open(os.path.join(HERE, 'RESULTS.json')) as fh: + res = json.load(fh) + for d, recs in res.pop('cap_bands').items(): + res['dims'][d]['cap_bands'] = recs + return res + + +def check_run(procs, full=False): + res = load_results() + with open(os.path.join(HERE, 'RESULTS.json')) as fh: + require(fh.read() == results_text(res), 'RESULTS.json is not in canonical layout') + require(res['parameters'] == params(), 'parameters differ') + require(res['mutant'] is None and res['certified'] is True, 'header') + for d in (2, 3): + rd = res['dims'][str(d)] + tml = tm_law(d) + require(tm_summary(tml) == rd['tm_law'], 'Taylor-model law differs (d = %d)' % d) + rate = rate_sweep(d, tml) + require(rate == rd['rate'], 'rate sweep differs (d = %d)' % d) + ref = reference(d) + require(ref == rd['reference'], 'reference integrals differ (d = %d)' % d) + c_lo, c_hi = Fr(ref['c_ref'][0]), Fr(ref['c_ref'][1]) + cx = C_REF_197 if d == 2 else C_REF_205 + require(c_lo <= cx <= c_hi, 'reference c_{B,K} does not contain the value of Math-#197/#205 (d = %d)' % d) + require(rd['cap_boxes'] == [[str(bb[0]), str(bb[1]), str(kb[0]), str(kb[1])] for bb, kb in CAP_BOXES], 'cap boxes') + bl = ENG[d].bands() + require([rec['r'] for rec in rd['cap_bands']] == [[str(a), str(b)] for a, b in bl], 'cap band layout (d = %d)' % d) + which = None if full else cap_sample(d) + recs = cap_sweep(d, procs, which) + for i, rec in zip(range(len(bl)) if which is None else which, recs): + require(rec == rd['cap_bands'][i], 'cap band %d differs (d = %d)' % (i, d)) + require(assemble(d, rate, ref, rd['cap_bands']) == rd['table'], 'assembled table differs (d = %d)' % d) + print('CHECK PASSED (%s)' % ('full' if full else 'sample')) + + +# ============================================================================= 7. floating controls (not part of the certificate) +def controls(): + """Floating estimates of the reference-kernel cap-route benchmarks Gamma_{B,K} = |S| int A_0^ref c_G / (3 k^(5/3)) for comparison with C_{B,K}: + planar c_G = R(b) J(k) of Math-#203 with J(k) = 2359296 k^3 (1 + delta(k)), delta(1/2) = 0.0232, delta(3/4) = 1.3e-4; + d = 3 from the floating R_3(b) R(b) and J^(3)/J ratios of Math-#208.""" + from math import exp, pi, sqrt, erf + out = {} + K2 = 2 * pi * 144 / (2 * pi) ** 3 / sqrt(12) + Ib2 = sqrt(pi) / 2 * erf(1) / (2 * sqrt(pi)) # int_0^1 e^{-b^2} db / (2 sqrt(pi)) (m R = p_w(0)) + def J2(k): + dl = 0.0232 + (k - 0.5) / 0.25 * (1.3e-4 - 0.0232) if k < 0.75 else 1.3e-4 * max(0.0, (1 - k) / 0.25) + return 2359296 * k ** 3 * (1 + dl) + n = 200000 + Ik2 = sum(((0.5 + (i + 0.5) * 1.5 / n) ** (1 / 3)) * exp(-12 * (0.5 + (i + 0.5) * 1.5 / n) ** 2) * J2(0.5 + (i + 0.5) * 1.5 / n) + * 1.5 / n for i in range(n)) + out['Gamma_2'] = K2 * Ib2 * Ik2 + bs = [0, 0.25, 0.5, 0.75, 1.0] + m3 = [0.6715728752625627, 1.195532362807149, 2.05794812257308, 3.43079921823, 5.54806027367761] + r3r = [1.1880833639847637, 0.9085531055205767, 0.6853903187865128, 0.5096151204445851, 0.37314989157732215] + f = [exp(-0.75 * b * b) * m * r for b, m, r in zip(bs, m3, r3r)] + Ib3 = 0.25 / 3 * (f[0] + 4 * f[1] + 2 * f[2] + 4 * f[3] + f[4]) + K3 = 4 * pi * 144 / (2 * pi) ** 4 / sqrt(12) + def J3(k): + dl = 0.1617 + (k - 0.5) / 0.25 * (0.0012 - 0.1617) if k < 0.75 else 0.0012 * max(0.0, (1 - k) / 0.25) + return 2359296 * k ** 3 * (1 + dl) + Ik3 = sum(((0.5 + (i + 0.5) * 1.5 / n) ** (1 / 3)) * exp(-12 * (0.5 + (i + 0.5) * 1.5 / n) ** 2) * J3(0.5 + (i + 0.5) * 1.5 / n) + * 1.5 / n for i in range(n)) + out['Gamma_3'] = K3 * Ib3 * Ik3 + return out + + +def mc_ratio(d, n, seed=20261001): + """Floating Monte Carlo of (A_r/A_0^ref - 1)/r^2 for the reference kernel at r = 1/8, 1/16: the exact law at r from the + series (all r, not only r <= 1/64), Z_r/r^2 by sampling W_r/r^2 = P1 P2 1{typed}, and the exact pin ratio.""" + import random + M = ENG[d] + names = ('w', 'v', 'om', 'T', 'tau', 'nu') if d == 2 else \ + ('a11', 'a22', 'a12', 'v1', 'v2', 'o11', 'o22', 'o12', 'T', 'tau', 'nu1', 'nu2') + pins, targets = M.functionals() + P = [pins[x] for x in M.PIN_NAMES] + idx = [M.TARGET_NAMES.index(x) for x in names] + Tf = [targets[x] for x in names] + tml = tm_law(d) + def cov(F, G, key, r): + ser, w, _ = M.pair_series(F, G, key) + return sum(c * r ** p for p, c in ser.items()) + def law(r, b, k): + n_ = len(P) + S = [[None] * n_ for _ in range(n_)] + for i in range(n_): + for j in range(i, n_): + S[i][j] = S[j][i] = cov(P[i], P[j], ('p', i, j), r) + Stp = [[cov(Tf[a], P[j], ('tp', idx[a], j), r) for j in range(n_)] for a in range(len(Tf))] + Mx = [list(S[i]) + [Stp[a][i] for a in range(len(Tf))] for i in range(n_)] + for col in range(n_): + for i in range(col + 1, n_): + f = Mx[i][col] / Mx[col][col] + for j in range(col, len(Mx[i])): + Mx[i][j] -= f * Mx[col][j] + W = [] + for a in range(len(Tf)): + x = [None] * n_ + for i in reversed(range(n_)): + x[i] = (Mx[i][n_ + a] - sum(Mx[i][j] * x[j] for j in range(i + 1, n_))) / Mx[i][i] + W.append(x) + kv = [-r ** 3 / 2, -r ** 2, Fr(12)] + [Fr(0)] * (n_ - 3) + mu = [float(b * W[a][0] + k * sum(W[a][j] * kv[j] for j in range(n_))) for a in range(len(Tf))] + def stt(a, c): + i, j = idx[a], idx[c] + return cov(Tf[a], Tf[c], ('tt', i, j), r) if i <= j else cov(Tf[c], Tf[a], ('tt', j, i), r) + C = [[float(stt(a, c) - sum(W[a][l] * Stp[c][l] for l in range(n_))) for c in range(len(Tf))] for a in range(len(Tf))] + L = [[0.0] * len(C) for _ in C] + for i in range(len(C)): + for j in range(i + 1): + s = C[i][j] - sum(L[i][l] * L[j][l] for l in range(j)) + L[i][j] = math.sqrt(max(s, 0.0)) if i == j else (s / L[j][j] if L[j][j] > 0 else 0.0) + return mu, L + rng = random.Random(seed) + out = [] + for r in (Fr(1, 8), Fr(1, 16)): + rf = float(r) + for b, k in ((Fr(0), Fr(1, 2)), (Fr(0), Fr(2)), (Fr(1), Fr(1, 2)), (Fr(1), Fr(2))): + mu, L = law(r, b, k) + kf = float(k) + tot = tot2 = 0.0 + for _ in range(n): + z = [rng.gauss(0.0, 1.0) for _ in mu] + x = [mu[i] + sum(L[i][j] * z[j] for j in range(i + 1)) for i in range(len(mu))] + if d == 2: + w, v, om, T, tau, nu = x + lam = -w + c1, c2 = 6 * kf - rf * T, 6 * kf + rf * T + rf * rf * tau + P1 = c1 * lam - rf * v * v + P2 = c2 * (lam - rf * om) + rf * (rf * nu - v) ** 2 + Wv = P1 * P2 if (c1 > 0 and P1 > 0 and P2 > 0) else 0.0 + else: + a11, a22, a12, v1, v2, o11, o22, o12, T, tau, nu1, nu2 = x + B = (-a11, -a22, -a12) + S_ = (B[0] - rf * o11, B[1] - rf * o22, B[2] - rf * o12) + detB, detS = B[0] * B[1] - B[2] ** 2, S_[0] * S_[1] - S_[2] ** 2 + c1, c2 = 6 * kf - rf * T, 6 * kf + rf * T + rf * rf * tau + e1, e2 = rf * nu1 - v1, rf * nu2 - v2 + P1 = c1 * detB - rf * (B[1] * v1 * v1 - 2 * B[2] * v1 * v2 + B[0] * v2 * v2) + P2 = c2 * detS + rf * (S_[1] * e1 * e1 - 2 * S_[2] * e1 * e2 + S_[0] * e2 * e2) + Wv = 0.0 + if B[0] > 0 and detB > 0 and P1 > 0 and P2 > 0: + # index(H_S) = 2: sign changes of the leading minors (-S_11, det S, r P2) of H_S reordered (y1, y2, x) + m1, m2, m3 = -S_[0], detS, rf * P2 + if m1 != 0 and m2 != 0: + neg = (m1 < 0) + (m2 / m1 < 0) + (m3 / m2 < 0) + Wv = P1 * P2 if neg == 2 else 0.0 + tot += Wv + tot2 += Wv * Wv + mean = tot / n + se = math.sqrt(max(tot2 / n - mean * mean, 0.0) / n) + mref = float(m_ref(d, b).mid()) + zr, zse = mean / (36 * kf * kf * mref), se / (36 * kf * kf * mref) + ev = lambda t: sum(float(c) * rf ** j for j, c in enumerate(t.frac_coeffs())) + E = ev(tml.e_bb) * float(b) ** 2 + 2 * ev(tml.e_bk) * float(b) * kf + ev(tml.e_kk) * kf * kf + pr = math.sqrt(12 / ev(tml.det)) * math.exp(-(E - (1.5 * float(b) ** 2 + 24 * kf * kf)) / 2) + out.append({'d': d, 'r': str(r), 'b': str(b), 'k': str(k), 'n': n, 'Z_ratio': round(zr, 5), 'Z_se': round(zse, 5), + 'pin_ratio': round(pr, 6), 'coef': round((pr * zr - 1) / rf ** 2, 3), 'coef_se': round(pr * zse / rf ** 2, 3)}) + return out + + +def main(): + global MUT + ap = argparse.ArgumentParser() + ap.add_argument('--check', action='store_true') + ap.add_argument('--check-full', action='store_true') + ap.add_argument('--mutant', choices=MUTANTS) + ap.add_argument('--procs', type=int, default=4) + ap.add_argument('--controls', action='store_true') + args = ap.parse_args() + MUT = args.mutant + if args.controls: + out = controls() + out['mc'] = mc_ratio(2, 200000) + mc_ratio(3, 100000) + print(json.dumps(out, indent=1)) + return + if args.check or args.check_full: + check_run(args.procs, full=args.check_full) + return + require(MUT is None, 'a mutant cannot regenerate RESULTS.json') + res = full_run(args.procs) + with open(os.path.join(HERE, 'RESULTS.json'), 'w') as fh: + fh.write(results_text(res)) + print(json.dumps({d: {R: row['C_BK'] for R, row in res['dims'][d]['table'].items()} for d in res['dims']})) + + +if __name__ == '__main__': + main() diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/theorem_v.py b/frontiers/c8_torus_vanishing_remainder_20261001/theorem_v.py new file mode 100644 index 000000000..61e664559 --- /dev/null +++ b/frontiers/c8_torus_vanishing_remainder_20261001/theorem_v.py @@ -0,0 +1,849 @@ +#!/usr/bin/env python3 +"""Theorem L' (NOTE.md): the radial ledger of [LP] Theorem B on the C8 band with a VANISHING remainder, d = 2 and d = 3. + +For the torus kernel (every side L >= 10, every frame) and for the reference kernel, b in [0, 1], k in [1/2, 2], 0 < r <= R: + + -(a_dn r^2 + beta_dn r) <= A_r(b, k, u) / A_0(b, k, u) - 1 <= a_up r^2 + beta_up r, + +where A_0 = lim_{r -> 0} A_r is the contact integrand of THE SAME kernel (for the torus: its own A_0^(L), not the Gaussian +A_0^ref). Math-#215 (Theorem L) proved the same shape against A_0^ref with an additive, r-independent eps_d; here every +remainder carries a positive power of r. The first-order coefficient beta collects the torus corrections only (about +1e-12); for the reference kernel it is a rounding residue below 1e-80. + +Method (NOTE.md sections 2-5): +- Taylor models with INTERVAL coefficients and a remainder proportional to r^(N+1) (class TV): every rounding goes into a + coefficient radius, every truncation into the r^(N+1) remainder. The value at r = 0 is the constant coefficient, so + t(r) - t(0) is controlled by the coefficients of r^1, r^2, ... only. +- The torus part of each covariance has the coefficient majorant |tau_p| <= T0 e^p (Cauchy estimates and + g_x!/(g_x - A)! <= w^A e^(g_x - w), as in the engines of Math-#206/#212); it enters as radii T0 e^p on the coefficients + p <= N and as T0 e^(N+1)/(1 - e R) r^(N+1). +- Increments against r = 0: the pin-density ratio pi_r/pi_0 (the kernel normalization Theta multiplies increments only), + the main normalizer term by the derivative bound dF = 2 E[lambda_+] dmu + Phi(mu/sigma) d(sigma^2) (d = 2) and by a Gaussian + gradient coupling with |grad det(B)^2 1{B > 0}| <= ||B||_F^3 (d = 3); first-order terms carry an explicit factor r; the + thin set and the far set G_0^c by Markov at a high moment (r^2). +- The interval band laws, the second-order term bounds and the reference integrals are those of Math-#215's theorem_b.py and + the engines of Math-#206/#212, as byte-identical pinned copies (SOURCE_MAP.json). + + python3 -B -S theorem_v.py --write regenerate RESULTS.json + python3 -B -S theorem_v.py --check replay and compare with RESULTS.json (exact) + python3 -B -S theorem_v.py --mutant NAME a semantic mutation; the replay must then fail (exit 1) +""" +import argparse +import importlib.util +import json +import math +import os +import sys +from fractions import Fraction as Fr + +HERE = os.path.dirname(os.path.abspath(__file__)) +OBJECT = 'CL-C8-TORUS-VANISHING-REMAINDER-20261001-v1' +MUTANTS = ('torus-const-only', 'exp-radius', 'no-lin', 'no-om-term', 'coupling-half') +MUT = None + + +def require(cond, msg): + if not cond: + raise SystemExit('FAIL: ' + msg) + + +def _load(name, fn): + spec = importlib.util.spec_from_file_location(name, os.path.join(HERE, fn)) + mod = importlib.util.module_from_spec(spec) + spec.loader.exec_module(mod) + return mod + + +TB = _load('theorem_b', 'theorem_b.py') # Math-#215, pinned byte-identical copy (loads engine_e2.py, engine_e3.py) +E2, E3, ENG = TB.E2, TB.E3, TB.ENG +R_STARS = TB.R_STARS +SUBBOXES, SUB_K = TB.SUBBOXES, TB.SUB_K +TM_S, TM_N, TM_P = TB.TM_S, TB.TM_N, TB.TM_P +TINY = TB.TINY +E_UP = Fr(27183, 10000) # > e +SQRT2_UP = Fr(14143, 10000) # > sqrt(2) +MARKOV_P = 64 # moment order of the far-set bound +dec_up, dec_down, cdiv = TB.dec_up, TB.dec_down, TB.cdiv + + +# ============================================================================= 1. interval-coefficient Taylor models +class TV: + """f(r) = sum_{j <= N} c_j r^j + rho(r) for 0 <= r <= 2^-s, with FIXED (r-independent) coefficients + c_j in [(C_j - D_j)/2^P, (C_j + D_j)/2^P] and |rho(r)| <= (E/2^P) r^(N+1). In particular f(0) = c_0.""" + __slots__ = ('C', 'D', 'E', 's', 'N') + + def __init__(self, C, D, E, s, N): + self.C, self.D, self.E, self.s, self.N = C, D, E, s, N + + @classmethod + def build(cls, centers, radii, e, s, N): + """centers, radii: {power: rational}; e: rational coefficient of r^(N+1) in the remainder bound.""" + C, D = [0] * (N + 1), [0] * (N + 1) + R = Fr(1, 2 ** s) + E = Fr(e) + for p, x in centers.items(): + x = Fr(x) + if x == 0: + continue + if p <= N: + v = round(x * (1 << TM_P)) + C[p] += v + if x * (1 << TM_P) != v: + D[p] += 1 + else: + E += abs(x) * R ** (p - N - 1) + for p, x in radii.items(): + x = Fr(x) + if p <= N: + D[p] += cdiv((x * (1 << TM_P)).numerator, (x * (1 << TM_P)).denominator) + else: + E += x * R ** (p - N - 1) + return cls(C, D, cdiv(E.numerator << TM_P, E.denominator), s, N) + + @classmethod + def const(cls, x, s, N, rad=0): + return cls.build({0: Fr(x)}, {0: Fr(rad)} if rad else {}, 0, s, N) + + def zero(self): + return TV([0] * (self.N + 1), [0] * (self.N + 1), 0, self.s, self.N) + + def polybound(self): + """Upper bound (units 2^-P) of |sum c_j r^j| on [0, 2^-s].""" + return sum(cdiv(abs(c) + d, 1 << (self.s * j)) for j, (c, d) in enumerate(zip(self.C, self.D)) if c or d) + + def __add__(self, o): + if not isinstance(o, TV): + o = TV.const(o, self.s, self.N) + return TV([a + b for a, b in zip(self.C, o.C)], [a + b for a, b in zip(self.D, o.D)], self.E + o.E, self.s, self.N) + __radd__ = __add__ + + def __neg__(self): + return TV([-a for a in self.C], list(self.D), self.E, self.s, self.N) + + def __sub__(self, o): + if not isinstance(o, TV): + o = TV.const(o, self.s, self.N) + return self + (-o) + + def scale(self, x): + x = Fr(x) + C, D = [], [] + for c, d in zip(self.C, self.D): + v = c * x + w = round(v) + C.append(w) + D.append(cdiv((d * abs(x)).numerator, (d * abs(x)).denominator) + (1 if v != w else 0)) + e = self.E * abs(x) + return TV(C, D, cdiv(e.numerator, e.denominator), self.s, self.N) + + def __mul__(self, o): + if not isinstance(o, TV): + return self.scale(o) + N, s = self.N, self.s + S = [0] * (2 * N + 1) + Rd = [0] * (2 * N + 1) + for i, (a, da) in enumerate(zip(self.C, self.D)): + if a or da: + for j, (b, db) in enumerate(zip(o.C, o.D)): + if b or db: + S[i + j] += a * b + Rd[i + j] += abs(a) * db + da * abs(b) + da * db + C, D, E = [], [], 0 + for k in range(N + 1): + v = S[k] >> TM_P + C.append(v) + D.append(cdiv(Rd[k], 1 << TM_P) + (1 if S[k] - (v << TM_P) else 0)) + for k in range(N + 1, 2 * N + 1): + if S[k] or Rd[k]: + E += cdiv(abs(S[k]) + Rd[k], 1 << (TM_P + s * (k - N - 1))) + E += cdiv(self.E * o.polybound() + o.E * self.polybound(), 1 << TM_P) + E += cdiv(self.E * o.E, 1 << (TM_P + s * (N + 1))) + return TV(C, D, E, s, N) + __rmul__ = __mul__ + + def c0(self): + return Fr(self.C[0] - self.D[0], 1 << TM_P), Fr(self.C[0] + self.D[0], 1 << TM_P) + + def lin_bound(self): + """G with |f(r) - f(0)| <= G r on [0, 2^-s] (exact rational).""" + R = Fr(1, 2 ** self.s) + g = sum(Fr(abs(c) + d, 1 << TM_P) * R ** (j - 1) for j, (c, d) in enumerate(zip(self.C, self.D)) if j >= 1) + return g + Fr(self.E, 1 << TM_P) * R ** self.N + + def inv(self): + lo, hi = self.c0() + require(lo > 0 or hi < 0, 'TV inverse at a constant term containing 0') + # centered at 1/C_0 (so that centers stay the reference values), radius covering [1/hi, 1/lo] + uc = Fr(1 << TM_P, self.C[0]) + u = TV.build({0: uc}, {0: max(abs(1 / hi - uc), abs(1 / lo - uc))}, 0, self.s, self.N) + h = TV([0] + self.C[1:], [0] + self.D[1:], self.E, self.s, self.N) + g = h * u + require(g.C[0] == 0 and g.D[0] == 0, 'TV inverse: g(0) = 0') + G1 = g.lin_bound() + R = Fr(1, 2 ** self.s) + require(G1 * R < Fr(1, 2), 'TV inverse: |g| too large') + acc = TV.const(1, self.s, self.N) + p = TV.const(1, self.s, self.N) + mg = -g + for _ in range(self.N): + p = p * mg + acc = acc + p + tail = G1 ** (self.N + 1) / (1 - G1 * R) + acc = TV(acc.C, acc.D, acc.E + cdiv(tail.numerator << TM_P, tail.denominator), self.s, self.N) + return acc * u + + # ----------------------------------------------------------------------------- readers + def coeff(self, j): + return Fr(self.C[j] - self.D[j], 1 << TM_P), Fr(self.C[j] + self.D[j], 1 << TM_P) + + def high(self, R): + """H with |sum_{j >= 3} c_j r^j + rho(r)| <= H r^2 on (0, R].""" + h = sum(Fr(abs(c) + d, 1 << TM_P) * R ** (j - 2) for j, (c, d) in enumerate(zip(self.C, self.D)) if j >= 3) + return h + Fr(self.E, 1 << TM_P) * R ** (self.N - 1) + + def inc(self, R): + """(l_lo, l_hi, q_lo, q_hi): f(r) - f(0) in [l_lo r + q_lo r^2, l_hi r + q_hi r^2] for 0 < r <= R.""" + l_lo, l_hi = self.coeff(1) + q_lo, q_hi = self.coeff(2) + H = self.high(R) + if MUT == 'no-lin': + l_lo = l_hi = Fr(0) + return l_lo, l_hi, q_lo - H, q_hi + H + + def absinc(self, R): + """(a, b): |f(r) - f(0)| <= a r^2 + b r.""" + l_lo, l_hi, q_lo, q_hi = self.inc(R) + return max(abs(q_lo), abs(q_hi)), max(abs(l_lo), abs(l_hi)) + + def absval(self, R): + """(s, c): |f(r)| <= c + s r on [0, R], c = sup |c_0|.""" + lo, hi = self.c0() + s = sum(Fr(abs(c) + d, 1 << TM_P) * R ** (j - 1) for j, (c, d) in enumerate(zip(self.C, self.D)) if j >= 1) + return s + Fr(self.E, 1 << TM_P) * R ** self.N, max(abs(lo), abs(hi)) + + def enclose(self, R): + """Interval of f(r) over r in [0, R].""" + lo, hi = self.c0() + for j in range(1, self.N + 1): + a, b = self.coeff(j) + Rj = R ** j + lo += min(a, 0) * Rj + hi += max(b, 0) * Rj + e = Fr(self.E, 1 << TM_P) * R ** (self.N + 1) + return lo - e, hi + e + + def center(self, j): + return Fr(self.C[j], 1 << TM_P) + + +def cov_tv(M, F, G, key, s, N, M1): + """TV on [0, 2^-s] of the covariance of F, G under the unnormalized torus kernel: the exact reference series (centers), + the Taylor truncation tail (gx > NX, proportional to r^(NX+1-w)) and the torus part sum_p tau_p r^p with + |tau_p| <= T0 e^p (radii T0 e^p for p <= N, remainder T0 e^(N+1)/(1 - e R) r^(N+1)).""" + R = Fr(1, 2 ** s) + ser, w, absterms = M.pair_series(F, G, key) + A = max((t[1] for t in absterms), default=0) + tail = T0 = Fr(0) + for t in absterms: + acc, aa, gys = t[0], t[1], t[2:] + hprod = fprod = 1 + for g in gys: + hprod *= M.hcoef(g) + fprod *= math.factorial(g) + tail += acc * abs(hprod) + T0 += acc * fprod * Fr(max(w, 1)) ** aa + require(M.NX + 1 - w >= N + 1, 'truncation tail below the Taylor-model order') + tail_e = 2 * Fr(M.NX + 1) ** A * tail * R ** (M.NX + 1 - w - (N + 1)) + T0 = T0 * M1 + eu = Fr(1) if MUT == 'exp-radius' else E_UP + if MUT == 'torus-const-only': + radii = {0: T0} + tor_e = Fr(0) + else: + radii = {p: T0 * eu ** p for p in range(N + 1)} + tor_e = T0 * eu ** (N + 1) / (1 - eu * R) + return TV.build(ser, radii, tail_e + tor_e, s, N) + + +def solve_tv(A, B): + n = len(A) + Mx = [[A[i][j] for j in range(n)] + [B[c][i] for c in range(len(B))] for i in range(n)] + det, invs = None, [] + for col in range(n): + piv = Mx[col][col] + det = piv if det is None else det * piv + ip = piv.inv() + invs.append(ip) + for i in range(col + 1, n): + f = Mx[i][col] * ip + for j in range(col, len(Mx[i])): + Mx[i][j] = Mx[i][j] - f * Mx[col][j] + X = [] + for c in range(len(B)): + x = [None] * n + for i in reversed(range(n)): + acc = Mx[i][n + c] + for j in range(i + 1, n): + acc = acc - Mx[i][j] * x[j] + x[i] = acc * invs[i] + X.append(x) + return X, det + + +class Law: + pass + + +def tv_law(d, torus=True, s=TM_S, N=TM_N, L=10): + """The TV law over [0, 2^-s] of the targets TB.TM_NAMES[d]: as TB.tm_law, with interval-coefficient models.""" + M = ENG[d] + names = TB.TM_NAMES[d] + pins, targets = M.functionals() + P = [pins[nm] for nm in M.PIN_NAMES] + idx = [M.TARGET_NAMES.index(nm) for nm in names] + T = [targets[nm] for nm in names] + M1 = M.torus_M1(L) if torus else Fr(0) + n, nt = len(P), len(T) + Spp = [[None] * n for _ in range(n)] + for i in range(n): + for j in range(i, n): + Spp[i][j] = Spp[j][i] = cov_tv(M, P[i], P[j], ('p', i, j), s, N, M1) + Stp = [[cov_tv(M, T[a], P[j], ('tp', idx[a], j), s, N, M1) for j in range(n)] for a in range(nt)] + Stt = [[None] * nt for _ in range(nt)] + for a in range(nt): + for b in range(a, nt): + i, j = idx[a], idx[b] + if i <= j: + Stt[a][b] = Stt[b][a] = cov_tv(M, T[a], T[b], ('tt', i, j), s, N, M1) + else: + Stt[a][b] = Stt[b][a] = cov_tv(M, T[b], T[a], ('tt', j, i), s, N, M1) + zero = TV.const(0, s, N) + kvec = [TV.build({3: Fr(-1, 2)}, {}, 0, s, N), TV.build({2: Fr(-1)}, {}, 0, s, N), TV.const(12, s, N)] + [zero] * (n - 3) + e1 = [TV.const(1, s, N)] + [zero] * (n - 1) + X, det = solve_tv(Spp, Stp + [e1, kvec]) + W, xs0, xs1 = X[:nt], X[nt], X[nt + 1] + law = Law() + law.d, law.M1, law.s, law.N = d, M1, s, N + law.th1 = M.theta_minus_one(L) if torus else Fr(0) + law.alpha = {nm: W[a][0] for a, nm in enumerate(names)} + law.beta = {nm: sum((W[a][j] * kvec[j] for j in range(n)), zero) for a, nm in enumerate(names)} + law.Cu = {} + for a, na in enumerate(names): + for b, nb in enumerate(names): + law.Cu[(na, nb)] = Stt[a][b] - sum((W[a][l] * Stp[b][l] for l in range(n)), zero) + law.e_bb = xs0[0] + law.e_bk = sum((kvec[j] * xs0[j] for j in range(n)), zero) + law.e_kk = sum((kvec[j] * xs1[j] for j in range(n)), zero) + law.det = det + if d == 3: + an = ('a11', 'a22', 'a12') + Xg, _ = solve_tv([[law.Cu[(x, y)] for y in an] for x in an], [[law.Cu[(o, y)] for y in an] for o in ('o11', 'o22', 'o12')]) + law.G = {o: [-Xg[c][l] for l in range(3)] for c, o in enumerate(('o11', 'o22', 'o12'))} + return law + + +def tv_summary(law): + """Low-order centers and radii (decimal, 30 places) and remainder coefficients of the TV law, for the record.""" + def row(t): + out = {'c': [], 'rad': []} + for j in range(4): + x = t.center(j) + out['c'].append(dec_down(x, 30) if x >= 0 else dec_up(x, 30)) + out['rad'].append(dec_up(Fr(t.D[j], 1 << TM_P), 30)) + out['rem_r9'] = dec_up(Fr(t.E, 1 << TM_P), 30) + return out + out = {'det': row(law.det), 'e_bb': row(law.e_bb), 'e_bk': row(law.e_bk), 'e_kk': row(law.e_kk)} + for nm in law.alpha: + out['alpha_' + nm] = row(law.alpha[nm]) + out['beta_' + nm] = row(law.beta[nm]) + names = TB.TM_NAMES[law.d] + for i, a in enumerate(names): + for b in names[i:]: + out['C_%s_%s' % (a, b)] = row(law.Cu[(a, b)]) + return out + + +# ============================================================================= 2. (a, b) forms: a r^2 + b r +def ab_add(*xs): + return (sum(x[0] for x in xs), sum(x[1] for x in xs)) + + +def ab_mul(x, c): + return (x[0] * c, x[1] * c) + + +def ab_at(x, R): + return x[0] * R * R + x[1] * R + + +def ab_norm(xs): + """Minkowski: || (a_i r^2 + b_i r)_i ||_2 <= sqrt(sum a_i^2) r^2 + sqrt(sum b_i^2) r.""" + return (E3.sqrt_up(sum(x[0] * x[0] for x in xs)), E3.sqrt_up(sum(x[1] * x[1] for x in xs))) + + +def exp_up_ab(M, x, R): + """exp(a r^2 + b r) - 1 <= (a r^2 + b r) exp(a R^2 + b R) on (0, R].""" + f = TB.exp_minus(M, -ab_at(x, R)).hi + return (x[0] * f, x[1] * f) + + +def flat(t, ref, what): + """The reference part (centers) of t is the constant ref for every r (Lemma 1 of Math-#215).""" + require(abs(t.center(0) - ref) <= TINY and all(abs(t.center(j)) <= TINY for j in range(1, t.N + 1)), + 'the reference law of %s is not flat in r' % what) + + +def mean_inc(law, nm, b1, k1, R): + """|mu(r) - mu(0)| for mu = b alpha + k beta (b <= b1, k <= k1), in (a, b) form.""" + return ab_add(ab_mul(law.alpha[nm].absinc(R), b1), ab_mul(law.beta[nm].absinc(R), k1)) + + +def pi_ratio_v(law, Q, R): + """log(pi_r / pi_0) in [-dn, up] for the SAME kernel: pi_r ~ det(S_r/Theta)^(-1/2) exp(-Theta E_r/2), so + log(pi_r/pi_0) = -(1/2) log(det S_r/det S_0) - Theta (E_r - E_0)/2; Theta in [1, 1 + th1] multiplies increments only.""" + b0, b1, k0, k1 = Q + th1 = law.th1 + l_lo, l_hi, q_lo, q_hi = law.det.inc(R) + d0lo = law.det.c0()[0] + require(d0lo > 0, 'pin determinant at r = 0') + xm = (max(abs(q_lo), abs(q_hi)) * R * R + max(abs(l_lo), abs(l_hi)) * R) / d0lo + require(xm < Fr(1, 2), 'pin determinant increment') + up = (max(Fr(0), -q_lo) / (2 * d0lo * (1 - xm)), max(Fr(0), -l_lo) / (2 * d0lo * (1 - xm))) + dn = (max(Fr(0), q_hi) / (2 * d0lo), max(Fr(0), l_hi) / (2 * d0lo)) + parts = ((law.e_bb, (b0 * b0, b1 * b1)), (law.e_bk, (2 * b0 * k0, 2 * b1 * k1)), (law.e_kk, (k0 * k0, k1 * k1))) + EL = EH = EQ = EP = Fr(0) + for t, (wlo, whi) in parts: + a_lo, a_hi, c_lo, c_hi = t.inc(R) + EL += min(wlo * a_lo, whi * a_lo) + EH += max(wlo * a_hi, whi * a_hi) + EQ += min(wlo * c_lo, whi * c_lo) + EP += max(wlo * c_hi, whi * c_hi) + f = (1 + th1) / 2 + up = ab_add(up, (f * max(Fr(0), -EQ), f * max(Fr(0), -EL))) + dn = ab_add(dn, (f * max(Fr(0), EP), f * max(Fr(0), EH))) + return up, dn + + +def combine_v(M, p_up, p_dn, z_up, z_dn, R): + """A_r/A_0 = (pi_r/pi_0) (z_r/z_0): (1 + pu)(1 + zu) - 1 <= pu + zu + pu zu(R); 1 - (1 - pd)(1 - zd) <= pd + zd.""" + pu = exp_up_ab(M, p_up, R) + zR = ab_at(z_up, R) + up = (pu[0] + z_up[0] + pu[0] * zR, pu[1] + z_up[1] + pu[1] * zR) + return up, ab_add(p_dn, z_dn) + + +def far_set(M, ma_T, sd_T, ma_tau, sd_tau, k0, R): + """q4 with P(G_0^c) <= q4 r^4 on (0, R]: G_0^c = {|T| > 3k/(2r)} u {|tau| > 3k/(2r^2)}, Markov at order p = MARKOV_P: + P(|X| > t) <= (||X||_p / t)^p, and r^p <= R^(p-4) r^4, r^(2p) <= R^(2p-4) r^4.""" + p = MARKOV_P + nT = M.gauss_norm(ma_T, sd_T, p) + nt = M.gauss_norm(ma_tau, sd_tau, p) + c = Fr(2) / (3 * k0) + q4 = (nT * c) ** p * R ** (p - 4) + (nt * c) ** p * R ** (2 * p - 4) + return q4 + + +# ============================================================================= 3. the ratio A_r / A_0 on a (b, k) box +def bounds_d2_v(Q, R, law, il): + """(up, dn) in (a, b) form: A_r/A_0 - 1 <= up(r), 1 - A_r/A_0 <= dn(r) on the box, 0 < r <= R (NOTE sections 3-4).""" + M = E2 + b0, b1, k0, k1 = Q + th1 = law.th1 + flat(law.alpha['w'], -1, 'alpha_w') + flat(law.beta['w'], 0, 'beta_w') + flat(law.Cu[('w', 'w')], 2, 'Var w') + # law of lambda = -w at r and at 0: mean mu = -(b alpha_w + k beta_w), variance Cu_ww / Theta + sA, cA = (law.alpha['w'] + 1).absval(R) + sB, cB = law.beta['w'].absval(R) + Dm = b1 * (cA + sA * R) + k1 * (cB + sB * R) # sup |mu - b| over [0, R] + sC, cC = (law.Cu[('w', 'w')] - 2).absval(R) + Sm = cC + sC * R + s_lo = M.isqrt_iv(M.IV((2 - Sm) / (1 + th1))).lo + s_hi = M.isqrt_iv(M.IV(2 + Sm)).hi + m_lo, m_hi = b0 - Dm, b1 + Dm + mmax = max(abs(m_lo), abs(m_hi)) + Fmin = M.elam2_pos(m_lo, s_lo).lo # <= F_r and <= F_0 + Fmax = M.elam2_pos(m_hi, s_hi).hi + F1max = M.elam_pos(m_hi, s_hi).hi + # main term: |F(mu_r, s_r) - F(mu_0, s_0)| <= 2 E[lambda_+] |dmu| + Phi(mu/s) |d(s^2)| along the segment (both <= sup) + dmu = mean_inc(law, 'w', b1, k1, R) + dS = law.Cu[('w', 'w')].absinc(R) # |d(s^2)| <= |dCu| / Theta, Theta >= 1 + Fae = ab_mul(ab_add(ab_mul(dmu, 2 * F1max), dS), 1 / Fmin) + norm = 36 * k0 * k0 * Fmin # 36 k^2 F_0 >= norm + b, k = M.IV(b0, b1), M.IV(k0, k1) + ma = {nm: M.absup(b * il.alpha[nm] + k * il.beta[nm]) for nm in M.TARGET_NAMES} + gn = lambda nm, p: M.gauss_norm(ma[nm], il.sdQ[nm].hi, p) + lam = lambda p: M.gauss_norm(mmax, s_hi, p) + T, tau, om, v, nu = (lambda p, x=x: gn(x, p) for x in ('T', 'tau', 'om', 'v', 'nu')) + kap = lambda p: 6 * k1 * tau(p) + T(2 * p) ** 2 + R * T(2 * p) * tau(2 * p) + # first order: r E[lambda_+ om] = r [mu_om F_1 - bw_om (F - m F_1)], |mu_om| <= c + s r, |bw_om| <= |Cu_om,w| / Cu_ww + sa_, ca_ = law.alpha['om'].absval(R) + sb_, cb_ = law.beta['om'].absval(R) + sc_, cc_ = law.Cu[('om', 'w')].absval(R) + cwlo = (2 - Sm) / (1 + th1) # Var w = Cu_ww / Theta >= cwlo + # relative to 36 k^2 r^2 F_0: the 36 k^2 cancels, so the normalizer is F_0 >= Fmin + T2 = (((b1 * sa_ + k1 * sb_) * F1max + sc_ / cwlo * (Fmax + mmax * F1max)) / Fmin, + ((b1 * ca_ + k1 * cb_) * F1max + cc_ / cwlo * (Fmax + mmax * F1max)) / Fmin) + if MUT == 'no-om-term': + T2 = (Fr(0), Fr(0)) + # second order (explicit r^2), as in Math-#215 + mt = M.IV(6 * k0, 6 * k1) * (b * il.alpha['tau'] + k * il.beta['tau']) * M.IV(Fmin, Fmax) + side = 6 * k1 * M.absup(il.bw['tau']) * s_hi * lam(4) ** 2 + R * T(4) * tau(4) * lam(4) ** 2 + T3_up = max(Fr(0), mt.hi + side) / norm + T3_dn = max(Fr(0), -mt.lo + side + T(4) ** 2 * lam(4) ** 2) / norm + T4 = R * kap(4) * lam(4) * om(2) + xi2 = 6 * k1 * (R * nu(4) ** 2 + 2 * nu(4) * v(4)) + T(4) * (R * R * nu(8) ** 2 + 2 * R * nu(8) * v(8) + 2 * v(8) ** 2) \ + + R * tau(4) * v(8) ** 2 + T5 = lam(2) * xi2 + T6 = v(8) ** 2 * (6 * k1 + R * T(4) + R * R * tau(4)) * om(4) + v(4) ** 2 * (R * nu(4) + v(4)) ** 2 + Yn = ('v', 'om', 'nu') + prec = 1 / cwlo + TB.wq_bound(M, il.Cg, Yn, [[M.absup(il.bw[nm])] for nm in Yn]) + pbar = Fr(2, 5) * M.sqrt_up(prec) + h = lambda p: Fr(2) / (9 * k0) * v(2 * p) ** 2 + om(p) + bad = R * pbar * h(2) * (Fr(15, 2) * k1 * h(4) + v(8) ** 2) * (9 * k1 * (h(4) + om(4)) + (R * nu(8) + v(8)) ** 2) + # G_0^c: E[W 1{G_0^c}]/r^2 <= P1n P2n P(G_0^c)^(1/2) <= P1n P2n sqrt(q4) r^2 + q4 = far_set(M, ma['T'], il.sdQ['T'].hi, ma['tau'], il.sdQ['tau'].hi, k0, R) + P1n = (6 * k1 + R * T(8)) * lam(8) + R * v(8) ** 2 + P2n = (6 * k1 + R * T(8) + R * R * tau(8)) * (lam(8) + R * om(8)) + R * (R * nu(8) + v(8)) ** 2 + g0c = P1n * P2n * M.sqrt_up(q4) + X = ab_add(T2, ((T4 + T5 + T6 + bad + g0c) / norm, Fr(0))) + p_up, p_dn = pi_ratio_v(law, Q, R) + z_up = ab_add(Fae, X, (T3_up, Fr(0))) + z_dn = ab_add(Fae, X, (T3_dn, Fr(0))) + return combine_v(M, p_up, p_dn, z_up, z_dn, R) + + +def bounds_d3_v(Q, R, law, il): + """(up, dn) in (a, b) form in d = 3 (NOTE sections 3-4).""" + M = E3 + b0, b1, k0, k1 = Q + th1 = law.th1 + names = ('a11', 'a22', 'a12') + for nm, ref in (('a11', -1), ('a22', -1), ('a12', 0)): + flat(law.alpha[nm], ref, 'alpha_' + nm) + flat(law.beta[nm], 0, 'beta_' + nm) + for (x, y), ref in ((('a11', 'a11'), 2), (('a22', 'a22'), 2), (('a12', 'a12'), 1), (('a11', 'a22'), 0), + (('a11', 'a12'), 0), (('a22', 'a12'), 0)): + flat(law.Cu[(x, y)], ref, 'Cov(%s, %s)' % (x, y)) + Fw = (Fr(1), Fr(1), M.SQRT2) + # Loewner coupling with the contact law over the whole band [0, R] (as in Math-#215): moment bounds and m_0 >= m0lo + comps = [] + for nm, ref, f in (('a11', -1, 1), ('a22', -1, 1), ('a12', 0, 2)): + sA, cA = (law.alpha[nm] - ref).absval(R) + sB, cB = law.beta[nm].absval(R) + comps.append((b1 * (cA + sA * R) + k1 * (cB + sB * R)) * f) + eps = M.sqrt_up(sum(x * x for x in comps)) + Cb = [] + for i in range(3): + row = [] + for j in range(3): + lo, hi = law.Cu[(names[i], names[j])].enclose(R) + row.append(M.IV(lo, hi) * Fw[i] * Fw[j] * M.IV(1 / (1 + th1), 1)) + Cb.append(row) + Lb = M.cholesky(Cb) + eta = M.sqrt_up(sum(M.absup(Lb[i][j] - (M.SQRT2 if i == j else 0)) ** 2 for i in range(3) for j in range(i + 1))) + Dl = M.rup_rel(eps + eta * TB.R0C) + cm = M.contact_moment + bA, bB = TB.grid_dn(b0 - Dl), TB.grid_up(b1 + Dl) + m3b0 = cm(2, 0, b0).lo + L3up = 2 * cm(1, 1, bB).hi / m3b0 + L3dn = 2 * cm(1, 1, b1).hi / cm(2, 0, bA).lo + pR0 = M.chi_tail(3, TB.R0C) + muB = M.rup_rel(b1 * M.sqrt_up(Fr(2)) + eps) + sig = M.sqrt_up(max(x.hi for x in (Cb[0][0], Cb[1][1], Cb[2][2])) + + 2 * max(M.absup(Cb[i][j]) for i in range(3) for j in range(3) if i != j)) + U = M.chi_upper_moments(3, TB.R0C, 4) + out_D2 = sum((U[i] * (M.binom(4, i) * muB ** (4 - i) * sig ** i) for i in range(5)), M.IV(0)).hi / 4 + tail_c = M.sqrt_up(cm(4, 0, b1).hi) * M.sqrt_up(pR0) + nBF16 = M.rup_rel(muB + sig * M.chi_norm(3, 16)) + out_ball = M.sqrt_up(pR0) + + def up_moment(a, c): + return M.rup_rel(cm(a, c, bB).hi + 2 ** c * M.rup_rel(nBF16 ** (2 * a + c)) * out_ball) + ED2, ED4, ED2T2, ET4, DT = up_moment(2, 0), up_moment(4, 0), up_moment(2, 2), up_moment(0, 4), up_moment(1, 1) + trCb = sum(x.hi for x in (Cb[0][0], Cb[1][1], Cb[2][2])) + BF = lambda p: M.frob_norm(muB, trCb, p) + rho_lo = TB.exp_minus(M, Dl * L3dn).lo - tail_c / m3b0 # m3_r / m3(b) over [0, R], r = 0 included + rho_hi = TB.exp_minus(M, -(Dl * L3up)).hi + out_D2 / m3b0 + require(rho_lo > 0, 'contact moment floor') + m0lo = m3b0 * rho_lo # m_0 = E_0[D^2 1{B > 0}] >= m0lo + rel = 1 / rho_lo # bounds relative to m3(b) -> relative to m_0 + # main term: gradient coupling B_r = mu_r + C_r^(1/2) Z, B_0 = mu_0 + C_0^(1/2) Z (natural coordinates), + # |g(B_r) - g(B_0)| <= max(||B_r||, ||B_0||)^3 ||B_r - B_0||, g = det(B)^2 1{B > 0}; + # ||C_r^(1/2) - C_0^(1/2)||_F <= ||C_r - C_0||_F / (2 sqrt(lmB)) (Sylvester equation in the eigenbases) + Fu = (Fr(1), Fr(1), SQRT2_UP) # rational upper bounds of the weights Fw + dmuB = ab_norm([ab_mul(mean_inc(law, nm, b1, k1, R), f) for nm, f in zip(names, Fu)]) + dC = ab_norm([ab_mul(law.Cu[(names[i], names[j])].absinc(R), Fu[i] * Fu[j]) for i in range(3) for j in range(3)]) + lmB = TB.gersh_min(Cb) + require(lmB > 0, 'Gershgorin floor of the transverse covariance') + dS = ab_mul(dC, 1 / (2 * M.sqrt_down(lmB))) + K3 = M.sqrt_up(Fr(2)) * BF(8) ** 3 / m0lo + if MUT == 'coupling-half': + K3 = K3 / 2 + main = ab_mul(ab_add(dmuB, dS), K3) + # secondary law (engine interval law over [0, R]) + b, k = M.IV(b0, b1), M.IV(k0, k1) + ma = {nm: M.absup(b * il.alpha[nm] + k * il.beta[nm]) for nm in M.TARGET_NAMES} + T = lambda p: M.gauss_norm(ma['T'], il.sdQ['T'], p) + tau = lambda p: M.gauss_norm(ma['tau'], il.sdQ['tau'], p) + mu_o = M.sqrt_up(ma['o11'] ** 2 + ma['o22'] ** 2 + 2 * ma['o12'] ** 2) + trO = sum(m * il.C[(nm, nm)].hi for nm, m in zip(M.ONAMES, (1, 1, 2))) + Om = lambda p: M.frob_norm(mu_o, trO, p) + mu_v = M.sqrt_up(ma['v1'] ** 2 + ma['v2'] ** 2) + v = lambda p: M.frob_norm(mu_v, il.C[('v1', 'v1')].hi + il.C[('v2', 'v2')].hi, p) + mu_nu = M.sqrt_up(ma['nu1'] ** 2 + ma['nu2'] ** 2) + nu = lambda p: M.frob_norm(mu_nu, il.C[('nu1', 'nu1')].hi + il.C[('nu2', 'nu2')].hi, p) + e = lambda p: R * nu(p) + v(p) + kap = lambda p: 6 * k1 * tau(p) + T(2 * p) ** 2 + R * T(2 * p) * tau(2 * p) + c1 = lambda p: 6 * k1 + R * T(p) + c2 = lambda p: 6 * k1 + R * T(p) + R * R * tau(p) + norm36 = 36 * k0 * k0 * m3b0 + # mean of Omega': ||mu_Omega'(r)||_F <= c_mu + s_mu r + s_mu2 = c_mu2 = Fr(0) + for nm, mult in zip(M.ONAMES, (1, 1, 2)): + sa_, ca_ = law.alpha[nm].absval(R) + sb_, cb_ = law.beta[nm].absval(R) + s_mu2 += mult * (b1 * sa_ + k1 * sb_) ** 2 + c_mu2 += mult * (b1 * ca_ + k1 * cb_) ** 2 + s_mu, c_mu = M.sqrt_up(s_mu2), M.sqrt_up(c_mu2) + muO = s_mu * R + c_mu + # regression of Omega' on B: E[Omega' - mu | B] = beta (B - mu_B) + DY + G = law.G + beta_t = G['o11'][0] + for (o, l) in (('o22', 1), ('o12', 2)): + flat(G[o][l] - beta_t, 0, 'regression of %s on B' % o) + for o, ls in (('o11', (1, 2)), ('o22', (0, 2)), ('o12', (0, 1))): + for l in ls: + flat(G[o][l], 0, 'off-diagonal regression of %s on B' % o) + require(abs(beta_t.center(0)) <= TINY, 'regression of Omega on B at r = 0 (reference part)') + dGs2 = dGc2 = Fr(0) + for o, mult in zip(('o11', 'o22', 'o12'), (1, 1, 2)): + for l in range(3): + t = G[o][l] - beta_t if (o, l) in (('o11', 0), ('o22', 1), ('o12', 2)) else G[o][l] + s_, c_ = t.absval(R) + dGs2 += mult * s_ ** 2 + dGc2 += mult * c_ ** 2 + dGs, dGc = M.sqrt_up(dGs2), M.sqrt_up(dGc2) + b1c = beta_t.center(1) # = 1/2 + require(b1c > 0, 'sign of the first-order regression coefficient of Omega on B') + # r beta(r) - b1c r^2 = c0 r + (c1 - b1c) r^2 + ...: |.| <= e_beta r + sb_beta r^2 + b0lo, b0hi = beta_t.c0() + e_beta = max(abs(b0lo), abs(b0hi)) + c1lo, c1hi = beta_t.coeff(1) + sb_beta = max(abs(c1lo - b1c), abs(c1hi - b1c)) + (max(abs(x) for x in beta_t.coeff(2)) + beta_t.high(R)) * R + DTlo = cm(1, 1, bA).lo - M.sqrt_up(cm(2, 2, b1).hi) * M.sqrt_up(pR0) + m3up = cm(2, 0, b1).hi + tau_lo = b0 * max(DTlo, Fr(0)) / m3up + tau_hi = b1 * DT / m3b0 + X3lo = 2 * rho_lo - tau_hi - eps * DT / m3b0 + X3hi = 2 * rho_hi - tau_lo + eps * DT / m3b0 + X3m = max(abs(X3lo), abs(X3hi)) + # r |R1|/m3(b) <= r (c_mu + s_mu r) DT/m3 + r (dGc + dGs r) sqrt(E[D^2 tr^2 1]) sqrt(tr Cb)/m3 + R1a = (s_mu * DT + dGs * M.sqrt_up(ED2T2) * M.sqrt_up(trCb)) / m3b0 + R1b = (c_mu * DT + dGc * M.sqrt_up(ED2T2) * M.sqrt_up(trCb)) / m3b0 + U3_up = (max(Fr(0), -b1c * X3lo) + sb_beta * X3m + R1a, e_beta * X3m + R1b) + U3_dn = (max(Fr(0), b1c * X3hi) + sb_beta * X3m + R1a, e_beta * X3m + R1b) + if MUT == 'no-om-term': + U3_up = U3_dn = (Fr(0), Fr(0)) + + def cov_Oz(o, z): + if z in names: + s_, c_ = law.Cu[(o, z)].absval(R) + return s_ * R + c_ + return M.absup(il.C[(o, z)]) + + def explained(zs): + rows = [[cov_Oz(o, z) for z in zs] for o in M.ONAMES] + D = [il.C[(z, z)].lo for z in zs] + Ef2 = Fr(0) + for i in range(len(zs)): + for j in range(len(zs)): + if i != j: + c = M.absup(il.C[(zs[i], zs[j])]) + Ef2 += c * c / (D[i] * D[j]) + Ef = M.sqrt_up(Ef2) + require(Ef < Fr(1, 2), 'diagonal scaling (explained part of Omega)') + return sum(mult * sum(x * x / Dz for x, Dz in zip(row, D)) for row, mult in zip(rows, (1, 1, 2))) / (1 - Ef) + zA = names + tuple(M.VNAMES) + ('nu1', 'nu2') + tB, tK = explained(names), explained(zA) + YB4, YK4 = M.frob_norm(0, tB, 4), M.frob_norm(0, tK, 4) + detX = M.absup(il.Cg[('o11', 'o22')] - il.Cg[('o12', 'o12')]) + E1 = up_moment(1, 0) + ED2lo = max(Fr(0), cm(2, 0, bA).lo - tail_c) + mt = M.IV(6 * k0, 6 * k1) * (b * il.alpha['tau'] + k * il.beta['tau']) * M.IV(ED2lo, ED2) + side = 6 * k1 * il.bnorm['tau'] * M.sqrt_up(trCb) * M.sqrt_up(ED4) + R * T(4) * tau(4) * M.sqrt_up(ED4) + U2_up = max(Fr(0), mt.hi + side) / norm36 + U2_dn = max(Fr(0), -mt.lo + side + T(4) ** 2 * M.sqrt_up(ED4)) / norm36 + U3r = R * kap(4) * M.sqrt_up(ED2T2) * Om(4) + U4rel = (detX * E1 + M.sqrt_up(ED2) * (muO + YB4) ** 2 / 2) / m3b0 + U4 = R * R * kap(4) * M.sqrt_up(ED2) * Om(8) ** 2 / 2 + U5 = M.sqrt_up(ED2T2) * ((2 * T(4) + R * tau(4)) * v(8) ** 2 + 2 * c1(4) * nu(8) * v(8) + R * c1(4) * nu(8) ** 2) + U67rel = (M.sqrt_up(ED2) * (muO * e(4) ** 2 + e(8) ** 2 * YK4) + M.sqrt_up(ET4) * (muO * v(4) ** 2 + v(8) ** 2 * YK4)) / (6 * k0 * m3b0) + U6 = R * T(8) * M.sqrt_up(ED2) * Om(8) * e(8) ** 2 + U7 = R * (T(8) + R * tau(8)) * M.sqrt_up(ET4) * Om(8) * v(8) ** 2 + U8 = R * c2(8) * Om(16) ** 2 / 2 * M.iroot(ET4, 4) * v(16) ** 2 + U9 = M.sqrt_up(ET4) * v(8) ** 2 * e(8) ** 2 + U10 = R * M.iroot(ET4, 4) * v(8) ** 2 * Om(4) * e(8) ** 2 + Yn = tuple(M.VNAMES) + tuple(M.ONAMES) + ('nu1', 'nu2') + rows = [] + for nm in Yn: + if nm in M.ONAMES: + cov = [] + for an, f in zip(names, (1, 1, 2)): + s_, c_ = law.Cu[(nm, an)].absval(R) + cov.append((s_ * R + c_) * f) + rows.append([M.sqrt_up(sum(x * x for x in cov)) / lmB]) + else: + rows.append([M.absup(x) for x in il.bt[nm]]) + Wt = TB.wq_bound(M, il.Cg, Yn, rows) + c_hi = sig * sig + cl = 1 / (1 / lmB + Wt) + KBp = M.rup_rel((M.PI * M.SQRT2 / (2 * M.PI * M.SQRT2PI)).hi / (cl * M.sqrt_down(cl))) + m1 = muB + c_hi * M.sqrt_up(Wt) * M.chi_norm(7, 8) + Hs = TB.half_moments(M, c_hi) + Ja = m1 ** 4 / 4 + sum(M.binom(3, i) * m1 ** (3 - i) * Hs[i] for i in range(4)) + Jb = m1 ** 3 / 3 + sum(M.binom(2, i) * m1 ** (2 - i) * Hs[i] for i in range(3)) + h = lambda p: Fr(2) / (9 * k0) * v(2 * p) ** 2 + Om(p) + nhg = h(4) * (Fr(15, 2) * k1 * h(8) + v(16) ** 2) * (9 * k1 * (h(8) + Om(8)) + e(16) ** 2) + nhgO = h(8) * (Fr(15, 2) * k1 * h(16) + v(32) ** 2) * (9 * k1 * (h(16) + Om(16)) + e(32) ** 2) * Om(8) + bad = R * KBp * (Ja * nhg + R * Jb * nhgO) + q4 = far_set(M, ma['T'], il.sdQ['T'], ma['tau'], il.sdQ['tau'], k0, R) + P1n = c1(8) * BF(16) ** 2 + R * BF(8) * v(16) ** 2 + P2n = c2(8) * (BF(16) + R * Om(16)) ** 2 + R * (BF(8) + R * Om(8)) * e(16) ** 2 + g0c = P1n * P2n * M.sqrt_up(q4) + Xa = (U3r + U4 + U5 + U6 + U7 + U8 + U9 + U10 + bad + g0c) / norm36 + U4rel + U67rel + # everything above except `main` is relative to m3(b); m3(b)/m_0 <= 1/rho_lo + z_up = ab_add(main, ab_mul(ab_add(U3_up, (Xa, Fr(0)), (U2_up, Fr(0))), rel)) + z_dn = ab_add(main, ab_mul(ab_add(U3_dn, (Xa, Fr(0)), (U2_dn, Fr(0))), rel)) + p_up, p_dn = pi_ratio_v(law, Q, R) + return combine_v(M, p_up, p_dn, z_up, z_dn, R) + + +def rate_sweep_v(d, law): + """For every R and every sub-box: (a_up, beta_up, a_dn, beta_dn), decimal strings rounded up.""" + M = ENG[d] + fn = bounds_d2_v if d == 2 else bounds_d3_v + out = {} + for R in R_STARS: + il = M.band_law(Fr(0), R) + rows = [] + for bb, kb in SUBBOXES: + up, dn = fn((bb[0], bb[1], kb[0], kb[1]), R, law, il) + rows.append([dec_up(up[0], 6), dec_up(up[1], 24), dec_up(dn[0], 6), dec_up(dn[1], 24)]) + out[str(R)] = {'boxes': rows, + 'a_up': max((r[0] for r in rows), key=Fr), 'beta_up': max((r[1] for r in rows), key=Fr), + 'a_dn': max((r[2] for r in rows), key=Fr), 'beta_dn': max((r[3] for r in rows), key=Fr)} + return out + + +# ============================================================================= 4. reference integrals and the corollary +def k_weights(d): + """int_box A_0^ref/(3k) on every sub-box (the r^1 weight), as decimal upper bounds.""" + M = ENG[d] + cells = TB.b_cells(d) + K = TB.K_const(d) + J11 = {kb: TB.k_integral(M, 1, 1, kb[0], kb[1]) for kb in SUB_K} + Ibs = {bb: TB.b_integral(M, cells, bb[0], bb[1]) for bb in set(bb for bb, _ in SUBBOXES)} + return [dec_up((K * Ibs[bb] * J11[kb]).hi, 24) for bb, kb in SUBBOXES] + + +def assemble_v(d, rate, ref, w1, eps0): + """Corollary L'1: for 0 < ell < R^3/2, + ell^(1/3) nu_cand(ell) - c_{B,K} in [-(c2_dn ell^(2/3) + c1_dn ell^(1/3)), c2_up ell^(2/3) + c1_up ell^(1/3)], + c_{B,K} = |S| int A_0/(3 k^(2/3)) of the same kernel; the weights are int A_0/(3k^(4/3)), int A_0/(3k), bounded by + (1 + eps0) times the reference weights (eps0 = Math-#215's |A_0/A_0^ref - 1| bound).""" + c_lo = Fr(ref['c_ref'][0]) + w2 = [Fr(x) for x in ref['w_2']] + w1 = [Fr(x) for x in w1] + f = 1 + eps0 + out = {} + for R in R_STARS: + rr = rate[str(R)] + c2_up = f * sum(Fr(row[0]) * w for row, w in zip(rr['boxes'], w2)) + c2_dn = f * sum(Fr(row[2]) * w for row, w in zip(rr['boxes'], w2)) + c1_up = f * sum(Fr(row[1]) * w for row, w in zip(rr['boxes'], w1)) + c1_dn = f * sum(Fr(row[3]) * w for row, w in zip(rr['boxes'], w1)) + cl = c_lo * (1 - eps0) + out[str(R)] = {'c2_up': dec_up(c2_up, 12), 'c2_dn': dec_up(c2_dn, 12), + 'c1_up': dec_up(c1_up, 30), 'c1_dn': dec_up(c1_dn, 30), + 'rel2_up': dec_up(c2_up / cl, 6), 'rel2_dn': dec_up(c2_dn / cl, 6), + 'rel1_up': dec_up(c1_up / cl, 30), 'rel1_dn': dec_up(c1_dn / cl, 30), + 'ell_max': '%s' % (R ** 3 / 2)} + return out + + +def params(): + return {'R_STARS': [str(x) for x in R_STARS], 'TV': {'s': TM_S, 'N': TM_N, 'P': TM_P}, 'E_UP': str(E_UP), + 'MARKOV_P': MARKOV_P, 'R0_COUPLING': str(TB.R0C), 'L_min': 10, + 'SUB_B': [[str(a), str(b)] for a, b in TB.SUB_B], 'SUB_K': [[str(a), str(b)] for a, b in SUB_K], + 'band': {'b': ['0', '1'], 'k': ['1/2', '2']}} + + +def eps0_215(d): + """Math-#215 Theorem L at r -> 0: |A_0/A_0^ref - 1| <= max(eps_up, eps_dn) over R (pinned RESULTS.json of #215, + recomputed here by TB.rate_sweep for R = 1/4096 and compared).""" + tml = TB.tm_law(d) + M = ENG[d] + R = R_STARS[0] + il = M.band_law(Fr(0), R) + fn = TB.bounds_d2 if d == 2 else TB.bounds_d3 + e = Fr(0) + for bb, kb in SUBBOXES: + up, dn = fn((bb[0], bb[1], kb[0], kb[1]), R, tml, il) + e = max(e, up[1], dn[1]) + return e + + +def compute(d): + law_t = tv_law(d, torus=True) + law_r = tv_law(d, torus=False) + rate_t = rate_sweep_v(d, law_t) + rate_r = rate_sweep_v(d, law_r) + ref = TB.reference(d) + w1 = k_weights(d) + e0 = eps0_215(d) + return {'tv_law_torus': tv_summary(law_t), 'rate_torus': rate_t, 'rate_reference_kernel': rate_r, + 'reference': {'c_ref': ref['c_ref'], 'w_c': ref['w_c'], 'w_2': ref['w_2'], 'w_1': w1}, + 'eps0_215': dec_up(e0, 24), + 'corollary_torus': assemble_v(d, rate_t, ref, w1, Fr(dec_up(e0, 24))), + 'corollary_reference_kernel': assemble_v(d, rate_r, ref, w1, Fr(0))} + + +def full_run(): + res = {'schema': 1, 'object': OBJECT, 'scientific_effect': 'NONE', 'certified': True, 'mutant': MUT, + 'parameters': params(), 'dims': {}} + for d in (2, 3): + res['dims'][str(d)] = compute(d) + return res + + +def text(res): + return json.dumps(res, indent=1, sort_keys=True) + '\n' + + +def main(): + global MUT + ap = argparse.ArgumentParser(description=__doc__, formatter_class=argparse.RawDescriptionHelpFormatter) + ap.add_argument('--write', action='store_true') + ap.add_argument('--check', action='store_true') + ap.add_argument('--mutant', choices=MUTANTS) + a = ap.parse_args() + MUT = a.mutant + res = full_run() + path = os.path.join(HERE, 'RESULTS.json') + if a.write: + require(MUT is None, 'no --write under a mutant') + with open(path, 'w') as fh: + fh.write(text(res)) + print('RESULTS.json written') + return + with open(path) as fh: + stored = fh.read() + if MUT is not None: + res['mutant'] = None + require(text(res) == stored, 'replay differs from RESULTS.json') + print(json.dumps({'object': OBJECT, 'check': 'passed', 'scientific_effect': 'NONE'})) + + +if __name__ == '__main__': + main() From 75b5fced3f6a2ede43b85273f1eb02e1fb524393 Mon Sep 17 00:00:00 2001 From: Dylan Roy Date: Thu, 1 Oct 2026 19:32:00 +0000 Subject: [PATCH 2/6] Theorem L' v1.1: explicit angular integrals (OA-227-C-01) Notation and docstring change in response to review 5384474393. - Corollary L'1 and section 5 now write c_{B,K} and ell^(1/3) nu_cand as explicit integrals over B x K x S^(d-1) with dsigma(u), as in [LP] (11.2)-(11.3). The box weights are integrals over box x S^(d-1). - Theorem L' is uniform in u, so no isotropy is assumed. The pinned reference weights already are full angular integrals, since K_const includes |S^(d-1)|. - theorem_v.py: the assemble_v docstring matches. The header comment now says the reference-kernel beta is exactly 0 (torus=False gives M1 = 0, and the upward-rounded stored values are all 0), replacing the stale "rounding residue below 1e-80". The code AST with docstrings blanked is identical, and RESULTS.json is unchanged. Manifest refreshed. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .../c8_torus_vanishing_remainder_20261001/NOTE.md | 13 +++++++++---- .../SOURCE_FILES.json | 12 ++++++------ .../theorem_v.py | 9 ++++++--- 3 files changed, 21 insertions(+), 13 deletions(-) diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md b/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md index 138521e82..d26916fa9 100644 --- a/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md +++ b/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md @@ -56,8 +56,9 @@ What changes against [B]: estimates at `M_1(10) < 4.46 x 10^-15`. **Corollary L′1 (the densities with a vanishing remainder).** Let `nu_cand`, `nu_eld` be the compact-window densities of [LP] -Theorem B in the versions (11.2), for the torus kernel or the Gaussian kernel, and `c_{B,K} = |S^(d-1)| int A_0/(3 k^(2/3))` -the leading constant **of the same kernel**. For `0 < ell < R^3 / 2`: +Theorem B in the versions (11.2), for the torus kernel or the Gaussian kernel, and +`c_{B,K} = int_{B x K x S^(d-1)} A_0(b, k, u)/(3 k^(2/3)) db dk dsigma(u)` the leading constant **of the same kernel**. The +angular integral uses ordinary surface measure `dsigma`, as in [LP] (11.2)–(11.3). The torus is not assumed isotropic. For `0 < ell < R^3 / 2`: -(c2_dn ell^(2/3) + c1_dn ell^(1/3)) <= ell^(1/3) nu_cand(ell) - c_{B,K} <= c2_up ell^(2/3) + c1_up ell^(1/3), ell^(1/3) nu_cand(ell) - C_{B,K}(R) ell <= ell^(1/3) nu_eld(ell) <= ell^(1/3) nu_cand(ell), @@ -212,8 +213,12 @@ so this term is `O(r^2)`. This gives Theorem L′ on each of the 96 boxes `[i/8, (i+1)/8] x [1/2 + j/8, 1/2 + (j+1)/8]` (as in [B]) and each `R`. **Corollary L′1.** As in [B] Corollary L1: -- `ell^(1/3) nu_cand(ell) = |S| int A_r / (3 k^(2/3))` with `r = (ell/k)^(1/3) <= R`. -- Integrate Theorem L′ box by box: `a r^2` gives the weight `int A_0 / (3 k^(4/3))`, and `beta r` gives `int A_0 / (3 k)`. +- `ell^(1/3) nu_cand(ell) = int_{B x K x S^(d-1)} A_r(b, k, u) / (3 k^(2/3)) db dk dsigma(u)`, with `r = (ell/k)^(1/3) <= R` + ([LP] (11.2)). +- Theorem L′ holds uniformly in the direction `u`. Integrate it box by box, over each box and all of `S^(d-1)`: + `a r^2` gives the weight `int_{box x S^(d-1)} A_0 / (3 k^(4/3))`, and `beta r` gives `int_{box x S^(d-1)} A_0 / (3 k)`. + No isotropy is used. The reference weights of the pinned `theorem_b.py` are these full angular integrals for `A_0^ref` + (its `K_const` includes `|S^(d-1)|`), so the program needs no extra sphere factor (OA-227-C-01, review 5384474393). - The weights of the torus `A_0^(L)` are at most `(1 + eps_0)` times the reference weights. Here `eps_0` is [B]'s `|A_0/A_0^ref - 1|` bound, recomputed by the pinned `theorem_b.py` at `R = 1/4096`: `eps_0 <= 3.95 x 10^-10` (`d = 2`), `1.78 x 10^-8` (`d = 3`). diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json b/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json index 65e873d9d..fa78b0603 100644 --- a/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json +++ b/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json @@ -3,9 +3,9 @@ "files": [ { "path": "NOTE.md", - "bytes": 17185, - "sha256": "f7ee8cdbbf6a551092b4abcb482b63addfde1f9d49fc855282a8115f9cd9b1bf", - "git_blob": "138521e82d6c4088a61b97a89020164dbbae6c40" + "bytes": 17749, + "sha256": "bf69df7b21ab282ddf1c76ed4fa3867862af72aaeb97292d154a041696179dd3", + "git_blob": "d26916fa9f652725eee349f7523930c16e2ae608" }, { "path": "README.md", @@ -45,9 +45,9 @@ }, { "path": "theorem_v.py", - "bytes": 39529, - "sha256": "b41772a2762214d8ce2022ce81a589a0b07b0e5e473d2e13a82574d3b7d0ba9c", - "git_blob": "61e664559bf3fdc5892388d5f9710348eb24c94a" + "bytes": 39859, + "sha256": "3e659f640dfbc5c2f92499056cbad6a69e73b45ed49b60f2bade204211b7d493", + "git_blob": "503fe83b4e252bbdc4ae2c655c540ec26c7cc8cf" } ] } diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/theorem_v.py b/frontiers/c8_torus_vanishing_remainder_20261001/theorem_v.py index 61e664559..503fe83b4 100644 --- a/frontiers/c8_torus_vanishing_remainder_20261001/theorem_v.py +++ b/frontiers/c8_torus_vanishing_remainder_20261001/theorem_v.py @@ -8,7 +8,8 @@ where A_0 = lim_{r -> 0} A_r is the contact integrand of THE SAME kernel (for the torus: its own A_0^(L), not the Gaussian A_0^ref). Math-#215 (Theorem L) proved the same shape against A_0^ref with an additive, r-independent eps_d; here every remainder carries a positive power of r. The first-order coefficient beta collects the torus corrections only (about -1e-12); for the reference kernel it is a rounding residue below 1e-80. +1e-12); for the reference kernel it is exactly 0 (torus=False gives M1 = 0, and the computed beta is the rational 0: its +upward-rounded stored values are all 0). Method (NOTE.md sections 2-5): - Taylor models with INTERVAL coefficients and a remainder proportional to r^(N+1) (class TV): every rounding goes into a @@ -750,8 +751,10 @@ def k_weights(d): def assemble_v(d, rate, ref, w1, eps0): """Corollary L'1: for 0 < ell < R^3/2, ell^(1/3) nu_cand(ell) - c_{B,K} in [-(c2_dn ell^(2/3) + c1_dn ell^(1/3)), c2_up ell^(2/3) + c1_up ell^(1/3)], - c_{B,K} = |S| int A_0/(3 k^(2/3)) of the same kernel; the weights are int A_0/(3k^(4/3)), int A_0/(3k), bounded by - (1 + eps0) times the reference weights (eps0 = Math-#215's |A_0/A_0^ref - 1| bound).""" + c_{B,K} = int_{B x K x S^(d-1)} A_0(b, k, u)/(3 k^(2/3)) db dk dsigma(u) of the same kernel; the weights are + int_{box x S^(d-1)} A_0/(3k^(4/3)) and int_{box x S^(d-1)} A_0/(3k), bounded by (1 + eps0) times the reference + weights (eps0 = Math-#215's |A_0/A_0^ref - 1| bound). The reference weights w_2, w_1 are full angular integrals + (theorem_b's K_const includes |S^(d-1)|); Theorem L' is uniform in u, and no isotropy is assumed.""" c_lo = Fr(ref['c_ref'][0]) w2 = [Fr(x) for x in ref['w_2']] w1 = [Fr(x) for x in w1] From fc976ede12fbe8c659838b632dc6f04a48cb4831 Mon Sep 17 00:00:00 2001 From: Dylan Roy Date: Fri, 2 Oct 2026 00:26:21 +0000 Subject: [PATCH 3/6] Theorem L' v1.2: review cycle C56-C61 recorded (documentation only) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit - NOTE: 'six pins' -> six in d = 2, eight in d = 3 (C57); §0 lists the four result scopes (main rate; Gaussian candidate, eps0 = 0; torus candidate, + eps0; elder, + C_BK); §5 records C60's direct contact comparison for eps0 and C61's validation of C_BK; §§7-8 the corrected parent set and the current review status; §9 Revisions. - SOURCE_MAP: [E1], [E2], [REC] current_required (C61); [B], [E], [E3] repinned to #215 v1.5 (1aa250d), #206 v1.2 (13f932c), #212 v1.1 (250b907); workflow paths extended. - theorem_v.py: docstrings/comments only: TV preconditions (C56), combine_v's positivity case split, the exact Theta cancellation in the planar omega regression, the diagonal surplus in explained (C58). - theorem_b.py, engine_e2.py, engine_e3.py: byte-identical copies of the new heads (docstrings/comments only: sqrt_up 2^-198, box_bound domain, wq_bound surplus, 3/4 comment). AST with docstrings removed is unchanged in all four Python files; RESULTS.json and every constant unchanged; --check passes in both interpreter modes. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .../c8-torus-vanishing-remainder.yml | 3 + .../NOTE.md | 73 +++++++++++++++---- .../SOURCE_FILES.json | 36 ++++----- .../SOURCE_MAP.json | 59 +++++++++++---- .../engine_e2.py | 9 ++- .../engine_e3.py | 9 ++- .../theorem_b.py | 6 +- .../theorem_v.py | 14 +++- 8 files changed, 157 insertions(+), 52 deletions(-) diff --git a/.github/workflows/c8-torus-vanishing-remainder.yml b/.github/workflows/c8-torus-vanishing-remainder.yml index db74dc54a..8ec60df05 100644 --- a/.github/workflows/c8-torus-vanishing-remainder.yml +++ b/.github/workflows/c8-torus-vanishing-remainder.yml @@ -6,6 +6,9 @@ name: C8 torus vanishing remainder (certified replay) - 'frontiers/c8_torus_vanishing_remainder_20261001/**' - '.github/workflows/c8-torus-vanishing-remainder.yml' - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md' + - 'imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md' + - 'reviews/d1_section9_borel_repair_20260925/REPAIR.md' + - 'reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md' workflow_dispatch: permissions: diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md b/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md index d26916fa9..637257af3 100644 --- a/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md +++ b/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md @@ -77,6 +77,13 @@ and `c1/c <= 1.531 x 10^-10` (`d = 2`), `<= 3.985 x 10^-8` (`d = 3`). For the Ga So, on the torus, both densities are `c_{B,K} ell^(-1/3) + O(1) + O(ell^(1/3))` about the **torus** constant, with explicit constants and a vanishing remainder. [B] gave the same expansion only about `c^ref_{B,K}`, up to `eps'_d ell^(-1/3)`. +**Four result scopes, and what each consumes.** Every scope also retains the corrected parent [LP] + [CAP] + [E1] + [E2] + +[REC] (§8) and its Gaussian-field, typed-weight and Kac–Rice interpretation. +1. *Theorem L′*, the main radial rate, for both kernels: §§2–4 and the band-law, primitive and moment contracts of [E], [E3]. +2. *The Gaussian-kernel candidate corollary*: the rate and the reference integrals; `eps_0 = 0`. +3. *The torus candidate corollary*: additionally the finite contact comparison `eps_0` (§5). +4. *The elder corollary, for either kernel*: additionally [B]'s cap-derived `C_{B,K}(R)` (§5). + ## 1. Where [B]'s remainder came from [B]'s `eps_d` collects five things, each `r`-independent: @@ -130,7 +137,7 @@ The reference series gives the centers exactly. The truncation tail beyond `g_x checked). `M_1(10) < 4.46 x 10^-15` (`d = 2`) and `< 1.94 x 10^-13` (`d = 3`) decrease in `L`, so the models hold for every `L >= 10` and every frame; with `M_1 = 0` they are the Gaussian kernel's. -The six pins, the targets of [B] §2 and Gaussian elimination without pivoting give `TV`s for `det S_r`, the energies `e_bb`, +The pins (six in `d = 2`, eight in `d = 3`), the targets of [B] §2 and Gaussian elimination without pivoting give `TV`s for `det S_r`, the energies `e_bb`, `e_bk`, `e_kk`, the regression coefficients `alpha`, `beta` of the transverse Hessian and of `Omega'`, their covariances, and in `d = 3` the regression of `Omega'` on `B = -A`. The certificate checks that the centers are [B]'s reference laws. In particular the law of the transverse Hessian is flat in `r` ([B] Lemma 1): every center of `alpha_A + 1`, `beta_A` and @@ -222,7 +229,19 @@ This gives Theorem L′ on each of the 96 boxes `[i/8, (i+1)/8] x [1/2 + j/8, 1/ - The weights of the torus `A_0^(L)` are at most `(1 + eps_0)` times the reference weights. Here `eps_0` is [B]'s `|A_0/A_0^ref - 1|` bound, recomputed by the pinned `theorem_b.py` at `R = 1/4096`: `eps_0 <= 3.95 x 10^-10` (`d = 2`), `1.78 x 10^-8` (`d = 3`). -- The elder line is [B]'s. + + Only the contact (`r -> 0`) value of that bound is used. Review C60 (5386707554) proves it directly at contact: + - it uses the C57 contact intervals, Gaussian log-moment derivatives (`d = 2`), a Loewner and chi-tail comparison + (`d = 3`), the normalized pin density, and dominated convergence; + - it gives `|A_0^(L)/A_0^ref - 1| <= 3.943 x 10^-10` (`d = 2`) and `<= 1.778 x 10^-8` (`d = 3`) for `L >= 10`, every + frame and the compact marks. + + So the torus corollary does not rest on the positive-radius parts of [B]'s Theorem L. +- The elder line uses [B]'s `C_{B,K}(R)`. Review C61 (5386924911, at Math-#215 `b41eb28`) validates all fourteen published + `C_{B,K}` and `C_121` upper constants by a cellwise transfer. The transfer uses this packet's reviewed rate and C60's + contact comparison, the two cap engines at [B]'s caller, and the corrected parent set. C61's scope is + `0 <= nu_cand - nu_eld <= C_{B,K}(R) ell^(2/3)` for `ell < min(r_pop, R)^3/2` on the compact window. C61 does not claim + that every short bar is a candidate, nor an unrestricted difference rate. ## 6. Verification @@ -242,7 +261,7 @@ This gives Theorem L′ on each of the 96 boxes `[i/8, (i+1)/8] x [1/2 + j/8, 1/ - `coupling-half`: halve Lemma V5's constant. - The workflow `.github/workflows/c8-torus-vanishing-remainder.yml` checks: - the manifest; - - the main-resident pin [LP]; + - the main-resident pins [LP], [E1], [E2], [REC]; - the packet copies [B], [E], [E3], byte for byte; - both modes; - the mutants; @@ -257,25 +276,51 @@ deviations from `A_0^ref`. - **Still the compact window `B x K`, `L >= 10`, `d = 2, 3`, `R <= 1/64`.** These are [B]'s ranges. - **`beta` is not the true first-order coefficient.** It bounds the torus part of the `r^1` coefficients through `M_1`. Whether the torus `A_r` has a nonzero `r^1` term at all (of order `e^-50`) is not decided. -- **Consumed, not reviewed:** - - [LP]'s sections 8–12, as in [B]; - - [B]'s §§2–5 algebra (exact expansions, first-order cancellations, Hölder bounds, Loewner coupling for the moment - bounds); - - the interval band laws and transcendentals of [E], [E3]. - - [B] is author-side with an OpenAI amendment ACCEPT (5378815703). [E] and [E3] are author-side and unmerged. +- **Consumed, with their review status (OpenAI / Codex, delegated; same account, zero organizational-independence credit):** + - [LP]'s sections 8–12, as in [B], read with [E1], [E2], [REC]; + - [B]'s §§2–5 algebra (exact expansions, first-order cancellations, Hölder bounds, Loewner coupling): scoped reads + 5378815703, 5379687608, 5379747967, 5379848224, 5384256400. Its positive-radius Theorem L is not accepted as a whole; + this packet uses only its contact value (C60) and its `C_{B,K}` (C61); + - the band laws, interval and transcendental primitives and moment routines of [E], [E3]: reviewed at the contracts used + here (C57, C59). + + [B], [E] and [E3] are author-side and unmerged. - No `C`, `r_*`, `z_*` beyond [B], #206, #212 and #195. No new count, moment or elder statement beyond Corollary L′1. - Same GitHub account as every lane; zero organizational-independence credit. **I will not merge.** ## 8. Provenance - **Sources** (`SOURCE_MAP.json`): - - [LP] `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` (on `main`, verified by the workflow); - - [B] Math-#215 `theorem_b.py` at `d77cabf` (packet copy `theorem_b.py`) and its `NOTE.md` and `RESULTS.json` (provenance); - - [E] Math-#206 `theorem_a.py` (packet copy `engine_e2.py`); - - [E3] Math-#212 `theorem_a3.py` (packet copy `engine_e3.py`). + - [LP] `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` (on `main`, verified by the workflow), read + with its mandatory amendments, also on `main` and verified: + - [E1] `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md`; + - [E2] `reviews/d1_section9_borel_repair_20260925/REPAIR.md`; + - [REC] `reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md`; + - [B] Math-#215 `theorem_b.py` at `1aa250d` (v1.5; packet copy `theorem_b.py`), and its `NOTE.md` and `RESULTS.json` at + `d77cabf` (provenance); + - [E] Math-#206 `theorem_a.py` at `13f932c` (v1.2; packet copy `engine_e2.py`); + - [E3] Math-#212 `theorem_a3.py` at `250b907` (v1.1; packet copy `engine_e3.py`). - **Files:** - `NOTE.md`, `README.md`; - `theorem_v.py` (new); - the copies `theorem_b.py`, `engine_e2.py`, `engine_e3.py`; - `RESULTS.json`, `SOURCE_MAP.json`, `SOURCE_FILES.json`. + +## 9. Revisions + +- **v1.1 (`75b5fce`):** the explicit angular integrals (OA-227-C-01). +- **v1.2:** documentation only, recording the review cycle C56–C61. In every changed Python file the AST with docstrings + removed is unchanged, and so are `RESULTS.json` and every constant. + - **NOTE.** + - "six pins" becomes "six in `d = 2`, eight in `d = 3`" (C57). + - §0 gains the four result scopes. + - §5 records C60's direct contact comparison and C61's `C_{B,K}` disposition. + - §§7–8 add the corrected parent set [E1], [E2], [REC] (C61) and the current review status. + - **`theorem_v.py`.** + - The `TV` docstring states its preconditions (C56). + - `combine_v` states the positivity case split of the lower bound (C58). + - A comment at the planar `omega` regression says that `Theta` cancels exactly and `cwlo` is conservative (C58). + - A comment in `explained` keeps the diagonal surplus (C58). + - **The copies follow Math-#215 v1.5, Math-#206 v1.2 and Math-#212 v1.1.** These change docstrings and comments only: + `sqrt_up` (`2^-198`, C59-DOC-01), `box_bound`'s parameter domain (C61), `wq_bound`'s diagonal surplus (C58), and the + `3/4` multiplier comment (C60). diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json b/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json index fa78b0603..93f6ca6c0 100644 --- a/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json +++ b/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json @@ -3,9 +3,9 @@ "files": [ { "path": "NOTE.md", - "bytes": 17749, - "sha256": "bf69df7b21ab282ddf1c76ed4fa3867862af72aaeb97292d154a041696179dd3", - "git_blob": "d26916fa9f652725eee349f7523930c16e2ae608" + "bytes": 21356, + "sha256": "8295401f5858c67c611ee034b0021c15dc188218016b3a2c6263a40f1432d3bd", + "git_blob": "637257af38e03fe17ee65a93820bad83aa83aa9d" }, { "path": "README.md", @@ -21,33 +21,33 @@ }, { "path": "SOURCE_MAP.json", - "bytes": 5157, - "sha256": "1dea8c3d45b8693f6301fbeb988b7d51f3e33a466ff675e696b990816b171ba4", - "git_blob": "5623ee6052d2e53497423d333c89106a18f9a990" + "bytes": 6771, + "sha256": "cec379ce90a370097396dd83432c6934ada6f8439d2febd35917c9fcd244f750", + "git_blob": "bef853ce1c5d69d644b75dfa070fd392eaab7826" }, { "path": "engine_e2.py", - "bytes": 52401, - "sha256": "4e811b9f7e047b2f851a4be9c69ad69476f100ef30664dae252b659cfd63efbe", - "git_blob": "d74542ff66f4d953bc25819a4d0e439082078006" + "bytes": 52864, + "sha256": "c5ea8bf87c54761866315f28f6c93c6ce5c1e0f99d47c5a03f14d095d06871cc", + "git_blob": "ae279e9015483a666ece8deb2544cbc458a5e270" }, { "path": "engine_e3.py", - "bytes": 75753, - "sha256": "b0e1b3b4f43e140b51609332fe2e54f4f6b50d10f9f6af48d40ff548bbe742c0", - "git_blob": "ac809dbd36981d1b214fb2f8626417b50afb41f3" + "bytes": 76209, + "sha256": "310d4ab3d3a38a377d753753f3b814591a9090a947f87d6843ab7389389572d3", + "git_blob": "267890232119b35206be7a8cc74d730de9a68644" }, { "path": "theorem_b.py", - "bytes": 52707, - "sha256": "b79df0880e9690f4ac5e35b2b717bb65c1cbf1a323585165e5745a17104d6f12", - "git_blob": "f0e788cb01d9f2a434cbfef1d248a5b98dbf0681" + "bytes": 53141, + "sha256": "1e510cc385b2ed6502e2864cfdbd60514d92ed4217b035083dbf822d6e1543a1", + "git_blob": "36b39dbd33ef2e6d277ce1c3820b3f0d6659d779" }, { "path": "theorem_v.py", - "bytes": 39859, - "sha256": "3e659f640dfbc5c2f92499056cbad6a69e73b45ed49b60f2bade204211b7d493", - "git_blob": "503fe83b4e252bbdc4ae2c655c540ec26c7cc8cf" + "bytes": 40848, + "sha256": "f5d87b547a174f8abb42b8a13926829b5c3c0903f8a30de16852b654c7020f7f", + "git_blob": "b492b7166314eccb10341ce0a08dd236706adcdb" } ] } diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_MAP.json b/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_MAP.json index 5623ee605..bef853ce1 100644 --- a/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_MAP.json +++ b/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_MAP.json @@ -7,7 +7,7 @@ "source_repository": "d6g8k5htny-coder/Math-", "main_at_creation": "c2f1270d28192b3c06dd31f728ebf04d03d4246e", "delivered_under": "explicit owner instructions of Dylan Roy in Claude Code session session_015wNj8LPTKXsaT68G3DgPPh, given after OWNER_STOP (2026-09-27): 2026-09-28 \"you have my full permisions and approval to make the proper changes and uploads\"; 2026-09-30 \"Work on closing as many mathematical items as possible while working with and supporting the other models in the GitHub. Also advance the mathematics in every place that's justified based on what we currently know. Ultra tokens, ultra depth @GitHub @Superpowers\" (repeated 2026-09-30 as \"Continue to Work on closing ...\"). Pull requests only, never merged by their author; no register, GRAPH, STATUS, lemma_closed, prize or premise edits. OWNER_STOP.md addresses Cursor agents and automations and is untouched.", - "note": "current_required sources ([LP]) are verified on the tree by the workflow; packet_copy sources ([B], [E], [E3]) are verified against the packet copies byte for byte; [B-NOTE] and [B-RESULTS] are read-only provenance pinned by identity", + "note": "current_required sources ([LP] and the mandatory parent amendments [E1], [E2], [REC]) are verified on the tree by the workflow; packet_copy sources ([B], [E], [E3]) are verified against the packet copies byte for byte; [B-NOTE] and [B-RESULTS] are read-only provenance pinned by identity", "sources": [ { "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", @@ -20,13 +20,46 @@ "on_main": true, "role": "sections 1, 3, 8-12: the model, the preconditioned pins, the radial ledger A_r = 12 pi_r Z_r / r^2 (10.2) and its limit A_0 (10.3), the lifetime pushforward (11.1)-(11.3); section 11 records that A_r -> A_0 was not assigned a numerical rate. Consumed as statements, as in Math-#215" }, + { + "path": "imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md", + "bytes": 1782, + "sha256": "bad7ef609c4ad8c41ad6af562c1b6807921e19a9d556ed793ad1a0db6e202028", + "git_blob": "213594d6ca6a86fb938110f4d166d9ce275a02d0", + "id": "E1", + "commit": "6f74f7a77bae15d6aa07506bab38977e34dcc449", + "current_required": true, + "on_main": true, + "role": "mandatory erratum to [LP]: the preconditioning congruence is diag(r^(-1/2), I) (corrected parent P + CAP + E1 + E2 + REC; review C61)" + }, + { + "path": "reviews/d1_section9_borel_repair_20260925/REPAIR.md", + "bytes": 9062, + "sha256": "845abf9f9c99d672c2a10a887b5a2e7206a3d2de3d876f35f75ff6e2dc13e62f", + "git_blob": "fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a", + "id": "E2", + "commit": "6f74f7a77bae15d6aa07506bab38977e34dcc449", + "current_required": true, + "on_main": true, + "role": "mandatory repair of [LP] section 9: the marked Kac-Rice identity for the Borel elder indicator (replaces LP section 9 wherever sections 9-11 are consumed)" + }, + { + "path": "reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md", + "bytes": 23312, + "sha256": "451b9d7ffee072a73fc904cab891b6693e3f233808b89df10cce1b57036b65da", + "git_blob": "75da2597971510f843f8d90c743950cb8c177342", + "id": "REC", + "commit": "6f74f7a77bae15d6aa07506bab38977e34dcc449", + "current_required": true, + "on_main": true, + "role": "chain reconciliation and reading rule for the parent: consumers cite P + CAP + E1 + E2 + REC; embedding condition r < L/(4 sqrt 2), met by r <= 1/64, L >= 10" + }, { "path": "frontiers/c8_lifetime_difference_constant_20261001/theorem_b.py", - "bytes": 52707, - "sha256": "b79df0880e9690f4ac5e35b2b717bb65c1cbf1a323585165e5745a17104d6f12", - "git_blob": "f0e788cb01d9f2a434cbfef1d248a5b98dbf0681", + "bytes": 53141, + "sha256": "1e510cc385b2ed6502e2864cfdbd60514d92ed4217b035083dbf822d6e1543a1", + "git_blob": "36b39dbd33ef2e6d277ce1c3820b3f0d6659d779", "id": "B", - "commit": "d77cabf4fee5d36de2a85a810359c1c8812fc28c", + "commit": "1aa250df3020dd71db8b091d1d4516bc5054e2b8", "pull_request": 215, "packet_copy": "theorem_b.py", "current_required": false, @@ -59,11 +92,11 @@ }, { "path": "frontiers/c8_theorem_a_constants_planar_20260930/theorem_a.py", - "bytes": 52401, - "sha256": "4e811b9f7e047b2f851a4be9c69ad69476f100ef30664dae252b659cfd63efbe", - "git_blob": "d74542ff66f4d953bc25819a4d0e439082078006", + "bytes": 52864, + "sha256": "c5ea8bf87c54761866315f28f6c93c6ce5c1e0f99d47c5a03f14d095d06871cc", + "git_blob": "ae279e9015483a666ece8deb2544cbc458a5e270", "id": "E", - "commit": "0cb8ad3ca1fe1733e737570ab682f469b66d76e4", + "commit": "13f932c2f6ec07d8c4c9996bd030ab22eaa32108", "pull_request": 206, "packet_copy": "engine_e2.py", "current_required": false, @@ -72,11 +105,11 @@ }, { "path": "frontiers/c8_theorem_a_constants_d3_20261001/theorem_a3.py", - "bytes": 75753, - "sha256": "b0e1b3b4f43e140b51609332fe2e54f4f6b50d10f9f6af48d40ff548bbe742c0", - "git_blob": "ac809dbd36981d1b214fb2f8626417b50afb41f3", + "bytes": 76209, + "sha256": "310d4ab3d3a38a377d753753f3b814591a9090a947f87d6843ab7389389572d3", + "git_blob": "267890232119b35206be7a8cc74d730de9a68644", "id": "E3", - "commit": "c4b8ec3e0232a6dcfc29c8d938268d51693b46b0", + "commit": "250b90722e8c47adf8f7e7c41a61e8a8085d9741", "pull_request": 212, "packet_copy": "engine_e3.py", "current_required": false, diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/engine_e2.py b/frontiers/c8_torus_vanishing_remainder_20261001/engine_e2.py index d74542ff6..ae279e901 100644 --- a/frontiers/c8_torus_vanishing_remainder_20261001/engine_e2.py +++ b/frontiers/c8_torus_vanishing_remainder_20261001/engine_e2.py @@ -298,7 +298,9 @@ def rup_rel(x, bits=200): def sqrt_up(x): - """Rational upper bound of sqrt(x), x >= 0 rational, relative error below 2^-200 (no absolute rounding floor).""" + """Rational upper bound of sqrt(x), x >= 0 rational, relative excess below 2^-198 (no absolute rounding floor): the root + step adds less than 2^-209 and rup_rel(., 200) less than 2^-199. (The earlier '2^-200' was too sharp: at x = 49/100 the + excess is (8/7) 2^-200, review C59-DOC-01.)""" x = Fr(x) if x == 0: return Fr(0) @@ -682,7 +684,10 @@ def elam2_pos(m, s): def box_bound(law, bb, kb, r0, r1, last_band=False, x0=None, theta=None): """Certified upper bound of sup_{r in [r0, r1]} Q^W(G_r^c) / r^3 over b in bb, k in kb, with its pieces, for the near-branch - threshold x0 (X_0 of NOTE section 4) and the split parameter theta (section 5); every choice gives a valid bound.""" + threshold x0 (X_0 of NOTE section 4) and the split parameter theta (section 5). The bound is proved for 0 < x0 < 2/9 and + 0 < theta < 1: then 3 x0 < 1, x0 <= 1/4 (root bounds), kappa = 3/(1 - 3 x0), c = 1 - kappa x0/2 > 0 and + (1 + kappa x0) c^2 <= 1. No claim is made outside that domain. Both COMBOS pairs, and the (1/64, 1/32) of Math-#215's + caller, lie inside it (review C61).""" x0 = X0 if x0 is None else Fr(x0) theta = THETA if theta is None else Fr(theta) b = IV(bb[0], bb[1]) diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/engine_e3.py b/frontiers/c8_torus_vanishing_remainder_20261001/engine_e3.py index ac809dbd3..267890232 100644 --- a/frontiers/c8_torus_vanishing_remainder_20261001/engine_e3.py +++ b/frontiers/c8_torus_vanishing_remainder_20261001/engine_e3.py @@ -319,7 +319,9 @@ def rdown_rel(x, bits=200): def sqrt_up(x): - """Rational upper bound of sqrt(x), x >= 0 rational, relative error below 2^-200 (no absolute rounding floor).""" + """Rational upper bound of sqrt(x), x >= 0 rational, relative excess below 2^-198 (no absolute rounding floor): the root + step adds less than 2^-209 and rup_rel(., 200) less than 2^-199. (The earlier '2^-200' was too sharp: at x = 49/100 the + excess is (8/7) 2^-200, review C59-DOC-01.)""" x = Fr(x) if x == 0: return Fr(0) @@ -1053,7 +1055,10 @@ def det_norm(p, bpp, muB, sig): def box_bound(law, bb, kb, r0, r1, last_band=False, x0=None, theta=None): """Certified upper bound of sup_{r in [r0, r1]} Q^W(G_r^c) / r^3 over b in bb, k in kb, with its pieces, for the near-branch - threshold x0 (X_0 of NOTE section 4) and the split parameter theta (section 5); every choice gives a valid bound.""" + threshold x0 (X_0 of NOTE section 4) and the split parameter theta (section 5). The bound is proved for 0 < x0 < 2/9 and + 0 < theta < 1: then 3 x0 < 1, x0 <= 1/4 (root bounds), kappa = 3/(1 - 3 x0), c = 1 - kappa x0/2 > 0 and + (1 + kappa x0) c^2 <= 1. No claim is made outside that domain. COMBO = (1/64, 1/32), also used by Math-#215's caller, + lies inside it (review C61).""" x0 = X0 if x0 is None else Fr(x0) theta = THETA if theta is None else Fr(theta) b = IV(bb[0], bb[1]) diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/theorem_b.py b/frontiers/c8_torus_vanishing_remainder_20261001/theorem_b.py index f0e788cb0..36b39dbd3 100644 --- a/frontiers/c8_torus_vanishing_remainder_20261001/theorem_b.py +++ b/frontiers/c8_torus_vanishing_remainder_20261001/theorem_b.py @@ -405,7 +405,9 @@ def gersh_min(C): def wq_bound(M, Cg, rows, vecs): - """Upper bound of ||V' Cg^-1 V|| (rows of V given as magnitude lists): Cg = D^(1/2)(I + E)D^(1/2), Cg^-1 <= D^-1/(1 - ||E||).""" + """Upper bound of ||V' Cg^-1 V|| (rows of V given as magnitude lists). With D the lower endpoints of the diagonal, + Cg = D^(1/2)(I + H + E)D^(1/2) exactly, where H >= 0 is diagonal (the actual diagonal's surplus over D) and E has zero + diagonal with ||E||_F <= Ef; so Cg >= (1 - Ef) D and Cg^-1 <= D^-1/(1 - Ef) (review C58: the surplus H only helps).""" n = len(rows) D = [Cg[(rows[i], rows[i])].lo for i in range(n)] Ef2 = Fr(0) @@ -479,6 +481,8 @@ def bounds_d2(Q, R, tml, il): Fmin = M.elam2_pos(m_lo, sa).lo Fmax = M.elam2_pos(m_hi, sb).hi F1max = M.elam_pos(m_hi, sb).hi + # the variance multiplier (3/4) 2 sb = (3/2) sb only needs to be >= 1 (dF/d(sigma^2) = Phi(mu/sigma) <= 1); it is not + # a claim that Phi <= 3/4, which fails at mu = 1, sigma^2 = 2 (review C60) Fae = ae_add(ae_mul(Dae, 2 * F1max / Fmin), ae_mul(Sae, 2 * sb / Fmin * Fr(3, 4))) Fref = M.elam2_pos(b0, r2.lo).lo # secondary law: the engine's interval law over [0, R] diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/theorem_v.py b/frontiers/c8_torus_vanishing_remainder_20261001/theorem_v.py index 503fe83b4..b492b7166 100644 --- a/frontiers/c8_torus_vanishing_remainder_20261001/theorem_v.py +++ b/frontiers/c8_torus_vanishing_remainder_20261001/theorem_v.py @@ -70,7 +70,11 @@ def _load(name, fn): # ============================================================================= 1. interval-coefficient Taylor models class TV: """f(r) = sum_{j <= N} c_j r^j + rho(r) for 0 <= r <= 2^-s, with FIXED (r-independent) coefficients - c_j in [(C_j - D_j)/2^P, (C_j + D_j)/2^P] and |rho(r)| <= (E/2^P) r^(N+1). In particular f(0) = c_0.""" + c_j in [(C_j - D_j)/2^P, (C_j + D_j)/2^P] and |rho(r)| <= (E/2^P) r^(N+1). In particular f(0) = c_0. + Preconditions (the reuse boundary; review C56): C and D are integer lists of length N + 1 with every D_j >= 0, and + E >= 0; the operands of +, -, * share (s, N); powers are nonnegative integers; the readers are called with + 0 < R <= 2^-s; the lists are not mutated after construction. The certificate meets all of them; inputs outside + them are not covered.""" __slots__ = ('C', 'D', 'E', 's', 'N') def __init__(self, C, D, E, s, N): @@ -426,7 +430,9 @@ def pi_ratio_v(law, Q, R): def combine_v(M, p_up, p_dn, z_up, z_dn, R): - """A_r/A_0 = (pi_r/pi_0) (z_r/z_0): (1 + pu)(1 + zu) - 1 <= pu + zu + pu zu(R); 1 - (1 - pd)(1 - zd) <= pd + zd.""" + """A_r/A_0 = (pi_r/pi_0) (z_r/z_0): (1 + pu)(1 + zu) - 1 <= pu + zu + pu zu(R); 1 - (1 - pd)(1 - zd) <= pd + zd. + The lower sum is valid even when one lower estimate exceeds 1 (review C58): if pd + zd >= 1 it holds because both ratios + are positive; otherwise 0 <= 1 - pd, 1 - zd and the two lower factors may be multiplied.""" pu = exp_up_ab(M, p_up, R) zR = ab_at(z_up, R) up = (pu[0] + z_up[0] + pu[0] * zR, pu[1] + z_up[1] + pu[1] * zR) @@ -482,6 +488,8 @@ def bounds_d2_v(Q, R, law, il): sb_, cb_ = law.beta['om'].absval(R) sc_, cc_ = law.Cu[('om', 'w')].absval(R) cwlo = (2 - Sm) / (1 + th1) # Var w = Cu_ww / Theta >= cwlo + # the regression coefficient Cu_om,w / Cu_ww is free of Theta (it cancels exactly); dividing |Cu_om,w| by the + # normalized floor cwlo is deliberately conservative (review C58) # relative to 36 k^2 r^2 F_0: the 36 k^2 cancels, so the normalizer is F_0 >= Fmin T2 = (((b1 * sa_ + k1 * sb_) * F1max + sc_ / cwlo * (Fmax + mmax * F1max)) / Fmin, ((b1 * ca_ + k1 * cb_) * F1max + cc_ / cwlo * (Fmax + mmax * F1max)) / Fmin) @@ -654,6 +662,8 @@ def cov_Oz(o, z): return M.absup(il.C[(o, z)]) def explained(zs): + # with D the lower diagonal endpoints, C_zz = D^(1/2)(I + H + E)D^(1/2): H >= 0 diagonal (surplus), E zero-diagonal + # with ||E||_F <= Ef, so C_zz >= (1 - Ef) D; the surplus H only helps (review C58) rows = [[cov_Oz(o, z) for z in zs] for o in M.ONAMES] D = [il.C[(z, z)].lo for z in zs] Ef2 = Fr(0) From 2f6b8bd954bdac6db49b2ec854d189f7023c5587 Mon Sep 17 00:00:00 2001 From: Dylan Roy Date: Fri, 2 Oct 2026 00:45:01 +0000 Subject: [PATCH 4/6] Theorem L' v1.2.1: pin [CAP] as current_required (C63 source-contract finding) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit NOTE §0 states that every scope retains P + CAP + E1 + E2 + REC, but this consumer's map did not pin CAP, so a changed CAP was guarded only by #215. Add imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md (15160 B, blob 0633aca3) as current_required, its workflow path trigger, and the §6/§8 entries. A tampered copy now fails the workflow's pin check (tested locally). No Python file or RESULTS.json changes. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .github/workflows/c8-torus-vanishing-remainder.yml | 1 + .../c8_torus_vanishing_remainder_20261001/NOTE.md | 8 +++++--- .../SOURCE_FILES.json | 12 ++++++------ .../SOURCE_MAP.json | 13 ++++++++++++- 4 files changed, 24 insertions(+), 10 deletions(-) diff --git a/.github/workflows/c8-torus-vanishing-remainder.yml b/.github/workflows/c8-torus-vanishing-remainder.yml index 8ec60df05..f01604880 100644 --- a/.github/workflows/c8-torus-vanishing-remainder.yml +++ b/.github/workflows/c8-torus-vanishing-remainder.yml @@ -6,6 +6,7 @@ name: C8 torus vanishing remainder (certified replay) - 'frontiers/c8_torus_vanishing_remainder_20261001/**' - '.github/workflows/c8-torus-vanishing-remainder.yml' - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md' + - 'imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md' - 'imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md' - 'reviews/d1_section9_borel_repair_20260925/REPAIR.md' - 'reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md' diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md b/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md index 637257af3..f821fcbb6 100644 --- a/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md +++ b/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md @@ -261,7 +261,7 @@ This gives Theorem L′ on each of the 96 boxes `[i/8, (i+1)/8] x [1/2 + j/8, 1/ - `coupling-half`: halve Lemma V5's constant. - The workflow `.github/workflows/c8-torus-vanishing-remainder.yml` checks: - the manifest; - - the main-resident pins [LP], [E1], [E2], [REC]; + - the main-resident pins [LP], [CAP], [E1], [E2], [REC]; - the packet copies [B], [E], [E3], byte for byte; - both modes; - the mutants; @@ -292,7 +292,8 @@ deviations from `A_0^ref`. - **Sources** (`SOURCE_MAP.json`): - [LP] `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` (on `main`, verified by the workflow), read - with its mandatory amendments, also on `main` and verified: + with [CAP] `imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md` and its mandatory amendments, all on + `main` and verified: - [E1] `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md`; - [E2] `reviews/d1_section9_borel_repair_20260925/REPAIR.md`; - [REC] `reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md`; @@ -315,7 +316,8 @@ deviations from `A_0^ref`. - "six pins" becomes "six in `d = 2`, eight in `d = 3`" (C57). - §0 gains the four result scopes. - §5 records C60's direct contact comparison and C61's `C_{B,K}` disposition. - - §§7–8 add the corrected parent set [E1], [E2], [REC] (C61) and the current review status. + - §§7–8 add the corrected parent set [E1], [E2], [REC] (C61) and the current review status, and §8 pins [CAP] as well, + so that this consumer verifies the whole parent set it cites (C63, 5943439412). - **`theorem_v.py`.** - The `TV` docstring states its preconditions (C56). - `combine_v` states the positivity case split of the lower bound (C58). diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json b/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json index 93f6ca6c0..027af8dc0 100644 --- a/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json +++ b/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json @@ -3,9 +3,9 @@ "files": [ { "path": "NOTE.md", - "bytes": 21356, - "sha256": "8295401f5858c67c611ee034b0021c15dc188218016b3a2c6263a40f1432d3bd", - "git_blob": "637257af38e03fe17ee65a93820bad83aa83aa9d" + "bytes": 21554, + "sha256": "345c13aa8253dfbd77fe9d6cba2ae77e787a191b80dbdd7d7042588142ec3b9b", + "git_blob": "f821fcbb6bc0038bb7da7039de80c245f5addea0" }, { "path": "README.md", @@ -21,9 +21,9 @@ }, { "path": "SOURCE_MAP.json", - "bytes": 6771, - "sha256": "cec379ce90a370097396dd83432c6934ada6f8439d2febd35917c9fcd244f750", - "git_blob": "bef853ce1c5d69d644b75dfa070fd392eaab7826" + "bytes": 7411, + "sha256": "5d249d94e292413ac0cf7d4e3270a0c39d3284b2159cfc0253e111110a26e616", + "git_blob": "ffc6bbafbcb3c05140f021600c8ba9e5b806705e" }, { "path": "engine_e2.py", diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_MAP.json b/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_MAP.json index bef853ce1..ffc6bbafb 100644 --- a/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_MAP.json +++ b/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_MAP.json @@ -7,7 +7,7 @@ "source_repository": "d6g8k5htny-coder/Math-", "main_at_creation": "c2f1270d28192b3c06dd31f728ebf04d03d4246e", "delivered_under": "explicit owner instructions of Dylan Roy in Claude Code session session_015wNj8LPTKXsaT68G3DgPPh, given after OWNER_STOP (2026-09-27): 2026-09-28 \"you have my full permisions and approval to make the proper changes and uploads\"; 2026-09-30 \"Work on closing as many mathematical items as possible while working with and supporting the other models in the GitHub. Also advance the mathematics in every place that's justified based on what we currently know. Ultra tokens, ultra depth @GitHub @Superpowers\" (repeated 2026-09-30 as \"Continue to Work on closing ...\"). Pull requests only, never merged by their author; no register, GRAPH, STATUS, lemma_closed, prize or premise edits. OWNER_STOP.md addresses Cursor agents and automations and is untouched.", - "note": "current_required sources ([LP] and the mandatory parent amendments [E1], [E2], [REC]) are verified on the tree by the workflow; packet_copy sources ([B], [E], [E3]) are verified against the packet copies byte for byte; [B-NOTE] and [B-RESULTS] are read-only provenance pinned by identity", + "note": "current_required sources ([LP], [CAP] and the mandatory parent amendments [E1], [E2], [REC]) are verified on the tree by the workflow; packet_copy sources ([B], [E], [E3]) are verified against the packet copies byte for byte; [B-NOTE] and [B-RESULTS] are read-only provenance pinned by identity", "sources": [ { "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", @@ -20,6 +20,17 @@ "on_main": true, "role": "sections 1, 3, 8-12: the model, the preconditioned pins, the radial ledger A_r = 12 pi_r Z_r / r^2 (10.2) and its limit A_0 (10.3), the lifetime pushforward (11.1)-(11.3); section 11 records that A_r -> A_0 was not assigned a numerical rate. Consumed as statements, as in Math-#215" }, + { + "path": "imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md", + "bytes": 15160, + "sha256": "0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc", + "git_blob": "0633aca3c2a2882b0de4399da0a75d64c2e6b2e1", + "id": "CAP", + "commit": "6f74f7a77bae15d6aa07506bab38977e34dcc449", + "current_required": true, + "on_main": true, + "role": "section 1: the marked-cylinder good event G_r and the cap criterion; part of the corrected parent P + CAP + E1 + E2 + REC that every scope of Corollary L'1 retains (review C63), and through [B] and [E], [E3] the cap route behind the elder line's C_{B,K}" + }, { "path": "imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md", "bytes": 1782, From 9b8eddb5591085f4b0636fb2a616916adccd9fe7 Mon Sep 17 00:00:00 2001 From: Dylan Roy Date: Fri, 2 Oct 2026 00:51:29 +0000 Subject: [PATCH 5/6] Theorem L' v1.2.2: C63 wording findings (documentation only) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit - TV docstring: adds C56's integer E, integer s, N >= 0 with N >= 2 for the second-order readers (inc/absinc read coeff(2)), and exact rational reader/scalar inputs; the certificate uses s = 6, N = 8. - Corollary L'1: the window is ell < min(r_pop, R)^3/2, as in §5. - §7: 'uses only these assembled conclusions of Theorem L' (contact value, C_BK), the scoped [B] algebra and reference routines remaining consumed. AST with docstrings removed unchanged; RESULTS.json unchanged; --check passes in both modes. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .../c8_torus_vanishing_remainder_20261001/NOTE.md | 10 ++++++++-- .../SOURCE_FILES.json | 12 ++++++------ .../theorem_v.py | 9 +++++---- 3 files changed, 19 insertions(+), 12 deletions(-) diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md b/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md index f821fcbb6..6257851fa 100644 --- a/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md +++ b/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md @@ -58,7 +58,7 @@ What changes against [B]: **Corollary L′1 (the densities with a vanishing remainder).** Let `nu_cand`, `nu_eld` be the compact-window densities of [LP] Theorem B in the versions (11.2), for the torus kernel or the Gaussian kernel, and `c_{B,K} = int_{B x K x S^(d-1)} A_0(b, k, u)/(3 k^(2/3)) db dk dsigma(u)` the leading constant **of the same kernel**. The -angular integral uses ordinary surface measure `dsigma`, as in [LP] (11.2)–(11.3). The torus is not assumed isotropic. For `0 < ell < R^3 / 2`: +angular integral uses ordinary surface measure `dsigma`, as in [LP] (11.2)–(11.3). The torus is not assumed isotropic. For `0 < ell < min(r_pop, R)^3 / 2`, with `r_pop` [LP]'s population radius: -(c2_dn ell^(2/3) + c1_dn ell^(1/3)) <= ell^(1/3) nu_cand(ell) - c_{B,K} <= c2_up ell^(2/3) + c1_up ell^(1/3), ell^(1/3) nu_cand(ell) - C_{B,K}(R) ell <= ell^(1/3) nu_eld(ell) <= ell^(1/3) nu_cand(ell), @@ -280,7 +280,8 @@ deviations from `A_0^ref`. - [LP]'s sections 8–12, as in [B], read with [E1], [E2], [REC]; - [B]'s §§2–5 algebra (exact expansions, first-order cancellations, Hölder bounds, Loewner coupling): scoped reads 5378815703, 5379687608, 5379747967, 5379848224, 5384256400. Its positive-radius Theorem L is not accepted as a whole; - this packet uses only its contact value (C60) and its `C_{B,K}` (C61); + this packet uses only these assembled conclusions of Theorem L, namely its contact value (C60) and its `C_{B,K}` (C61), + while the scoped [B] algebra and reference routines above remain consumed; - the band laws, interval and transcendental primitives and moment routines of [E], [E3]: reviewed at the contracts used here (C57, C59). @@ -318,6 +319,11 @@ deviations from `A_0^ref`. - §5 records C60's direct contact comparison and C61's `C_{B,K}` disposition. - §§7–8 add the corrected parent set [E1], [E2], [REC] (C61) and the current review status, and §8 pins [CAP] as well, so that this consumer verifies the whole parent set it cites (C63, 5943439412). + - **v1.2.2 (C63, 5943465973).** Three wording changes: + - the `TV` docstring adds C56's integer `E`, integer `s, N >= 0` with `N >= 2` for the second-order readers, and + exact rational reader inputs; + - Corollary L′1's window becomes `ell < min(r_pop, R)^3/2`, as in §5; + - §7 says "these assembled conclusions of Theorem L". - **`theorem_v.py`.** - The `TV` docstring states its preconditions (C56). - `combine_v` states the positivity case split of the lower bound (C58). diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json b/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json index 027af8dc0..3e06dbda6 100644 --- a/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json +++ b/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json @@ -3,9 +3,9 @@ "files": [ { "path": "NOTE.md", - "bytes": 21554, - "sha256": "345c13aa8253dfbd77fe9d6cba2ae77e787a191b80dbdd7d7042588142ec3b9b", - "git_blob": "f821fcbb6bc0038bb7da7039de80c245f5addea0" + "bytes": 22079, + "sha256": "7bcd8f58834f1177e9e05f80984931d016388ff8b2e8c197a0898653ff92a0f7", + "git_blob": "6257851faa73af348dcf6ac498c59bdaa31b9a84" }, { "path": "README.md", @@ -45,9 +45,9 @@ }, { "path": "theorem_v.py", - "bytes": 40848, - "sha256": "f5d87b547a174f8abb42b8a13926829b5c3c0903f8a30de16852b654c7020f7f", - "git_blob": "b492b7166314eccb10341ce0a08dd236706adcdb" + "bytes": 41036, + "sha256": "1a3bc1c82bab6ea2b5a96cd7ad395fc3c6a7d4bbedd2cee12cda5812d4e8699d", + "git_blob": "5fd6b75df48940f293f0c112c1005418dcd75bde" } ] } diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/theorem_v.py b/frontiers/c8_torus_vanishing_remainder_20261001/theorem_v.py index b492b7166..5fd6b75df 100644 --- a/frontiers/c8_torus_vanishing_remainder_20261001/theorem_v.py +++ b/frontiers/c8_torus_vanishing_remainder_20261001/theorem_v.py @@ -71,10 +71,11 @@ def _load(name, fn): class TV: """f(r) = sum_{j <= N} c_j r^j + rho(r) for 0 <= r <= 2^-s, with FIXED (r-independent) coefficients c_j in [(C_j - D_j)/2^P, (C_j + D_j)/2^P] and |rho(r)| <= (E/2^P) r^(N+1). In particular f(0) = c_0. - Preconditions (the reuse boundary; review C56): C and D are integer lists of length N + 1 with every D_j >= 0, and - E >= 0; the operands of +, -, * share (s, N); powers are nonnegative integers; the readers are called with - 0 < R <= 2^-s; the lists are not mutated after construction. The certificate meets all of them; inputs outside - them are not covered.""" + Preconditions (the reuse boundary; reviews C56, C63): C and D are integer lists of length N + 1 with every D_j >= 0; + E is an integer >= 0; s >= 0 and N >= 0 are integers, with N >= 2 for the second-order readers (inc, absinc and + their callers read coeff(2)); the operands of +, -, * share (s, N); powers are nonnegative integers; reader radii + and scalars are exact rationals, with 0 < R <= 2^-s; the lists are not mutated after construction. The certificate + meets all of them (s = TM_S, N = TM_N = 8); inputs outside them are not covered.""" __slots__ = ('C', 'D', 'E', 's', 'N') def __init__(self, C, D, E, s, N): From 6dcb7e3a89d6235aae60de68ec049165fca0b049 Mon Sep 17 00:00:00 2001 From: Dylan Roy Date: Fri, 2 Oct 2026 17:01:59 +0000 Subject: [PATCH 6/6] Theorem L' v1.2.3: Corollary L'2, the expected selected-count bound (derived in review C63) Integrating Corollary L'1 and its elder line over (0, t] gives, for 0 < t < min(r_pop, R)^3/2, max(0, (3/2)c t^(2/3) - c1_dn t - (3/4)c2_dn t^(4/3) - (3/5)C_BK t^(5/3)) <= n_sel(t) <= (3/2)c t^(2/3) + c1_up t + (3/4)c2_up t^(4/3), per unit volume. In relative form n_sel(t) = (3/2)c t^(2/3)(1 + delta), -dn <= delta <= up, with an outward table at t = ell_* (e.g. d = 2, R = 1/4096: up 1.18e-7, dn 5.75e-7). The integration was checked symbolically; the numbers use corollary_torus here and #215's C_BK (RESULTS blob 894fae2e) with exact rationals. NOTE and manifest only; no code or RESULTS.json change. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .../NOTE.md | 30 +++++++++++++++++++ .../SOURCE_FILES.json | 6 ++-- 2 files changed, 33 insertions(+), 3 deletions(-) diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md b/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md index 6257851fa..8758ddf22 100644 --- a/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md +++ b/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md @@ -77,6 +77,33 @@ and `c1/c <= 1.531 x 10^-10` (`d = 2`), `<= 3.985 x 10^-8` (`d = 3`). For the Ga So, on the torus, both densities are `c_{B,K} ell^(-1/3) + O(1) + O(ell^(1/3))` about the **torus** constant, with explicit constants and a vanishing remainder. [B] gave the same expansion only about `c^ref_{B,K}`, up to `eps'_d ell^(-1/3)`. +**Corollary L′2 (expected selected counts; derived in review C63, 5387311588).** Let `n_sel(t)` be the expected number per +unit volume of selected elder pairs of the population with lifetime in `(0, t]`. Multiply by `L^d` for the expected total on +the torus. For `0 < t < ell_* = min(r_pop, R)^3/2`, integrating Corollary L′1 and its elder line over `(0, t]` gives + + max(0, (3/2) c t^(2/3) - c1_dn t - (3/4) c2_dn t^(4/3) - (3/5) C_{B,K}(R) t^(5/3)) + <= n_sel(t) <= (3/2) c t^(2/3) + c1_up t + (3/4) c2_up t^(4/3), + +with `c = c_{B,K}` of the same kernel. Equivalently, `n_sel(t) = (3/2) c t^(2/3) (1 + delta)` with +`-dn <= delta <= up`. Every term is increasing in `t`, so the values at `t = ell_*` bound the whole window. The ratios +use `corollary_torus` here and `C_{B,K}(R)` of [B] (`RESULTS.json` blob `894fae2e`), divided by `c >= c_ref,lo (1 - eps_0)`; +all are rounded up: + +| `d` | `R` | `ell_*` (`r_pop >= R`) | `up` | `dn` | +|---|---|---|---|---| +| 2 | `1/4096` | `2^-37` | `1.18e-7` | `5.75e-7` | +| 2 | `1/1024` | `2^-31` | `1.90e-6` | `3.37e-5` | +| 2 | `1/512` | `2^-28` | `7.60e-6` | `2.73e-4` | +| 3 | `1/4096` | `2^-37` | `4.35e-7` | `4.03e-6` | +| 3 | `1/1024` | `2^-31` | `7.50e-6` | `2.51e-4` | +| 3 | `1/512` | `2^-28` | `3.29e-5` | `3.84e-3` | + +Each scope applies: +- the upper bound has the torus candidate corollary's scope; +- the lower bound additionally has the elder corollary's scope, through `C_{B,K}`. + +The bound is per unit volume, for the compact-window population only. It is not a count of all short bars. + **Four result scopes, and what each consumes.** Every scope also retains the corrected parent [LP] + [CAP] + [E1] + [E2] + [REC] (§8) and its Gaussian-field, typed-weight and Kac–Rice interpretation. 1. *Theorem L′*, the main radial rate, for both kernels: §§2–4 and the band-law, primitive and moment contracts of [E], [E3]. @@ -319,6 +346,9 @@ deviations from `A_0^ref`. - §5 records C60's direct contact comparison and C61's `C_{B,K}` disposition. - §§7–8 add the corrected parent set [E1], [E2], [REC] (C61) and the current review status, and §8 pins [CAP] as well, so that this consumer verifies the whole parent set it cites (C63, 5943439412). + - **v1.2.3.** It adds Corollary L′2 (§0), the expected selected-count bound derived in review C63, with its relative + error table. The integration is checked symbolically, and the numbers are computed with exact rationals from the cited + tables. - **v1.2.2 (C63, 5943465973).** Three wording changes: - the `TV` docstring adds C56's integer `E`, integer `s, N >= 0` with `N >= 2` for the second-order readers, and exact rational reader inputs; diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json b/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json index 3e06dbda6..a6c4c960e 100644 --- a/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json +++ b/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_FILES.json @@ -3,9 +3,9 @@ "files": [ { "path": "NOTE.md", - "bytes": 22079, - "sha256": "7bcd8f58834f1177e9e05f80984931d016388ff8b2e8c197a0898653ff92a0f7", - "git_blob": "6257851faa73af348dcf6ac498c59bdaa31b9a84" + "bytes": 23886, + "sha256": "5ac08f6418750db3b1d0c156f8ed37b2eda904e3b0a9ac08bd7a842b4f23e2d8", + "git_blob": "8758ddf22870302db178f66d8265d946179e0705" }, { "path": "README.md",