diff --git a/.github/workflows/c8-lifetime-difference-constant.yml b/.github/workflows/c8-lifetime-difference-constant.yml new file mode 100644 index 000000000..6f20ce0b7 --- /dev/null +++ b/.github/workflows/c8-lifetime-difference-constant.yml @@ -0,0 +1,75 @@ +name: C8 lifetime difference constant and second-order ledger (certified replay) + +'on': + pull_request: + paths: + - 'frontiers/c8_lifetime_difference_constant_20261001/**' + - '.github/workflows/c8-lifetime-difference-constant.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' + - 'frontiers/c8_cap_failure_coefficient_dims_20261001/RESULTS.json' + workflow_dispatch: + +permissions: + contents: read + +jobs: + certified-replay: + runs-on: ubuntu-latest + timeout-minutes: 240 + 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 pins and engine copies; replay the certificate in both interpreter modes; reject mutants + env: + PYTHONDONTWRITEBYTECODE: '1' + run: | + set -euo pipefail + python -B -S - <<'PY' + import hashlib, json, os, pathlib, subprocess, sys + root = pathlib.Path('frontiers/c8_lifetime_difference_constant_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)) + 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']: + raise ValueError('source identity mismatch: ' + entry['path']) + def blob(data): + return hashlib.sha1(b'blob %d\0' % len(data) + data).hexdigest() + 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']) + script = str(root / 'theorem_b.py') + procs = str(max(2, os.cpu_count() or 2)) + # every cap band in one mode, the sampled replay in the other + for flags, mode in ((['-B', '-S'], '--check-full'), (['-B', '-O', '-S'], '--check')): + out = subprocess.run([sys.executable, *flags, script, mode, '--procs', procs], cwd=root, capture_output=True, timeout=10800) + if out.returncode != 0: + raise RuntimeError(mode + ' failed under ' + ' '.join(flags) + ': ' + out.stdout.decode()[-400:] + out.stderr.decode()[-400:]) + print(flags, mode, out.stdout.decode().strip()[-600:]) + for m in ('no-om-term', 'no-bad-set', 'no-det', 'no-torus', 'cap-half'): + rc = subprocess.run([sys.executable, '-B', '-S', script, '--check', '--procs', procs, '--mutant', m], cwd=root, capture_output=True, timeout=3600).returncode + if rc != 1: + raise RuntimeError('mutant ' + m + ' not rejected') + print(json.dumps({'scientific_effect': 'NONE', 'cap_bands_replayed': 274, 'mutants_rejected': 5})) + PY + git diff --exit-code diff --git a/frontiers/c8_lifetime_difference_constant_20261001/NOTE.md b/frontiers/c8_lifetime_difference_constant_20261001/NOTE.md new file mode 100644 index 000000000..845112c90 --- /dev/null +++ b/frontiers/c8_lifetime_difference_constant_20261001/NOTE.md @@ -0,0 +1,557 @@ +# Theorem B of [LP] with explicit constants on the C8 band (`d = 2, 3`): the radial ledger to second order, two-sided lifetime densities, and the difference constant `C_{B,K}` + +**Object:** `CL-C8-LIFETIME-DIFFERENCE-CONSTANT-20261001-v1`. **Author lane:** Anthropic / Claude. + +**Kind.** An author-side theorem with explicit constants, certified by exact rational interval arithmetic (standard library +only). It uses Taylor models in the radius for the pin law, and the box-bound engines of Math-#206 and Math-#212 for the cap +route. + +**Scientific effect:** NONE. Catalog entry C8 (`reviews/candidates_pending_20260928/CANDIDATES.md`) stays OPEN. No STATUS, +PROOF_INDEX, GRAPH or catalog edit. Same GitHub account as every lane; zero organizational-independence credit. The author will +not merge. Scope claim: Math-#206 comment 5924133127. + +**Sources** (pinned in `SOURCE_MAP.json`): +- [LP] `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md`: + - section 1 (the model in every `d`, Theorem B with (1.2)); + - section 3 (the preconditioned pins (3.1)–(3.3)); + - section 8 (`1 - p_r <= Q^W(G_r^c)`); + - sections 9–11 (the weighted Kac–Rice interface, the radial ledger (10.1)–(10.2), the lifetime pushforward (11.1)–(11.3)); + - section 12 ((12.1), (12.3)). + Sections 8–12 are consumed as statements; nothing else of [LP] is used. +- [CAP] `imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md`, section 1 (the good event `G_r`), through the + engines. +- **The corrected parent reading set.** [LP] is read together with three mandatory amendments, all on `main` and pinned + `current_required`. The parent object is P + CAP + E1 + E2 + REC, as REC section 1 requires of every consumer (review C61): + - [E1] `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md`: the preconditioning congruence is `diag(r^(-1/2), I)`; + - [E2] `reviews/d1_section9_borel_repair_20260925/REPAIR.md`: [LP] section 9 is replaced by the marked Kac–Rice identity + for the Borel elder indicator (finite-measure equality on a cylinder algebra, then a monotone-class extension); + - [REC] `reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md`: the reading rule. It uses joint convergence in law + and the embedding condition `r < L/(4 sqrt 2)`, which `r <= 1/64` and `L >= 10` satisfy. +- [E] Math-#206 at `13f932c` (v1.2): the planar certificate `theorem_a.py`, carried byte for byte as `engine_e2.py`. +- [E3] Math-#212 at `250b907` (v1.1): the `d = 3` certificate `theorem_a3.py`, carried byte for byte as `engine_e3.py`. +- Read-only cross-checks and comparisons: + - [W] Math-#197: `c_{B,K}` in the plane. + - [W3] Math-#205: `c_{B,K}` in `d = 3`. + - [G] Math-#203: `c_G = R(b) J(k)`. + - [G3] Math-#208: `c_G^(3)`. + +## 0. Statement + +**Setting** ([LP] sections 1, 10, 11 with `d in {2, 3}`, `m = d - 1`). +- The torus `X = (R / L Z)^d` with the periodized Gaussian kernel `K_L`, or the reference kernel `exp(-|z|^2/2)`. +- For an orthonormal frame `R` with axial vector `u`: the pins `M = -(r/2) u`, `S = (r/2) u`, `f(M) = b`, `f(S) = b - k r^3`, + `grad f(M) = grad f(S) = 0`, the regression law `Q`, the typed weight `W_r`, and `Z_r = E_Q W_r`. +- [LP] (10.2) writes the candidate intensity per unit volume as `r A_r(b, k, u) dr db dk dsigma(u)` with + + A_r(b, k, u) = 12 pi_r(R; v_r) Z_r / r^2, + + where `pi_r(R; v_r)` is the density of the preconditioned pins (3.1) at their prescribed value (3.3). +- The band is `B = [0, 1]`, `K = [1/2, 2]`. +- The **reference contact integrand** is the closed form + + A_0^ref(b, k) = 432 (2 pi)^-(d+1) 12^(-1/2) k^2 exp(-12 k^2) exp(-3 b^2/4) m_d(b), + + m_2(b) = (b^2 + 2) Phi(b/sqrt2) + sqrt2 b phi(b/sqrt2), m_3(b) = E[det(B)^2 1{B > 0}], B = b I + GOE_2. + + This is the integrand of [LP] (11.3) for the reference kernel: `pi_0 = (2 pi)^-(d+1) 12^(-1/2) exp(-3b^2/4 - 12k^2)` and + `z_0 = 36 k^2 m_d(b)`. `m_3` has the closed form of [E3] section 6. + +**Theorem L (the radial ledger to second order).** 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 + eps_d) <= A_r(b, k, u) / A_0^ref(b, k) - 1 <= a_up(R) r^2 + eps_d. + +The same holds for the reference kernel. The constants are exact rationals (`RESULTS.json`), shown here 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` | `26.256537` | `26.108063` | +| `1/1024` | `13.830901` | `3.655573` | `27.446808` | `27.301701` | +| `1/512` | `13.841872` | `3.680046` | `29.869053` | `29.730582` | +| `1/256` | `13.864854` | `3.729062` | `34.883076` | `34.757367` | +| `1/128` | `13.915029` | `3.827402` | `45.611590` | `45.508415` | +| `1/64` | `14.032636` | `4.025510` | `70.060400` | `69.981330` | + +with `eps_2 <= 3.96 x 10^-10` and `eps_3 <= 1.79 x 10^-8`. These collect the torus, truncation and rounding remainders and +do not depend on `r`. + +What Theorem L gives, and what it does not: +- **It gives** the second-order coefficient of the deviation of the ledger from the reference contact integrand, `a(R)`, + up to the radius-independent `eps_d`. [LP] proves the convergence ((10.3)) and states in section 11 that "Convergence of + `A_r` to `A_0` was not assigned a numerical rate." +- **It does not give** a vanishing-remainder rate. `eps_d` does not tend to `0` with `r`. On the torus, Theorem L + therefore does not prove `A_r = A_0 (1 + O(r^2))` for the torus limit `A_0`. With [LP] (10.3) it gives only + `|A_0 / A_0^ref - 1| <= eps_d` for that limit. A vanishing-remainder rate would need the torus corrections tracked as + functions of `r`, which the certificate does not do. + +There is no first-order term: +- for the Gaussian kernel, the law of the transverse Hessian at `M` is the contact law at every `r` (Lemma 1); +- the first-order parts of `Z_r / r^2` cancel exactly (Lemma 3). + +In the plane `a_up` is attained at `k = 2`, where the genuine coefficient is about `13.2`–`13.4` (Monte Carlo, section 9). + +**Corollary L1 (lifetime densities to second order).** Let `nu_cand`, `nu_eld` be the compact-window densities of [LP] +Theorem B for pairs at distance at most `r_pop`, in the versions (11.2). For `0 < ell < min(r_pop, R)^3 / 2`: + + | ell^(1/3) nu_cand(ell) - c^ref_{B,K} | <= c'_up(R) ell^(2/3) + eps'_d, (upper and lower sides with c'_up, c'_dn) + ell^(1/3) nu_cand(ell) - C_{B,K}(R) ell <= ell^(1/3) nu_eld(ell) <= ell^(1/3) nu_cand(ell). + +- `c^ref_{B,K} = |S^(d-1)| int_{B x K} A_0^ref / (3 k^(2/3))` is certified. Its enclosures contain the values of Math-#197 and + Math-#205: + - `d = 2`: `[0.002386938000079198, 0.002386938059531751]`; + - `d = 3`: `[0.000974382342936858, 0.000974382390951253]`. +- `c'_up(R) = |S^(d-1)| sum_boxes a_up,box int_box A_0^ref / (3 k^(4/3))` is the weighted second-order constant, and `c'_dn` + is the same with `a_dn`. As ratios to `c^ref`: + + | `R` | `c'_up/c`, `d = 2` | `c'_dn/c`, `d = 2` | `c'_up/c`, `d = 3` | `c'_dn/c`, `d = 3` | + |---|---|---|---|---| + | `1/4096` | `6.2826` | `4.4646` | `23.154` | `22.428` | + | `1/512` | `6.3221` | `4.5056` | `27.378` | `26.662` | + | `1/64` | `6.6431` | `4.8364` | `67.061` | `66.397` | + + (Rounded up. All seven `R` are in `RESULTS.json`.) +- `eps'_2 <= 1.2 x 10^-13` and `eps'_3 <= 2.5 x 10^-12`, absolute. + +Hence, up to an additive `eps'_d ell^(-1/3)`, both compact-window densities are `c^ref_{B,K} ell^(-1/3) + O(ell^(1/3))` +with explicit constants. [LP] remarks that (1.2) "is an O(ell^(2/3)) difference, not a claim that either density separately +has a second-order expansion with that remainder". +- **What Corollary L1 is.** A two-sided expansion of each density about the reference constant. +- **What it is not.** Its remainder `eps'_d ell^(-1/3)` does not vanish as `ell -> 0`. For the torus constant `c_{B,K}` + (any `L >= 10`) it gives `|c_{B,K} - c^ref_{B,K}| <= eps'_d`, not a second-order expansion about `c_{B,K}` with a + vanishing remainder. + +For example, in the plane, for `ell < 2^-19`: + + -4.84 ell^(2/3) - 10^-10 <= ell^(1/3) nu_cand(ell) / c_{B,K} - 1 <= 6.65 ell^(2/3) + 10^-10, + +which is a relative error below `1.1 x 10^-3`. + +**Corollary L2 (explicit [LP] (1.2), (12.1), (12.3)).** For `0 < ell, t < min(r_pop, r_*)^3 / 2` and `q > -5/3`: + + 0 <= nu_cand(ell) - nu_eld(ell) <= C_{B,K}(r_*) ell^(2/3), + 0 <= E[N_cand(0, t] - N_eld(0, t]] <= (3/5) C_{B,K}(r_*) t^(5/3), + E sum_{candidate, not selected, ell <= t} ell^q <= C_{B,K}(r_*) t^(q + 5/3) / (q + 5/3), + +and the selected count satisfies + + (3/2)(c^ref - eps') t^(2/3) - (3/4) c'_dn t^(4/3) - (3/5) C_{B,K} t^(5/3) <= E N_eld(0, t] <= (3/2)(c^ref + eps') t^(2/3) + (3/4) c'_up t^(4/3). + +The certified difference constants (exact rationals in `RESULTS.json`, rounded up), with the relative size of the bound +at `ell = r_*^3/2`, `C_{B,K} ell / c^ref_{B,K}` (rounded up): + +| `r_*` | `C_{B,K}`, `d = 2` | `(3/5) C_{B,K}` | `C ell/c` at `r_*^3/2` | `C_{B,K}`, `d = 3` | `(3/5) C_{B,K}` | `C ell/c` at `r_*^3/2` | +|---|---|---|---|---|---|---| +| `1/4096` | `402.5297` | `241.5178` | `1.3e-06` | `1205.7089` | `723.4254` | `9.1e-06` | +| `1/2048` | `406.7323` | `244.0394` | `1.0e-05` | `1223.4762` | `734.0857` | `7.4e-05` | +| `1/1024` | `413.9359` | `248.3615` | `8.1e-05` | `1270.6952` | `762.4171` | `6.1e-04` | +| `1/512` | `428.0180` | `256.8108` | `6.7e-04` | `2488.0639` | `1492.8383` | `9.6e-03` | +| `1/256` | `3384.7882` | `2030.8729` | `4.3e-02` | `1807674.3925` | `1084604.6355` | `5.6e+01` | +| `1/128` | `178525.1111` | `107115.0667` | `1.8e+01` | `13273148.1212` | `7963888.8728` | `3.3e+03` | +| `1/64` | `198574.3344` | `119144.6007` | `1.6e+02` | `2950751318.4742` | `1770450791.0845` | `5.8e+06` | + +[LP] section 11 gives (1.2) in the form `C ell^(2/3) integral A^*/(3 k^(5/3))` with no value for either factor, and its +section 16 lists "a numerical r_* or C on a prescribed band" among the items not supplied. + +**What the refinement and the cap route contribute** (`r_* = 1/4096`): +- **Box resolution.** On their own boxes, the per-band bounds of [E] and [E3] give `2325.98` in the plane and `4999.57` in + `d = 3`. The refined boxes improve these by factors `5.8` and `4.1`. [E] has `28` boxes, the coarsest `k`-box being + `[1/2, 1]`; [E3] has `39`. +- **A single constant.** Using one sup constant `C(r_*)` of [E] or [E3] times the total weight gives `22467.8` and `38970.6`. +- **The reference-kernel cap-route benchmark.** `Gamma_{B,K} = |S^(d-1)| int A_0^ref c_G / (3 k^(5/3))` is built from + the reference-kernel cap-failure coefficients `c_G` of [G] and [G3]. It is the reference-kernel asymptotic lower + benchmark for any cap-route bound that is uniform over the present family. That family contains the reference + kernel, so as `r_* -> 0` such a bound can give no less than `Gamma_{B,K}`. In floating point, `Gamma_2 = 307.5` and + `Gamma_3 = 541` (section 9). + - `C_{B,K}(1/4096)` is within a factor `1.31` of that benchmark in the plane and `2.23` in `d = 3`. These floating + ratios compare our finite-radius bounds with the benchmark. They do not identify the exact cap-route limit of each + finite torus, which would use the torus `A_0^L` and `c_G^(L,u)`; [G] says the torus deviation from `c_G^ref` is not + enclosed. + - The `d = 3` excess comes from the [E3] box bounds near `k = 1/2`, which are about `2.2 c_G^(3)` there. +- **The cap route itself is loose.** By [G], the cap criterion overstates the actual elder-pairing failure rate by + `7.4 x 10^4` at `k = 1/2`. `C_{B,K}` bounds the [LP] quantity as [LP] defines it; it is not an estimate of the true + difference. +- **Above `r_* = 1/512`** the certified per-band bounds of [E] and [E3] grow quickly; their own `C(r_*)` tables show the same + jump. The useful range is `r_* <= 1/512`. + +## 1. The radial ledger and what is consumed + +[LP] sections 8–11 are consumed as statements: +- **Section 9, as repaired by [E2].** The weighted Kac–Rice identity on the near-diagonal pair domain. +- **Section 10, the ledger (10.1)–(10.2).** The candidate intensity is `r A_r dr db dk dsigma(u)`. The selected intensity + carries the extra factor `p_r`. `A_r` does not depend on the transverse frame. +- **Section 11, the pushforward (11.1)–(11.2).** For `ell < k_- r_pop^3`, + + nu_cand(ell) = ell^(-1/3) int_{B x K x S^(d-1)} A_{(ell/k)^(1/3)}(b, k, u) / (3 k^(2/3)) db dk dsigma(u), + + and `nu_eld` has the extra factor `p_r`. +- **Section 8.** `0 <= 1 - p_r <= Q^W(G_r^c)`. + +Theorem L is a statement about `A_r` alone. The corollaries follow by integration. +- **Corollary L1.** + - At fixed `ell < R^3/2`, every `k in K` has `r = (ell/k)^(1/3) <= (2 ell)^(1/3) < R`, and `r^2 = ell^(2/3) k^(-2/3)`. + - Integrate Theorem L box by box against `A_0^ref / (3 k^(2/3))`. This gives `c^ref + ell^(2/3) c'_up + eps'` above and the + mirror image below. + - `A_0^ref` does not depend on `u`, so the sphere contributes `|S^(d-1)|`. + - The integrals of `ell^(-1/3)`, `ell^(1/3)` and `ell^(2/3)` over `(0, t]` give the bounds on `E N_eld(0, t]`. +- **Corollary L2.** + - `nu_cand - nu_eld = ell^(-1/3) int A_r (1 - p_r) / (3 k^(2/3))`. + - On each cap box `1 - p_r <= Q^W(G_r^c) <= C_box r^3 = C_box ell / k`, so the difference is at most + `ell^(2/3) sum_box C_box int_box A_r / (3 k^(5/3))`. + - Theorem L bounds `A_r` by `(1 + a_up r_*^2 + eps) A_0^ref`. + - Integrating over `(0, t]` gives (12.1) and (12.3). + +## 2. The laws: Taylor models in `r` (Lemmas 1–2) + +**Taylor models.** Each covariance of two preconditioned functionals is the exact series of [E] / [E3] section 2, +`sum_gamma kappa_gamma pi_gamma r^(g_x - w)`. Added to it are: +- the truncation tail beyond `g_x = 60`; +- the torus remainder, by Cauchy estimates on the polydisc: `M_1(10) < 4.46 x 10^-15` in the plane and `< 1.94 x 10^-13` + in `d = 3`. + +The certificate keeps each one on `[0, 1/64]` as a **Taylor model**: +- `f(r) = sum_{j <= 8} c_j r^j + theta e`, with `|theta| <= 1`; +- the coefficients are fixed-point integers with `288` fractional bits; +- every rounding, every truncated power `r^j <= 64^-j`, both tails and the torus remainder go into `e`. + +Products, sums and reciprocals (a geometric series with an explicit tail) stay in this class. Gaussian elimination without +pivoting then gives Taylor models for the following, all exact in their polynomial parts up to `2^-288`: +- `det Sigma_r`; +- the pin energies `e_bb`, `e_bk`, `e_kk`, with `v_r' Sigma_r^-1 v_r = e_bb b^2 + 2 e_bk b k + e_kk k^2`; +- the regression coefficients `alpha` and `beta` (mean `= b alpha + k beta`) of the transverse Hessian `A = D_y^2 f(M)` and of + `Omega' = (D_y^2 f(S) - D_y^2 f(M))/r`; +- their covariances; +- in `d = 3`, the regression matrix of `Omega'` on `B = -A`. + +This removes the dependence loss of interval elimination. For example, the band laws of [E] give the `k`-coefficient of the +mean of `A` on `[0, r]` only as `+-3 r`, while the Taylor model gives `0` up to `10^-11`. + +**Lemma 1 (the transverse Hessian is exactly contact-distributed).** For the reference kernel, every `r > 0`, every `b` and +every `k`, the `Q`-law of `A = D_y^2 f(M)` is that of `-b I_m + G`. Here `G` is the `GOE_m` of [G3], with `N(0, 2)` diagonal +and `N(0, 1)` off-diagonal entries; the law does not depend on `r` or `k`. + +*Proof.* The kernel factorizes, `exp(-|z|^2/2) = exp(-x^2/2) exp(-|y|^2/2)`. Every pin is either axial (`partial_x^a f` on the +axis) or a first transverse derivative on the axis. +- For an axial functional `g`, `Cov(f_{y_i y_j}(x, 0), g) = -delta_ij Cov(f(x, 0), g)`, because + `partial_{y_i} partial_{y_j} exp(-|y|^2/2) = -delta_ij` at `y = 0`. +- For a transverse pin, `Cov(f_{y_i y_j}(x, 0), f_{y_l}(x', 0)) = Cov(f(x, 0), f_{y_l}(x', 0)) = 0`, because odd + `y`-derivatives of the kernel vanish at `y = 0`. + +So `A + f(M) I` is uncorrelated with every pin, hence independent of all of them. Under `Q`, `f(M) = b`, so +`A = -b I + (A + f(M) I)` with `A + f(M) I` keeping its unconditional law: +- `Var(f_yy + f) = 3 - 2 + 1 = 2`; +- `Cov(f_{y_1 y_1} + f, f_{y_2 y_2} + f) = 1 - 1 - 1 + 1 = 0`; +- `Var f_{y_1 y_2} = 1`. + +∎ + +The certificate checks Lemma 1 on the Taylor models: +- every polynomial coefficient of `alpha_A + 1`, `beta_A` and `Cov(A) - Cov(GOE)` vanishes to within `10^-60`; +- the code requires this, and only the remainder survives: at most `1.9 x 10^-12` in the plane and `8.1 x 10^-11` in `d = 3`. + +So on the torus the law is the contact law up to these remainders. + +**Lemma 2 (pin law, energies, `Omega'`).** For the reference kernel, the Taylor models give (coefficients exact up to +`2^-288`): + + d = 2: det Sigma_r = 12 - 18 r^2 + 14.4 r^4 + O(r^6), d = 3: det Sigma_r = 12 - 24 r^2 + 25.4 r^4 + O(r^6), + both: e_bb = 3/2 + r^2/8 - r^4/96 + ..., e_bk = -(3/4) r^3 + ..., e_kk = 24 - 6 r^2 + (6/5) r^4 + ... + +For `Omega'`: +- **Mean.** `alpha = 0`, and `beta = r^2` on the diagonal entries. +- **Covariance with the Hessian.** + - Plane: `Cov(om, w) = -r + r^3/4`. + - `d = 3`: `Cov(A_ii, O_ii) = -r + r^3/4` and `Cov(A_12, O_12) = -r/2 + r^3/8`; all other cross-covariances vanish. + - Hence in `d = 3`, `E[Omega' | B] = mu_Omega + beta(r) (B - mu_B)` with `beta(r) = (r - r^3/4)/2 + ...`, the same scalar for + all three entries. +- **Covariance.** `Var om = 2 - r^2/2 + r^4/12`; `Cov(O_11, O_22) = 0`; `Var O_12 = 1 - r^2/4 + ...`. + +At `r = 0` this is the contact law of [LP] section 15: +- `det = 12 = det Cov(f, f_xx) det Cov(f_x, f_xxx) = 2 . 6`; +- the energy is `3b^2/2 + 24 k^2`, which gives `pi_0 ∝ exp(-3b^2/4 - 12 k^2)`. + +## 3. The pin-density ratio + + pi_r / pi_0^ref = (12 Theta^n / det S_r)^(1/2) exp(-(E_r - E_0)/2), + +with: +- `n = 2(d + 1)`; +- `S_r` the unnormalized pin covariance and `Theta in [1, 1 + theta_1(10)]` the kernel normalization; +- `E_r = Theta (e_bb b^2 + 2 e_bk b k + e_kk k^2)` and `E_0 = 3 b^2/2 + 24 k^2`. + +The Taylor models give `det S_r - 12 in [d_lo r^2 - e, d_hi r^2 + e]`, and the same for each energy coefficient. On a box +`b in [b_0, b_1]`, `k in [k_0, k_1]` (`b, k >= 0`), `log(pi_r/pi_0^ref)` lies between + + -max(0, d_hi) r^2/24 - max(0, E_hi/2) r^2 - (remainders) and max(0, -d_lo) r^2 / (24 (1 - x_m)) + max(0, -E_lo/2) r^2 + (remainders), + +using `log(1 + x) >= x/(1 + x)` and `log(1 + x) <= x`. The dominant genuine term is `+3 k^2 r^2` from `e_kk = 24 - 6 r^2`. At +`k = 2` this is `12 r^2`, the largest part of `a_up` in the plane. + +## 4. The normalizer: exact expansion and the first-order cancellation (Lemma 3) + +**Plane.** Use the notation of [E] section 1: `lambda = -w`, `c1 = 6k - rT`, `c2 = 6k + rT + r^2 tau`, `e = (r nu - v)^2`. +The typed event is `{c1 > 0, P1 > 0, P2 > 0}`, which lies inside `{lambda > 0}`, and on it `W_r = r^2 P1 P2`. Expanding +exactly, + + P1 P2 = c1 c2 lambda^2 - r c1 c2 lambda om + r^2 lambda xi + r^2 v^2 (c2 om - e), + + c1 c2 = 36 k^2 + r^2 kappa, kappa = 6 k tau - T^2 - r T tau, + xi = 6k (r nu^2 - 2 nu v) - T (r^2 nu^2 - 2 r nu v + 2 v^2) - r tau v^2. + +Three cancellations remove every first-order term: +- **In `c1 c2`.** The first-order parts of `c1` and `c2` cancel. +- **Between `P1` and `P2`.** The terms `r c1 lambda e` (from `P2`) and `-r c2 lambda v^2` (from `P1`) cancel, because + `e - v^2 = r^2 nu^2 - 2 r nu v`. +- **The `om` term.** The remaining first-order term `-36 k^2 r E[lambda_+ om]` has an `O(r)` coefficient. Regress `om` on `w`: + `om = om_g + bw (w - mu_w)` with `om_g` independent of `w`. Then + + E[lambda_+ om] = mu_om E[lambda_+] - bw (E[lambda_+^2] - mu_lambda E[lambda_+]), mu_lambda = -mu_w, + + and Lemma 2 gives `mu_om = k r^2` and `bw = -r/2 + O(r^3)`. + +So `Z_r / (36 k^2 r^2) = E[lambda_+^2] + O(r^2)`, and by Lemma 1 `E[lambda_+^2] = m_2(b)` exactly for the reference kernel. + +The second-order terms are bounded by Hölder's inequality with the Gaussian norms of the band law of [E] on `[0, R]`. The one +exception is `kappa`. Since `E[tau | w] = mu_tau + bw_tau (w - mu_w)` with `mu_tau = -2k + O(r^2)`, the term +`6k E[tau lambda_+^2] = 6k mu_tau E[lambda_+^2] + ...` is negative. The certificate therefore bounds the `kappa` term with its +sign: +- above by `max(0, 6k mu_tau F + |side terms|)`; +- below by `6k mu_tau F - E[T^2 lambda_+^2] - |side terms|`. + +**`d = 3`.** Write `D = det B` (`B = -A`), `A* = adj B`, `B_S = B - r Omega'`, and `e = r nu - v` (a vector). [E3] (I2) gives +`P1 = c1 D - r v' A* v` and `P2 = c2 det B_S + r e' adj(B_S) e`. In two dimensions `det B_S = D - r tr(A* Omega') + r^2 det Omega'` +and `adj(B_S) = A* - r adj(Omega')`, so + + P1 P2 = c1 c2 D^2 - r c1 c2 D tr(A* Omega') + r^2 c1 c2 D det Omega' + r^2 D xi_3 - r^2 c1 D e' adj(Omega') e + + r^2 c2 tr(A* Omega') v' A* v - r^3 c2 det Omega' v' A* v - r^2 (v' A* v)(e' A* e) + r^3 (v' A* v)(e' adj(Omega') e), + + xi_3 = -(2T + r tau) v' A* v - 2 c1 nu' A* v + r c1 nu' A* nu. + +Again `c1 e' A* e - c2 v' A* v = r xi_3`. The main term is `36 k^2 E[D^2 1{B > 0}]`, which equals `36 k^2 m_3(b)` for the +reference kernel (Lemma 1). On the torus the Loewner coupling of [E3] section 6 compares it with `m_3`, at the same `b` and with +shift `eps + 12 eta` (remainders only). + +Four terms whose expectations are of lower order than their absolute values are treated through conditional expectations: +- **The first-order term `-36 k^2 r E[D tr(A* Omega') 1]`.** By Lemma 2, `E[Omega' | B] = mu_Omega + beta (B - mu_B) + DY`, so + + E[D tr(A* Omega') 1] = beta (2 E[D^2 1] - b E[D tr B 1]) + R_1, + + with `|R_1| <= ||mu_Omega|| E[D tr B 1] + ||DG|| ||D tr B 1||_2 (tr Cb)^(1/2)` (`DY = DG (B - mu_B)`, remainder only). Since + `beta = r/2 + O(r^3)`, write `T_Omega = -36 k^2 r E[D tr(A* Omega') 1]`, `M = E[D^2 1{B > 0}]` and + `J = E[D tr B 1{B > 0}]`. `T_Omega` is a term of `E[P1 P2] = Z_r / r^2`, so its raw `Z_r` contribution is `r^2 T_Omega`, and + after division by `36 k^2 r^2` it contributes `T_Omega / (36 k^2) = -r beta (2M - bJ) - r R_1` to `Z_r / (36 k^2 r^2)`. For + the reference kernel `beta = a(r) = (1 - e^(-r^2/2))/r`, `mu_Omega = k r^2 I` and `DY = 0`, so this is exactly + + T_Omega / (36 k^2) = -r a(r) (2M - bJ) - k r^3 J = -(1/2) r^2 (2M - bJ) + O(r^3), + + uniformly on the band (OA-215-A-01, successor read 5934120553). Relative to the main term `M = m_3`, divide by `M`. It is + signed and enters `a_dn`, with only `O(r)` slack in `a_up`. +- **`r^2 E[c1 c2 D det Omega' 1]`.** `E[det Omega' | B] = det(E[Omega' | B]) + E[det X']`, where `X' = Omega' - E[Omega' | B]` + is independent of `B`. Here `E[det X'] = Cov(X'_11, X'_22) - Var X'_12` is about `-1`, and `det(E[Omega' | B]) = O(r^2)`. +- **`r^2 E[c2 tr(A* Omega') v' A* v 1]` and `r^2 E[c1 D e' adj(Omega') e 1]`.** Condition on `zeta = (B, v, nu)`. Then + `E[Omega' | zeta] = mu_Omega + Y_K`, and the explained part `Y_K` has `E||Y_K||^2 <= sum_O c_O' D^-1 c_O / (1 - ||E||)`, which is + `O(r^2)`. So both terms are `O(r^3)`. +- **The `kappa` term**, with its sign as in the plane. + +The moments of `B` are the closed-form contact moments `E[det^a tr^c 1{B > 0}]` at the shifted mean, plus outside-ball parts. + +## 5. The thin typed shell (Lemma 4) + +The typed event differs from `{lambda > 0}` (plane) or `{B > 0}` (`d = 3`) only where the smallest eigenvalue is `O(r)`. On +`G_0 = {r |T| <= 3k/2, r^2 |tau| <= 3k/2}` one has `c1 >= 9k/2` and `3k <= c2 <= 9k`. + +**Plane.** +- `{lambda > 0}` minus the typed event lies in `{0 < lambda <= r h}`, with `h = 2 v^2/(9k) + |om|`. +- There `|P1 P2| <= r^2 (15 k h/2 + v^2)(9k (h + |om|) + e)`. +- Given `(v, om, nu)`, `lambda` is Gaussian. Its variance is at least `1/(1/Var w + bw' Cg^-1 bw)` (Schur complement), with + `Cg^-1 <= D^-1/(1 - ||E||_F)` by diagonal scaling. +- So `E[G 1{0 < lambda <= r h}] <= r pbar E[G h]`, and the shell contributes `O(r^3)`. + +**`d = 3`.** +- The shell is `{0 < lambda_1 <= r h}`, with `h = 2 ||v||^2/(9k) + ||Omega'||_F`. +- There `|P1 P2| <= r^2 lambda_2 (lambda_2 + r ||Omega'||)(15 k h/2 + ||v||^2)(9k (h + ||Omega'||) + ||e||^2)`. +- Given `Y = (v, Omega', nu)`, `B` is Gaussian with covariance `C' >= cl I`, where `1/cl <= 1/lambda_min(Cb) + ||Bt' Cg^-1 Bt||`. +- In eigenvalue coordinates (`dB = sqrt2 (lambda_2 - lambda_1) d lambda_1 d lambda_2 d theta`, `theta in [0, pi)`): + + E[lambda_2 (lambda_2 + r Omega) 1{0 < lambda_1 <= s} | Y] <= s KB' J(||mu'||, r Omega), + + where: + - `KB' = pi sqrt2 (2 pi)^(-3/2) cl^(-3/2)`; + - `J(m, x) = int_0^inf lambda^2 (lambda + x) exp(-(lambda - m)_+^2/(2 c_hi)) d lambda`, a polynomial in `m` and `x` built from + half-Gaussian moments; + - `||mu' - mu_B|| <= ||C_BY C_YY^(-1/2)|| chi_7`. +- The rows of `Bt` for `Omega'` come from the Taylor models and are `O(r)`; those for `v` and `nu` are below `10^-11`. + +**Both.** The complement of `G_0` has probability below `10^-50` for `R <= 1/64`. It enters through +`||P1 P2||_2 P(G_0^c)^(1/2)`. + +The sweep evaluates these bounds on `96` boxes (`b`-step `1/8`, `k`-step `1/8`) for each `R`. The box constants feed the +weighted `c'`; their maxima are Theorem L's `a_up` and `a_dn`. + +## 6. Reference integrals + +`A_0^ref / (3 k^p) = K_d k^(2-p) exp(-12 k^2) exp(-3b^2/4) m_d(b)`, with `K_d = |S^(d-1)| 144 (2 pi)^-(d+1) / sqrt12`. + +**The `k`-integrals** are incomplete gamma functions, + + int_{k0}^{k1} k^q exp(-12 k^2) dk = (1/2) 12^(-(q+1)/2) [gamma((q+1)/2, 12 k1^2) - gamma((q+1)/2, 12 k0^2)], + +with the powers `x^a` from exact integer roots and the series closed by a geometric tail. + +**The `b`-integrals** use the composite midpoint rule on `4096` cells: +- the error per cell is at most `h^3 |g''|_max / 24`; +- `|g''|` is bounded per block of `16` cells through `m`, `m'`, `m''`, all increasing in `b`; +- plane: `m' = 2 E[(b + sqrt2 z)_+]` and `m'' = 2 Phi(b/sqrt2) <= 2`; +- `d = 3`: `m_3' = 2 E[det tr 1]` and `m_3'' = 2 E[tr^2 1] + 4 E[det 1]` (contact moments). + +**Cross-checks** (exact, in the certificate): the enclosures of `c^ref_{B,K}` contain the values of Math-#197 +(`0.0023869380211529483`, Owen-T closed form) and Math-#205 (`0.00097438236602425`). + +## 7. The cap sweep and `C_{B,K}` + +The cap route needs `sup_{r <= r_*} Q^W(G_r^c)/r^3` on boxes that resolve `k` near `1/2`. There the weight +`A_0^ref / k^(5/3) ∝ k^(1/3) e^(-12 k^2)` concentrates, and `c_G` grows like `k^3`. + +The engines of [E] and [E3] (`box_bound`, with their first parameter pair `(X_0, theta)`) are run on: +- every band of [E] (`153`) and of [E3] (`121`); +- `56` boxes per band: `b`-boxes `[i/4, (i+1)/4]`, and `k`-boxes of width `1/32` on `[1/2, 3/4]`, width `1/16` on `[3/4, 1]`, + then `[1, 3/2]` and `[3/2, 2]`. + +The certified per-box bounds are stored to four decimals, rounded up. For each `r_*`, `C_box(r_*)` is the maximum over the bands +with `r_1 <= r_*`, and + + C_{B,K}(r_*) = (1 + a_up(r_*) r_*^2 + eps) sum_box C_box(r_*) |S^(d-1)| int_box A_0^ref / (3 k^(5/3)). + +**Reviewed disposition of `C_{B,K}` (C61, review 5386924911 at `b41eb28`).** C61 accepts the planar and `d = 3` weighted +cap engines at this caller, `(X_0, theta) = (1/64, 1/32)`. It then validates all fourteen published `C_{B,K}` and `C_121` +upper constants by a different, cellwise transfer. That transfer uses Math-#227's reviewed finite-radius rate (C56–C59) and +C60's direct contact comparison in place of the factor `(1 + a_up r_*^2 + eps)` above. The published constants dominate the +alternative bound. + +The cap-derived candidate-minus-selected-elder error estimates no longer rest on those unaccepted positive-radius parts; +C61 does not accept the older individual-density expansions or the selected-count bounds that also use `c'` and `eps'`. +C61's scope is +the compact window `b in [0, 1]`, `k in [1/2, 2]`, `L >= 10`, `d = 2, 3`, and the corrected parent set above. It gives +`0 <= nu_cand - nu_eld <= C_{B,K}(R) ell^(2/3)` and the cumulative `C_121(R) t^(5/3)` for `ell, t < min(r_pop, R)^3/2`. + +C61 does not claim: +- that every short bar is a candidate; +- an unrestricted difference rate; +- the shrinking multiple-witness node; +- C8. + +## 8. Arithmetic + +- Exact rationals throughout. +- The Taylor models are fixed-point integers with explicit rounding. +- Interval enclosures of `exp`, `Phi` and the Gaussian and contact moments come from the engines. +- Fractional powers use integer `p`-th roots. +- Every stored number is rounded outward to a decimal and replayed exactly. + +## 9. Controls (floating point; not part of the certificate) + +**Monte Carlo of the genuine coefficient.** `theorem_b.py --controls` estimates `(A_r / A_0^ref - 1) / r^2` for the reference +kernel at `r = 1/8` and `1/16` and at the four corners of the band: +- the law at `r` comes from the exact series, which is valid at every `r` (the Taylor models stop at `1/64`); +- the pin ratio is exact; +- `Z_r / r^2` is sampled as `P1 P2 1{typed}` (seed `20261001`; `2 x 10^5` samples per point in the plane, `10^5` in `d = 3`). + +Values are shown with one standard error: + +| `d` | `r` | `(b, k) = (0, 1/2)` | `(0, 2)` | `(1, 1/2)` | `(1, 2)` | +|---|---|---|---|---|---| +| `2` | `1/8` | `0.08 ± 0.32` | `13.19 ± 0.39` | `0.64 ± 0.22` | `13.43 ± 0.26` | +| `2` | `1/16` | `0.16 ± 1.28` | `12.74 ± 1.34` | `1.09 ± 0.86` | `13.35 ± 0.90` | +| `3` | `1/8` | `2.24 ± 1.61` | `10.89 ± 1.75` | `0.84 ± 0.73` | `12.88 ± 0.87` | +| `3` | `1/16` | `-1.95 ± 5.71` | `17.29 ± 6.16` | `2.64 ± 2.99` | `17.45 ± 3.15` | + +The certified `a_up` at `R = 1/4096` on the four corner boxes (`b`- and `k`-width `1/8`): + +| `d` | `(0, 1/2)` | `(0, 2)` | `(1, 1/2)` | `(1, 2)` | +|---|---|---|---|---| +| `2` | `4.818760` | `13.822900` | `3.918911` | `13.742115` | +| `3` | `25.666574` | `16.876052` | `12.525134` | `14.762963` | + +What the comparison shows: +- **Plane, `k = 2`.** The coefficient is `13.2`–`13.4`, and `a_up = 13.823` is attained here, about `4%` above it. + - Most of `a_up` is the exact pin-density limit `12.75 r^2` (`0.75` from `det`, `12` from `e_kk`). + - The `Z`-part is small and negative in the Monte Carlo (about `-0.7`); the certificate bounds it above by `1.07`. +- **Plane, `k = 1/2`.** The coefficient is `0.1`–`1.1`, against box constants `3.9`–`4.8`. + - Here the `Z`-part is about `-0.8` to `-1.4`, while the certificate bounds it above by `2.0`–`2.9`. + - This region carries most of the weight of `c'`, so `c'_up` is several times the genuine weighted coefficient. +- **`d = 3`.** At `k = 2` the coefficient is about `11`–`17`, against box constants `14.8`–`16.9`. + - At `(0, 1/2)` the coefficient is about `0`–`2`, against the box constant `25.67`, which is the global `a_up`. + - There `23.5` of that constant is the `Z`-part. It consists mostly of the Hölder bounds of `r^2 E[D xi_3 1]` (`13.9`) and + `r^2 E[(v' A* v)(e' A* e) 1]` (`8.7`), relative to `36 k^2 m_3`. + +**Reference-kernel cap-route benchmarks** (section 0). In floating point, `Gamma_{B,K} = |S^(d-1)| int A_0^ref c_G / +(3 k^(5/3))` is: +- `Gamma_2 = 307.51`, with the planar `c_G = R(b) J(k)` of [G] (`J(k) = 2359296 k^3 (1 + delta(k))`); +- `Gamma_3 = 540.97`, with the `d = 3` ratios of [G3]. + +So `C_{B,K}(1/4096)` is `1.31 Gamma_2` and `2.23 Gamma_3`. These are comparisons with the reference-kernel benchmark, +not with the exact cap-route limit of a finite torus. + +## 10. What this does not do + +- **`d = 2, 3` and the band `B x K` only.** + - The method needs a compact `K` away from `0`; the shell bound uses `k >= 1/2`. + - The unrestricted densities of [LP] Theorem C are not addressed. There the constant term `B_{d,L}` (Math-#191, Math-#211) + comes from pairs this window excludes. +- **No vanishing-remainder rate on the torus.** The remainders `eps_d` and `eps'_d` depend on neither `r` nor `ell`. + Theorem L and Corollary L1 are expansions about the reference contact integrand and constant. They give no rate for + `A_r -> A_0`, nor for `ell^(1/3) nu -> c_{B,K}`, on the torus. +- **The Kac–Rice interface is consumed, not reviewed.** [LP] sections 8–12 are used as statements. +- **The cap route only for `C_{B,K}`.** + - `C_{B,K}` bounds `nu_cand - nu_eld` through `1 - p_r <= Q^W(G_r^c)`. + - By [G] the cap criterion overstates the pairing failure by `7.4 x 10^4` at `k = 1/2`, so the true difference coefficient + is far smaller. + - No lower bound for it is given. +- **Precision** (section 9). + - In the plane `a_up` is within about `4%` at `k = 2`, where it is attained. Near `k = 1/2`, where most of the weight of + `c'` lies, the box constants exceed the genuine coefficient by about `3`–`5`. + - In `d = 3`, `a_up` and `a_dn` are attained near `(b, k) = (0, 1/2)`, where the genuine coefficient is about `0`–`2`. The + excess comes from Hölder bounds of `r^2 E[D xi_3 1]` and `r^2 E[(v' A* v)(e' A* e) 1]`. These are large relative to + `36 k^2 m_3` at small `k`, and conditional moments would sharpen them further. + - At `k = 2` the `d = 3` box constants (`14.8`–`16.9`) are comparable to the Monte Carlo values (`11`–`17`). +- **Not a review** of [LP], [CAP], [E] or [E3]. The engines are used as certified box-bound and law functions; their + correctness is the subject of Math-#206 and Math-#212. +- **Status.** C8 stays OPEN. +- **Independence.** Same GitHub account as every lane; zero organizational-independence credit. The author will not merge. + +## 11. Verification + +- `python3 -B -S theorem_b.py --check --procs N` replays exactly: + - the Taylor-model laws and their recorded coefficients; + - the rate sweep (`96` boxes, seven `R`, both dimensions); + - the reference integrals and the cross-checks against Math-#197 and Math-#205; + - a sample of the cap bands (the band ending at each `r_*`, at `2^-20`, and the last band); + - the assembled tables. +- `--check` takes about 4 minutes on four cores. +- `--check-full` replays all `274` cap bands, in about 28 minutes on four cores. +- Mutants `no-om-term`, `no-bad-set`, `no-det`, `no-torus` and `cap-half` make the replay fail. +- The workflow checks: + - the manifest; + - the main-resident pins ([LP], [CAP], [E1], [E2], [REC]); + - the engine copies against `SOURCE_MAP.json`; + - `--check-full` and `--check` in the two interpreter modes; + - the mutants; + - a clean tree. + +## 12. Revisions + +**v1.5 (documentation only).** It records review C61 and the corrected parent set: +- [E1], [E2] and [REC] become `current_required` pins (§0), and §1 reads [LP] section 9 through [E2]. +- §7 records C61's disposition of `C_{B,K}`. +- Two clarifications in `theorem_b.py`: + - `wq_bound`'s docstring keeps the nonnegative diagonal surplus `H` (C58); + - a comment at the `Fae` line says that `(3/2) sb >= 1` is all the variance multiplier needs, and that it is not a claim + `Phi <= 3/4`, which is false at `mu = 1, sigma^2 = 2` (C60). +- The engine copies follow Math-#206 v1.2 and Math-#212 v1.1, which are docstring-only: `sqrt_up`'s bound is now `2^-198` + (C59-DOC-01), and `box_bound` now states its proved parameter domain `0 < x0 < 2/9`, `0 < theta < 1` (C61). + +In every changed Python file the AST with docstrings removed is unchanged, and `RESULTS.json` and every constant are unchanged. + +**v1.5.1 (wording; C63, 5943465973).** §7 is narrowed. C61's transfer covers the cap-derived candidate-minus-selected-elder +error estimates. It covers neither Corollary L1's individual-density expansions nor Corollary L2's selected-count bounds, +both of which also use `c'` and `eps'` from the full positive-radius Theorem L. diff --git a/frontiers/c8_lifetime_difference_constant_20261001/README.md b/frontiers/c8_lifetime_difference_constant_20261001/README.md new file mode 100644 index 000000000..d14fafdfa --- /dev/null +++ b/frontiers/c8_lifetime_difference_constant_20261001/README.md @@ -0,0 +1,42 @@ +# Theorem B of [LP] with explicit constants (C8; `d = 2, 3`): the radial ledger to second order, two-sided lifetime densities, and `C_{B,K}` + +**Object:** `CL-C8-LIFETIME-DIFFERENCE-CONSTANT-20261001-v1`. **Author lane:** Anthropic / Claude. **Kind:** author-side theorem +with explicit constants, certified by exact rational interval arithmetic (standard library only). **Scientific effect:** NONE: no +register, catalog, GRAPH or STATUS change, and catalog entry C8 stays OPEN. + +**Setting.** The compact-window lifetime densities of [LP] Theorem B on the band `b in [0, 1]`, `k in [1/2, 2]`, uniformly over +frames and torus sides `L >= 10` (and the reference kernel), through the radial ledger `A_r = 12 pi_r(R; v_r) Z_r / r^2` of [LP] +(10.2) and the pushforward (11.2), which are consumed as statements. + +**Results** (`NOTE.md` Theorem L and Corollaries L1, L2): +- **The ledger to second order.** For `0 < r <= R`, `|A_r / A_0^ref - 1| <= a(R) r^2 + eps` with the closed-form contact + integrand `A_0^ref`; `eps` is a radius-independent remainder (torus, truncation and rounding), so this is not a rate for + `A_r -> A_0` on the torus. In the plane `a_up(1/4096) = 13.823`, `a_dn = 3.638`, `eps <= 4.0 x 10^-10`; in `d = 3` + `a_up(1/4096) = 25.67`, `a_dn = 25.52`, `eps <= 1.8 x 10^-8`. [LP] assigns no rate to `A_r -> A_0`. +- **Densities to second order.** For `ell < min(r_pop, R)^3 / 2`, + `|ell^(1/3) nu_cand(ell) / c^ref_{B,K} - 1| <= 6.29 ell^(2/3) + 5 x 10^-11` in the plane and + `<= 23.2 ell^(2/3) + 3 x 10^-9` in `d = 3` (`R = 1/4096` constants). The same holds for `nu_eld` up to + `C_{B,K} ell / c^ref_{B,K}`. So both densities are `c^ref_{B,K} ell^(-1/3) + O(ell^(1/3))` up to an additive + `eps'_d ell^(-1/3)` that does not vanish as `ell -> 0` (`eps'_d <= 1.2 x 10^-13` in the plane, `2.5 x 10^-12` in + `d = 3`). For the torus constant `c_{B,K}` this gives `|c_{B,K} - c^ref_{B,K}| <= eps'_d`, not a vanishing-remainder + expansion. +- **The difference constant.** `0 <= nu_cand - nu_eld <= C_{B,K}(r_*) ell^(2/3)` for `ell < min(r_pop, r_*)^3 / 2`. + - `C_{B,K}(1/4096) = 402.53` in the plane and `1205.71` in `d = 3`. + - At `r_* = 1/512` the values are `428.02` and `2488.07` (rounded up). + - With these, (12.1) holds with `(3/5) C_{B,K}` and (12.3) with `C_{B,K} / (q + 5/3)`. + - [LP] gives (1.2) with no value. +- `c^ref_{B,K}` is certified in both dimensions; the enclosures contain the values of Math-#197 and Math-#205. + +**Files.** +- `NOTE.md`: statement; the ledger and what is consumed; Lemma 1 (for the Gaussian kernel the transverse Hessian at `M` is + exactly contact-distributed at every `r`); the Taylor-model laws; the pin-density ratio; the exact expansion of `P1 P2` and + the first-order cancellation; the thin typed shell; reference integrals; the cap sweep; controls; non-claims. +- `theorem_b.py`: the certificate. `--check --procs N` replays the Taylor models, the rate sweep, the reference integrals, a + sample of cap bands and the tables exactly against `RESULTS.json`; `--check-full` replays every cap band; + `--mutant {no-om-term, no-bad-set, no-det, no-torus, cap-half}` must fail; `--controls` prints floating controls. +- `engine_e2.py`, `engine_e3.py`: byte-identical copies of the certificates of Math-#206 (`0cb8ad3`) and Math-#212 + (`c4b8ec3`), used as libraries (band laws, contact moments, box bounds); pinned in `SOURCE_MAP.json`. +- `RESULTS.json`, `SOURCE_MAP.json`, `SOURCE_FILES.json`. +- Workflow `.github/workflows/c8-lifetime-difference-constant.yml`. + +Same GitHub account as every other lane; zero organizational-independence credit. 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[REC], and the read-only comparison [G3]) are verified on the tree by the workflow; packet_copy sources ([E], [E3]) are verified against the packet's byte-identical engine copies; the remaining sources ([W], [W3], [G]) are read-only cross-checks and comparisons pinned for provenance", + "sources": [ + { + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "id": "LP", + "commit": "3e0a91b69892b9717564c592e7a966492a81566a", + "current_required": true, + "on_main": true, + "role": "section 1: the model in every dimension and Theorem B with (1.2); section 3: the preconditioned pins (3.1)-(3.3); section 8: 1 - p_r <= Q^W(G_r^c); sections 9-11: the weighted Kac-Rice interface, the radial ledger (10.1)-(10.2) with A_r = 12 pi_r Z_r/r^2, and the lifetime pushforward (11.1)-(11.3); section 12: (12.1) and (12.3). Sections 8-12 are consumed as statements; section 11 records that the convergence A_r -> A_0 was not assigned a numerical rate, and section 16 lists numerical constants among the items not supplied" + }, + { + "path": "imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md", + "bytes": 15160, + "sha256": "0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc", + "git_blob": "0633aca3c2a2882b0de4399da0a75d64c2e6b2e1", + "id": "CAP", + "commit": "3e0a91b69892b9717564c592e7a966492a81566a", + "current_required": true, + "on_main": true, + "role": "section 1: the cylinder D, the partial-block norms M_3, M_4 and condition (1) defining G_r; consumed through the box-bound engines of Math-#206 and Math-#212" + }, + { + "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] (corrected parent reading set P + CAP + E1 + E2 + REC, per REC section 1): the preconditioning congruence is diag(r^(-1/2), I)" + }, + { + "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 (not necessarily continuous) elder indicator, by equality of finite measures on a generating cylinder algebra and a monotone-class extension; it 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; joint convergence in law; the embedding condition r < L/(4 sqrt 2), which r <= 1/64 and L >= 10 satisfy" + }, + { + "path": "frontiers/c8_theorem_a_constants_planar_20260930/theorem_a.py", + "bytes": 52864, + "sha256": "c5ea8bf87c54761866315f28f6c93c6ce5c1e0f99d47c5a03f14d095d06871cc", + "git_blob": "ae279e9015483a666ece8deb2544cbc458a5e270", + "id": "E", + "commit": "13f932c2f6ec07d8c4c9996bd030ab22eaa32108", + "pull_request": 206, + "packet_copy": "engine_e2.py", + "current_required": false, + "on_main": false, + "role": "the planar certificate of Theorem E: exact covariance series and torus remainder, interval band laws, box_bound for sup Q^W(G_r^c)/r^3 on (b, k) boxes, band list; carried byte for byte as engine_e2.py and used as a library" + }, + { + "path": "frontiers/c8_theorem_a_constants_d3_20261001/theorem_a3.py", + "bytes": 76209, + "sha256": "310d4ab3d3a38a377d753753f3b814591a9090a947f87d6843ab7389389572d3", + "git_blob": "267890232119b35206be7a8cc74d730de9a68644", + "id": "E3", + "commit": "250b90722e8c47adf8f7e7c41a61e8a8085d9741", + "pull_request": 212, + "packet_copy": "engine_e3.py", + "current_required": false, + "on_main": false, + "role": "the d = 3 certificate of Theorem E3: exact 3D covariance series, band laws with the regression on the transverse Hessian, contact moments E[det^a tr^c 1{B > 0}], box_bound, band list; carried byte for byte as engine_e3.py" + }, + { + "path": "frontiers/c8_window_coefficient_planar_20260930/NOTE.md", + "bytes": 23551, + "sha256": "ebdda42ce51cd43a364cbd303abf314d4592042b7391f43d2fc0ebb736a6d8fa", + "git_blob": "4bd311630eed0d5f7f23a5c7b73b4f56d17664e3", + "id": "W", + "commit": "3dcf612df0dfac4e6d6f0daa075d88e9fae0ebb4", + "pull_request": 197, + "current_required": false, + "on_main": false, + "role": "read-only cross-check: the reference planar c_{B,K} = 0.0023869380211529483 on B x K (Owen-T closed form), which the certified enclosure of c^ref_{B,K} must contain; Math-#197 has since merged at e08a50d with NOTE revisions v1.2-v1.2.1 (blob 8d73ae38), RESULTS.json unchanged, and the cross-check value is unchanged" + }, + { + "path": "frontiers/c8_window_coefficient_d3_torus_20260930/NOTE.md", + "bytes": 17401, + "sha256": "67839353382c8011627cdad08b96ac446fbc4ff01d1e5aff39ab326d8e32d071", + "git_blob": "380bbc7aa3a16f5c8497bd202c51dcd85963ae4a", + "id": "W3", + "commit": "c0d8e925955d7a23c58221006510f804ef3631fe", + "pull_request": 205, + "current_required": false, + "on_main": false, + "role": "read-only cross-check: the reference d = 3 c_{B,K} = 0.00097438236602425 on B x K; Math-#205 has since merged at 044d42d with a wording-only NOTE revision (blob 549dc7cc), and the cross-check value is unchanged" + }, + { + "path": "frontiers/c8_cap_failure_coefficient_planar_20260930/NOTE.md", + "bytes": 28432, + "sha256": "0cb75475d9ee387a4045ff01ab9edb43a73c8c4edb6fc78d30518ced5bb02c5a", + "git_blob": "0b64f1c2828dfae7e6c104bac044004fc72c221d", + "id": "G", + "commit": "a388580eee356a88ce31aeed3999ab3fdc302312", + "pull_request": 203, + "current_required": false, + "on_main": false, + "role": "read-only comparison: c_G = R(b) J(k) with J(k) = 2359296 k^3 (1 + delta(k)), for the floating reference-kernel cap-route benchmark of C_{B,K}" + }, + { + "path": "frontiers/c8_cap_failure_coefficient_dims_20261001/RESULTS.json", + "bytes": 6837, + "sha256": "1406d5138ac3d33bcb0358f6a3005a67e3a4a2dbfc2c43bc3ac2593667550608", + "git_blob": "e9fd6f377d1fd9dca9d09cc884d2bad2422e8b17", + "id": "G3", + "commit": "da483732664d128728950dcad6d4fdbe5181f7cd", + "pull_request": 208, + "current_required": true, + "on_main": true, + "role": "read-only comparison: the floating R_3(b) R(b) values and the J^(3)/J ratios, for the floating d = 3 reference-kernel cap-route benchmark; Math-#208 merged at b55b8f3 with this RESULTS.json byte-identical, so the pin is checked on the tree" + } + ] +} diff --git a/frontiers/c8_lifetime_difference_constant_20261001/engine_e2.py b/frontiers/c8_lifetime_difference_constant_20261001/engine_e2.py new file mode 100644 index 000000000..ae279e901 --- /dev/null +++ b/frontiers/c8_lifetime_difference_constant_20261001/engine_e2.py @@ -0,0 +1,1249 @@ +#!/usr/bin/env python3 +"""Explicit constants for Theorem A of [LP] in the plane (d = 2), 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) <= C(r_*) r^3, + +with G_r the good event of [LP] section 7 / [CAP] (1) and C(r_*) certified here (NOTE.md Theorem E); the same at the single +fixed-axis SIDE24 point (b, k) = (6/5, 1/6) of [CAP] section 1 (Theorem E'). 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 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) + 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). 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]) + 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_lifetime_difference_constant_20261001/engine_e3.py b/frontiers/c8_lifetime_difference_constant_20261001/engine_e3.py new file mode 100644 index 000000000..267890232 --- /dev/null +++ b/frontiers/c8_lifetime_difference_constant_20261001/engine_e3.py @@ -0,0 +1,1714 @@ +#!/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 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) + 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). 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]) + 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_lifetime_difference_constant_20261001/theorem_b.py b/frontiers/c8_lifetime_difference_constant_20261001/theorem_b.py new file mode 100644 index 000000000..36b39dbd3 --- /dev/null +++ b/frontiers/c8_lifetime_difference_constant_20261001/theorem_b.py @@ -0,0 +1,1166 @@ +#!/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). 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) + 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 + # 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] + 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()