diff --git a/.github/workflows/c6-remote-coefficient-numerics.yml b/.github/workflows/c6-remote-coefficient-numerics.yml new file mode 100644 index 000000000..fdb6f61ce --- /dev/null +++ b/.github/workflows/c6-remote-coefficient-numerics.yml @@ -0,0 +1,67 @@ +name: C6 remote coefficient numerics (replay) + +'on': + pull_request: + paths: + - 'frontiers/c6_remote_coefficient_numerics_20260930/**' + - '.github/workflows/c6-remote-coefficient-numerics.yml' + - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md' + - 'frontiers/remote_window_20260924/PROOF.md' + - 'frontiers/c6_cluster_law_20260929/PROOF.md' + - 'frontiers/remote_collision_20260928/PROOF.md' + - 'frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-controls-and-replay: + runs-on: ubuntu-latest + timeout-minutes: 20 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + persist-credentials: false + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - name: Verify manifest and main-resident pins, replay exact controls and sample kernels, reject mutants + env: + PYTHONDONTWRITEBYTECODE: '1' + run: | + set -euo pipefail + python -B -S - <<'PY' + import hashlib, json, pathlib, subprocess, sys + root = pathlib.Path('frontiers/c6_remote_coefficient_numerics_20260930').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']) + pins = json.loads((root / 'SOURCE_MAP.json').read_text()) + for entry in pins['sources']: + if not entry['current_required']: + continue + data = pathlib.Path(entry['path']).read_bytes() + blob = hashlib.sha1(b'blob %d\0' % len(data) + data).hexdigest() + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or blob != entry['git_blob']: + raise ValueError('consumed source identity mismatch on this tree: ' + entry['path']) + script = str(root / 'remote.py') + for flags in (['-B', '-S'], ['-B', '-O', '-S']): + out = subprocess.run([sys.executable, *flags, script, '--check'], cwd=root, capture_output=True, timeout=900) + if out.returncode != 0: + raise RuntimeError('check failed under ' + ' '.join(flags) + ': ' + out.stdout.decode()[-400:]) + for m in ['hermite-sign', 'det-abs', 'weight-indicator']: + rc = subprocess.run([sys.executable, *flags, script, '--check', '--mutant', m], cwd=root, capture_output=True, timeout=900).returncode + if rc != 1: + raise RuntimeError('mutant ' + m + ' not rejected') + print(json.dumps({'scientific_effect': 'NONE', 'certified': False})) + PY + git diff --exit-code diff --git a/.github/workflows/extreme-cluster-sampling.yml b/.github/workflows/extreme-cluster-sampling.yml new file mode 100644 index 000000000..d47c6fcba --- /dev/null +++ b/.github/workflows/extreme-cluster-sampling.yml @@ -0,0 +1,38 @@ +name: Extreme whole-cluster sampling checks + +'on': + pull_request: + paths: + - 'frontiers/extreme_cluster_sampling_20260930/**' + - '.github/workflows/extreme-cluster-sampling.yml' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-and-source: + runs-on: ubuntu-latest + timeout-minutes: 10 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + persist-credentials: false + fetch-depth: 0 + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - name: Fetch exact public source objects + run: | + set -euo pipefail + git fetch --no-tags origin 80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337 + git fetch --no-tags origin 382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf + git fetch --no-tags origin bfcc67dc8957b3ed1fbb789e5a0305e98d7359c8 + - name: Verify historical sources, packet and exact controls + env: + PYTHONDONTWRITEBYTECODE: '1' + GIT_NO_REPLACE_OBJECTS: '1' + run: | + set -euo pipefail + python -B -S frontiers/extreme_cluster_sampling_20260930/verify.py + git diff --exit-code diff --git a/.github/workflows/fixed-r-inverse-lifetime.yml b/.github/workflows/fixed-r-inverse-lifetime.yml new file mode 100644 index 000000000..9216a6e52 --- /dev/null +++ b/.github/workflows/fixed-r-inverse-lifetime.yml @@ -0,0 +1,36 @@ +name: Fixed-r inverse-lifetime checks + +'on': + pull_request: + paths: + - 'frontiers/fixed_r_inverse_lifetime_20260930/**' + - '.github/workflows/fixed-r-inverse-lifetime.yml' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-and-source: + runs-on: ubuntu-latest + timeout-minutes: 10 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + persist-credentials: false + fetch-depth: 0 + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - name: Fetch exact public parent model + run: | + set -euo pipefail + git fetch --no-tags origin 7a1cb09a9d58d179393e6d29146252f714b9c9bf + - name: Verify source, packet and both Python modes + env: + PYTHONDONTWRITEBYTECODE: '1' + GIT_NO_REPLACE_OBJECTS: '1' + run: | + set -euo pipefail + python -B -S frontiers/fixed_r_inverse_lifetime_20260930/verify.py + git diff --exit-code diff --git a/.github/workflows/radial-rate.yml b/.github/workflows/radial-rate.yml new file mode 100644 index 000000000..fa1a9abe5 --- /dev/null +++ b/.github/workflows/radial-rate.yml @@ -0,0 +1,38 @@ +name: Microscopic radial-rate checks + +'on': + pull_request: + paths: + - 'frontiers/radial_rate_20260930/**' + - '.github/workflows/radial-rate.yml' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-and-source: + runs-on: ubuntu-latest + timeout-minutes: 10 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + persist-credentials: false + fetch-depth: 0 + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - name: Fetch exact public source objects + run: | + set -euo pipefail + git fetch --no-tags origin 80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337 + git fetch --no-tags origin 382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf + git fetch --no-tags origin 7a1cb09a9d58d179393e6d29146252f714b9c9bf + - name: Verify sources, packet and both Python modes + env: + PYTHONDONTWRITEBYTECODE: '1' + GIT_NO_REPLACE_OBJECTS: '1' + run: | + set -euo pipefail + python -B -S frontiers/radial_rate_20260930/verify.py + git diff --exit-code diff --git a/frontiers/c6_remote_coefficient_numerics_20260930/NOTE.md b/frontiers/c6_remote_coefficient_numerics_20260930/NOTE.md new file mode 100644 index 000000000..47fc0fedf --- /dev/null +++ b/frontiers/c6_remote_coefficient_numerics_20260930/NOTE.md @@ -0,0 +1,234 @@ +# Numerical values of the remote singleton coefficient `k integral_X Lambda` (planar, parent kernel) + +**Object:** `CL-C6-REMOTE-COEFF-NUMERICS-20260930-v1`. **Author lane:** Anthropic / Claude. **Kind:** numerical +note. **Scientific effect:** NONE. **Certified:** no (floating-point quadrature with convergence checks; not an +enclosure). Companion of the near-coefficient note `frontiers/c6_cluster_coefficients_numerics_20260930` +(Math-#168), which evaluated `nu_near(1) = alpha_1` and `nu(2) = alpha_2` and listed the remote part of `nu(1)` as +not evaluated. This note evaluates that part. + +## 1. What is evaluated + +[CL] (1.4) writes the one-point cluster coefficient as + + nu(1) = nu_near(1) + k integral_X Lambda(x; b, k, u) dx, Lambda = sum_j Lambda_j, + +with `Lambda_j` the contact kernel of [RM] (13): + + Lambda_j(x) = p_(Y_x | U_0 = v_0)(0, b) / z_0 . E[ w_0 F_j(H_x) | U_0 = v_0, Y_x = (0, b) ], + +`U_0 = (f, f_x, f_z, f_xx, f_xz, f_xxx)(0) = (b, 0, 0, 0, 0, 12k)` the contact observation, `A_0 = f_zz(0)` the soft +curvature, `w_0 = (6k)^2 A_0^2 1{A_0 < 0}` ([RM] (10) at `d = 2`), `z_0 = E[w_0 | U_0 = v_0] = 36 k^2 m_(2,b)` +([RM] (11), [LP] (5.4)) with `m_(2,b) = E[A_0^2 1{A_0 < 0} | U_0]` taken under the same kernel as the numerator +(continuum: `A_0 | U_0 ~ N(-b, 2)` exactly; `L = 6`: the periodized conditional law, `m_(2,0) = 1.0000175`), `Y_x = (f_x, f_z, f)(x)`, `H_x` the Hessian at `x`, `F_j(H) = |det H| 1{index H = j}`. The +field is the centered unit-variance Gaussian field on `X = R^2/(L Z^2)` with the exact kernel of [LP] section 2, +`K_L(z) = sum_n exp(-|z + Ln|^2/2) / sum_n exp(-|Ln|^2/2)`. Orientation: the continuum kernel is isotropic, so +continuum quantities (`Lambda_inf`, `c_hole`, sample kernels, profiles) hold for every pin direction `u`; the +square torus is not rotation-invariant, and every `L`-periodized value in this note is computed with the pin +direction `u = e_1` aligned with a lattice axis (the image lattice is written in pin-frame coordinates). No other +orientation is computed. Everything is a finite-dimensional Gaussian computation: the 13-vector of jets (seven at `0`, six at `x`) has covariance +`Cov(d^alpha f(s), d^beta f(t)) = (-1)^|alpha| (d^(alpha+beta) K_L)(t - s)`, with +`d^a_x d^c_z exp(-|z|^2/2) = (-1)^(a+c) He_a(x_1) He_c(z) exp(-|z|^2/2)` (probabilists' Hermite polynomials). + +[CL] section 5.1 shows `k integral_X Lambda = lim_(rho -> 0) k integral_(D_rho) Lambda` exists, with +`k integral_(|x| < rho) Lambda <= C rho^2`; so `Lambda` is integrable at the pin and the torus integral of the +kernel is the right object. A direct scaling check is consistent with this: at a critical point `x` of the contact +field at level `b` with `|x| = eps` small, the Euler identity `x . grad f = 3 C + A_0 z^2 + 4 q + ...` for the cubic +part `C`, the soft quadratic `A_0 z^2 / 2` and the quartic part `q` forces `A_0 z^2 = O(eps^4)`, i.e. `A_0 = O(eps^2)`; +the density of `Y_x` at `(0, b)` is `O(eps^(-6))` while `w_0 |det H_x| = O(eps^4 . eps^2)`, so `Lambda = O(1)` along +such a cone. (A first guess `A_0 = O(eps)` would give `Lambda ~ eps^(-2)` and a logarithmically divergent integral; +it is wrong exactly because of the Euler identity.) Two qualifications: the division by `z^2` is legitimate only on +cones `|z| >= c |x|`, so this is a scaling check on such cones and gives no pointwise bound towards the axis +(where the computed kernel is in any case exponentially small, section 4); and the integrability of `Lambda` at the +pin used by this note rests on [CL] section 5.1's integral bound `k integral_(|x| < rho) Lambda <= C rho^2`, not on +this heuristic. No uniform pointwise `Lambda = O(1)` is asserted. The computed values are bounded and, in fact, +`Lambda(x) -> 0` as `x -> 0` in every sampled direction (section 4). + +## 2. Exact structure + +**Far field.** As `|x| -> infinity` the two sites decouple: `p_(Y_x | U_0)(0, b) -> phi(b)/(2 pi)` (`f`, `f_x`, +`f_z` independent with unit variance), `E[w_0 | U_0] / z_0 -> 1`, and `H_x | (f = b, grad f = 0)` has `h_1 = f_xx`, +`h_3 = f_zz ~ N(-b, 2)` independent (`Var f_xx = 3`, `Cov(f_xx, f_zz) = 1`, `Cov(f, f_xx) = -1`, regression on +`f = b` removes the common part), `h_2 = f_xz ~ N(0, 1)`. Hence + + Lambda_inf,j(b) = phi(b)/(2 pi) . E[ F_j(H) | f = b, grad f = 0 ], + +the ordinary height-resolved density of index-`j` critical points at level `b`, per unit height and area. It is +`k`-free; the `k`-dependence of `Lambda(x)` enters only through the conditioning on `f_xxx(0) = 12k` and dies with +the correlations. It is also closed-form: with `h_1 = a + c`, `h_3 = a - c` (`a ~ N(-b, 1)`, `c ~ N(0, 1)`, `h_2 ~ +N(0, 1)` independent) `det H = a^2 - rho^2`, `rho^2 ~ Exp(mean 2)`, and + + E F_1 = sqrt2 e^(-b^2/4), + E F_2 = (b^2+1) Phi(b) + b phi(b) - 2 Phi(b) + sqrt2 e^(-b^2/4) Phi(b/sqrt2), E F_0 = E F_2 |_(b -> -b), + E|det H_b| = b^2 - 1 + 2 sqrt2 e^(-b^2/4), Lambda_inf(b) = phi(b) E|det H_b| / (2 pi), + +so `Lambda_inf(0) = (2 sqrt2 - 1)/(2 pi)^(3/2) = 0.1160934862...`, and `integral Lambda_inf(b) db = 2/(pi sqrt3)`, the +total critical-point density. The quadrature reproduces these closed forms to `9e-13` (absolute), a +second exact end-to-end control of the Hessian integrator with a non-centered mean. + +**Exact control of the Hessian integrator.** For the unconditional planar Hessian (`Var h_1 = Var h_3 = 3`, +`Cov(h_1, h_3) = 1`, `Var h_2 = 1`) write `h_1 = a + c`, `h_3 = a - c` with `a ~ N(0, 2)`, `c ~ N(0, 1)` independent; +then `det H = a^2 - (c^2 + h_2^2) = a^2 - rho^2` with `rho^2 ~ Exp(mean 2)`, so +`E|det H| = 2 E[(rho^2 - a^2)_+] = 4 E exp(-a^2/2) = 4/sqrt3`, `E F_1 = 2/sqrt3`, `E F_0 = E F_2 = 1/sqrt3`, +`E det H = 0`. (Total critical-point density `2/(pi sqrt3) = 0.3676`, maxima `1/(2 pi sqrt3) = 0.0919`: the +Longuet-Higgins values for this spectrum.) The integrator reproduces these to `9.7e-11` (absolute). + +**Coordinates adapted to the cone `det H = 0`.** The integrals `E[A_0^2 1{A_0 < 0} |det H| 1{index j}]` over the +conditional Gaussian law of `(A_0, H_x)` are computed with `A_0` integrated out in closed form +(`E[A^2 1{A < 0}] = (m^2 + s^2) Phi(-m/s) - m s phi(m/s)` for `A | H ~ N(m, s^2)`, `m` affine in `H`) and the +three Hessian coordinates written as `a = tr H / 2` and polar `(rho, theta)` in the traceless plane `(c, h_2)`. +Then `det H = a^2 - rho^2`, the kink of `|det H|` is the coordinate surface `rho = |a|`, and the index is read off +from `rho < |a|` (definite; sign of `a` separates minimum and maximum) or `rho > |a|` (saddle). The `a`-range is +split at `0`, the `rho`-range at `|a|`, the `theta` nodes are those of the periodic trapezoid rule under an +analytic map concentrating them at the angular position of the (generally offset and anisotropic) conditional +mean. Apart from two truncations, each piece of the integrand is then smooth: the Gaussian ranges are cut at +`cut = 9` (the `a`-range at `cut` standard deviations; the `rho`-range endpoint `rho_max = |mean| + cut . max +marginal sd` of the transverse pair, which the `|a|` split can cross; and a Mahalanobis cut `q > 2 cut^2` inside +the `rho`-loop). These are numerical truncations whose omitted weighted mass depends on the actual conditional +covariance, mean and normalization; no universal relative size is claimed. What can be said in general is that +the transverse cap sits at standardized distance at least `cut/sqrt2 = 6.4` (the largest eigenvalue of a `2 x 2` +covariance is at most twice its largest marginal variance), so the omitted Gaussian mass is at most of order +`e^(-20)` times polynomial factors, and the `a`-cut and the Mahalanobis cut are at `9` and `sqrt2 . 9` +standard deviations; an absolute bound for the omitted Gaussian domain of the exact integral is the Maxwell-tail +majorant of the OpenAI lane (Math-#174 comment 5907931352, `3e-16` for the `b = 0` control). No +geometric-convergence theorem is claimed for the implemented rule; what is claimed is the observed behaviour on +the exact controls (`E|det H| = 4/sqrt3` to `1e-10` at the default order and `3e-15` at the higher order; the +closed-form far field to `1e-12`) and the order comparisons of section 5. + +**Ill-conditioned points.** Near the pin the conditional law of `Y_x` given `U_0` has variances down to `|x|^6`, +and the double-precision regression can lose positivity, either in the Hessian block (near the axis, where +`Lambda` is below `10^(-20)`) or in the Schur complement `Var(A_0 | H_x)` (near the pin in the transverse sector, +where `Lambda` is small but not negligible: at `r = 0.05` transverse a clamped `Var(A_0 | H_x)` would misreport +`Lambda` by a factor ten). The full `4 x 4` conditional covariance of `(A_0, H_x)` is therefore Cholesky-factored, +and any nonpositive pivot sends the point (`54` in the whole run) to a 50-digit decimal regression. + +## 3. Method + +`Lambda_j(x)` is evaluated on midpoint grids (spacing `h`, reflection symmetry `z -> -z` used) and integrated by +the midpoint rule: + +- continuum kernel, hole constant `c_hole,j = integral_(R^2) (Lambda_j - Lambda_inf,j) dx` on `[-6, 6]^2` + (`h = 0.15`). The sixth-order jet correlations decay only like `He_6(|x|) e^(-|x|^2/2)` (`0.32` at distance 4, + `4.4e-4` at 6), so the cutoff is checked directly: `Lambda/Lambda_inf - 1` is `3.6e-5` (axis), `3.1e-6` (45 deg), + `4.4e-7` (transverse) at `|x| = 5`, and `4.8e-9`, `4.4e-10`, `3.8e-11` at `|x| = 6`; from these sampled + profiles the neglected exterior is estimated at less than `10^(-7)` on `c_hole` (an estimate from the sampled + rays, not an integrated tail bound); +- periodized kernel at `L = 6`, direct integral over the fundamental domain `[-3, 3]^2` (`h = 0.15`); +- periodized kernel at `L = 12`, direct integral (`h = 0.2`), compared with `L^2 Lambda_inf + c_hole`. + +Then `integral_X Lambda ~ L^2 Lambda_inf + c_hole` for `L = 12, 24` (the periodization correction being +`O(e^(-18))` at `L >= 12`), and the remote singleton mass is `k` times that. + +## 4. Values + +**Far field** (`k`-free), per unit height and area: + +| `b` | `Lambda_inf` | minima `j = 0` | saddles `j = 1` | maxima `j = 2` | `p_inf = phi(b)/(2 pi)` | +|---|---|---|---|---|---| +| `0.0` | 0.116093 | 0.01315 | 0.0897936 | 0.01315 | 0.0634936 | +| `1.0` | 0.0848309 | 0.000850618 | 0.0424155 | 0.0415649 | 0.0385108 | + +(At `b = 0`: `E|det H_0| = 1.82843` with `h_1, h_3 ~ N(0, 2)`, `h_2 ~ N(0, 1)`.) + +**Correlation hole** `c_hole = integral_(R^2) (Lambda - Lambda_inf) dx` (dimension of an area times a density +per height; negative means depletion): + +| `k` | `b` | `c_hole` | `j = 0` | `j = 1` | `j = 2` | `c_hole / Lambda_inf` (effective depleted area) | +|---|---|---|---|---|---|---| +| `0.5` | `0.0` | -1.08414 | -0.302824 | -0.88442 | 0.103108 | -9.338 | +| `1.0` | `0.0` | -1.63115 | -0.313786 | -1.3325 | 0.0151367 | -14.05 | +| `2.0` | `0.0` | -2.1382 | -0.330282 | -1.75684 | -0.0510768 | -18.42 | +| `0.5` | `1.0` | -0.750629 | -0.00830817 | -0.260797 | -0.481524 | -8.849 | +| `1.0` | `1.0` | -1.18448 | -0.0055866 | -0.464948 | -0.713947 | -13.96 | +| `2.0` | `1.0` | -1.56814 | -0.00268732 | -0.636506 | -0.928945 | -18.49 | + +The pinned pair depletes its neighbourhood: along the pin axis the contact field is `b + 2k x_1^3 + ...` to leading +order, a monotone ramp, so window-level critical points near the pins live in the transverse sector only +(`Lambda(x)/Lambda_inf` at `k = 1, b = 0`: on the axis `2e-22` at `|x_1| = 1.5`, `1.6e-6` at `2`, `0.037` at +`2.5`, `0.68` at `3`, `1.02` at `4`; transversally `0.21` at `|z| = 1`, `0.68` at `2`, `1.00` at `3`; the ray +profiles at angles `0, 30, 45, 60, 90` degrees are in `RESULTS.json`, `ray_profile`). Near the pin `Lambda -> 0` +(`near_zero_profile`: `Lambda(0.05, 60 deg) = 0.001`, `Lambda(0.05, 90 deg) = 0.000104`, `Lambda(0.4, 90 deg) = 0.0079` at `k = 1, b = 0`; on the +axis and at `30` degrees it is below `10^(-20)`). + +**Remote singleton mass** `R(k, b, L) = k integral_X Lambda dx` and the assembled `nu(1)`, with `alpha_1` from the +companion note ([NUM] = Math-#168 `RESULTS.json`, merged, pinned; GH60 values; `alpha_2` likewise): + +| `k` | `b` | `L` | `integral_X Lambda` | `R = k integral_X Lambda` | `alpha_1` (#168) | `nu(1) = alpha_1 + R` | `nu(2) = alpha_2` | `nu(2)/nu(1)` | +|---|---|---|---|---|---|---|---|---| +| `0.5` | `0.0` | `6` (direct) | 3.05732 | 1.52866 | 1.11821 | 2.64687 | 0.0347454 | 0.0131 | +| `0.5` | `0.0` | `12` | 15.6333 | 7.81666 | 1.11821 | 8.93487 | 0.0347454 | 0.00389 | +| `0.5` | `0.0` | `24` | 65.7857 | 32.8929 | 1.11821 | 34.0111 | 0.0347454 | 0.00102 | +| `1.0` | `0.0` | `6` (direct) | 2.54206 | 2.54206 | 1.32906 | 3.87112 | 0.0255091 | 0.00659 | +| `1.0` | `0.0` | `12` | 15.0863 | 15.0863 | 1.32906 | 16.4154 | 0.0255091 | 0.00155 | +| `1.0` | `0.0` | `24` | 65.2387 | 65.2387 | 1.32906 | 66.5678 | 0.0255091 | 0.000383 | +| `2.0` | `0.0` | `6` (direct) | 2.17973 | 4.35945 | 2.1621 | 6.52155 | 0.0169281 | 0.0026 | +| `2.0` | `0.0` | `12` | 14.5793 | 29.1585 | 2.1621 | 31.3206 | 0.0169281 | 0.00054 | +| `2.0` | `0.0` | `24` | 64.7316 | 129.463 | 2.1621 | 131.625 | 0.0169281 | 0.000129 | +| `0.5` | `1.0` | `6` (direct) | 2.23698 | 1.11849 | 0.320155 | 1.43865 | 0.00994791 | 0.00691 | +| `0.5` | `1.0` | `12` | 11.465 | 5.73251 | 0.320155 | 6.05267 | 0.00994791 | 0.00164 | +| `0.5` | `1.0` | `24` | 48.112 | 24.056 | 0.320155 | 24.3762 | 0.00994791 | 0.000408 | +| `1.0` | `1.0` | `6` (direct) | 1.8121 | 1.8121 | 0.380523 | 2.19262 | 0.00730347 | 0.00333 | +| `1.0` | `1.0` | `12` | 11.0312 | 11.0312 | 0.380523 | 11.4117 | 0.00730347 | 0.00064 | +| `1.0` | `1.0` | `24` | 47.6781 | 47.6781 | 0.380523 | 48.0587 | 0.00730347 | 0.000152 | +| `2.0` | `1.0` | `6` (direct) | 1.53019 | 3.06037 | 0.619027 | 3.6794 | 0.00484668 | 0.00132 | +| `2.0` | `1.0` | `12` | 10.6475 | 21.295 | 0.619027 | 21.9141 | 0.00484668 | 0.000221 | +| `2.0` | `1.0` | `24` | 47.2945 | 94.589 | 0.619027 | 95.208 | 0.00484668 | 5.09e-05 | + +For `L = 6` the direct periodized integral and the approximation `36 Lambda_inf + c_hole` are both given +(`RESULTS.json`, `remote_total`); they differ by between `2.4e-3` and `6.4e-2` relative, most at `k = 2`. At +`L = 6` the depleted region (radius about `3`) fills the fundamental domain, so the depletions around the image +pins overlap and the additive approximation `L^2 Lambda_inf + c_hole` is not valid; the table uses the direct +periodized integral for `L = 6`. (The periodization also changes the one-site covariances by up to `8.8e-4` at +`L = 6`; that is included in the direct value.) For `L = 12` the direct integral at `k = 1, b = 0` is `15.0866` +against `15.0863` from the far field plus hole, so the additive form is accurate to `2e-5` there. No direct +`L = 24` integral was run: at `L = 24` the additive form is expected, not measured, to be at least as accurate. + +**Reading.** For the parent torus sizes the remote singletons dominate `nu(1)` (`R ~ L^2 Lambda_inf`: by a factor +`50` at `L = 24`, `k = 1, b = 0`), so the two-point cluster probability relative to a nonempty window is +`nu(2)/(nu(1) + nu(2)) ~ 3.83e-04` at `k = 1, b = 0, L = 24`, against the near-only share `0.019` reported in +#168. The near share is the right object for the cluster geometry; the global share is the right object for the +finite-torus count. The pins do not only deplete: along the `45`-`60` degree rays `Lambda/Lambda_inf` overshoots to +`1.24`-`1.48` at `|x| = 2`-`2.5` (`k = 1, b = 0`) before settling, so the displaced critical points pile up in a +ring; the net hole is nevertheless negative for every `(k, b)` computed, and grows with `k` (steeper ramp +`2k x_1^3`, stronger `f_xxx` conditioning) and shrinks with `b`. + +## 5. Precision (empirical, not bounds) + +- Quadrature: at the sample points the default orders `(24, 32, 24)` differ from `(32, 48, 32)` by at most `3e-9` + relative wherever `Lambda > 10^(-2) Lambda_inf`, by `5e-4` at `(0.15, 0.15)` where `Lambda = 2e-5 Lambda_inf`, and + by up to `4.0e-01` at points where `Lambda < 10^(-16) Lambda_inf` (the recorded maximum, carried entirely by such + points: the conditional law of `H_x` is extremely anisotropic there and the value is exponentially small anyway). + On the exact controls the default order is accurate to `9.7e-11` (unconditional Hessian) and + `9e-13` (far field, both `b`). The `h = 0.3` hole constant changes by `4e-08` between the + two orders. +- Grid: at `k = 1, b = 0` the hole constant is `-1.629661` (`h = 0.3`), `-1.631148` (`h = 0.15`), + `-1.631335` (`h = 0.1`). Measured: the tabulated `h = 0.15` value differs from the `h = 0.1` value by + `1.88e-4`. Model (NONRIGOROUS, this one parameter set only): a three-point power law `A(h) = A_0 + C h^p` + fitted to the non-nested spacings `0.3, 0.15, 0.1` gives `p = 2.63` (`Lambda` has a conical, + direction-dependent limit at the pin), `A_0 = -1.63143`, and model residuals `2.9e-4` at `h = 0.15` and + `9.8e-5` at `h = 0.1`; these are extrapolations, not error bounds, and nothing is inferred from them for the + other `(k, b)` rows. The `h = 0.3` run on `[-5, 5]^2` differs from the `[-6, 6]^2` run by `9.5e-5`; because + the cell count is rounded (`n = round(5/0.3) = 17`, effective spacing `5/17 = 0.294` against `0.3`), that + difference mixes grid placement with the exterior annulus and is not a pure exterior-tail measurement (the + exterior estimate rests on the sampled profiles of section 3). `--check` replays this summary arithmetic + from the stored numbers. +- The periodized `L = 12` direct integral (`h = 0.2`) agrees with far field plus hole to `2.0e-05` relative. +- The Monte Carlo cross-check of the unconditional control (`2 . 10^5` samples, seed 2026) gives + `E|det H| = 2.3032 +- 0.0058`, `4/sqrt3 = 2.3094` lying at `1.1` standard errors. + +These are observed discrepancies. No error bound is claimed. + +## 6. What the numbers are not + +Not a proof and not an enclosure. They are values of the formula [RM] (13) integrated as [CL] (1.4) prescribes; +[RM] is an accepted source, [CL] nonauthor-accepted and merged; the numbers do not re-review either. Planar only +(`d = 2`); no rate, no finite-`r` statement; the assembled `nu(1)` inherits the empirical precision of #168's +`alpha_1` (about `0.3%` or better). No register, catalog or STATUS change. Same GitHub account as every lane; zero +organizational-independence credit. The author will not merge. + +## 7. Provenance + +`remote.py` (standard library; `--check` replays the exact controls, the far field, sixteen sample kernels at two +parameter pairs for the continuum and the `L = 6` kernel, one decimal-regression point and the near-zero profile +against `RESULTS.json`; three mutants `hermite-sign`, `det-abs`, `weight-indicator` must exit 1), `RESULTS.json` +(full run, about 36 minutes), `SOURCE_MAP.json` (pins [LP], [RM], [CL], [RC] on `main` +`7d0c89a`), `SOURCE_FILES.json`, workflow `c6-remote-coefficient-numerics.yml`. diff --git a/frontiers/c6_remote_coefficient_numerics_20260930/README.md b/frontiers/c6_remote_coefficient_numerics_20260930/README.md new file mode 100644 index 000000000..79b060628 --- /dev/null +++ b/frontiers/c6_remote_coefficient_numerics_20260930/README.md @@ -0,0 +1,23 @@ +# Remote singleton coefficient numerics (planar, parent kernel) + +**Object:** `CL-C6-REMOTE-COEFF-NUMERICS-20260930-v1`. **Author lane:** Anthropic / Claude. **Kind:** numerical +note (floating-point quadrature with convergence checks; no proof, no enclosure). **Scientific effect:** NONE. + +Evaluates the contact kernel `Lambda_j(x; b, k, u)` of [RM] (13) in the plane for the parent kernel +`K(z) = exp(-|z|^2/2)` and its `L`-periodization, its far-field value, the correlation-hole constant +`c_hole = integral_(R^2) (Lambda - Lambda_inf) dx`, and the remote singleton mass `k integral_X Lambda dx` +that enters `nu(1)` by [CL] (1.4). Companion of the near-coefficient note (Math-#168), whose values of +`alpha_1 = nu_near(1)` are quoted to assemble `nu(1)`. + +- `NOTE.md` — statement of what is evaluated, exact structure, method, values, precision, non-claims. +- `remote.py` — standard-library script (Python `>= 3.11`): `--check` (exact controls and replay against + `RESULTS.json`, point-wise tolerance `1e-6` relative to cover interpreter differences), full run regenerates + `RESULTS.json` (about 36 minutes); `--mutant {hermite-sign, det-abs, weight-indicator}` must exit 1. +- Orientation: continuum values hold for every pin direction `u`; all `L`-periodized values are for `u = e_1` + aligned with a lattice axis. +- `RESULTS.json` — all numbers, grids and convergence checks. +- `SOURCE_MAP.json` — pinned sources on `main` (blob identities); `SOURCE_FILES.json` — packet manifest. +- Workflow `.github/workflows/c6-remote-coefficient-numerics.yml` replays manifest, pins, both check modes and the + mutants. + +Same GitHub account as every other lane; zero organizational-independence credit. The author will not merge. diff --git a/frontiers/c6_remote_coefficient_numerics_20260930/RESULTS.json b/frontiers/c6_remote_coefficient_numerics_20260930/RESULTS.json new file mode 100644 index 000000000..3e4a7ddb8 --- /dev/null +++ b/frontiers/c6_remote_coefficient_numerics_20260930/RESULTS.json @@ -0,0 +1,5438 @@ +{ + "certified": false, + "controls": { + "far_field": { + "0.0": { + "E_F_j": [ + 0.2071067811866195, + 1.4142135623732415, + 0.20710678118661952 + ], + "lambda": 0.11609348619094405, + "lambda_j": [ + 0.013149962564175739, + 0.08979356106259258, + 0.01314996256417574 + ], + "rho_j": [ + 0.013149962564175728, + 0.08979356106259259, + 0.01314996256417573 + ] + }, + "1.0": { + "E_F_j": [ + 0.022087746264714315, + 1.1013906298184966, + 1.079302883553834 + ], + "lambda": 0.08483094979588068, + "lambda_j": [ + 0.0008506175936846622, + 0.042415474897939356, + 0.04156485730425666 + ], + "rho_j": [ + 0.0008506175936846623, + 0.04241547489793938, + 0.04156485730425671 + ] + } + }, + "mc_absdet_unconditional": { + "mean": 2.3031855264703522, + "n": 200000, + "se": 0.005762874124208739, + "seed": 2026 + }, + "one_site_exact": { + "Cov_f_fxx": "-1", + "Cov_fx_fxxx": "-3", + "Cov_fxx_fzz": "1", + "Var_fxx": "3", + "Var_fxxx": "15", + "Var_fxz": "1" + }, + "one_site_lattice_max_dev": { + "L12": 7.003755912451762e-58, + "L24": 5.551522854989634e-244, + "L6": 0.0008783433650769723, + "continuum": 0.0 + }, + "reflection_symmetry_max_rel_dev": 8.38164154450512e-16, + "unconditional_hessian": { + "E_F_j": [ + 0.5773502692137803, + 1.1547005384275606, + 0.5773502692137802 + ], + "E_absdet": 2.309401076855121, + "E_det": 0.0 + } + }, + "exact_regression_points": 54, + "far_field": { + "0.0": { + "E_F_j_given_level": [ + 0.2071067811866195, + 1.4142135623732415, + 0.20710678118661952 + ], + "lambda_inf": 0.11609348619094405, + "lambda_inf_j": [ + 0.013149962564175739, + 0.08979356106259258, + 0.01314996256417574 + ], + "p_inf": 0.06349363593424098 + }, + "1.0": { + "E_F_j_given_level": [ + 0.022087746264714315, + 1.1013906298184966, + 1.079302883553834 + ], + "lambda_inf": 0.08483094979588068, + "lambda_inf_j": [ + 0.0008506175936846622, + 0.042415474897939356, + 0.04156485730425666 + ], + "p_inf": 0.038510836890748947 + } + }, + "grid_convergence": { + "k=1.0,b=0.0": { + "hole_h=0.1,Rc=6": -1.6313351977268296, + "hole_h=0.15,Rc=6": -1.6311476645638006, + "hole_h=0.3,Rc=5": -1.6297553083021523, + "hole_h=0.3,Rc=6": -1.6296607043116234, + "hole_h=0.3,Rc=6,high_order": -1.629660659845337, + "hole_j_h=0.1,Rc=6": [ + -0.31378570687015506, + -1.332596279425655, + 0.015046788568980531 + ], + "lambda_inf_high_order": 0.11609348619092598, + "torus_L=12_approx": 15.086314346932143, + "torus_L=12_direct_h=0.2": 15.086621885475793 + } + }, + "hole": { + "k=0.5,b=0.0": { + "grid": { + "Rc": 6.0, + "h": 0.15, + "points": 3200, + "symmetry": "z -> -z" + }, + "hole": -1.084135453753016, + "hole_j": [ + -0.3028235079276278, + -0.8844195042052593, + 0.10310755837987123 + ], + "hole_unweighted": 6.6558328680387655, + "hole_unweighted_j": [ + 1.6907231338002304, + 3.2743866004383944, + 1.6907231338001405 + ], + "lambda_max_on_grid": 0.1867399995903017, + "lambda_min_on_grid": 1.3675393485847551e-126 + }, + "k=0.5,b=1.0": { + "grid": { + "Rc": 6.0, + "h": 0.15, + "points": 3200, + "symmetry": "z -> -z" + }, + "hole": -0.7506293381103409, + "hole_j": [ + -0.008308169817322056, + -0.2607974077845683, + -0.4815237605084506 + ], + "hole_unweighted": 6.30991417071404, + "hole_unweighted_j": [ + 0.5638736743837274, + 3.4979879412468344, + 2.2480525550834787 + ], + "lambda_max_on_grid": 0.14996841549563256, + "lambda_min_on_grid": 1.6639091561363451e-127 + }, + "k=1.0,b=0.0": { + "grid": { + "Rc": 6.0, + "h": 0.15, + "points": 3200, + "symmetry": "z -> -z" + }, + "hole": -1.6311476645638006, + "hole_j": [ + -0.3137857115939324, + -1.3324986709058282, + 0.015136717935960147 + ], + "hole_unweighted": 1.7266312061970344, + "hole_unweighted_j": [ + 0.5963757449363564, + 0.5338797163242601, + 0.5963757449364179 + ], + "lambda_max_on_grid": 0.17946929470083037, + "lambda_min_on_grid": 0.0 + }, + "k=1.0,b=1.0": { + "grid": { + "Rc": 6.0, + "h": 0.15, + "points": 3200, + "symmetry": "z -> -z" + }, + "hole": -1.184482355875255, + "hole_j": [ + -0.005586604498269274, + -0.46494835381060384, + -0.713947397566382 + ], + "hole_unweighted": 1.850309671857266, + "hole_unweighted_j": [ + 0.23644269280226493, + 1.2280733401923594, + 0.38579363886264156 + ], + "lambda_max_on_grid": 0.13172552616590733, + "lambda_min_on_grid": 0.0 + }, + "k=2.0,b=0.0": { + "grid": { + "Rc": 6.0, + "h": 0.15, + "points": 3200, + "symmetry": "z -> -z" + }, + "hole": -2.1382024573116403, + "hole_j": [ + -0.3302820124213753, + -1.7568436474476312, + -0.051076797442633874 + ], + "hole_unweighted": 0.7685792561786426, + "hole_unweighted_j": [ + 0.47398232526493733, + -0.17938539435123557, + 0.4739823252649408 + ], + "lambda_max_on_grid": 0.19321540570437315, + 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amendments, applied" + }, + { + "lane": "OpenAI/GPT-6 Astra", + "review_id": 5364176857, + "scope": "reassessment at a04fc81; formulation ACCEPT stands; truncation wording repaired" + } + ] +} diff --git a/frontiers/c6_remote_coefficient_numerics_20260930/SOURCE_MAP.json b/frontiers/c6_remote_coefficient_numerics_20260930/SOURCE_MAP.json new file mode 100644 index 000000000..0d3f8b889 --- /dev/null +++ b/frontiers/c6_remote_coefficient_numerics_20260930/SOURCE_MAP.json @@ -0,0 +1,99 @@ +{ + "schema": 1, + "object": "CL-C6-REMOTE-COEFF-NUMERICS-20260930-v1", + "scientific_effect": "NONE", + "source_repository": "d6g8k5htny-coder/Math-", + "main_at_creation": "7d0c89a5bb77a52a7a661d7292ba5d955bd0a66d", + "note": "Numerical evaluation only. The formula evaluated is the contact kernel Lambda_j(x; b, k, u) of [RM] (13), summed and integrated over the torus as in [CL] (1.4)-(1.5) (the remote singleton part k integral_X Lambda of nu(1)), at d = 2 with the exact kernel of [LP] section 2. No source theorem is consumed as accepted beyond its recorded disposition; the numbers are deterministic floating-point quadrature values with reported convergence checks, not enclosures. The companion near-coefficient note (Math-#168) merged to main on 30 September 2026 (02772ec) and is now pinned as [NUM]. OpenAI Sol review 5363739676 (30 September 2026, head f4d8bb9) accepts the closed-form structure of section 2 as identities; recorded under nonauthor_reads without changing review_status. OpenAI Astra review 5364077220 (head f4d8bb9) accepts the Slice B formulation with two wording amendments, applied.", + "sources": [ + { + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "id": "LP", + "commit": "7d0c89a5bb77a52a7a661d7292ba5d955bd0a66d", + "role": "section 2: the exact periodized Gaussian kernel K_L (periodization of exp(-|z|^2/2)); section 3: contact frame U_0 and target v_0 = (b, 0, 0, 12k, 0, 0); (5.4)-(5.5): the full normalizer z_0 = lim Z_r / r^2", + "current_required": true, + "on_main": true + }, + { + "path": "frontiers/remote_window_20260924/PROOF.md", + "bytes": 18355, + "sha256": "a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7", + "git_blob": "b383bfcc88ec4ad497dff01fb6640e429ba24a84", + "id": "RM", + "commit": "7d0c89a5bb77a52a7a661d7292ba5d955bd0a66d", + "role": "Theorem A (fixed-remote mean measure) and the contact kernel Lambda_j(x; b, k, u) of (13): p_(Y_x | U_0 = v_0)(0, b) / z_0 times E[w_0 F_j(H_x) | U_0 = v_0, Y_x = (0, b)] with w_0 = (6k)^2 (det A_0)^2 1{A_0 < 0} ((10)) and z_0 = E[w_0 | U_0 = v_0] ((11)); this is the object evaluated", + "current_required": true, + "on_main": true + }, + { + "path": "frontiers/c6_cluster_law_20260929/PROOF.md", + "bytes": 70131, + "sha256": "96b8e2d59defee6059bf151a896b7fc7c91fc529b385a3db8a87574901197b38", + "git_blob": "ba492c8e62e58bfc055fc8254c790e893d346ba5", + "id": "CL", + "commit": "7d0c89a5bb77a52a7a661d7292ba5d955bd0a66d", + "role": "(1.4): nu(1) = nu_near(1) + k integral_X Lambda dx, nu(2) = nu_near(2); section 5.1 (5.2): existence of k integral_X Lambda = lim_(rho -> 0) k integral_(D_rho) Lambda with error <= C rho^2, which is what makes the torus integral of the bounded kernel the right object; Math-#159, nonauthor-accepted (OpenAI review 5360192822), merged", + "current_required": true, + "on_main": true + }, + { + "path": "frontiers/remote_collision_20260928/PROOF.md", + "bytes": 26002, + "sha256": "b9b8b58fd8266db7ffe6537078003888ef138b445d3289ec5588f116af9050c2", + "git_blob": "7b48a88e2af54e759e89a8c8219573bb450a65ea", + "id": "RC", + "commit": "7d0c89a5bb77a52a7a661d7292ba5d955bd0a66d", + "role": "Corollary E: Q_r^W{N_j(E) >= 1} = k r^3 integral_E Lambda_j + O(r^4 |E|) on fixed remote regions, the probability reading of the same kernel (context only)", + "current_required": true, + "on_main": true + }, + { + "path": "frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json", + "bytes": 4095, + "sha256": "9c88ea85b6ba08199996c9b5ff74ecd06afc0f458e08c28b4c3af6191f437fcb", + "git_blob": "2648314f8c11f0637ce301ba76360142a8309e8f", + "id": "NUM", + "commit": "02772ec0427f4dda1f2a6b9e980a031804e276a6", + "role": "near coefficients alpha_1(k, b) (GH60) and alpha_2(k, b) quoted in NOTE.md section 4 to assemble nu(1) = alpha_1 + k integral_X Lambda; Math-#168, merged to main on 30 September 2026 (02772ec) with RESULTS.json bytes unchanged from the quoted head 0b35e76; xAI read 5360559751 accepted its Slices A and B as identities", + "current_required": true, + "on_main": true + } + ], + "nonauthor_acceptance": false, + "governing_status_changed": false, + "main_at_update": "02772ec0427f4dda1f2a6b9e980a031804e276a6", + "nonauthor_reads": [ + { + "lane": "OpenAI/GPT-5.6 Sol", + "review_id": 5363739676, + "head": "f4d8bb9c30b5848996fceb18d213433cfe9956b9", + "date": "2026-09-30", + "disposition": "ACCEPT Slice A (closed-form far field, covariance rule, index split, unconditional controls, cone-adapted coordinates); the near-pin Euler paragraph read as a scaling heuristic only, with [CL] section 5.1 supplying the integrability bound; explicitly not an acceptance of the grids, the then-stale RESULTS.json, the precision claims or merge readiness; same account, credit 0" + }, + { + "lane": "OpenAI/GPT-6 Astra", + "review_id": 5364077220, + "head": "f4d8bb9c30b5848996fceb18d213433cfe9956b9", + "date": "2026-09-30", + "disposition": "ACCEPT Slice B (cone-coordinate conditional-Gaussian formulation, closed-form A_0 integration, corrected full-normalizer interface, 4x4 positivity check) with two wording amendments (truncation breakpoints in the cubature; angular qualification of the Euler scaling check), applied in the successor commit; not an acceptance of numerical precision or merge readiness; same account, credit 0" + }, + { + "lane": "OpenAI/GPT-6 Astra", + "review_id": 5364176857, + "head": "a04fc81a8753769b8228419d33998aed0679fd30", + "date": "2026-09-30", + "disposition": "successor reassessment: the two Slice B amendments resolved, the formulation ACCEPT stands; one further wording repair for the truncation sentence (no universal relative bound; exterior figure an estimate), applied in the successor commit; Slice C still open; same account, credit 0" + }, + { + "lane": "OpenAI GPT-6 Astra Pro", + "review_id": 5364643055, + "head": "a8c6d7918061c62a2e523b6cda34e9b78c9ac320", + "date": "2026-09-30", + "disposition": "AMEND (reporting only): two stale NOTE section 5 sentences; applied in the successor commit (Rc = 5 minus Rc = 6 at h = 0.3 is 9.46e-5, not 2.1e-3; fitted-error figures labelled nonrigorous model extrapolations for k = 1, b = 0 only with the measured 1.88e-4 finest-grid difference; unequal effective Rc = 5 spacing 5/17 noted; L = 24 improvement stated as expectation, not measurement). Does not certify quadrature accuracy or the table.", + "slice": "C (empirical precision reporting)" + } + ] +} diff --git a/frontiers/c6_remote_coefficient_numerics_20260930/remote.py b/frontiers/c6_remote_coefficient_numerics_20260930/remote.py new file mode 100644 index 000000000..9553e3a95 --- /dev/null +++ b/frontiers/c6_remote_coefficient_numerics_20260930/remote.py @@ -0,0 +1,805 @@ +#!/usr/bin/env python3 +"""Numerical evaluation of the remote contact kernel Lambda_j(x; b, k, u) of [RM] (13) for the +parent kernel K(z) = exp(-|z|^2/2) (and its L-periodization) in the plane d = 2, together with +its far-field value, the correlation-hole constant and the torus integral k * integral_X Lambda dx +that enters nu(1) = nu_near(1) + k integral_X Lambda ([CL] (1.4)). + +Standard library only. Deterministic nested Gauss quadrature; no random numbers except in the +optional Monte Carlo cross-check (fixed seed). Nothing here is a proof or an enclosure. + + python -B -S remote.py # full run, writes RESULTS.json (long: tens of minutes) + python -B -S remote.py --check # exact controls + replay of a small subset against RESULTS.json + # (Python >= 3.11; point-wise replay tolerance 1e-6 relative covers + # interpreter/libm differences between 3.11 and 3.14) + python -B -S remote.py --check --mutant NAME # must exit 1 + +Conventions. Coordinates x = (x1, z): x1 along the pin axis u, z transverse. Jets are indexed by +derivative orders (a, c) = d^a/dx1^a d^c/dz^c. Cov(d^alpha f(s), d^beta f(t)) = (-1)^|alpha| +(d^(alpha+beta) K)(t - s). Contact observation U_0 = (f, f_x, f_z, f_xx, f_xz, f_xxx)(0) = +(b, 0, 0, 0, 0, 12k); A_0 = f_zz(0); Y_x = (f_x, f_z, f)(x) = (0, 0, b); H_x = (f_xx, f_xz, f_zz)(x). +w_0 = 36 k^2 A_0^2 1{A_0 < 0}; z_0 = E[w_0 | U_0] = 36 k^2 m_(2,b). +Lambda_j(x) = p_(Y_x | U_0)(0, 0, b) * E[w_0 F_j(H_x) | U_0, Y_x] / z_0, F_j(H) = |det H| 1{index j}. +""" +import argparse +import json +import math +import os +import random +import sys +from decimal import Decimal, getcontext +from fractions import Fraction + +getcontext().prec = 50 + +HERE = os.path.dirname(os.path.abspath(__file__)) +MUTANTS = ('hermite-sign', 'det-abs', 'weight-indicator') +MUT = None + +SITE0 = [(0, 0), (1, 0), (0, 1), (2, 0), (1, 1), (0, 2), (3, 0)] # f, f_x, f_z, f_xx, f_xz, f_zz, f_xxx at 0 +SITEX = [(0, 0), (1, 0), (0, 1), (2, 0), (1, 1), (0, 2)] # f, f_x, f_z, f_xx, f_xz, f_zz at x +OBS0 = [0, 1, 2, 3, 4, 6] # U_0 within the 13-vector +OBSX = [8, 9, 7] # Y_x = (f_x, f_z, f)(x) +TARGETS = [5, 10, 11, 12] # A_0 = f_zz(0), h1 = f_xx(x), h2 = f_xz(x), h3 = f_zz(x) + + +def require(cond, msg): + if not cond: + print('FAIL: ' + msg) + sys.exit(1) + + +# ----------------------------------------------------------------------------- Hermite / kernel +def hermite(n, t): + """Probabilists' Hermite polynomial He_n(t): (d/dt)^n exp(-t^2/2) = (-1)^n He_n(t) exp(-t^2/2).""" + h0, h1 = 1, t + if n == 0: + return h0 + for m in range(1, n): + h0, h1 = h1, t * h1 - m * h0 + return h1 + + +def _exp(t): + return t.exp() if isinstance(t, Decimal) else math.exp(t) + + +def _sqrt(t): + return t.sqrt() if isinstance(t, Decimal) else math.sqrt(t) + + +def _log(t): + return t.ln() if isinstance(t, Decimal) else math.log(t) + + +def kernel_derivative(a, c, x1, z, L=None, images=2): + """(d^a/dx1^a d^c/dz^c) K_L(x1, z) for K(z) = exp(-|z|^2/2), K_L its normalized L-periodization. + L = None: continuum kernel (single image). Works in float or Decimal arithmetic (type of x1). + Orientation: (x1, z) are pin-frame coordinates and the image lattice is L Z^2 in these SAME coordinates, + i.e. every periodized value in this note is for the pin direction u = e_1 aligned with a lattice axis. + The continuum kernel is isotropic, so its values hold for every u; the periodized ones do not ([RM]: + Lambda_j depends on u through the frame), and no other orientation is computed here.""" + if L is None: + return (-1) ** (a + c) * hermite(a, x1) * hermite(c, z) * _exp(-(x1 * x1 + z * z) / 2) + tot = 0 * x1 + norm = 0 * x1 + for m1 in range(-images, images + 1): + for m2 in range(-images, images + 1): + s1, s2 = x1 + L * m1, z + L * m2 + tot += (-1) ** (a + c) * hermite(a, s1) * hermite(c, s2) * _exp(-(s1 * s1 + s2 * s2) / 2) + norm += _exp(-(L * L) * (m1 * m1 + m2 * m2) / 2 + 0 * x1) + return tot / norm + + +def cov_entry(alpha, beta, dx1, dz, L=None): + """Cov(d^alpha f(s), d^beta f(t)) with t - s = (dx1, dz).""" + sign = (-1) ** (alpha[0] + alpha[1]) + if MUT == 'hermite-sign': + sign = (-1) ** (beta[0] + beta[1]) + return sign * kernel_derivative(alpha[0] + beta[0], alpha[1] + beta[1], dx1, dz, L) + + +def joint_covariance(x1, z, L=None): + """13 x 13 covariance of the two-site jet vector (SITE0 at 0, SITEX at x); float or Decimal.""" + jets = [(0, a) for a in SITE0] + [(1, a) for a in SITEX] + n = len(jets) + zero = 0 * x1 + if L is not None and isinstance(x1, Decimal): + L = Decimal(L) + C = [[zero] * n for _ in range(n)] + for i in range(n): + si, ai = jets[i] + for j in range(n): + sj, aj = jets[j] + d1 = (x1 if sj == 1 else zero) - (x1 if si == 1 else zero) + d2 = (z if sj == 1 else zero) - (z if si == 1 else zero) + C[i][j] = cov_entry(ai, aj, d1, d2, L) + return C + + +# ----------------------------------------------------------------------------- exact one-site table +def hermite_zero_exact(n): + """He_n(0) exactly: 0 for odd n, (-1)^(n/2) (n-1)!! for even n.""" + if n % 2: + return Fraction(0) + v = Fraction(1) + for m in range(1, n, 2): + v *= m + return v * (-1) ** (n // 2) + + +def exact_one_site_cov(alpha, beta): + return (-1) ** (alpha[0] + alpha[1]) * hermite_zero_exact(alpha[0] + beta[0]) * hermite_zero_exact(alpha[1] + beta[1]) + + +# ----------------------------------------------------------------------------- linear algebra +def cholesky(A): + n = len(A) + zero = 0 * A[0][0] + Lm = [[zero] * n for _ in range(n)] + for i in range(n): + for j in range(i + 1): + s = A[i][j] - sum((Lm[i][k] * Lm[j][k] for k in range(j)), zero) + if i == j: + if s <= 0: + raise ValueError('matrix not positive definite (pivot %r at %d)' % (s, i)) + Lm[i][j] = _sqrt(s) + else: + Lm[i][j] = s / Lm[j][j] + return Lm + + +def chol_solve(Lm, b): + n = len(Lm) + zero = 0 * Lm[0][0] + y = [zero] * n + for i in range(n): + y[i] = (b[i] - sum((Lm[i][k] * y[k] for k in range(i)), zero)) / Lm[i][i] + x = [zero] * n + for i in reversed(range(n)): + x[i] = (y[i] - sum((Lm[k][i] * x[k] for k in range(i + 1, n)), zero)) / Lm[i][i] + return x + + +def submatrix(C, rows, cols): + return [[C[i][j] for j in cols] for i in rows] + + +def condition(C, obs, values, targets): + """Gaussian regression: law of C[targets] given C[obs] = values (zero prior means). + Returns (mean, cov, obs_logdensity_at_values).""" + Coo = submatrix(C, obs, obs) + zero = 0 * C[0][0] + values = [v + zero for v in values] + Lm = cholesky(Coo) + alpha = chol_solve(Lm, values) + logdet = 2 * sum((_log(Lm[i][i]) for i in range(len(obs))), zero) + quad = sum((values[i] * alpha[i] for i in range(len(obs))), zero) + logdens = float(-quad / 2 - logdet / 2) - 0.5 * len(obs) * math.log(2 * math.pi) + mean = [] + cov = [] + Cto = submatrix(C, targets, obs) + for r, t in enumerate(targets): + mean.append(sum((Cto[r][i] * alpha[i] for i in range(len(obs))), zero)) + solved = [chol_solve(Lm, Cto[r]) for r in range(len(targets))] + for r, t in enumerate(targets): + row = [] + for s, u in enumerate(targets): + row.append(C[t][u] - sum((Cto[r][i] * solved[s][i] for i in range(len(obs))), zero)) + cov.append(row) + return mean, cov, logdens + + +# ----------------------------------------------------------------------------- quadrature rules +def gauss_legendre(n): + nodes, weights = [], [] + for i in range(1, n + 1): + x = math.cos(math.pi * (i - 0.25) / (n + 0.5)) + for _ in range(100): + p0, p1 = 1.0, x + for m in range(2, n + 1): + p0, p1 = p1, ((2 * m - 1) * x * p1 - (m - 1) * p0) / m + dp = n * (x * p1 - p0) / (x * x - 1.0) + dx = p1 / dp + x -= dx + if abs(dx) < 1e-15: + break + p0, p1 = 1.0, x + for m in range(2, n + 1): + p0, p1 = p1, ((2 * m - 1) * x * p1 - (m - 1) * p0) / m + dp = n * (x * p1 - p0) / (x * x - 1.0) + nodes.append(x) + weights.append(2.0 / ((1.0 - x * x) * dp * dp)) + return nodes, weights + + +def gauss_hermite(n): # retained for the Monte Carlo-free control of the normalization; unused in the main integral + """Probabilists' Gauss-Hermite: sum w_i g(t_i) ~ E g(T), T ~ N(0, 1).""" + nodes, weights = [], [] + for i in range(1, n + 1): + if i == 1: + x = math.sqrt(2 * n + 1) - 1.85575 * (2 * n + 1) ** (-1.0 / 6.0) + elif i == 2: + x = nodes[-1] - 1.14 * n ** 0.426 / nodes[-1] + elif i == 3: + x = 1.86 * nodes[-1] - 0.86 * nodes[0] + elif i == 4: + x = 1.91 * nodes[-1] - 0.91 * nodes[-2] + else: + x = 2.0 * nodes[-1] - nodes[-2] + for _ in range(200): + h0, h1 = 1.0, x + for m in range(1, n): + h0, h1 = h1, x * h1 - m * h0 + dh = n * h0 + dx = h1 / dh + x -= dx + if abs(dx) < 1e-14: + break + h0, h1 = 1.0, x + for m in range(1, n): + h0, h1 = h1, x * h1 - m * h0 + nodes.append(x) + weights.append(math.factorial(n) / (n * n * h0 * h0)) + return nodes, weights + + +SQRT2PI = math.sqrt(2 * math.pi) + + +def phi(t): + return math.exp(-t * t / 2) / SQRT2PI + + +def Phi(t): + return 0.5 * math.erfc(-t / math.sqrt(2)) + + +def m2_negative(m, s): + """E[A^2 1{A < 0}] for A ~ N(m, s^2).""" + if MUT == 'weight-indicator': + return m * m + s * s + if s <= 0.0: + return m * m if m < 0 else 0.0 + t = m / s + return (m * m + s * s) * Phi(-t) - m * s * phi(t) + + +class Rules: + """Quadrature orders: n_a Gauss-Legendre nodes on each side of a = 0, n_theta trapezoid nodes on the circle, + n_rho Gauss-Legendre nodes on each of [0, |a|] and [|a|, rho_max]; Gaussian ranges cut at cut standard deviations.""" + + def __init__(self, n_a=24, n_theta=32, n_rho=24, cut=9.0): + self.gl_a = gauss_legendre(n_a) + self.n_theta = n_theta + self.gl_rho = gauss_legendre(n_rho) + self.cut = cut + self.orders = [n_a, n_theta, n_rho] + + +def weighted_determinant_integrals(mu, S, rules): + """(A, h1, h2, h3) ~ N(mu, S). Returns dict with + G[j] = E[ A^2 1{A<0} |det H| 1{index H = j} ], F[j] = E[ |det H| 1{index H = j} ], D = E det H (control). + Coordinates: a = (h1 + h3)/2, c = (h1 - h3)/2, so det H = a^2 - c^2 - h2^2 = a^2 - rho^2 with (c, h2) = + rho (cos theta, sin theta). The cone det H = 0 is the coordinate surface rho = |a|; the a-range is split at 0 + (index by the sign of the trace) and the rho-range at |a|. Apart from the truncations (a-range at cut standard + deviations; rho_max = |mean| + cut * max marginal sd of the transverse pair, i.e. at standardized distance at + least cut/sqrt2 since the largest eigenvalue is at most twice the largest marginal variance; the Mahalanobis cut + q > 2 cut^2 below) each piece of the integrand is smooth. The omitted weighted mass of these truncations depends + on the conditional covariance, mean and normalization (no universal relative bound is claimed); the accuracy of + the nested Gauss-Legendre / trapezoid rule is established empirically on the exact controls, not by a + convergence theorem. The general (non-isotropic) Gaussian density + of (a, c, h2) is evaluated directly at the nodes; A is integrated out in closed form through m2_negative.""" + # linear map (h1, h2, h3) -> w = (a, c, h2) + T = [[0.5, 0.0, 0.5], [0.5, 0.0, -0.5], [0.0, 1.0, 0.0]] + SHH = [[S[i][j] for j in (1, 2, 3)] for i in (1, 2, 3)] + muH = [mu[1], mu[2], mu[3]] + muw = [sum(T[i][k] * muH[k] for k in range(3)) for i in range(3)] + TS = [[sum(T[i][k] * SHH[k][j] for k in range(3)) for j in range(3)] for i in range(3)] + Sw = [[sum(TS[i][k] * T[j][k] for k in range(3)) for j in range(3)] for i in range(3)] + Lw = cholesky(Sw) + logdet = 2.0 * sum(math.log(Lw[i][i]) for i in range(3)) + norm = math.exp(-0.5 * logdet) / (2 * math.pi) ** 1.5 + # A | w regression: A = muA + wA . (w - muw), residual sd sA + SAH = [S[0][j] for j in (1, 2, 3)] + SAw = [sum(T[i][k] * SAH[k] for k in range(3)) for i in range(3)] + wA = chol_solve(Lw, SAw) + sA2 = S[0][0] - sum(SAw[i] * wA[i] for i in range(3)) + if sA2 <= 0.0: + raise ValueError('Var(A_0 | H) not positive: %r' % sA2) + sA = math.sqrt(sA2) + muA = mu[0] + # inverse via Cholesky for the quadratic form + Linv = [[0.0] * 3 for _ in range(3)] + for j in range(3): + e = [1.0 if i == j else 0.0 for i in range(3)] + col = chol_solve(Lw, e) + for i in range(3): + Linv[i][j] = col[i] + P = Linv # S_w^{-1} + cut = rules.cut + sa = math.sqrt(Sw[0][0]) + sc, s2 = math.sqrt(Sw[1][1]), math.sqrt(Sw[2][2]) + rho_max = math.hypot(muw[1], muw[2]) + cut * max(sc, s2) + G = [0.0, 0.0, 0.0] + F = [0.0, 0.0, 0.0] + # theta nodes: periodic trapezoid rule in u, theta = theta0 + u - beta sin u, concentrated at the angular + # position theta0 of the (c, h2) mean when the Gaussian is offset and narrow (analytic periodic map: the + # trapezoid rule keeps its geometric convergence; the Jacobian 1 - beta cos u is included in the weights) + rmu = math.hypot(muw[1], muw[2]) + smax = max(sc, s2) + width = min(1.0, smax / max(rmu, 1e-300)) + beta = max(0.0, 1.0 - rules.n_theta * width / (8 * math.pi)) + theta0 = math.atan2(muw[2], muw[1]) if rmu > 0 else 0.0 + cs = [] + for i in range(rules.n_theta): + u = -math.pi + 2 * math.pi * (i + 0.5) / rules.n_theta + th = theta0 + u - beta * math.sin(u) + cs.append((math.cos(th), math.sin(th), 1.0 - beta * math.cos(u))) + wtheta = 2 * math.pi / rules.n_theta + a_lo, a_hi = muw[0] - cut * sa, muw[0] + cut * sa + for lo, hi in ((a_lo, min(0.0, a_hi)), (max(0.0, a_lo), a_hi)): + if hi <= lo: + continue + mid, half = 0.5 * (lo + hi), 0.5 * (hi - lo) + for ta, wa in zip(*rules.gl_a): + a = mid + half * ta + wt_a = wa * half + aa = abs(a) + jpos = 0 if a > 0 else 2 # det > 0: sign of the trace decides minimum (0) / maximum (2) + segs = [] + if aa < rho_max: + segs.append((0.0, aa, jpos)) + segs.append((aa, rho_max, 1)) + else: + segs.append((0.0, rho_max, jpos)) + da = a - muw[0] + for lo_r, hi_r, j in segs: + if hi_r <= lo_r: + continue + mid_r, half_r = 0.5 * (lo_r + hi_r), 0.5 * (hi_r - lo_r) + accG = accF = 0.0 + for tr, wr in zip(*rules.gl_rho): + rho = mid_r + half_r * tr + adet = a * a - rho * rho + if MUT != 'det-abs': + adet = abs(adet) + base = wr * rho * adet + for cth, sth, jac in cs: + c = rho * cth + h2 = rho * sth + dc = c - muw[1] + d2 = h2 - muw[2] + q = (P[0][0] * da * da + P[1][1] * dc * dc + P[2][2] * d2 * d2 + + 2 * (P[0][1] * da * dc + P[0][2] * da * d2 + P[1][2] * dc * d2)) + if q > 2 * cut * cut: + continue + dens = math.exp(-0.5 * q) * jac + mA = muA + wA[0] * da + wA[1] * dc + wA[2] * d2 + accF += base * dens + accG += base * dens * m2_negative(mA, sA) + G[j] += wt_a * half_r * accG + F[j] += wt_a * half_r * accF + scale = norm * wtheta + G = [g * scale for g in G] + F = [f * scale for f in F] + D = (muw[0] ** 2 + Sw[0][0]) - (muw[1] ** 2 + Sw[1][1]) - (muw[2] ** 2 + Sw[2][2]) + return {'G': G, 'F': F, 'D': D} + + +# ----------------------------------------------------------------------------- Lambda +def m2b(b): + return m2_negative(-b, math.sqrt(2.0)) + + +def _conditioning(x1, z, b, k, L, exact): + """Density of Y_x at (0, 0, b) given U_0 (log) and the conditional law (mu, S) of (A_0, H_x) given + U_0 and Y_x. exact = True recomputes the Gaussian regression in 50-digit decimal arithmetic (used + automatically where the double-precision regression loses positivity; the near-axis points, where + Lambda is exponentially small).""" + if exact: + x1, z, bb, kk = Decimal(repr(x1)), Decimal(repr(z)), Decimal(repr(b)), Decimal(repr(k)) + else: + bb, kk = b, k + C = joint_covariance(x1, z, L) + u0 = [bb, 0 * bb, 0 * bb, 0 * bb, 0 * bb, 12 * kk] + # one-site contact law of A_0 = f_zz(0) given U_0 under the SAME kernel (continuum: exactly N(-b, 2); + # periodized: the L-dependent conditional law), so that z_0 = 36 k^2 E[A_0^2 1{A_0 < 0} | U_0] is the + # normalizer of the field actually used ([RM] (11)) + meanA, covA, _ = condition(C, OBS0, u0, [5]) + m2 = m2_negative(float(meanA[0]), math.sqrt(float(covA[0][0]))) + meanY, covY, _ = condition(C, OBS0, u0, OBSX) + LY = cholesky(covY) + dev = [0 - meanY[0], 0 - meanY[1], bb - meanY[2]] + a = chol_solve(LY, dev) + logp = float(-sum((dev[i] * a[i] for i in range(3)), 0 * bb) / 2 - sum((_log(LY[i][i]) for i in range(3)), 0 * bb)) - 1.5 * math.log(2 * math.pi) + obs = OBS0 + OBSX + vals = u0 + [0 * bb, 0 * bb, bb] + mu, S, _ = condition(C, obs, vals, TARGETS) + mu = [float(v) for v in mu] + S = [[float(v) for v in row] for row in S] + # the quadrature needs the (h1, h2, h3) block AND the Schur complement Var(A_0 | H_x) positive; a + # nonpositive pivot anywhere in the full 4 x 4 conditional covariance means the double-precision + # regression has lost the tiny residual variances and the caller must redo it in decimal arithmetic + cholesky(S) + return logp, mu, S, m2 + + +def lambda_at(x1, z, b, k, rules, L=None): + """Returns dict: Lambda_j (j = 0, 1, 2), Lambda (sum), the unweighted remote density rho_j = + p * E[F_j | obs] (what the same point would carry without the pin weight), logp, coupling.""" + try: + logp, mu, S, mm = _conditioning(x1, z, b, k, L, False) + exact = False + except ValueError: + logp, mu, S, mm = _conditioning(x1, z, b, k, L, True) + exact = True + EXACT_COUNT[0] += 1 + p = math.exp(logp) + W = weighted_determinant_integrals(mu, S, rules) + lam = [p * W['G'][j] / mm for j in range(3)] + rho = [p * W['F'][j] for j in range(3)] + fsum = W['F'][0] + W['F'][1] + W['F'][2] + coupling = (W['G'][0] + W['G'][1] + W['G'][2]) / (mm * fsum) if fsum > 0 else None + return {'lambda_j': lam, 'lambda': sum(lam), 'rho_j': rho, 'rho': sum(rho), 'logp': logp, 'exact_regression': exact, + 'coupling': coupling} + + +def lambda_far(b, rules): + """Far-field value: p = phi(b)/(2 pi), H | (f = b, grad f = 0): h1, h3 ~ N(-b, 2) independent, + h2 ~ N(0, 1); A_0 independent of H. Lambda_inf,j = phi(b)/(2 pi) E[F_j(H)].""" + mu = [-b, -b, 0.0, -b] + S = [[2.0, 0.0, 0.0, 0.0], [0.0, 2.0, 0.0, 0.0], [0.0, 0.0, 1.0, 0.0], [0.0, 0.0, 0.0, 2.0]] + W = weighted_determinant_integrals(mu, S, rules) + p = phi(b) / (2 * math.pi) + lam = [p * W['G'][j] / m2b(b) for j in range(3)] + return {'lambda_j': lam, 'lambda': sum(lam), 'rho_j': [p * W['F'][j] for j in range(3)], 'E_F_j': W['F']} + + +def far_field_closed_form(b): + """Exact far field. With h1 = a + c, h3 = a - c, a ~ N(-b, 1), c ~ N(0, 1), h2 ~ N(0, 1) independent: + det H = a^2 - rho^2, rho^2 ~ Exp(mean 2). E[(rho^2 - a^2)_+ | a] = 2 exp(-a^2/2), E[(a^2 - rho^2)_+ | a] = + a^2 - 2 + 2 exp(-a^2/2), and E[exp(-a^2/2) 1{a < 0}] = exp(-b^2/4) Phi(b/sqrt2)/sqrt2. Hence + E F_1 = sqrt2 e^(-b^2/4) (saddles) + E F_2 = (b^2+1) Phi(b) + b phi(b) - 2 Phi(b) + sqrt2 e^(-b^2/4) Phi(b/sqrt2) (maxima, a < 0) + E F_0 = the same with b -> -b (minima) + E|det H_b| = b^2 - 1 + 2 sqrt2 e^(-b^2/4), + and Lambda_inf,j(b) = phi(b) E F_j / (2 pi). At b = 0: Lambda_inf = (2 sqrt2 - 1)/(2 pi)^(3/2). Integrating + Lambda_inf over b gives 2/(pi sqrt3), the total critical density (Longuet-Higgins).""" + e = math.exp(-b * b / 4) + F1 = math.sqrt(2) * e + F2 = (b * b + 1) * Phi(b) + b * phi(b) - 2 * Phi(b) + math.sqrt(2) * e * Phi(b / math.sqrt(2)) + F0 = (b * b + 1) * Phi(-b) - b * phi(b) - 2 * Phi(-b) + math.sqrt(2) * e * Phi(-b / math.sqrt(2)) + p = phi(b) / (2 * math.pi) + return {'lambda': p * (b * b - 1 + 2 * math.sqrt(2) * e), 'lambda_j': [p * F0, p * F1, p * F2], 'E_absdet': b * b - 1 + 2 * math.sqrt(2) * e} + + +def unconditional_hessian_controls(rules): + """H unconditional: Var f_xx = Var f_zz = 3, Cov = 1, Var f_xz = 1. With h1 = a + c, h3 = a - c + (a ~ N(0, 2), c ~ N(0, 1) independent) det H = a^2 - (c^2 + h2^2) = a^2 - rho^2 with rho^2 ~ Exp(mean 2), + so E|det H| = 2 E[(rho^2 - a^2)_+] = 4 E exp(-a^2/2) = 4/sqrt3, E F_1 = 2/sqrt3, E F_0 = E F_2 = 1/sqrt3, + E det H = 0. Total critical density 2/(pi sqrt3), maxima 1/(2 pi sqrt3) (Longuet-Higgins).""" + mu = [0.0, 0.0, 0.0, 0.0] + S = [[2.0, 0.0, 0.0, 0.0], [0.0, 3.0, 0.0, 1.0], [0.0, 0.0, 1.0, 0.0], [0.0, 1.0, 0.0, 3.0]] + W = weighted_determinant_integrals(mu, S, rules) + return {'E_absdet': sum(W['F']), 'E_F_j': W['F'], 'E_det': W['D']} + + +def mc_absdet(n, seed=2026): + """Monte Carlo cross-check of E|det H| for the unconditional Hessian (fixed seed).""" + rng = random.Random(seed) + acc = acc2 = 0.0 + for _ in range(n): + g1, g2, g3 = rng.gauss(0, 1), rng.gauss(0, 1), rng.gauss(0, 1) + h1 = math.sqrt(2.0) * g1 + g3 # Var 3 + h3 = math.sqrt(2.0) * g2 + g3 # Var 3, Cov(h1, h3) = 1 + h2 = rng.gauss(0, 1) + v = abs(h1 * h3 - h2 * h2) + acc += v + acc2 += v * v + mean = acc / n + var = acc2 / n - mean * mean + return mean, math.sqrt(var / n) + + +# ----------------------------------------------------------------------------- grids +def hole_integral(b, k, rules, Rc=5.0, h=0.2, lam_inf=None): + """2 * sum over the midpoint grid on [-Rc, Rc] x (0, Rc] of (Lambda(x) - Lambda_inf) h^2, by index. + Continuum kernel.""" + n2 = int(round(Rc / h)) + h = Rc / n2 # effective spacing: the cells tile [-Rc, Rc] x (0, Rc] exactly + n1 = 2 * n2 + tot = [0.0, 0.0, 0.0] + tot_rho = [0.0, 0.0, 0.0] + minlam, maxlam = float('inf'), float('-inf') + for i in range(n1): + x1 = -Rc + (i + 0.5) * h + for j in range(n2): + z = (j + 0.5) * h + r = lambda_at(x1, z, b, k, rules) + for q in range(3): + tot[q] += r['lambda_j'][q] - lam_inf['lambda_j'][q] + tot_rho[q] += r['rho_j'][q] - lam_inf['rho_j'][q] + minlam = min(minlam, r['lambda']) + maxlam = max(maxlam, r['lambda']) + scale = 2.0 * h * h + return {'hole_j': [scale * t for t in tot], 'hole': scale * sum(tot), + 'hole_unweighted_j': [scale * t for t in tot_rho], 'hole_unweighted': scale * sum(tot_rho), + 'grid': {'Rc': Rc, 'h': h, 'points': n1 * n2, 'symmetry': 'z -> -z'}, + 'lambda_min_on_grid': minlam, 'lambda_max_on_grid': maxlam} + + +def torus_integral(b, k, rules, L, h=0.2): + """integral over the fundamental domain [-L/2, L/2]^2 of Lambda_L(x) dx (periodized kernel), by index. + The pin direction u is the lattice axis e_1 (see kernel_derivative).""" + n2 = int(round(L / (2 * h))) + h = L / (2 * n2) # effective spacing: the cells tile [-L/2, L/2] x (0, L/2] exactly + n1 = 2 * n2 + tot = [0.0, 0.0, 0.0] + for i in range(n1): + x1 = -L / 2 + (i + 0.5) * h + for j in range(n2): + z = (j + 0.5) * h + r = lambda_at(x1, z, b, k, rules, L) + for q in range(3): + tot[q] += r['lambda_j'][q] + scale = 2.0 * h * h + return {'integral_j': [scale * t for t in tot], 'integral': scale * sum(tot), + 'grid': {'L': L, 'h': h, 'points': n1 * n2, 'symmetry': 'z -> -z', 'u': 'e_1'}} + + +# ----------------------------------------------------------------------------- controls +EXACT_COUNT = [0] +# replay tolerance for point-wise kernel values: libm/interpreter differences (Python 3.11 versus 3.12-3.14 were +# observed at 8e-9 relative on a near-pin sample) stay far below this; the mutants move values by O(1) +REPLAY_TOL = 1e-6 +SAMPLE_POINTS = [(0.15, 0.15), (0.5, 0.3), (-0.7, 0.9), (1.0, 0.0), (0.0, 1.0), (2.0, 1.5), (-1.5, 0.5), (3.0, 3.0)] + + +def controls(rules): + out = {} + # (a) exact one-site Hermite table versus the floating kernel derivatives at 0 (continuum and L = 24, 12) + table = {} + maxdev = {'continuum': 0.0, 'L24': 0.0, 'L12': 0.0, 'L6': 0.0} + for a in SITE0: + for c in SITE0: + ex = exact_one_site_cov(a, c) + table['%d%d,%d%d' % (a + c)] = str(ex) + fl = cov_entry(a, c, 0.0, 0.0) + maxdev['continuum'] = max(maxdev['continuum'], abs(fl - float(ex))) + for L, key in ((24.0, 'L24'), (12.0, 'L12'), (6.0, 'L6')): + maxdev[key] = max(maxdev[key], abs(cov_entry(a, c, 0.0, 0.0, L) - float(ex))) + require(table['20,20'] == '3' and table['30,30'] == '15' and table['10,30'] == '-3' and table['00,20'] == '-1' + and table['20,02'] == '1' and table['11,11'] == '1' and table['00,00'] == '1', 'exact Hermite one-site table') + require(maxdev['continuum'] < 1e-12 and maxdev['L24'] < 1e-9 and maxdev['L12'] < 1e-9, 'lattice one-site covariances at L = 24, 12') + require(1e-4 < maxdev['L6'] < 2e-3, 'L = 6 one-site deviation is of the expected size') + out['one_site_exact'] = {'Var_fxx': table['20,20'], 'Var_fxxx': table['30,30'], 'Cov_fx_fxxx': table['10,30'], + 'Cov_f_fxx': table['00,20'], 'Cov_fxx_fzz': table['20,02'], 'Var_fxz': table['11,11']} + out['one_site_lattice_max_dev'] = maxdev + # (b) unconditional Hessian: E|det H| = sqrt3, E F_2 = sqrt3/4, E F_1 = sqrt3/2, E det = 0 + U = unconditional_hessian_controls(rules) + require(abs(U['E_absdet'] - 4 / math.sqrt(3)) < 1e-9, 'E|det H| = 4/sqrt(3) (exact: a^2 - rho^2 decomposition)') + require(abs(U['E_F_j'][2] - 1 / math.sqrt(3)) < 1e-9 and abs(U['E_F_j'][0] - 1 / math.sqrt(3)) < 1e-9 + and abs(U['E_F_j'][1] - 2 / math.sqrt(3)) < 1e-9, 'index split 1/sqrt3, 2/sqrt3, 1/sqrt3') + require(abs(U['E_det']) < 1e-12, 'E det H = 0 (Euler characteristic of the torus)') + out['unconditional_hessian'] = U + # (c) far field: E[w_0 | U_0] / z_0 = 1 and Lambda_inf = phi(b) E|det H_b| / (2 pi) + far = {} + for b in (0.0, 1.0): + r = lambda_far(b, rules) + far[str(b)] = r + rr = lambda_at(8.0, 0.0, b, 1.0, rules) + require(abs(rr['lambda'] - r['lambda']) < 1e-9 * r['lambda'], 'Lambda(8, 0) equals the far-field value, b = %r' % b) + require(abs(rr['coupling'] - 1.0) < 1e-9, 'pin weight decouples at |x| = 8') + out['far_field'] = far + # (d) z -> -z symmetry and a few sample values + sym = 0.0 + for (x1, z) in SAMPLE_POINTS[:4]: + r1 = lambda_at(x1, z, 0.0, 1.0, rules) + r2 = lambda_at(x1, -z, 0.0, 1.0, rules) + sym = max(sym, abs(r1['lambda'] - r2['lambda']) / r1['lambda']) + require(sym < 1e-10, 'reflection symmetry z -> -z') + out['reflection_symmetry_max_rel_dev'] = sym + # (e) k-dependence enters only through the f_xxx = 12k conditioning: at |x| = 8 Lambda is k-free + a1 = lambda_at(8.0, 0.0, 0.0, 0.5, rules)['lambda'] + a2 = lambda_at(8.0, 0.0, 0.0, 2.0, rules)['lambda'] + require(abs(a1 - a2) < 1e-9 * a1, 'far field is k-free') + return out + + +def sample_table(b, k, rules, L=None): + rows = [] + for (x1, z) in SAMPLE_POINTS: + r = lambda_at(x1, z, b, k, rules, L) + rows.append({'x': [x1, z], 'lambda': r['lambda'], 'lambda_j': r['lambda_j'], 'rho': r['rho'], 'coupling': r['coupling'], 'exact_regression': r['exact_regression']}) + return rows + + +def ray_profile(b, k, rules): + """Lambda(x)/Lambda_inf along rays from the pin at angles 90 (transverse), 60, 45, 30, 0 degrees.""" + far = lambda_far(b, rules)['lambda'] + rows = [] + for ang_deg in (90, 60, 45, 30, 0): + th = math.radians(ang_deg) + for rad in (0.25, 0.5, 0.75, 1.0, 1.5, 2.0, 2.5, 3.0, 4.0): + x1, z = rad * math.cos(th), rad * math.sin(th) + if ang_deg == 0: + z = 0.0 + r = lambda_at(x1, z, b, k, rules) + rows.append({'angle_deg': ang_deg, 'r': rad, 'ratio': r['lambda'] / far, 'ratio_unweighted': r['rho'] / far}) + return rows + + +def near_zero_profile(b, k, rules): + rows = [] + for rad in (0.05, 0.1, 0.2, 0.4): + for ang_deg in (0, 30, 60, 90, 120, 150, 180): + th = math.radians(ang_deg) + x1, z = rad * math.cos(th), rad * math.sin(th) + if ang_deg in (0, 180): + z = 0.0 + r = lambda_at(x1, z, b, k, rules) + rows.append({'r': rad, 'angle_deg': ang_deg, 'lambda': r['lambda'], 'lambda_j': r['lambda_j'], 'exact_regression': r['exact_regression']}) + return rows + + +# ----------------------------------------------------------------------------- main +def full_run(fast=False): + rules = Rules() + hi_rules = Rules(32, 48, 32) + res = {'object': 'CL-C6-REMOTE-COEFF-NUMERICS-20260930-v1', 'scientific_effect': 'NONE', 'certified': False, + 'orientation': 'continuum values (Lambda_inf, c_hole, samples, profiles) hold for every pin direction u (isotropic kernel); every L-periodized value is for u = e_1 aligned with a lattice axis', + 'python_requirement': '>= 3.11; point-wise replay tolerance 1e-6 relative', + 'quadrature': {'coordinates': 'a = tr H / 2, (c, h2) polar; splits at a = 0 and rho = |a|', + 'a_gauss_legendre_per_side': 24, 'theta_trapezoid': 32, 'rho_gauss_legendre_per_side': 24, 'cut_sd': 9.0, + 'high_order_check': [32, 48, 32]}} + res['controls'] = controls(rules) + mc = mc_absdet(200000) + res['controls']['mc_absdet_unconditional'] = {'mean': mc[0], 'se': mc[1], 'n': 200000, 'seed': 2026} + # quadrature-order check at the sample points + qdev = 0.0 + for (x1, z) in SAMPLE_POINTS: + a = lambda_at(x1, z, 0.0, 1.0, rules)['lambda'] + c = lambda_at(x1, z, 0.0, 1.0, hi_rules)['lambda'] + qdev = max(qdev, abs(a - c) / c) + res['quadrature']['max_rel_dev_vs_high_order_at_samples'] = qdev + res['samples'] = {} + res['samples_L6'] = {} + res['near_zero_profile'] = {} + res['ray_profile'] = {} + res['far_field'] = {} + res['hole'] = {} + res['torus'] = {} + res['remote_total'] = {} + ks = (0.5, 1.0, 2.0) + bs = (0.0, 1.0) + grid_h = 0.4 if fast else 0.15 + for b in bs: + far = lambda_far(b, rules) + res['far_field'][str(b)] = {'lambda_inf': far['lambda'], 'lambda_inf_j': far['lambda_j'], 'E_F_j_given_level': far['E_F_j'], + 'p_inf': phi(b) / (2 * math.pi)} + for k in ks: + key = 'k=%s,b=%s' % (k, b) + res['samples'][key] = sample_table(b, k, rules) + res['samples_L6'][key] = sample_table(b, k, rules, 6.0) + res['near_zero_profile'][key] = near_zero_profile(b, k, rules) + res['ray_profile'][key] = ray_profile(b, k, rules) + H = hole_integral(b, k, rules, Rc=6.0, h=grid_h, lam_inf=far) + res['hole'][key] = H + T6 = torus_integral(b, k, rules, 6.0, h=grid_h) + res['torus'][key] = {'L=6': T6} + rt = {} + for L in (6.0, 12.0, 24.0): + approx = L * L * far['lambda'] + H['hole'] + rt['L=%g' % L] = {'integral_X_Lambda_approx': approx, 'k_integral': k * approx} + rt['L=6']['integral_X_Lambda_periodized_direct'] = T6['integral'] + rt['L=6']['k_integral_direct'] = k * T6['integral'] + res['remote_total'][key] = rt + print('done', key, 'hole', H['hole'], 'L6 direct', T6['integral'], 'approx', 36 * far['lambda'] + H['hole'], flush=True) + # grid-convergence, quadrature-order and tail checks for one parameter set; direct periodized L = 12 integral + far0 = lambda_far(0.0, rules) + far0_hi = lambda_far(0.0, hi_rules) + Hc = hole_integral(0.0, 1.0, rules, Rc=6.0, h=0.3, lam_inf=far0) + Hc_hi = hole_integral(0.0, 1.0, hi_rules, Rc=6.0, h=0.3, lam_inf=far0_hi) + Hf = hole_integral(0.0, 1.0, rules, Rc=6.0, h=0.1, lam_inf=far0) if not fast else Hc + Ht = hole_integral(0.0, 1.0, rules, Rc=5.0, h=0.3, lam_inf=far0) + T12 = torus_integral(0.0, 1.0, rules, 12.0, h=0.2) + res['grid_convergence'] = {'k=1.0,b=0.0': {'hole_h=0.3,Rc=6': Hc['hole'], 'hole_h=0.3,Rc=6,high_order': Hc_hi['hole'], + 'hole_h=0.3,Rc=5': Ht['hole'], 'hole_h=0.1,Rc=6': Hf['hole'], + 'hole_h=%s,Rc=6' % grid_h: res['hole']['k=1.0,b=0.0']['hole'], + 'hole_j_h=0.1,Rc=6': Hf['hole_j'], + 'torus_L=12_direct_h=0.2': T12['integral'], + 'torus_L=12_approx': 144 * far0['lambda'] + res['hole']['k=1.0,b=0.0']['hole'], + 'lambda_inf_high_order': far0_hi['lambda']}} + res['exact_regression_points'] = EXACT_COUNT[0] + return res + + +def check_run(): + rules = Rules() + ctl = controls(rules) + path = os.path.join(HERE, 'RESULTS.json') + require(os.path.exists(path), 'RESULTS.json present') + with open(path) as fh: + ref = json.load(fh) + for b in (0.0, 1.0): + far = lambda_far(b, rules) + require(abs(far['lambda'] - ref['far_field'][str(b)]['lambda_inf']) < 1e-10, 'far-field replay b = %r' % b) + closed = far_field_closed_form(b) + require(abs(far['lambda'] - closed['lambda']) < 1e-10 and max(abs(far['lambda_j'][j] - closed['lambda_j'][j]) for j in range(3)) < 1e-10, + 'far field equals its closed form, b = %r' % b) + require(abs(ref['far_field']['0.0']['lambda_inf'] - (2 * math.sqrt(2) - 1) / (2 * math.pi) ** 1.5) < 1e-10, 'Lambda_inf(0) = (2 sqrt2 - 1)/(2 pi)^(3/2)') + for key, k in (('k=1.0,b=0.0', 1.0), ('k=0.5,b=1.0', 0.5)): + b = 0.0 if key.endswith('0.0') else 1.0 + rows = sample_table(b, k, rules) + for r, rr in zip(rows, ref['samples'][key]): + require(abs(r['lambda'] - rr['lambda']) <= REPLAY_TOL * max(abs(rr['lambda']), 1e-6), 'sample replay %s at %r' % (key, r['x'])) + rows6 = sample_table(b, k, rules, 6.0) + for r, rr in zip(rows6, ref['samples_L6'][key]): + require(abs(r['lambda'] - rr['lambda']) <= REPLAY_TOL * max(abs(rr['lambda']), 1e-6), 'L = 6 sample replay %s at %r' % (key, r['x'])) + # a near-axis point that needs the decimal regression + r = lambda_at(0.1, 0.0, 0.0, 1.0, rules) + require(r['exact_regression'] and r['lambda'] < 1e-20, 'near-axis point handled by the decimal regression and exponentially small') + for row in ref['near_zero_profile']['k=1.0,b=0.0']: + if row['r'] == 0.1 and row['angle_deg'] == 0: + require(abs(row['lambda'] - r['lambda']) <= REPLAY_TOL * max(abs(r['lambda']), 1e-30), 'near-zero replay') + gs = grid_summary(ref) + print(json.dumps({'check': 'ok', 'controls': list(ctl.keys()), 'grid_summary': gs, 'mutant': MUT})) + + +def grid_summary(ref): + """Replay of the NOTE section 5 summary arithmetic from the stored grid_convergence block (k = 1, b = 0): + the Rc = 5 / Rc = 6 difference at h = 0.3, the finest-grid difference, the three-point power-law model + A(h) = A_0 + C h^p on the non-nested spacings 0.3, 0.15, 0.1 (a floating extrapolation, not a bound), the + default/high-order difference and the L = 12 direct/additive discrepancy.""" + g = ref['grid_convergence']['k=1.0,b=0.0'] + a3, a15, a1 = g['hole_h=0.3,Rc=6'], g['hole_h=0.15,Rc=6'], g['hole_h=0.1,Rc=6'] + d_rc = g['hole_h=0.3,Rc=5'] - a3 + d_fine = a15 - a1 + ratio = (a3 - a15) / (a15 - a1) + f = lambda p: (0.3 ** p - 0.15 ** p) / (0.15 ** p - 0.1 ** p) - ratio + lo, hi = 1.0, 5.0 + require(f(lo) * f(hi) < 0, 'power-law exponent bracketed in (1, 5)') + for _ in range(200): + mid = 0.5 * (lo + hi) + if f(lo) * f(mid) <= 0: + hi = mid + else: + lo = mid + p = 0.5 * (lo + hi) + C = (a3 - a15) / (0.3 ** p - 0.15 ** p) + A0 = a15 - C * 0.15 ** p + out = {'rc5_minus_rc6_h0.3': d_rc, 'h0.15_minus_h0.1': d_fine, 'model_p': p, 'model_A0': A0, + 'model_residual_h0.15': abs(a15 - A0), 'model_residual_h0.1': abs(a1 - A0), + 'default_minus_high_order_h0.3': a3 - g['hole_h=0.3,Rc=6,high_order'], + 'L12_direct_minus_additive_rel': (g['torus_L=12_direct_h=0.2'] - g['torus_L=12_approx']) / g['torus_L=12_direct_h=0.2'], + 'certified': False} + require(abs(d_rc + 9.46e-5) < 5e-8, 'Rc = 5 minus Rc = 6 at h = 0.3 is -9.46e-5 (NOTE section 5)') + require(abs(d_fine - 1.875e-4) < 5e-7, 'h = 0.15 minus h = 0.1 is 1.875e-4 (NOTE section 5)') + require(abs(p - 2.63) < 5e-3 and abs(A0 + 1.63143) < 5e-5, 'power-law model p = 2.63, A_0 = -1.63143 (NOTE section 5)') + require(abs(out['model_residual_h0.15'] - 2.86e-4) < 5e-6 and abs(out['model_residual_h0.1'] - 9.83e-5) < 5e-7, + 'model residuals 2.9e-4 (h = 0.15) and 9.8e-5 (h = 0.1) (NOTE section 5)') + require(abs(out['default_minus_high_order_h0.3']) < 5e-8, 'default/high-order difference below 5e-8 (NOTE section 5)') + require(abs(out['L12_direct_minus_additive_rel'] - 2.04e-5) < 1e-7, 'L = 12 direct/additive discrepancy 2.0e-5 relative (NOTE section 4)') + return out + + +def main(): + global MUT + ap = argparse.ArgumentParser() + ap.add_argument('--check', action='store_true') + ap.add_argument('--fast', action='store_true', help='coarser grid (development only)') + ap.add_argument('--mutant', choices=MUTANTS) + args = ap.parse_args() + MUT = args.mutant + if args.check: + check_run() + return + res = full_run(fast=args.fast) + with open(os.path.join(HERE, 'RESULTS.json'), 'w') as fh: + json.dump(res, fh, indent=1, sort_keys=True) + fh.write('\n') + print('wrote RESULTS.json') + + +if __name__ == '__main__': + main() diff --git a/frontiers/extreme_cluster_sampling_20260930/COMPANION_RATIO.md b/frontiers/extreme_cluster_sampling_20260930/COMPANION_RATIO.md new file mode 100644 index 000000000..56348f676 --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/COMPANION_RATIO.md @@ -0,0 +1,236 @@ +# Addendum: an explicit unit-square law for the whole extreme doublet + +Object: OA-EXTREME-CLUSTER-SAMPLING-20260930-D1. +Author: OpenAI / GPT-6 Astra Pro. New conditional derivation; nonauthor review OPEN. +Scientific effect NONE. PROOF.md remains byte-identical. This addendum consumes +that manuscript's E12-E19/E29 as candidate inputs; their review is not inferred +from finite tests. It introduces no additional Gaussian source assumption. + +## 1. The law and its exact domain + +Consider the limiting ordinary extreme cluster of PROOF Theorem2, conditioned +on being a doublet. Its outer-root shape has density Q/J on S2_outer, with +J=1083417/280. Write p=-u, A=12p^2-3 and ell=v/A. Define + + rho=|Z_companion/Z_outer| in (0,1), + w=2ell, + z=(1+rho)(w-1)/(2rho). (D1) + +The map is a smooth bijection, up to the boundary null sets already stated in +PROOF, between S2_outer and the OPEN UNIT SQUARE 0p_c, equivalent to w>1 for 01,00 directly, +and eta_in(u,v)=(-p,A ell) has absolute Jacobian A. At fixed w, + + |partial p/partial rho|/2=(w^2-1)/(8rho^2 w). + +Together with Q=A^3(4ell^2-1)(4p ell-1), the density in (rho,w) is + + 81(1+rho)(w-1)^7(w+1)^7 + *(rho*w-rho+w+1)^4(rho*w+rho+w-1)^4 + /(4096rho^11 w^9). (D6) + +Insert w=1+2rho*z/(1+rho) and its Jacobian2rho/(1+rho) to obtain D2. These +identities can be checked by direct substitution; test_ratio_law.py also uses +independent exact two-variable differentiation of the inverse map to check +L=Q|det d(u,v)/d(rho,z)| on72 rational interior cases. Integration of D2 is J +because this is a bijective change of variables from the source outer sector, +not because of numerical normalization. + +## 3. A new small-companion asymptotic + +Let f_rho be the marginal density of rho under the doublet law. Set + + H0=integral_0^1 z^7(z+1)^4 dz=6401/3960, + Csmall=82944 H0/J=137647104/3972529. (D7) + +The rational factor multiplying165888 rho z^7(z+1)^4 in D2 tends uniformly +to1 as rho->0. Its denominator is bounded away from zero on the closed square. +Therefore + + f_rho(rho) ~ (275294208/3972529) rho, + P(rho0, then the microscopic extreme threshold->infinity, then epsilon->0. +It is not a uniform triple-scale assertion for the original field. + +More precisely, conditional on rho Zsmall^2, + eta_in/rho^3 -> 2 Zsmall^2(Zsmall+1). (D10) + +Equivalently eta_in/epsilon^3 has limit2 Qsmall^3 Zsmall^2(Zsmall+1). The small +companion thus approaches the upper height boundary with a CUBIC radius-ratio +scale while the outer height remains nonzero. This describes normalized height +marks, not an elder-pairing or actual finite-r persistence-lifetime theorem. + +The limiting outer-height density in D10 is particularly simple: + + e^3(1+sqrt(e))^4/(2H0), 01-, continuity of D2 and integration on the bounded +z interval give + + f_rho(1-)=857223/2889112-(945/361139)log2. (D12) + +For verification L(1,z)=81 z^7(z+2)^7/[8(z+1)]. Put x=z+1 and expand +(x^2-1)^7/x on[1,2]: its integral is2571669/2240-(81/8)log2. Dividing by J +gives D12. The equal-radius boundary itself still has probability zero. + +## 4. A rejection sampler with a certified ideal-real acceptance rate + +Define + + a(rho,z)=(rho*z+1)^4(rho*z+rho+1)^7 + /[(rho+1)^5(2rho*z+rho+1)^9]. (D13) + +It satisfies00 limit is taken before the new radius->infinity limit. + +Isolation: dedicated new runtime directory; upstream repository checkout is +not available (public raw-host DNS failed). GitHub writes will be additive on +a new branch. No frozen packet or another writer branch is edited. + +Execution: foreground inline; no subagent or independent-provider activity is +invented. Dylan's explicit autonomous continuation supplies the task authority. + +1. [x] Write fail-first rational tests of the root involution, anchor and constants. +2. [x] Implement stdlib Fraction controls with a separate Laurent integration + certificate and rational lower/upper bounds for log(2). +3. [x] Prove and document the root-sector partition, once-per-cluster change of + variables, maximum/minimum tail constants, joint anchored law, height + ordering, and deliberately independent-replica maximum consequence. +4. [x] Test source custody against real Git fixtures, including replacement refs, + noncommit objects, symlinks, nested cwd and duplicate/empty inventories. +5. [x] Verify both Python modes, deterministic outputs, semantic negative controls + and complete flat packet inventory. Publish all exact source pins. +6. [x] Create PR169, read back exact bytes/identities, request nonauthor + algebra/integration and probability reviews, and observe initial-head checks. + +Review focus: strict height endpoints; denominator-zero and equal-radius sets; +point versus configuration multiplicity; physical radius versus |Z|; the +order of limits; latent parameter/orientation bias; finite controls versus +analytic proofs; all source and provider-exposure limitations. + +Scientific effect NONE. No self-acceptance, self-merge, global-register or prize +change. No finite-r spatial rate, growing-volume result, numerical Gaussian +coefficient or independence inside the original field is promised. + +Ruling: no duplicate review of PR166 A/B/C. Its explicit maximum-coefficient +and ordinary-cluster sampling nonclaims define this separate consumer. +Ruling: preserve source commits even when a metadata-only peer successor appears; +new target source authentication remains pinned to the consumed mathematical bytes. + +7. [x] Derive the separate companion-ratio unit-square law, its small-ratio/height + asymptotic and ideal-real rejection sampler; add fail-first exact controls. +8. [ ] Publish the additive successor with original main proof/code unchanged, + request separate Slice D review, and read its actual hosted results. + +Ruling: the companion sampler is explicitly ordinary floating point, while all +normalization/envelope/acceptance constants are checked by exact rational algebra. +Ruling: adopt the peer's physical-order counterexample as credited support only; +it validates the scope boundary and is not an independent mathematical verdict. diff --git a/frontiers/extreme_cluster_sampling_20260930/PROOF.md b/frontiers/extreme_cluster_sampling_20260930/PROOF.md new file mode 100644 index 000000000..19a6044f2 --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/PROOF.md @@ -0,0 +1,486 @@ +# Extreme whole-cluster sampling: exact maximum/minimum tails and the anchored doublet law + +Object: OA-EXTREME-CLUSTER-SAMPLING-20260930-v1. +Author: OpenAI / GPT-6 Astra Pro. Foreground continuation authorized by Dylan. +Disposition: AUTHOR-SIDE CONDITIONAL PROOF CANDIDATE; NONAUTHOR REVIEW OPEN. +Scientific effect NONE. No earlier proof, review, global register, Boolean, +prize or integration disposition is changed. Same-provider/source exposure is +not organizational independence. Finite checks below are not analytic acceptance. + +## 1. Exact object and source boundary + +Fix d>=2, L>0, b in R, k>0 and one axial/transverse orthonormal frame. This +paper concerns the ALREADY-FORMED limiting microscopic cubic measure M, not +the full finite-r Gaussian field and not its remote singleton component. +The limit that forms M is taken BEFORE the radius t tends to infinity. + +Sources and their complete identities are in SOURCES.json: + +[R] PR166 RADIAL_TAIL.md, exact head80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337, +blob0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4, 12533 bytes. Consume the exact +root-parametrized integral R4-R12, its Gaussian majorant, the point tail R13, +and the height moment identities H3-H4. Its separate reviews B/C bind these +unchanged bytes. This paper adds a once-per-cluster selector; R does NOT already +claim the coefficient of the maximum or an ordinary cluster-selected H1 law. + +[M] PR166 TWO_SCALE_LAW.md at the same head, bloba32fd5f7d941bbe1fe943df045b1e0fbec8d691c, +17139 bytes: only the definition M1-M2 of the explicit microscopic measure and +its embedding in the fixed original frame. The new two-scale convergence theorem +of that file is NOT a premise of the calculation here. + +[SC] PR162 PROOF.md at382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf, +blob16c56821b52fd76b0be791622b9c3809eafde75a: the explicit Gaussian/spectral M, +its finite nonempty mass, and the contact-density bound. This conversation +AUTHORED SC; citing it does not independently revalidate its parent premises. + +[CUB] PR158 PROOF.md atbfcc67dc8957b3ed1fbb789e5a0305e98d7359c8, +blobbb446d08db8a944537a743ad550b88c1c2ad5758: the exact deterministic cubic +classifier only. Its Gaussian-sector statement is not consumed. This conversation +authored CUB. R and M are also OpenAI-authored, in a separate active lane. + +All Gaussian conditions remain conditional on those exact sources. No whole +review disposition is inferred from these source-local uses. + +For theta=(s,h,O,tau), the hard curvatures satisfy 00 is finite, with total mass +A_micro=a1+a2>0, and n belongs to {0,1,2} almost everywhere. + +The unsheared cubic and its physical limiting root position are + + P(X,Z)=2kX^3-3kX/2-k/2+sZ^2/2+(a/2)(X^2-1/4)Z + +(beta/2)XZ^2+(c/6)Z^3, + z_i=X_i e_axis+Z_i e_soft(O), R_i=|z_i|, + eta_i=-P(X_i,Z_i)/k in (0,1). (E2) + +Only additional strict-window roots are included, not the pins. Every such +root has full limiting index d-1. Define R_max on nonempty configurations; +on doublets also define R_min. Let + + F(t)=integral sum_i 1{R_i>t} dM, + G(t)=M(n>0,R_max>t), + G2(t)=M(n=2,R_max>t), B2(t)=M(n=2,R_min>t). (E3) + +F is a POINT intensity. G is a CLUSTER tail. Exactly, for every t, + + F(t)=G(t)+B2(t). (E4) + +This elementary multiplicity identity is the distinction missing from a +naive transfer of the point-selected law to whole clusters. + +## 2. Main constants and results + +Let J_cusp and gamma(a) be the positive finite integral and physical-shear +factor from R12: + + gamma(a)=sqrt(1+(a/(12k))^2), + J_cusp=integral H(h) gamma(a)^11 + h0(A0,Rot_O tau(a,a^2/(12k),a^3/(144k^2),xi)) da dxi dh dO. + +Set + + C0=(216/11) k^7 (c_m/z0) J_cusp, + I=246528/35, + J=1083417/280, + D=27066286003/223205220-(79298560/4782969) log(2), + I_max=I-D. (E5) + +Here D is a universal scalar shape integral, NOT a hard Hessian or the cubic +coefficient called D in CUB/R. In Sections3-4 the latter is written D_cub. + +**Theorem 1 (whole-cluster tail coefficients).** All the constants multiplying +C0 below are strictly positive, and + + F(t) ~ C0 I t^-11, [the prior R result] + G(t) ~ C0 (I-D) t^-11, + G2(t) ~ C0 J t^-11, + B2(t) ~ C0 D t^-11, + M(n=1,R_max>t) ~ C0 (I-D-J) t^-11. (E6) + +In particular the maximum/point coefficient ratio is the exact universal number + + theta_star=(I-D)/I + =1545114756173/1572181042176 + +(10841600/4605999147)log(2). (E7) + +This ratio is NOT asserted to be a temporal extremal index of the original +field. There is no stationary sequence of pin pairs or mixing assertion here. + +Under the ordinary microscopic cluster probability law +pi=A_micro^-1 1{n>0}dM, conditioning on R_max>t gives + + P_pi(n=2 | R_max>t) -> J/(I-D), + P_pi(K_t=2 | R_max>t) -> D/(I-D), + E_pi[K_t | R_max>t] -> I/(I-D), (E8) + +where K_t is the number of points above threshold t. A double cluster need +not have both of its points above the threshold. Exact rational/logarithmic +interval certificates, not numerical Gaussian integration, give + + 0.984415773209943375 < theta_star < 0.984415773209943376, + 0.558034234065747870 < J/(I-D) < 0.558034234065747871, + 0.015830939745347846 < D/(I-D) < 0.015830939745347847. (E9) + +**Theorem 2 (anchored extreme configuration).** Section6 specifies the full +weak limit of a cluster under pi(. | R_max>t), with its positions divided by t, +its original normalized heights, and its maximum-radius root distinguished. +The radial ratio has Pareto exponent11 and is independent of its anchored shape, +sign and cusp parameters. The limiting doublet is antipodal, has radius ratio +rho in (0,1), and its farther saddle has strictly GREATER downward height eta, +i.e. it lies lower in the height window. Shape/count/height/radius-ratio laws +are universal for these fixed-parameter microscopic extremes; the orientation +and tail amplitude generally retain model dependence. + +Theorems1-2 are new consumers, not part of R's already reviewed point theorem. + +## 3. Companion-root involution and the once-per-cluster shape domain + +Use R's shear invariants + + u=X+aZ/(12k), x=-s>0, + b0(u)=3-12u^2, v=xZ^2/k, + B=k b0(u)/Z^2, D_cub=k(v-b0(u)u)/Z^3. (E10) + +A strict typed window root is parametrized exactly by Z!=0 and + + S={-3/20. (E11) + +The letter S denotes this bounded SHAPE region, not the conditioned saddle. + +Put b=b0(u) temporarily and define + + L=4v^2-8buv+b^2, + t_c=(4v^2-b^2)/L, + u_c=[2bv-u(4v^2+b^2)]/L, v_c=v t_c^2. (E12) + +Off L=0, (u_c,v_c,t_c Z) is the other extra stationary root. Its membership +in S determines whether it is also counted. Thus n=2 exactly when (u_c,v_c) +is in S. L=0 is the degree-drop locus: the second root in this parametrization +is at infinity rather than a finite second point. It is an algebraic null set. +No contribution is assigned to a fictitious point there. + +Proof of E12. At Z'=tZ the stationary line and conic reduce to + + b u'+(v-bu)t=v, 12(u')^2+b t^2=3. + +When b!=0, substitute u'=(v-(v-bu)t)/b. The resulting quadratic has one root +1; its other root is the ratio of its constant to its leading coefficient: + + t_c=(12v^2-3b^2)/(12(v-bu)^2+b^3) + =(4v^2-b^2)/(4v^2-8buv+b^2). + +Substitution gives u_c and then v_c. At b=0 the direct equations give the +same formula wherever L!=0; the b=0 slice is null in any case. Since the +construction swaps the two roots, applying it twice returns the original +(u,v), with t_c t_cc=1. Direct useful identities are + + b0(u_c)=b t_c^2, + v_c-b0(u_c)u_c=(v-bu)t_c^3. (E13) + +These preserve the same x,B,D_cub under E10 with Z'=t_cZ. + +### 3.1 Explicit doublet and anchor domains + +For doublets b<0. Write p=-u, A=12p^2-3=-b>0. Define + + V_+(p)=[36p^2-12p-3+3 sqrt((2p-1)^3(18p+7))]/4, + L_p=A/2, U_p=3+6p. + +Up to boundary null sets, the doublet region splits into + + S2_outer: + 1/2=0 cannot meet v<-b. +For b<0, u=-p with 1/20. If p<1, +H_p(A)=18(p-1)(2p-1)^2(2p+1)<0 and A=1, U_p<=A and +H_p(U_p)=36(p-1)(2p+1)^2>=0, so the interval is (L_p,U_p). + +Both counted roots are on the negative branch of the hyperbola. The stationary +line minus that strictly concave branch is strictly convex and is negative +at Z=0 (its line intercept is -v/A<-1/2). If it has two zeros, zero lies strictly +between them: a strictly convex function with two zeros is positive outside +their interval. Hence t_c<0 on doublets. Since 4v^2-b^2>0, L<0 there. Consequently + + |t_c|>1 iff v>b u=A p, + |t_c|<1 iff v0, so +L_p=1, Ap-U_p=3(p-1)(2p+1)^2>=0. These +observations prove E14. Thus S_star contains exactly one asymptotic maximum +anchor per cluster, apart from the harmless tie curve. + +### 3.2 Height ordering and the intensity-change identity + +On a doublet put ell=v/A (>1/2). A direct substitution in E12 gives + + eta_c-eta=2(ell-p)(4p ell-1)^3/(4ell^2-8p ell+1)^2. (E17) + +Here p>1/2 and 4p ell>1. The outer anchor has ell

0 over a nonempty open set. I-D-J is positive because the single-root shape +sector is also nonempty and open. + +For later height moments define D_j=integral_(S2_inner) eta^j Q du dv. An explicit +primitive for the same rationalization is + + P_j(p,v)=sum_(ell=0..j) binom(j,ell)(1/2-p)^(j-ell)/6^ell + *[16p v^(ell+4)/(ell+4)-4A v^(ell+3)/(ell+3) + -4p A^2 v^(ell+2)/(ell+2)+A^3 v^(ell+1)/(ell+1)]. (E24) + +E22-E23 then give + + D_0=27066286003/223205220-(79298560/4782969)log2, + D_1=1259212131941593/15982386572880 + -(3186360320/387420489)log2, + D_2=72706616008642669/1315122095139840 + -(15686696960/3486784401)log2. (E25) + +For certified decimal bounds use log2=2 sum_(j>=0)3^(-(2j+1))/(2j+1). +After N terms the omitted positive remainder is bounded above by + + 2*3^(-(2N+1))/[(2N+1)(1-1/9)]. (E26) + +All interval endpoints are rational. The code uses60 terms, propagates coefficient +signs and positive divisions, and rounds the displayed lower/upper decimals +OUTWARD. No floating-point quadrature is used to establish E9 or E25. + +## 5. Proof of the maximum and minimum tails + +R's exact root-resolved integral is, with the physical radius retained, + + (108 k^7 c_m/z0) Q(u,v) H(h) |Z|^-12 + h0(A0,Rot_O tau(a,a^2/(12k)+k b0/Z^2, + a^3/(144k^2)+a b0/(4Z^2)+2k(v-b0u)/Z^3,xi)) + du dv dZ da dxi dh dO. (E27) + +It counts a cluster once for each of its qualifying roots. For G(t), multiply +by 1{R_i>t} times the indicator that this root has the greatest ACTUAL physical +radius, dividing by the number of ties if necessary. The sum over roots is +exactly 1{R_max>t}. This is a Borel operation on a finite configuration and +introduces no extra determinant or normalizer. + +Put Z=t z. For almost every shape, off the algebraic exceptional sets L=0, +eta_c in{0,1} and |t_c|=1, the actual maximum indicator tends to 1_(S_star). +Indeed, R_i/t->gamma(a)|z| and R_c/t->gamma(a)|t_c z| if a counted companion +exists. The bounded u and u_c offsets disappear, but a is NOT integrated away: +its gamma factor remains inside the radius condition. + +The dominating argument is exactly the source R11-R12 argument with one more +factor between0 and1. For t>=3, |u|<=3/2 and R_i>t imply +|z|>1/(2gamma(a)). Orthogonal Gaussian norm equivalence bounds the density by +C exp[-c(|h|^2+a^2+|xi|^2)]. The z-integral of |z|^-12 over that set costs +only C gamma(a)^11; S is bounded, H(h) is polynomial, and the resulting +Gaussian integral is finite. Thus dominated convergence applies, including +the unbounded a and hard/tensor coordinates. No inverse hard eigenvalue enters. + +The limiting z-integral on either sign, with the radius condition retained, +is gamma(a)^11/11. Both signs, the source factors108 and the cusp integral +yield C0. The shape integral is I_max=integral_(S_star)Q=I-D, proving +G(t)~C0(I-D)t^-11. Multiplying by 1{n=2} gives shape S2_outer and coefficient +C0 J for G2. E4 and the known point tail then give the B2 coefficient C0 D. +Subtract G2 from G to get the singleton coefficient. This proves E6, and all +ratios E7-E9 follow since the denominators have strictly positive limits. + +There is no assumption that the seed root's |Z| ordering equals physical +radius ordering at finite t. It need only agree almost everywhere in the limit; +the exact physical selector was used before taking that limit. + +Moment consequences: under the appropriate normalized nonempty/doublet measure, +R_max and R_min both have moment threshold11. For R_min the doublet condition is +understood. The capped unnormalized moments have exact boundary coefficients + + integral_(n>0) min(R_max,t)^11 dM ~11 C0(I-D)log t, + integral_(n=2) min(R_min,t)^11 dM ~11 C0 D log t. (E28) + +These follow from Tonelli and E6. They are not uniform finite-r spatial moments. + +## 6. Full anchored extreme law and its universal consequences + +Distinguish the actual maximum-radius root, using any Borel tie rule. Under +pi(. | R_max>t), retain its seed shape(u,v), q=R_max/t, sign epsilon=sign Z, +and remaining parameters(a,h,O,xi). The limiting joint probability is the product +of + + Q(u,v)1_(S_star)/I_max du dv; + 11q^-12 dq on q>1; + the uniform sign law on{-1,1}; + H(h)gamma(a)^11 h0(A0,Rot_O tau_cusp)/J_cusp da dxi dh dO. (E29) + +Proof: insert a bounded continuous function of these variables into E27 with +the exact maximum selector. The same majorant works. The limiting change +q=gamma(a)|z| produces gamma(a)^11 q^-12 dq, so normalization by the maximum +coefficient gives E29. The maps and the selector are continuous off the +already excluded null curves. This proves weak convergence, not total variation +of spatial laws. No uniformity in escaping model parameters is asserted. + +Let + + e_*(a,O)=[-a e_axis/(12k)+e_soft(O)]/gamma(a). + +If the selected shape is a singleton, the limiting marked configuration is +one point at epsilon q e_* with height eta=u+1/2+v/6. If it is a doublet, +put rho=|t_c|<1, eta_c=u_c+1/2+v_c/6. The full configuration, in positions +divided by the threshold t, is + + {(epsilon q e_*, eta, d-1), + (-epsilon q rho e_*, eta_c, d-1)}. (E30) + +E17 proves eta_c1. +Combining E31 with P(n=2)=J/I_max gives D/I_max in E8 once more. This reconciles +the two-point-cluster frequency with the much smaller double-exceedance frequency. + +### 6.1 The maximum anchor's height is not the old intensity-selected height + +Let H_eta(e) be R's nonnegative height numerator (H1-H2), with integral I. +The height density for the maximum anchor is instead + + [H_eta(e)-6 integral_(1/2)^1 Q(-p,6(e+p-1/2)) + 1{A(p)p<6(e+p-1/2)0, E6 gives + + P(max_(i<=n) R_max(C_i) <= a_n x) + =[1-P_pi(R_max>a_n x)]^n -> exp(-x^-11). (E34) + +This is the standard iid extreme-value consequence with the newly identified +cluster coefficient. Normalizing by the POINT tail coefficient instead gives +exp(-theta_star x^-11); it is not the same normalization. No new coefficient +convergence rate, simultaneous n/r limit, within-field Poisson assertion or +stationary-sequence extremal-index theorem follows from E34. + +## 8. Review obligations and nonclaims + +A: E10-E19, companion involution, strict-window partition, tie/degree-drop +boundaries, exact Jacobian/multiplicity and the height ordering. +B: E20-E26 and E33, the rationalized Laurent integral, all logarithmic constants, +certified rational intervals and the independent finite integration controls. +C: E27-E34, actual physical maximum selection, domination, full anchored weak +law, the differing sampling measures, min/max/exceedance coefficients and iid +scope. These are not waived by tests or parent review acceptance. + +No global scientific-status change, numerical evaluation of J_cusp, finite-r +uniform spatial tail/moment, new elder-pairing law, dependence theorem across pin +pairs, growing-volume theorem, or Lean formalization is claimed. Counterexamples +to shortcuts: F counts both above-threshold roots whereas G counts the cluster +once; replacing G by F sets theta_star=1 incorrectly. Omitting the maximum +selector and summing over roots gives the prior point-intensity height law, not +E32. Uniformly choosing one root inside each uniformly selected cluster is yet +another sampling convention and must not be identified with point intensity. Dropping log2 from E21 changes a positive exact shape integral. A single +1/r rescaling still cannot retain the remote singleton population. diff --git a/frontiers/extreme_cluster_sampling_20260930/README.md b/frontiers/extreme_cluster_sampling_20260930/README.md new file mode 100644 index 000000000..e5c71f7ec --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/README.md @@ -0,0 +1,130 @@ +# Extreme whole-cluster sampling + +OpenAI / GPT-6 Astra Pro. New source-bound conditional proof candidate. +**Nonauthor analytic review OPEN. Scientific effect NONE. No self-merge.** + +PR166 identified the point-INTENSITY radius tail and extreme-height law. This +packet keeps the same limiting microscopic measure and selects each extreme +CLUSTER once. Those are different probability laws: a cluster with both roots +above a threshold contributes twice to point intensity but once to a maximum. + +Start with PROOF.md. It supplies an explicit companion-root involution, the +larger-radius anchor domain, exact maximum/minimum tail coefficients, a joint +anchored extreme law, strict height ordering in doublets, and certified universal +constants. Generic Palm/anchor and iid extreme-value principles are credited in +RECONNAISSANCE.md; no claim of inventing those general principles is made. + +## Main candidate results + +Let C0 be the common positive spectral/Gaussian amplitude defined in PROOF E5. +Put + + I=246528/35, + J=1083417/280, + D=27066286003/223205220-(79298560/4782969)log(2). + +The point tail is the prior C0 I t^-11. New proposed tails are + + cluster maximum: C0(I-D)t^-11; + doublet maximum: C0 J t^-11; + doublet minimum: C0 D t^-11. + +Thus maximum/point coefficient ratio is approximately0.9844157732099434; +a cluster conditioned to have an extreme maximum is a doublet with probability +about0.5580342340657479, whereas BOTH points exceed that same threshold with +probability about0.01583093974534785. These are not interchangeable statistics. +All displayed decimals have rational interval certificates, not quadrature fits. +No numerical evaluation of the Gaussian amplitude C0 is claimed. + +The maximum anchor's limiting downward height has mean about0.8195869887849185, +not the old point-intensity mean. On an extreme doublet its companion lies in +the opposite direction, is closer, and has a strictly smaller downward height. +The cluster radial maximum ratio is independently Pareto(11). See the full +parameter/shape law and qualifications in PROOF E29-E33. + +## Exact scope + +Every theorem here concerns the ALREADY-FORMED limiting microscopic measure: +**first r->0, then t->infinity**. Fixed d>=2,L>0,b real,k>0 and frame. The full +original endpoint normalizer is retained through its source z0. The remote +singleton population is not part of this measure. No finite-r uniform spatial +moment, convergence rate, original-field independence, numerical Gaussian +coefficient, new elder pairing, Lean formalization or global-register change. + +The source use is explicitly conditional: RADIAL root integral/majorant and +point theorem, MICRO definition M1-M2 only, SC's spectral measure/finite nonempty +mass, and CUB's deterministic classifier. MICRO's separate two-scale theorem, +PR159's alternative proof and PR157's exponential theorem are not premises. +The author of this thread authored SC/CUB; no independent revalidation is implied. + +## Reproduction + +Python standard library, compatible with Python3.11+. Full project source +verification requires a Git checkout with all four exact commit/path identities: + + python -B -S frontiers/extreme_cluster_sampling_20260930/verify.py + +The hosted workflow fetches the named public source commits, verifies every +historical path/blob/byte/SHA256, checks the complete packet, runs both Python +modes and the semantic negative controls. Git replacement refs are disabled, +annotated tags/trees cannot masquerade as commits, and source ancestors/leaves +are checked for actual regular-file/tree modes. + +Standalone replay without actual upstream Git objects: + + python -B -S verify.py --local-only + +This explicitly reports source_pins_checked=false. It still runs all real-Git +fixture regressions, but those authenticate test fixtures, not project sources. + +`extremes.py` is the independent exact control entry point. It uses Fraction +arithmetic for companion roots, strict domains, the anchor partition, Laurent +integration and outward log(2) enclosures. `test_extremes.py` includes an independent +line/conic root oracle, exact automatic differentiation of the involution Jacobian, +and dense interpolation checks of the Laurent integrand. Tests do not prove the +Gaussian domination, multiplicity change of variables, or weak convergence. + +## Review requested + +A: PROOF E10-E19, companion roots, type/window/degree-drop/tie boundaries, + exact anchor partition, weighted involution and height ordering. +B: E20-E26/E33, both strip integrals, Laurent rationalization, log(2) constants, + interval certificates and the distinct height sampling laws. +C: E27-E34, exact PHYSICAL maximum selector before the limit, Gaussian domination, + whole-cluster weak law, min/max/double-exceedance probabilities and iid scope. + +Pickups and verdicts must bind exact proof bytes and declare source exposure. +Neither a prior source verdict nor green code is acceptance of this new consumer. + +## Separate companion-ratio addendum + +COMPANION_RATIO.md adds a unit-square parametrization of the limiting extreme +DOUBLE cluster. Its explicit positive rational density determines both heights +and the companion/maximum radius ratio rho, without an unknown Gaussian integral. +It gives the further small-ratio law + + P(rho0, then extreme radius->infinity, THEN epsilon->0. No uniform triple-scale +statement about the original field is made. + +An ideal-real rejection algorithm has exact acceptance probability +3972529/137647104. `ratio_law.sample_doublet` implements its scalar shape/height +part with ordinary floating-point pseudorandom numbers and a bounded trial budget: + + import random + from ratio_law import sample_doublet + marks = sample_doublet(random.Random(20260930)) + +It does not simulate the Gaussian orientation/amplitude or a finite-r field, +and its rounding error is not interval-enclosed. Sixteen added test methods +bring the suite to82 per Python mode. An exact counterexample credited to the +supporting recovery lane pins the distinction between finite-scale physical +radius ordering and the |Z| ordering valid in the second limit. + +This addendum is a SEPARATE Slice D review obligation. It leaves PROOF.md, +extremes.py, the original RESULTS and all source identities unchanged. An +acceptance of A/B/C is not automatically an acceptance of the new addendum. diff --git a/frontiers/extreme_cluster_sampling_20260930/RECONNAISSANCE.md b/frontiers/extreme_cluster_sampling_20260930/RECONNAISSANCE.md new file mode 100644 index 000000000..12bfcc069 --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/RECONNAISSANCE.md @@ -0,0 +1,57 @@ +# Reconnaissance memo — 30 September 2026 UTC + +Scientific effect NONE. Limited literature reconnaissance, not a novelty or +priority audit. No retrieved paper supplies the Gaussian/pinned-field premises +of the new manuscript; those come from the exact GitHub sources. + +## Connector attempts + +Consensus.search was actually invoked on the two listed Primary and Gadget +accounts with queries about Gaussian critical-point clusters, Palm distributions, +regular variation and extreme cluster marks (years2015-2026/2018-2026). Both +returned the 30-search monthly quota error and an October1 reset. No search +result or paper record was obtained from that connector and none is fabricated. + +## Primary web searches and access depth + +Queries included: +- site.arxiv.org tail processes anchored cluster distributions extremal index regular variation +- site.arxiv.org Gaussian critical points repulsion Palm conditioning short range saddles + +Opened primary arXiv abstract/metadata records only: + +1. Hrvoje Planinic, "Palm theory for extremes of stationary regularly varying + time series and random fields", arXiv:2104.03810v2 (14 March2023). + https://arxiv.org/abs/2104.03810 + Relevant concept: exceedance-selected cluster laws carry a size bias relative + to an anchored/typical cluster law. This generic distinction predates this + packet. The new work is the exact companion involution, shape partition and + coefficients for the specified limiting pinned-Gaussian cubic measure. + +2. Gunter Last, "Tail processes and tail measures: An approach via Palm calculus", + arXiv:2112.15380v3 (9 January2023). + https://arxiv.org/abs/2112.15380 + Relevant concept: tail/exceedance measures, anchor maps and Palm calculus. + No temporal extremal index is inferred here: our original field is not turned + into a stationary regularly varying sequence of pin pairs by analogy. + +3. Dmitry Beliaev, Valentina Cammarota, Igor Wigman, "No repulsion between critical + points for planar Gaussian random fields", arXiv:1911.03455v3 (27 November2020). + https://arxiv.org/abs/1911.03455 + Related but different: short-range critical-point behavior in an unconditioned + planar isotropic model. It is not a proof of the present exact pinned, + determinant-tilted, already-formed microscopic extreme-cluster law. + +No full text of these papers was read for this memo. No exhaustive absence claim +is justified by these searches. The iid Frechet consequence in PROOF Section7 is +standard once its regularly varying tail has been derived; only its source-specific +coefficient is newly calculated here. + +## Source exposure and coordination + +This conversation authored PR158/CUB and PR162/SC. PR166/RADIAL and MICRO are +another OpenAI author lane, with their separate nonauthor records. Sharing a +provider or GitHub login confers no organizational independence. New consumers +require their own review. Main#95 pickup5901916663 and PR166 comment5902055460 +identify the nonoverlapping frontier and shared candidate formulas. No claimed +reviewer/writer lane was duplicated and no parent proof was changed. diff --git a/frontiers/extreme_cluster_sampling_20260930/RESULTS.json b/frontiers/extreme_cluster_sampling_20260930/RESULTS.json new file mode 100644 index 000000000..6cfef6d63 --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/RESULTS.json @@ -0,0 +1,69 @@ +{ + "constants": { + "companion_ratio_11th_moment_given_double": { + "lower": "0.028369119274289249", + "upper": "0.028369119274289250" + }, + "double_cluster_given_extreme": { + "lower": "0.558034234065747870", + "upper": "0.558034234065747871" + }, + "inner_double_shape_mass": { + "lower": "109.769950345687984746", + "upper": "109.769950345687984747" + }, + "maximum_shape_mass": { + "lower": "6933.887192511454872396", + "upper": "6933.887192511454872397" + }, + "maximum_to_point_tail_ratio": { + "lower": "0.984415773209943375", + "upper": "0.984415773209943376" + }, + "mean_exceedance_count": { + "lower": "1.015830939745347846", + "upper": "1.015830939745347847" + }, + "outer_double_shape_mass": { + "lower": "3869.346428571428571428", + "upper": "3869.346428571428571429" + }, + "outer_height_mean": { + "lower": "0.819586988784918536", + "upper": "0.819586988784918537" + }, + "outer_height_second_moment": { + "lower": "0.694438469047655310", + "upper": "0.694438469047655311" + }, + "point_shape_mass": { + "lower": "7043.657142857142857142", + "upper": "7043.657142857142857143" + }, + "two_exceedances_given_extreme": { + "lower": "0.015830939745347846", + "upper": "0.015830939745347847" + } + }, + "inner_integrals": { + "0": [ + "27066286003/223205220", + "-79298560/4782969" + ], + "1": [ + "1259212131941593/15982386572880", + "-3186360320/387420489" + ], + "2": [ + "72706616008642669/1315122095139840", + "-15686696960/3486784401" + ] + }, + "log2_terms": 60, + "mathematical_acceptance": false, + "passed": true, + "rational_shape_cases": 893, + "scientific_effect": "NONE", + "scope": "exact finite identities and rational/logarithmic integrals only", + "tail_power": 11 +} diff --git a/frontiers/extreme_cluster_sampling_20260930/SOURCES.json b/frontiers/extreme_cluster_sampling_20260930/SOURCES.json new file mode 100644 index 000000000..c990aa1c0 --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/SOURCES.json @@ -0,0 +1,44 @@ +{ + "schema": 1, + "repository": "d6g8k5htny-coder/Math-", + "scientific_effect": "NONE", + "new_consumer_review": "OPEN", + "sources": [ + { + "id": "RADIAL", + "commit": "80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337", + "path": "frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md", + "bytes": 12533, + "sha256": "250897858c8b314c0ff85ec1860a725efacfdae0ff7409fced4cadf7bfb85973", + "git_blob": "0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4", + "use": "Exact R4-R13 root integral, Gaussian domination, point tail; H3-H4 point height moments." + }, + { + "id": "MICRO", + "commit": "80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337", + "path": "frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md", + "bytes": 17139, + "sha256": "e81d7fe09d25c3266eb8e62922756f54761d7f74d692fb35d9a8071fffd73769", + "git_blob": "a32fd5f7d941bbe1fe943df045b1e0fbec8d691c", + "use": "M1-M2 definition and original-frame embedding ONLY, not the two-scale convergence theorem." + }, + { + "id": "SC", + "commit": "382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf", + "path": "frontiers/spectral_cluster_closure_20260929/PROOF.md", + "bytes": 25006, + "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", + "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a", + "use": "Explicit spectral/Gaussian micro measure, finite positive nonempty mass and raw Gaussian density bound. Author-side exposure." + }, + { + "id": "CUB", + "commit": "bfcc67dc8957b3ed1fbb789e5a0305e98d7359c8", + "path": "frontiers/planar_cubic_cluster_20260929/PROOF.md", + "bytes": 19889, + "sha256": "117e9299a71e6139270266889eb772ed97518e402d9af029bbdb67caf9d4a0f0", + "git_blob": "bb446d08db8a944537a743ad550b88c1c2ad5758", + "use": "Exact typed deterministic cubic classifier ONLY; no Gaussian-sector consumption. Author-side exposure." + } + ] +} diff --git a/frontiers/extreme_cluster_sampling_20260930/VALIDATION.md b/frontiers/extreme_cluster_sampling_20260930/VALIDATION.md new file mode 100644 index 000000000..f9220a2e0 --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/VALIDATION.md @@ -0,0 +1,126 @@ +# Validation record — author-side, not analytic acceptance + +Scientific effect NONE. Runtime: Python3.13.5 and Git2.47.3. Shipped code is +standard-library only and uses Python3.11-compatible syntax. The hosted workflow +requests Python3.11.16 and real historical project source objects. + +## Test-first mathematical work + +The initial ten-test scaffold produced ten expected assertion failures before +implementation. One proposed companion fixture was first recalculated from an +independent line/conic intersection and corrected BEFORE the implementation: +for u=-1,v=6 the correct companion is(-13/23,294/529,-7/23). + +After implementation, all ten tests passed. Subsequent mathematical tests bring +the suite to35 tests:893 rational shape/companion/classifier/anchor checks; +independent exact line/conic elimination;32 exact automatic-differentiation +checks of Q(T)|det DT|=Q|t_c|^11; direct polynomial-strip integrals; dense Lagrange +interpolation checks of the rationalized Laurent integrand at four rational +y-values and four height-moment orders; strict endpoint, degree-drop and tie +cases; outward rational log(2) intervals and normalized ratios. + +An exploratory generic symbolic integration timed out. The delivered integral +certificate instead uses the explicit rationalizing substitution and a finite +Laurent polynomial, integrated termwise by Fraction arithmetic. Exploratory CAS +scripts are not runtime dependencies or shipped proof evidence. + +## Test-first custody work + +A no-op verifier scaffold ran31 real-Git/inventory test methods and produced45 +expected assertion failures (including separate subtests). The completed verifier +passed all31. Hostile fixtures cover changed historical identities, source files +whose worktree contents differ, missing/duplicate/unexpected identities, path +traversal, duplicate JSON, ancestor and leaf symlinks, tree/blob/annotated-tag +objects offered as commits, and Git replacement refs that can forge or invalidate +ordinary git-show results. All Git reads disable replacements, and the nested +packet path test exercises --full-tree. No project source tampering is asserted. + +The hosted environment also sets GIT_NO_REPLACE_OBJECTS=1. Replaying the hostile +fixture under that environment first produced two expected failures because its +ordinary git-show precondition inherited the protection. The fixture now removes +that variable ONLY for its deliberately ordinary Git calls; the real verifier +always disables replacements. Replaying the same environment passes both controls. +This fixes the test harness, not the source-authentication policy. + +## Fresh complete local replay + +66 tests PASS under python -B -S and66 under python -B -O -S. Normal/optimized +baseline stdout is byte-identical to RESULTS.json with no stderr. The eight +semantic mutants reject with return code1 in EACH mode on the intended assertions: + +| Mutation | Intended rejected statement | +|---|---| +| drop-companion-sign | signed companion | +| point-is-cluster | one maximum anchor, not point counting | +| close-window | height endpoint must be open | +| drop-log2 | inner integral | +| double-count-inner | inner integral | +| lose-height-jacobian | point integral and height convention | +| wrong-tail-power | cusp exponent ledger | +| uniform-cluster-from-point | cluster law must be unbiassed | + +Unknown mutant labels return2. The complete packet inventory is authenticated +before and after replay. No temporary files or __pycache__ are added to it. + +The decimal enclosures use60 rational terms of the convergent log(2) series, +a rigorous rational remainder upper bound, signed interval arithmetic, positive +divisions and outward decimal rounding. No Gaussian coefficient is numerically +computed and no quadrature is used to certify a decimal. + +## Limitations and corrections + +The local environment could not resolve raw.githubusercontent.com; no full +upstream checkout was obtained. Therefore local-only replay explicitly reports +source_pins_checked=false. Real-Git fixtures test the verifier but do not replace +actual source authentication. The hosted target must fetch and verify all four +SOURCES.json identities, separately from the existing full downstream/formal gates. +No hosted pass is asserted by this author-time file; actual head/run/job readbacks +will be posted on the PR, with revalidation if head or base changes. + +During manuscript self-review, a loose sentence equating a uniformly chosen root +of a uniformly chosen cluster with point intensity was corrected before the +first publication. Point intensity sums over roots and biases cluster sampling; +uniformly choosing a root within a uniformly chosen cluster is a third convention. +The corrected PROOF Section8 explicitly distinguishes them. + +Finite exact tests support algebra and implementation. They do not independently +prove the source Gaussian assumptions, changes of variables with multiplicity, +physical-anchor domination or weak convergence. Those new arguments require +source-bound nonauthor analytic review. Existing Lean package success is not a +Lean formalization of this manuscript. No self-merge or scientific-register fold. + +## Separate unit-square companion addendum — successor evidence + +Original PROOF.md, extremes.py, test_extremes.py, RESULTS.json, SOURCES.json, +verify.py and test_verify.py remain byte-identical to published head7ab0d31. +The new files are COMPANION_RATIO.md, ratio_law.py and test_ratio_law.py. + +Test-first record: the initial ratio scaffold ran12 methods and produced15 +expected assertion failures, including subtests. The sampler scaffold extension +ran15 methods and produced20 assertion failures plus one KeyError because its +stub returned an empty mapping. The implemented15 methods then passed. One more +exact physical-order counterexample from peer support5902401772 was independently +recomputed and added as a regression; it does not confer independent acceptance. + +Fresh normal and optimized replay, both with the hosted replacement-disabling +environment, now each run82 tests and PASS. The16 new methods include72 independent +exact Jacobian comparisons,25 rational companion/height substitutions, explicit +positive height-difference/top-boundary factors, exact endpoint and small-ratio +constants, ideal proposal-density factorization/envelope, reproducible float +sampler/range tests and rejection-budget checks. The original semantic mutants +and deterministic RESULTS identity remain required by verify.py. + +A seeded FLOATING-POINT diagnostic (seed20260930,20000 accepted doublets) observed +mean proposal count35.0511 and mean rho^11=0.030448088971641796. Every sampled height +pair and radius ratio satisfied the strict ranges. The exact theoretical values +are137647104/3972529 for mean proposals and D/J for the eleventh moment. This is +an illustrative Monte Carlo sanity check, not a proof, interval guarantee, or +assertion of exact finite-sample agreement. The exact ideal-real acceptance and +moment identities are established analytically, independently of this diagnostic. + +Initial published-head hosted evidence was directly read: target36655624974 / +job109699228659 SUCCESS; full36655625213 with downstream109699229856, +formal109699230080 and required aggregate109699995661 all SUCCESS. These are +7ab0d31 results ONLY. An addendum successor must obtain its own actual hosted +source authentication and current-base full checks; these predecessor runs do +not substitute for them. No new analytic review is claimed by this record. diff --git a/frontiers/extreme_cluster_sampling_20260930/extremes.py b/frontiers/extreme_cluster_sampling_20260930/extremes.py new file mode 100644 index 000000000..38f3d4537 --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/extremes.py @@ -0,0 +1,352 @@ +"""Exact controls for the extreme whole-cluster candidate; standard library only. + +The rational identities and integral certificates are executable. Gaussian +conditioning, changes of variables and weak limits are analytic obligations, +not conclusions that this program certifies. +""" +from __future__ import annotations +import argparse +from fractions import Fraction as F +import json +from math import comb +from typing import Optional + +Poly = dict[int, F] +AffineLog = tuple[F, F] # a + b log(2) +Interval = tuple[F, F] +MUTANTS = ('drop-companion-sign', 'point-is-cluster', 'close-window', + 'drop-log2', 'double-count-inner', 'lose-height-jacobian', + 'wrong-tail-power', 'uniform-cluster-from-point') +_MUTANT: Optional[str] = None + + +def _r(x: F | int) -> F: + if isinstance(x, bool) or not isinstance(x, (F, int)): + raise TypeError('an integer or Fraction is required') + return F(x) + + +def b0(u: F | int) -> F: + u = _r(u) + return 3 - 12*u*u + + +def height(u: F | int, v: F | int) -> F: + return _r(u) + F(1, 2) + _r(v)/6 + + +def in_shape(u: F | int, v: F | int) -> bool: + u, v = _r(u), _r(v) + if _MUTANT == 'close-window': + return -F(3,2) <= u <= F(1,2) and abs(b0(u))/2 <= v <= 3-6*u + return -F(3,2) < u < F(1,2) and abs(b0(u))/2 < v < 3-6*u + + +def shape_weight(u: F | int, v: F | int) -> F: + u, v = _r(u), _r(v) + b = b0(u) + return (4*v*v-b*b)*(b-4*u*v) + + +def companion(u: F | int, v: F | int) -> Optional[tuple[F, F, F]]: + """Other root's (u',v',Z'/Z); None is the linear degree-drop case. + + This algebraic map is defined even outside the strict shape region; + membership must be checked separately. It is not a field root solver. + """ + u, v = _r(u), _r(v) + b = b0(u) + den = 4*v*v - 8*b*u*v + b*b + if den == 0: + return None + t = (4*v*v-b*b)/den + up = (2*b*v-u*(4*v*v+b*b))/den + vp = v*t*t + if _MUTANT == 'drop-companion-sign': + t = abs(t) + return up, vp, t + + +def is_double(u: F | int, v: F | int) -> bool: + if not in_shape(u,v): + return False + c = companion(u,v) + return c is not None and in_shape(c[0], c[1]) + + +def classifier_double(u: F | int, v: F | int) -> bool: + """Independent CUB threshold read on an already counted root.""" + u, v = _r(u), _r(v) + b = b0(u) + return in_shape(u,v) and v < -b and 12*(v-b*u)**2 < (v+b)**2*(2*v-b) + + +def anchor_weight(u: F | int, v: F | int) -> F: + """Asymptotic |Z|-maximum selector; ties have weight one half.""" + if not in_shape(u,v): + raise ValueError('strictly typed in-window root required') + if _MUTANT == 'point-is-cluster': + return F(1) + c = companion(u,v) + if c is None or not in_shape(c[0], c[1]): + return F(1) + ratio = abs(c[2]) + return F(1) if ratio < 1 else F(1,2) if ratio == 1 else F(0) + + +def inner_double(u: F | int, v: F | int) -> bool: + """Polynomial description; strict boundary, no square-root comparison.""" + u, v = _r(u), _r(v) + p = -u + a = 12*p*p-3 + f = -2*v*v+(36*p*p-12*p-3)*v-72*p**3+36*p*p+18*p-9 + return in_shape(u,v) and F(1,2)0 + + +def outer_double(u: F | int, v: F | int) -> bool: + u, v = _r(u), _r(v) + p = -u + a = 12*p*p-3 + return ((F(1,2) Poly: + out: Poly = {} + for poly in polys: + for k, value in poly.items(): + out[k] = out.get(k,F(0)) + value + return {k:v for k,v in out.items() if v} + + +def scale(poly: Poly, c: F | int) -> Poly: + c = _r(c) + return {k:v*c for k,v in poly.items() if v*c} + + +def multiply(left: Poly, right: Poly) -> Poly: + out: Poly = {} + for i,x in left.items(): + for j,y in right.items(): + out[i+j] = out.get(i+j,F(0)) + x*y + return {k:v for k,v in out.items() if v} + + +def power(poly: Poly, n: int) -> Poly: + if isinstance(n,bool) or not isinstance(n,int) or n<0: + raise ValueError('nonnegative integer power required') + result: Poly = {0:F(1)} + base = dict(poly) + while n: + if n & 1: + result = multiply(result,base) + base = multiply(base,base) + n //= 2 + return result + + +def evaluate(poly: Poly, x: F | int) -> F: + x = _r(x) + return sum((v*x**k for k,v in poly.items()),F(0)) + + +def integrate_polynomial(poly: Poly, left: F, right: F) -> F: + if any(k<0 for k in poly): + raise ValueError('ordinary polynomial required') + return sum((v*(right**(k+1)-left**(k+1))/F(k+1) + for k,v in poly.items()),F(0)) + + +def _moment_gap(u: Poly, b: Poly, hi: Poly, lo: Poly, j: int) -> Poly: + """Integrate Q(u,v)*(u+1/2+v/6)^j in v between polynomial bounds.""" + if isinstance(j,bool) or not isinstance(j,int) or not 0<=j<=12: + raise ValueError('moment order must be an integer between 0 and 12') + q = {3:scale(u,-16), 2:scale(b,4), + 1:scale(multiply(u,power(b,2)),4), 0:scale(power(b,3),-1)} + bias = add(u,{0:F(1,2)}) + total: Poly = {} + for ell in range(j+1): + eta = scale(power(bias,j-ell),F(comb(j,ell),6**ell)) + for order, coef in q.items(): + n = order+ell+1 + gap = add(power(hi,n),scale(power(lo,n),-1)) + total = add(total,scale(multiply(multiply(eta,coef),gap),F(1,n))) + return total + + +def point_moment(j: int) -> F: + u: Poly = {1:F(1)} + b = add({0:F(3)},scale(power(u,2),-12)) + hi = add({0:F(3)},scale(u,-6)) + value = F(0) + for left,right,sgn in ((F(-3,2),F(-1,2),-1),(F(-1,2),F(1,2),1)): + poly = _moment_gap(u,b,hi,scale(b,F(sgn,2)),j) + value += integrate_polynomial(poly,left,right) + return value/6 if _MUTANT=='lose-height-jacobian' else value + + +def inner_certificate(j: int) -> Poly: + """Laurent integrand on y in [1/4,1], after rationalizing V_+(p).""" + p: Poly = {-1:F(2,9), 0:F(1,18), 1:F(2,9)} + a = add(scale(power(p,2),12),{0:F(-3)}) + hi: Poly = {-2:F(8,9), -1:F(-4,3), 1:F(4,9)} + lo = multiply(a,p) + gap = _moment_gap(scale(p,-1),scale(a,-1),hi,lo,j) + return multiply(gap,{-2:F(2,9),0:F(-2,9)}) + + +def inner_moment(j: int) -> AffineLog: + poly = inner_certificate(j) + rat = sum((c*(1-F(1,4)**(k+1))/F(k+1) + for k,c in poly.items() if k!=-1),F(0)) + log = 2*poly.get(-1,F(0)) + if _MUTANT=='drop-log2': + log = F(0) + if _MUTANT=='double-count-inner': + rat,log = 2*rat,2*log + return rat,log + + +def outer_double_mass() -> F: + p: Poly = {1:F(1)} + a = add(scale(power(p,2),12),{0:F(-3)}) + u,b,lo = scale(p,-1),scale(a,-1),scale(a,F(1,2)) + first = _moment_gap(u,b,multiply(a,p),lo,0) + second = _moment_gap(u,b,add({0:F(3)},scale(p,6)),lo,0) + return (integrate_polynomial(first,F(1,2),F(1)) + + integrate_polynomial(second,F(1),F(3,2))) + + +def log_two_bounds(n: int=60) -> Interval: + """Rational atanh(1/3) lower sum and a proved geometric upper remainder.""" + if isinstance(n,bool) or not isinstance(n,int) or n<1: + raise ValueError('positive integer number of terms required') + z,power_z = F(1,3), F(1,3) + lower = F(0) + for j in range(n): + lower += 2*power_z/F(2*j+1) + power_z *= z*z + remainder = 2*power_z/(F(2*n+1)*(1-z*z)) + return lower,lower+remainder + + +def affine_bounds(value: AffineLog, log: Interval) -> Interval: + a,b = value + return (a+b*log[0],a+b*log[1]) if b>=0 else (a+b*log[1],a+b*log[0]) + + +def divide_positive(numerator: Interval, denominator: Interval) -> Interval: + if numerator[0]<0 or denominator[0]<=0: + raise ValueError('positive interval division required') + return numerator[0]/denominator[1],numerator[1]/denominator[0] + + +def subtract_fixed(x: F, interval: Interval) -> Interval: + return x-interval[1],x-interval[0] + + +def certified_constants(n: int=60) -> dict[str,Interval]: + log = log_two_bounds(n) + I,J = point_moment(0),outer_double_mass() + inner = affine_bounds(inner_moment(0),log) + max_mass = subtract_fixed(I,inner) + inner1 = affine_bounds(inner_moment(1),log) + inner2 = affine_bounds(inner_moment(2),log) + double_probability = divide_positive((J,J),max_mass) + if _MUTANT=='uniform-cluster-from-point': + double_probability = ((J+inner[0])/I,(J+inner[1])/I) + return { + 'point_shape_mass':(I,I), 'inner_double_shape_mass':inner, + 'maximum_shape_mass':max_mass, 'outer_double_shape_mass':(J,J), + 'maximum_to_point_tail_ratio':(max_mass[0]/I,max_mass[1]/I), + 'double_cluster_given_extreme':double_probability, + 'two_exceedances_given_extreme':divide_positive(inner,max_mass), + 'mean_exceedance_count':divide_positive((I,I),max_mass), + 'companion_ratio_11th_moment_given_double':(inner[0]/J,inner[1]/J), + 'outer_height_mean':divide_positive(subtract_fixed(point_moment(1),inner1),max_mass), + 'outer_height_second_moment':divide_positive(subtract_fixed(point_moment(2),inner2),max_mass), + } + + +def decimal_bounds(interval: Interval, digits: int=18) -> dict[str,str]: + if not 1<=digits<=60: + raise ValueError('digits must lie between 1 and 60') + unit = 10**digits + lo = interval[0]*unit + hi = interval[1]*unit + low = lo.numerator//lo.denominator + high = -((-hi.numerator)//hi.denominator) + def fmt(n: int) -> str: + sign='-' if n<0 else '' + n=abs(n) + return f'{sign}{n//unit}.{n%unit:0{digits}d}' + return {'lower':fmt(low),'upper':fmt(high)} + + +def _need(ok: bool, message: str) -> None: + if not ok: + raise AssertionError(message) + + +def finite_checks() -> int: + _need(not in_shape(F(-1),F(9)),'height endpoint must be open') + _need(companion(F(-1),F(6))==(F(-13,23),F(294,529),F(-7,23)), 'signed companion') + _need(anchor_weight(F(-3,4),F(3))==0,'one maximum anchor, not point counting') + _need(point_moment(0)==F(246528,35),'point integral and height convention') + _need(point_moment(1)/point_moment(0)==F(5771,7062),'point first moment') + _need(inner_moment(0)==(F(27066286003,223205220),-F(79298560,4782969)), 'inner integral') + _need(outer_double_mass()==F(1083417,280),'outer double integral') + const = certified_constants(50) + p2=const['double_cluster_given_extreme'] + _need(F(558,1000)=high:continue + for j in range(1,20): + v=low+(high-low)*F(j,20) + _need(in_shape(u,v) and shape_weight(u,v)>0,'open positive shape') + c=companion(u,v) + _need(is_double(u,v)==classifier_double(u,v),'companion/CUB disagreement') + if c is not None: + up,vp,t=c + _need(b0(up)==b*t*t,'conic invariance') + _need(b*up+(v-b*u)*t==v,'line invariance') + back=companion(up,vp) + _need(back is not None and back[0]==u and back[1]==v and back[2]*t==1,'involution') + if in_shape(up,vp): + _need(t<0,'doublet transverse signs') + _need(anchor_weight(u,v)+anchor_weight(up,vp)==1,'once per cluster') + if abs(t)!=1: + _need(inner_double(u,v)==(abs(t)>1),'inner region') + _need(outer_double(u,v)==(abs(t)<1),'outer region') + _need((height(up,vp)-height(u,v))*(abs(t)-1)>0,'outer saddle is deeper') + cases+=1 + return cases + + +def results() -> dict: + cases=finite_checks() + return {'passed':True,'scientific_effect':'NONE','mathematical_acceptance':False, + 'scope':'exact finite identities and rational/logarithmic integrals only', + 'rational_shape_cases':cases,'log2_terms':60, + 'tail_power':11, + 'inner_integrals':{str(j):[str(x) for x in inner_moment(j)] for j in range(3)}, + 'constants':{k:decimal_bounds(v) for k,v in sorted(certified_constants().items())}} + + +def main() -> None: + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--mutant',choices=MUTANTS) + args=parser.parse_args() + global _MUTANT + _MUTANT=args.mutant + print(json.dumps(results(),indent=2,sort_keys=True)) + + +if __name__=='__main__':main() diff --git a/frontiers/extreme_cluster_sampling_20260930/ratio_law.py b/frontiers/extreme_cluster_sampling_20260930/ratio_law.py new file mode 100644 index 000000000..49404bf07 --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/ratio_law.py @@ -0,0 +1,106 @@ +"""Universal extreme-doublet shape: exact rational controls and a float simulator. + +The density is for the already-formed limiting microscopic extreme doublet, +not finite-r field samples. sample_doublet uses ordinary floating-point random +numbers for experimentation; it is not an exact-real or interval sampler. +""" +from fractions import Fraction as F +from math import comb, sqrt +import random +import extremes as e + + +def _unit(x): + if isinstance(x,bool) or not isinstance(x,(int,F)): + raise TypeError('integer or Fraction required') + x=F(x) + if not 0<=x<=1: raise ValueError('unit interval required') + return x + + +def coordinates(rho,z): + """Continuous square extension of the outer root's (u,v); interior is typed.""" + rho,z=_unit(rho),_unit(z) + den=2*rho*z+rho+1 + p=(2*rho*z*z+2*rho*z+rho+2*z+1)/(2*den) + ell=(1+2*rho*z/(1+rho))/2 + return -p,(12*p*p-3)*ell + + +def acceptance(rho,z): + rho,z=_unit(rho),_unit(z) + return ((rho*z+1)**4*(rho*z+rho+1)**7 / + ((rho+1)**5*(2*rho*z+rho+1)**9)) + + +def density(rho,z): + """Unnormalized positive density Q(u,v)|d(u,v)/d(rho,z)|; integral is J.""" + rho,z=_unit(rho),_unit(z) + return 165888*rho*z**7*(z+1)**4*acceptance(rho,z) + + +def heights(rho,z): + """Downward heights of outer and companion roots, respectively.""" + rho,z=_unit(rho),_unit(z) + den=(rho+1)*(2*rho*z+rho+1) + outer=z*z*(rho*z+1)*(rho*z+2*rho+1)/den + inner=rho**3*z*z*(z+1)*(rho*z+rho+2)/den + return outer,inner + + +def z_normalizer(): + return sum((F(comb(4,j),8+j) for j in range(5)),F(0)) + + +def small_ratio_constant(): + """CDF coefficient P(rho=a:continue + den=(rho+1)*(2*rho*z+rho+1) + return {'radius_ratio':rho,'shape_coordinate':z, + 'height_maximum':z*z*(rho*z+1)*(rho*z+2*rho+1)/den, + 'height_companion':rho**3*z*z*(z+1)*(rho*z+rho+2)/den, + 'proposals':trial} + raise RuntimeError('extreme-doublet rejection sampler exhausted its trial budget') diff --git a/frontiers/extreme_cluster_sampling_20260930/test_extremes.py b/frontiers/extreme_cluster_sampling_20260930/test_extremes.py new file mode 100644 index 000000000..503ca3571 --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/test_extremes.py @@ -0,0 +1,213 @@ +"""Exact algebra controls; these tests do not certify the Gaussian arguments.""" +from fractions import Fraction as F +import unittest +import extremes as e + +class InitialTests(unittest.TestCase): + def test_strict_shape(self): + self.assertTrue(e.in_shape(F(-3,4),F(3))) + def test_companion_exact(self): + self.assertEqual(e.companion(F(-1),F(6)),(F(-13,23),F(294,529),F(-7,23))) + def test_outer_double_mass(self): + self.assertEqual(e.outer_double_mass(),F(1083417,280)) + def test_point_mass(self): + self.assertEqual(e.point_moment(0),F(246528,35)) + def test_point_mean(self): + self.assertEqual(e.point_moment(1)/F(246528,35),F(5771,7062)) + def test_inner_mass(self): + self.assertEqual(e.inner_moment(0),(F(27066286003,223205220),-F(79298560,4782969))) + def test_inner_first_moment(self): + self.assertEqual(e.inner_moment(1),(F(1259212131941593,15982386572880),-F(3186360320,387420489))) + def test_inner_second_moment(self): + self.assertEqual(e.inner_moment(2),(F(72706616008642669,1315122095139840),-F(15686696960,3486784401))) + def test_log_two_lower(self): + lo,hi=e.log_two_bounds(30) + self.assertGreater(lo,F(6931471805599453,10**16)) + def test_log_two_upper(self): + lo,hi=e.log_two_bounds(30) + self.assertLess(hi,F(6931471805599454,10**16)) + self.assertGreater(hi,lo) + + +class GeometryTests(unittest.TestCase): + def test_full_rational_grid(self): + self.assertEqual(e.finite_checks(),893) + def test_degree_drop(self): + self.assertTrue(e.in_shape(F(-5,8),F(27,16))) + self.assertIsNone(e.companion(F(-5,8),F(27,16))) + self.assertEqual(e.anchor_weight(F(-5,8),F(27,16)),1) + def test_equal_radius_doublet(self): + u,v=F(-3,4),F(45,16) + self.assertEqual(e.companion(u,v),(u,v,F(-1))) + self.assertEqual(e.anchor_weight(u,v),F(1,2)) + def test_zero_B_finite_companion_outside_window(self): + c=e.companion(F(-1,2),F(3)) + self.assertEqual(c,(F(1,2),F(3),F(1))) + self.assertFalse(e.is_double(F(-1,2),F(3))) + self.assertEqual(e.anchor_weight(F(-1,2),F(3)),1) + def test_height_and_type_boundaries_are_open(self): + for u,v in ((F(-1),F(9)),(F(-1),F(9,2)),(F(-1,2),F(0))): + self.assertFalse(e.in_shape(u,v)) + def test_invalid_numeric_types(self): + for x in (True,0.5,'1/2'): + with self.subTest(x=x),self.assertRaises(TypeError):e.b0(x) + def test_anchor_rejects_outside_shape(self): + with self.assertRaises(ValueError):e.anchor_weight(2,1) + def test_other_root_via_independent_line_conic(self): + for iu in range(-21,7): + u=F(iu,16);b=3-12*u*u + if not -F(3,2) Path: + """Check spelling before resolve; a symlinked ancestor must not disappear.""" + path=Path(os.path.abspath(path)) + for p in (path,*path.parents): + if p.is_symlink(): raise ValueError('symlink prohibited: '+str(p)) + return path + + +def read_json(path: Path) -> dict: + path=no_symlinks(path) + def unique(pairs): + result={} + for key,value in pairs: + if key in result: raise ValueError('duplicate JSON key: '+key) + result[key]=value + return result + obj=json.loads(path.read_text(encoding='utf-8'),object_pairs_hook=unique) + if not isinstance(obj,dict): raise ValueError('JSON object required: '+str(path)) + return obj + + +def source_path(text: str) -> PurePosixPath: + if not isinstance(text,str) or not text or '\\' in text or '\0' in text: + raise ValueError('invalid path') + p=PurePosixPath(text) + if p.is_absolute() or '..' in p.parts or str(p)!=text or str(p)=='.': + raise ValueError('noncanonical source path: '+text) + return p + + +def check_identity(data: bytes, entry: dict) -> None: + if not isinstance(entry,dict): raise ValueError('identity entry must be an object') + size=entry.get('bytes'); sha=entry.get('sha256'); blob=entry.get('git_blob') + if type(size) is not int or size<0 or not isinstance(sha,str) or not HEX64.fullmatch(sha): + raise ValueError('invalid size/hash metadata') + if not isinstance(blob,str) or not HEX40.fullmatch(blob): raise ValueError('invalid blob identity') + actual_blob=hashlib.sha1(b'blob '+str(len(data)).encode()+b'\0'+data).hexdigest() + if len(data)!=size or hashlib.sha256(data).hexdigest()!=sha or actual_blob!=blob: + raise ValueError('identity mismatch: '+str(entry.get('path'))) + + +def git(root: Path, *args: str) -> bytes: + env=os.environ.copy() + for key in ('GIT_DIR','GIT_WORK_TREE','GIT_INDEX_FILE','GIT_OBJECT_DIRECTORY', + 'GIT_ALTERNATE_OBJECT_DIRECTORIES','GIT_NAMESPACE'): + env.pop(key,None) + env['GIT_NO_REPLACE_OBJECTS']='1' + command=['git','--no-replace-objects','--literal-pathspecs',*args] + try: + return subprocess.run(command,cwd=root,env=env,capture_output=True, + check=True,timeout=60).stdout + except (subprocess.CalledProcessError,subprocess.TimeoutExpired) as exc: + raise ValueError('Git authentication failed: '+' '.join(args)) from exc + + +def verify_sources(root: Path, required_ids=None) -> int: + root=no_symlinks(root) + required=SOURCE_IDS if required_ids is None else frozenset(required_ids) + entries=read_json(root/'SOURCES.json').get('sources') + if not isinstance(entries,list) or not entries or not required: + raise ValueError('nonempty exact source inventory required') + ids=[]; paths=[] + for entry in entries: + if not isinstance(entry,dict) or not isinstance(entry.get('id'),str): + raise ValueError('invalid source entry') + ids.append(entry['id']) + commit=entry.get('commit'); path=entry.get('path') + if not isinstance(commit,str) or not HEX40.fullmatch(commit): + raise ValueError('literal full commit identity required') + p=source_path(path); paths.append((commit,path)) + # Reject tree/blob/annotated-tag objects, rather than peeling them. + if git(root,'cat-file','-t',commit)!=b'commit\n': + raise ValueError('declared source object is not a commit') + for count in range(1,len(p.parts)+1): + prefix='/'.join(p.parts[:count]) + raw=git(root,'ls-tree','--full-tree','-z',commit,'--',prefix) + records=[record for record in raw.split(b'\0') if record] + if len(records)!=1: raise ValueError('missing/ambiguous source path: '+prefix) + try: + header,name=records[0].split(b'\t',1) + mode,kind,object_sha=header.split() + except ValueError as exc: raise ValueError('malformed Git tree record') from exc + if name.decode('utf-8')!=prefix: raise ValueError('Git path mismatch') + if count int: + root=no_symlinks(root) + entries=read_json(root/'MANIFEST.json').get('files') + if not isinstance(entries,list) or not entries: raise ValueError('empty packet inventory') + names=[] + for entry in entries: + if not isinstance(entry,dict): raise ValueError('invalid packet identity') + text=entry.get('path'); p=source_path(text) + if len(p.parts)!=1 or text=='MANIFEST.json': raise ValueError('flat packet leaf required') + names.append(text) + path=no_symlinks(root/text) + if not path.is_file(): raise ValueError('missing regular packet file: '+text) + check_identity(path.read_bytes(),entry) + if len(names)!=len(set(names)): raise ValueError('duplicate packet path') + if sorted(p.name for p in root.iterdir())!=sorted(names+['MANIFEST.json']): + raise ValueError('complete packet inventory mismatch') + return len(entries) + + +def main() -> None: + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--local-only',action='store_true') + args=parser.parse_args() + root=no_symlinks(Path(__file__).absolute().parent) + verify_inventory(root) + n_sources=0 if args.local_only else verify_sources(root) + expected=(root/'RESULTS.json').read_bytes() + for flags in (['-B','-S'],['-B','-O','-S']): + subprocess.run([sys.executable,*flags,'-m','unittest','discover','-v'], + cwd=root,check=True,timeout=180) + command=[sys.executable,*flags,str(root/'extremes.py')] + result=subprocess.run(command,cwd=root,capture_output=True,check=True,timeout=60) + if result.stdout!=expected or result.stderr: + raise ValueError('deterministic mathematical output mismatch') + for name in MUTANTS+('unknown',): + result=subprocess.run(command+['--mutant',name],cwd=root, + capture_output=True,timeout=60) + if result.returncode!=(2 if name=='unknown' else 1): + raise ValueError('negative control did not reject: '+name) + verify_inventory(root) + print(json.dumps({'passed':True,'source_pins_checked':not args.local_only, + 'source_count':n_sources,'scientific_effect':'NONE', + 'mathematical_acceptance':False},sort_keys=True)) + + +if __name__=='__main__': main() diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/MANIFEST.json b/frontiers/fixed_r_inverse_lifetime_20260930/MANIFEST.json new file mode 100644 index 000000000..dd6871344 --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/MANIFEST.json @@ -0,0 +1,67 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "review_status": "NONAUTHOR_REVIEW_OPEN", + "files": [ + { + "path": "PROOF.md", + "bytes": 17213, + "sha256": "5adef725c1c696adae9bc73a525efe0bfe09d5c07e95ee3f067db9e877283630", + "git_blob": "0d4018777d13672c1de1696525512e80b23b7c3c" + }, + { + "path": "README.md", + "bytes": 3763, + "sha256": "cd167a13b3f7250b09635a61aec9b47f48da044c87bafeb86ce7486cb8d63d8b", + "git_blob": "dc87d46d95a7a5d6dcdd7debf1da7422a7d1d80c" + }, + { + "path": "RECONNAISSANCE.md", + "bytes": 2222, + "sha256": "38ca0b266398fcfcec2565f2bdd1b53b15494995fcd433f442090887e0a66158", + "git_blob": "47b0f54a2189efcfa9adbaae3ec4e8bb5fa744df" + }, + { + "path": "RESULTS.json", + "bytes": 516, + "sha256": "46ccc0f5e2d8cba11040ca1bc617e9e99e54ca7b19ac85ff4091e2601104efeb", + "git_blob": "ef9ff136e85198dad73d664d1250775baf3e1d7e" + }, + { + "path": "SOURCES.json", + "bytes": 651, + "sha256": "9ca481cd235a0e16ee8175538be0eea2477fbd8340833bc84d6ba995f20cf6e0", + "git_blob": "a7a4ebdd902a11f5daf87e0ea8fd93576aa403fa" + }, + { + "path": "VALIDATION.md", + "bytes": 2500, + "sha256": "b1528945a1d7610b2d3b27fbc907cf6538d887262f634d02b4fe0fc64fc0693c", + "git_blob": "7dc4ffe63a94db6e21164309f089b8ba0b8f1b85" + }, + { + "path": "inverse.py", + "bytes": 6501, + "sha256": "69c8e91e227bde716e237945358b0244d8a414eda9de9d4ce5e4fccb93ee7862", + "git_blob": "62698d649ff1eba92656fb555bf57d603f2bbc37" + }, + { + "path": "test_inverse.py", + "bytes": 6312, + "sha256": "0bcd93e078ee4800647f8af5aaf77894c63912d63dca9d74c49f732432bc5e6f", + "git_blob": "9695c472e7b0f4acc7946c310765d6ec204dc742" + }, + { + "path": "test_verify.py", + "bytes": 4083, + "sha256": "ee6aa8e6b589acb2eb32b482b4e9965d614d7c8416737624cd6df16eaa0b2960", + "git_blob": "1fccbe702825118906bfea6e266973a51225ca78" + }, + { + "path": "verify.py", + "bytes": 5190, + "sha256": "df2f54f44a95f1dc62022145feed2e22dca242bd37266e1eed82b12e519f6355", + "git_blob": "22db229cfc2be9d3486d7759c82c50bea83b14e0" + } + ] +} diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/PROOF.md b/frontiers/fixed_r_inverse_lifetime_20260930/PROOF.md new file mode 100644 index 000000000..0d4018777 --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/PROOF.md @@ -0,0 +1,332 @@ +# Fixed-separation lifetime tails and the sharp inverse-moment boundary + +Object OA-FIXED-R-INVERSE-LIFETIME-20260930-v1. +Author: OpenAI / GPT-6 Astra Pro. AUTHOR-SIDE PROOF CANDIDATE; new nonauthor +analytic review OPEN. Scientific effect NONE. No prior proof, status, graph, +Boolean, prize or review disposition is changed. + +## 1. Statement and the precise probability law + +Fix d>=2, L>0, b real, k>0, a unit vector u and 0b) min_t f(gamma(t)), + ell=b-d_f(M). (2) + +An empty class of paths has d_f=-infinity and ell=+infinity (essential class). +On finite deaths this is the actual lifetime, not a gap to a counted witness. +Let F denote failure of the prescribed saddle S to be the ordinary partner. +On the usual distinct-critical-value locus, (2) is the ordinary elder saddle +value. The estimates below apply directly to (2), without counting saddles +or assuming a particular local saddle is the global partner. + +**Theorem 1 (two-sided fixed-r small-lifetime order).** There are constants +00, depending on the fixed parameters, such that +for 0=0, + + E_QW[ell^(-p);ell finite] < infinity iff p<2/3, + E_QW[ell^(-p);F,ell finite] < infinity iff p<2/3. (4) + +For p=0 these denote the finite-event probabilities. The corresponding +conditional expectations have the same finiteness boundary. Multiplication by +the fixed positive scale Delta leaves the threshold unchanged, so (4) also +holds for the normalized lifetime X_r=ell/Delta. At critical and supercritical +powers the capped expectations obey + + E_QW[min(ell^(-1),T)^(2/3);F,ell finite]=Theta(log T), + E_QW[min(ell^(-1),T)^p;F,ell finite]=Theta(T^(p-2/3)) + for p>2/3. (5) + +The same holds without F or after the stated finite-event conditioning. The +constants are not evaluated. There is no claimed exact leading coefficient +or lifetime-density equivalent in (3). Together with elementary finiteness of +positive moments, the full finite-death power-moment domain is q>-2/3: +E[ell^q;finite] and E[ell^q;F,finite] are finite exactly on that range. + +**Scope.** These are statements at EACH FIXED positive r, with epsilon->0. +The constants and epsilon0 may degenerate arbitrarily as r->0. No uniform +integrability as r->0 or limit of inverse moments is asserted. In particular +this is NOT the distant-partner-conditioned law of PR181, where r->0 and a +second extreme-radius limit occur before a small-lifetime limit. Its exponent4 +and this exponent2/3 concern different laws and different limiting operations. + +[P] is the exact source in SOURCES.json. Only its model, observation, weighting +conventions and fixed-point Fourier facts in Sections1-2 are used; the latter +are reconstructed below. The uniform cap theorem, global count laws, elder +candidates PR170/175, positive-moment PR177 and extreme-mark PR181 are NOT proof +premises. The author is exposed to [P]. This is a new consumer, not independent +revalidation of that source's other theorems. + +## 2. Fixed-parameter Gaussian facts, with no contact uniformity + +Poisson summation gives Fourier weights + + a_n=exp(-2 pi^2 |n|^2/L^2)/sum_j exp(-2 pi^2 |j|^2/L^2)>0. + +For every integer m, sum_n sqrt(a_n)(1+|n|)^m is finite. A real sine/cosine +expansion and Minkowski's inequality show that every C^m norm of the field +has moments of every finite order. For orders below1 use a larger moment. +This also supplies a smooth version of the field. + +Any finite list of distinct derivative evaluations at distinct sites has +strictly positive covariance. Indeed a zero-variance linear combination +annihilates every Fourier mode since a_n>0. As a distribution it is a finite +sum of derivatives of delta masses, all of whose Fourier coefficients vanish. +The distribution is zero. Test functions supported near one site with arbitrary +finite jets show that all its coefficients vanish. Symmetric tensor entries +are listed once only; a change of orthonormal frame is invertible on these jets. + +This proves nondegeneracy of the original observation vector O and of the +following additional lists modulo O: + + U=(H_M,H_S), J=(H_M,H_S,D^3 f(M)). (6) + +All conditional covariances here are at FIXED r>0. None is replaced by its +singular r=0 counterpart. Their conditional Gaussian densities are strictly +positive on every compact set and bounded above by C exp(-c|U|^2), or the +analogous bound for J, with constants depending on the fixed observations. + +Full-field regression conditional on U has the form + + f=m_U+G_U, ||m_U||_C3 <= C(1+|U|), + G_U independent of U, E||G_U||_C3^q0. In particular K is NOT presumed +independent of the Hessians. The same argument conditional on J gives an +independent C4 residual and a uniform C4 mean bound on each compact J set. + +The Gaussian density of U is positive on the open product of the required +Hessian cones; W is strictly positive there. All its polynomial moments are +finite. Hence Z in (1) is positive and finite. No r^-2 normalizer estimate is +needed or used. + +For completeness, (2) is measurable: for hh is existence +of a path from M to a point above b with its minimum strictly above h. A +uniform approximation of such a path by torus piecewise-linear paths with +rational vertices retains both strict margins. Thus a countable set of paths +suffices; their endpoint values and compact-path minima are measurable field +functionals. An essential maximum simply has no eligible path. + +## 3. A deterministic local barrier gives a lower bound on lifetime + +Let f be any C3 field on the torus with f(M)=b, grad f(M)=0 and H_M<0. Write +lambda_min for the smallest eigenvalue of -H_M and use K above, so +0= c_L lambda_min^3/K^2, c_L=a_L^2/3>0. (9) + +This holds also as an inequality for ell=+infinity. It proves strict positivity +of every finite lifetime without any global Morse argument. The barrier does +not claim the partner is on that sphere; it bounds all older-reaching paths. + +## 4. Keep the determinant weight: the upper small-lifetime bound + +For any nonnegative measurable function psi, (7) and the Gaussian U density +imply a one-dimensional weighted eigenvalue estimate + + E_Q[W K^2 psi(lambda_min)] + <= C integral_0^infinity lambda psi(lambda) + (1+lambda)^N exp(-c lambda^2)d lambda. (10) + +Here is the full reduction. On H_M<0 diagonalize -H_M with ordered positive +eigenvalues 0=1. Apply (10) with +psi(lambda)=min(1,epsilon/lambda^3), and divide ONCE by Z. For epsilon=s^3<=1, +the portion lambda<=1 is bounded by a constant times the exact integral + + integral_0^1 lambda min(1,s^3/lambda^3)d lambda + =s^2/2+s^3(1/s-1)=(3/2)s^2-s^3. (12) + +The lambda>=1 portion is O(epsilon) by the Gaussian envelope. Therefore +Q^W(00. (13) + +Let C range over a compact box of strictly negative definite (d-1)-matrices, +v over a fixed small box of positive volume, and H_S over a compact box inside +the index-(d-1) cone. We may ensure eigenvalues of C stay between -2 and -1/2, +|det C|>=c_C>0 and |det H_S|>=c_S>0. The entire boxes can be chosen with +nonempty interiors. In these coordinates + + e^T H_M e=-lambda, H_M<0, + |det H_M|=lambda |det C|. (14) + +The change from the independent H_M entries to (lambda,v,C) has absolute +Jacobian1: only H_11 is replaced, with derivative -1 in lambda. We do not set +v=0 on a measure-zero slice, nor shrink its box with lambda. The direction e +adapts smoothly to v,C and has a uniform finite norm bound E. + +Let the third-derivative tensor T at M vary over a fixed compact positive-volume +box for which T[e,e,e]>=a>0 for every C,v in the chosen boxes. Such a box exists: +start with T_111=2a and all other entries zero, choose v small, and then a +sufficiently small full-dimensional tensor neighborhood. Initially allow +0<=lambda<=1. This defines a compact set of complete J targets. By Section2 its +raw conditional density has a strictly positive lower bound m_r on that set. + +Write the conditional full field as m_J+G_J. Choose a fixed residual C4 norm +bound with probability p0>0, possible because the residual has a finite norm +almost surely. It is independent of J. The conditional mean is bounded on the +compact targets; consequently there is a uniform directional fourth-derivative +bound M4>=1 along e on this event, accounting for |e|<=E. This uses a fixed +positive-probability residual event, not the probability of a shrinking Gaussian +ball. M4 and m_r may be very unfavorable as r decreases, which is allowed. + +For the straight path t -> M+t e, Taylor now gives + + f(M+t e) >= b-(lambda/2)t^2+(a/6)t^3-(M4/24)t^4. (15) + +Put B=6/a. If lambda<=6/(M4 B^2), then at t=B lambda the right side minus b is +at least (B^2/4)lambda^3>0. Over the entire interval 0<=t<=B lambda it is at +least -(3B^2/4)lambda^3, hence certainly at least -B^2 lambda^3. Reduce a +fixed lambda0>0 further so that B lambda0 Eb-Delta, not from a count of nearby critical +points. We need not identify the actual death saddle or forbid an even higher +alternative global path: either could only shorten ell further. + +For 0= c integral_0^((epsilon/B^2)^(1/3)) lambda d lambda + = (c/2)B^(-4/3)epsilon^(2/3). (17) + +This is the lower half of (3). It holds for all sufficiently small epsilon at +the fixed model parameters, not just along a sequence or finite numerical grid. + +## 6. Inverse moments and capped growth + +Both normalizing events {ell finite} and {F,ell finite} have positive probability +by (17). For epsilont) <= C t^(-2/3). (18) + +Endpoint strict versus weak inequalities cause no difficulty: upper bounds use +(3), and a lower bound at epsilon=1/(2t) is already enough. For p>0, Tonelli gives + + E[Y^p]=p integral_0^infinity t^(p-1) Q^W(Y>t)dt. + +The portion below a fixed large threshold is finite. The remaining integral +converges precisely when p<2/3. This proves (4). For positive powers, on finite deaths ell<=2||f||_infty; +all weighted positive moments of this norm are finite by the Gaussian norm +and Hessian moments of Section2 and Cauchy-Schwarz. This yields the stated +full power-moment domain without asserting any small-r moment convergence. +Truncating the same integral at T gives the logarithmic and power orders in (5). These are two-sided growth +orders, not exact coefficients; a density or precise regular-variation limit +was not proved by the two-sided comparison. + +## 7. What this closes, and what it does not + +The fixed-r inverse boundary is now determined at candidate-theorem level, +complementing PR177's separately derived positive finite-lifetime moments. +Neither the small-r marked law nor its moment-convergence hypotheses are used +here. In particular the new result cannot be reinterpreted as a failure of +PR181: that law further conditions on a distant replacement partner after +already taking r->0, whereas (3) includes all finite failures at fixed r. + +The exponent2/3 has two independent sources: local cubic escape costs a lifetime +of order lambda^3, while the original Palm weight and eigenvalue volume give +lambda d lambda. Dropping that weight changes the conclusion. As a comparison, +the same proof for the raw Gaussian law conditioned only on the two open Hessian +types, without either determinant tilt, gives exponent1/3; in its upper argument +integral_0^1 min(1,s^3/lambda^3)d lambda=(3/2)s-(1/2)s^3. +This comparison is not a change to the law (1) used in the main theorems. + +No exact lifetime density, leading coefficient, r-uniform constant, joint +r/epsilon limit, convergence of inverse moments as r->0, distant-partner inverse +threshold at finite r, or globally deduplicated persistence-bar intensity is +asserted. Essential-inclusive POSITIVE moments and their convention remain +separate. These proofs are not formalized in Lean and finite tests are not +analytic acceptance. + +Requested review A: fixed-r Gaussian nondegeneracy, regression and weighted +spectral reduction (Sections2-4), including the original full Z. Review B: +full-dimensional Schur sector, retained residual event, older-reaching path, +lower bound and the exact moment conclusions (Sections5-6). Prior source +verdicts or green workflows do not accept this new consumer. diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/README.md b/frontiers/fixed_r_inverse_lifetime_20260930/README.md new file mode 100644 index 000000000..dc87d46d9 --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/README.md @@ -0,0 +1,72 @@ +# Fixed-r lifetime moments: the inverse boundary is 2/3 + +OpenAI / GPT-6 Astra Pro. **New author-side theorem candidate; nonauthor review +OPEN. Scientific effect NONE. No self-merge.** + +For the exact determinant-weighted maximum/saddle pair-Palm law, hold the two +prescribed pins at any fixed positive embedded separation r. Let ell be the +ACTUAL ordinary elder lifetime, excluding the essential infinite lifetime. +The new proof gives two-sided small-lifetime bounds + + c_r epsilon^(2/3) <= P(0-2/3. Inverse moments of order p are finite exactly when p<2/3, both under +the finite-death restriction and conditional on failure and finite death. +The capped critical inverse moment grows as Theta(log T); above criticality +it grows as Theta(T^(p-2/3)). No exact leading coefficient is asserted. + +This fills the inverse-moment exclusion in the separately merged PR177. It +uses neither PR170/175's marked-limit candidates nor PR181's extreme-selected +law. The latter conditions on distant partners AFTER r->0 and has a different +small-lifetime exponent; it must not be substituted for this fixed-r law. + +## Why the exponent is 2/3 + +A deterministic sphere around the prescribed maximum forces +ell >= c lambda_min^3/K^2, where K controls the third derivative. Conditional +Gaussian norm moments retain K inside the endpoint determinant weight. The +remaining soft-eigenvalue integral is exactly + + integral_0^1 lambda min(1,s^3/lambda^3)d lambda=(3/2)s^2-s^3. + +An open Schur-coordinate sector gives the converse. The Hessian has a soft +curvature -lambda but a positive cubic derivative along an adapted direction. +A short straight path reaches a point above birth with only O(lambda^3) loss. +The original determinant weight supplies lambda, and full jet support supplies +d lambda. Thus the lower probability is of order epsilon^(2/3) as well. +The off-diagonal Hessian entries range over a fixed OPEN box, not a zero-measure +slice or a box shrinking with lambda. + +## Exact boundaries + +All constants can depend on fixed r,d,L,b,k and frame. No r-uniform rate, +interchange of limits, density equivalent, numerical amplitude, inverse-moment +convergence as r->0, or result conditional on a distant partner is claimed. +No source selection theorem, global count asymptotic or hard-fibre bridge is +required: maximin paths and local barriers control the actual lifetime directly. + +PROOF.md contains the full arguments. SOURCES.json binds the one parent model +carrier; only its model/weight conventions and fixed-jet Fourier facts are used, +and the latter are proved again here. Parent asymptotic theorems are not promoted. + +## Replay + + python -B -S frontiers/fixed_r_inverse_lifetime_20260930/verify.py + +The full mode requires the exact parent Git object. For a standalone extraction: + + python -B -S verify.py --local-only + +Local-only explicitly reports that historical project sources are not checked. +Thirty test methods run in each normal/optimized mode; six intentionally wrong +variants must fail per mode. Exact controls include36 Schur matrices across +four hard dimensions,36 Taylor-path cases/612 grid checks,36 local barriers, +and11 source/inventory fixture tests. These finite checks are not acceptance +of the Gaussian integration or the theorem. No new Lean formalization. + +The foreground work follows Dylan's NEW explicit September30 continuation +request, after the September27 stop. No stopped process or timer is restarted. +Review A covers Gaussian support/regression and the upper weighted bound; +review B covers the Schur sector, lower bound and moment domain. diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/RECONNAISSANCE.md b/frontiers/fixed_r_inverse_lifetime_20260930/RECONNAISSANCE.md new file mode 100644 index 000000000..47b0f54a2 --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/RECONNAISSANCE.md @@ -0,0 +1,37 @@ +# Narrow primary reconnaissance and scope — 30 September 2026 + +Queries actually executed: +- site.arxiv.org Gaussian random fields persistence lifetime distribution critical points small persistence +- site.arxiv.org Gaussian critical points finiteness moments Gass Stecconi 2023 +- site.arxiv.org Gaussian fields Palm small eigenvalue Hessian local maximum saddle persistence + +Primary abstract/metadata pages actually opened: + +1. Louis Gass and Michele Stecconi, The number of critical points of a Gaussian + field: finiteness of moments, arXiv:2305.17586v2, revised19 July2023. + https://arxiv.org/abs/2305.17586 + Context: count-moment finiteness under regularity and jet nondegeneracy. This + is not an inverse ordinary-elder lifetime theorem and is not a premise here. + +2. Dmitry Beliaev, Valentina Cammarota and Igor Wigman, No repulsion between + critical points for planar Gaussian random fields, arXiv:1911.03455v3, + revised27 November2020. https://arxiv.org/abs/1911.03455 + Context: near-diagonal critical-point correlations with index information. + No unconditioned count-correlation formula is substituted for the actual + fixed-pin lifetime law in the new manuscript. + +Access depth was abstract/metadata, not a full-text novelty audit. The cubic +fold scaling, Schur-complement algebra, Gaussian regression, spectral matrix +Jacobian and layer-cake formula are existing general tools. No claim of +inventing them or of exhaustive priority is made. The specific new in-project +argument combines the actual maximin barrier, fixed-r full-jet support and the +original determinant tilt to identify the two-sided inverse-moment boundary. + +Connected repository context read: PR177 README at + d768a1761aee8d6911c14bc7524089e14c8cd81c, blob508e3963304cf323099bc0e118559a1f86f04146, +explicitly excludes inverse moments. That positive-moment work is credited, not +consumed as a premise or independently revalidated. PR181's distant-selected +limit is a different conditioning and is also not a premise. Coordination +pickup main#95/5908596793 and technical notes177/5908600694,5908718757 separate +these lanes. The user requested GitHub/Superpowers; Consensus was not invoked +and no quota-status change is claimed. diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/RESULTS.json b/frontiers/fixed_r_inverse_lifetime_20260930/RESULTS.json new file mode 100644 index 000000000..ef9ff136e --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/RESULTS.json @@ -0,0 +1,15 @@ +{ + "critical_capped_growth": "Theta(log T)", + "extreme_partner_conditioning": false, + "fixed_r_small_lifetime_exponent": "2/3", + "inverse_moment_threshold": "2/3", + "mathematical_acceptance": false, + "passed": true, + "path_grid_checks": 612, + "path_parameter_cases": 36, + "scientific_effect": "NONE", + "scope": "finite rational Taylor, Schur and integral controls only", + "supercritical_capped_growth": "Theta(T^(p-2/3))", + "uniform_in_pin_separation": false, + "unweighted_comparison_exponent": "1/3" +} diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/SOURCES.json b/frontiers/fixed_r_inverse_lifetime_20260930/SOURCES.json new file mode 100644 index 000000000..a7a4ebdd9 --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/SOURCES.json @@ -0,0 +1,15 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "sources": [ + { + "id": "P", + "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "bytes": 40261, + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "use": "Sections1-2 model, exact original observations, full determinant weight and fixed-jet Fourier support. Gaussian facts reconstructed here. No Theorem A/B/C, contact limit, or marked elder law consumed. Author source exposure." + } + ] +} diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/VALIDATION.md b/frontiers/fixed_r_inverse_lifetime_20260930/VALIDATION.md new file mode 100644 index 000000000..7dc4ffe63 --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/VALIDATION.md @@ -0,0 +1,39 @@ +# Actual author validation, separate from analytic acceptance + +The initial eight-method mathematical scaffold failed all eight assertions. +After implementation all eight passed. The expanded mathematical suite has19 +methods; eleven real-Git/source/inventory tests bring the total to30. + +Fresh replay runs all30 methods in normal and optimized Python, including the +six CLI semantic mutants in each mode and the unknown label. Deterministic +baseline output must match RESULTS.json with no stderr. The exact controls +include36 full Schur matrices in hard dimensions1-4 with nonzero off-diagonal +entries,36 Taylor-path parameter sets and612 rational path points,36 sphere +barriers with360 interior points, exact cubic saddle gaps, weighted split +integrals, and the critical/supercritical scalar-model moment formulas. + +The verifier is reused from the author's PR181 driver, narrowed to the single +parent source and new mathematical CLI. Its byte-count field is required here. +The11 real-Git/inventory tests cover the actual nested working directory, byte +and hash drift, missing/duplicate source inventory, noncommit and unsafe paths, +symlink source/payload, historical replacement forgery, extra files and duplicate +JSON. The replacement fixture first verifies that an ordinary Git read sees +forged bytes, then that the protected reader authenticates the original bytes. +These fixtures are not evidence of tampering in the user's repository. + +The upper/lower Gaussian bounds and infinite-dimensional support are analytic +proof obligations, not conclusions of the scalar examples. No original field +is simulated and no Gaussian coefficient is numerically enclosed. No source +proof, review disposition or global status is changed. + +The local attempt to access the GitHub API failed with DNS resolution error. +Local-only replay therefore reports source_pins_checked=false. The hosted +workflow must fetch the ACTUAL parent commit and verify its full path, regular +mode, byte count, SHA256 and Git-blob identity before reporting source success. +Any hosted receipt belongs to the actual published head, not this author-time +record. Existing Lean checks do not formalize the new analytic proof. + +The new explicit owner instruction is the current September30 ChatGPT request +beginning "Continue to Work on closing as many mathematical items as possible". +It follows the September27 stop and authorizes foreground work, not timers. +No pre-stop autonomy grant or automated-loop restart is used as authorization. diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/inverse.py b/frontiers/fixed_r_inverse_lifetime_20260930/inverse.py new file mode 100644 index 000000000..62698d649 --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/inverse.py @@ -0,0 +1,162 @@ +"""Exact controls for the fixed-separation inverse-lifetime theorem candidate. + +Finite rational checks only: this does not simulate a conditioned Gaussian field +or certify the infinite-dimensional support argument. Standard library only. +""" +import argparse +from fractions import Fraction as F +import json + +MUTANTS=('lose-determinant-weight','wrong-cubic-power','wrong-soft-direction', + 'admit-critical-inverse','drop-endpoint-margin','unconditional-tail-shortcut') +_MUTANT=None + + +def rat(x): + if type(x) not in (int,F):raise TypeError('integer or Fraction required') + return F(x) + + +def determinant(matrix): + n=len(matrix) + if any(len(row)!=n for row in matrix):raise ValueError('square matrix required') + a=[[rat(x) for x in row] for row in matrix];out=F(1) + for j in range(n): + i=next((i for i in range(j,n) if a[i][j]),None) + if i is None:return F(0) + if i!=j:a[j],a[i]=a[i],a[j];out=-out + pivot=a[j][j];out*=pivot + for i in range(j+1,n): + q=a[i][j]/pivot + for k in range(j+1,n):a[i][k]-=q*a[j][k] + return out + + +def inverse(matrix): + n=len(matrix) + if any(len(row)!=n for row in matrix):raise ValueError('square matrix required') + a=[[rat(x) for x in row]+[F(i==j) for j in range(n)] for i,row in enumerate(matrix)] + for j in range(n): + i=next((i for i in range(j,n) if a[i][j]),None) + if i is None:raise ValueError('singular matrix') + a[j],a[i]=a[i],a[j];p=a[j][j];a[j]=[x/p for x in a[j]] + for i in range(n): + if i!=j: + q=a[i][j];a[i]=[x-q*y for x,y in zip(a[i],a[j])] + return [row[n:] for row in a] + + +def soft_chart(C,v,lam): + lam=rat(lam);C=[[rat(x) for x in row] for row in C];v=list(map(rat,v));n=len(C) + if not n or len(v)!=n or any(len(row)!=n for row in C) or lam<=0: + raise ValueError('positive soft curvature and a nonempty square hard block required') + if any(C[i][j]!=C[j][i] for i in range(n) for j in range(n)): + raise ValueError('hard block must be symmetric') + if any((-1)**j*determinant([row[:j] for row in C[:j]])<=0 for j in range(1,n+1)): + raise ValueError('hard block must be strictly negative definite') + inv=inverse(C);cv=[sum((inv[i][j]*v[j] for j in range(n)),F(0)) for i in range(n)] + top=sum((x*y for x,y in zip(v,cv)),F(0))-lam + H=[[top]+v]+[[v[i]]+C[i] for i in range(n)] + e=[F(1)]+[-x for x in cv] + if _MUTANT=='wrong-soft-direction':e=[F(1)]+cv + return H,e + + +def barrier(L,lam,K): + L,lam,K=map(rat,(L,lam,K)) + if L<=0 or not 0=1 required') + a=min(F(1),L/4);R=a*lam/K + return R,lam*R*R/3 + + +def path_constants(a,M): + """B=6/a, admissible lambda ceiling, loss bound C and endpoint-gain coefficient.""" + a,M=map(rat,(a,M)) + if a<=0 or M<0:raise ValueError('positive cubic lower bound and nonnegative fourth bound required') + B=6/a;cut=min(F(1),6/(M*B*B)) if M else F(1) + return B,cut,B*B,F(0) if _MUTANT=='drop-endpoint-margin' else B*B/4 + + +def cubic_gap(lam,g): + lam,g=map(rat,(lam,g)) + if lam<=0 or g<=0:raise ValueError('positive lambda and cubic required') + return 2*lam**(2 if _MUTANT=='wrong-cubic-power' else 3)/(3*g*g) + + +def upper_split(s): + """Integral_0^1 lambda min(1,s^3/lambda^3)d lambda, 0=0, 3p!=2 required') + n=2-three_p + return s**n+2*(1-s**n)/n + + +def check(): + def need(ok,msg): + if not ok:raise ValueError(msg) + need(tail_exponent(1)==F(2,3),'retain original maximum determinant weight') + need(cubic_gap(F(1,10),F(2))==F(1,6000),'cubic lifetime scale') + H,e=soft_chart([[F(-2)]],[F(1,3)],F(1,10)) + q=sum((e[i]*H[i][j]*e[j] for i in range(2) for j in range(2)),F(0)) + need(q==-F(1,10),'correct Schur soft direction') + need(not inverse_finite(F(2,3)),'critical inverse moment diverges') + need(path_constants(1,4)[3]==9,'strict endpoint above birth') + need(upper_split(F(1,10))==F(7,500),'retain weighted near-zero eigenvalue integral') + cases=0 + for a in (F(1,2),F(1),F(2)): + for M in (F(0),F(1),F(4),F(20)): + B,ceiling,C,gain=path_constants(a,M) + for frac in (F(1,10),F(1,2),F(1)): + lam=ceiling*frac + endpoint=-lam*(B*lam)**2/2+a*(B*lam)**3/6-M*(B*lam)**4/24 + need(endpoint>=gain*lam**3>0,'Taylor endpoint') + for j in range(17): + t=B*lam*F(j,16) + lower=-lam*t*t/2+a*t**3/6-M*t**4/24 + need(lower>=-C*lam**3,'Taylor path stays above lower barrier') + cases+=1 + return {'passed':True,'mathematical_acceptance':False,'scientific_effect':'NONE', + 'scope':'finite rational Taylor, Schur and integral controls only', + 'path_parameter_cases':cases,'path_grid_checks':17*cases, + 'fixed_r_small_lifetime_exponent':'2/3','inverse_moment_threshold':'2/3', + 'unweighted_comparison_exponent':'1/3', + 'critical_capped_growth':'Theta(log T)', + 'supercritical_capped_growth':'Theta(T^(p-2/3))', + 'uniform_in_pin_separation':False,'extreme_partner_conditioning':False} + + +def main(): + parser=argparse.ArgumentParser(description=__doc__);parser.add_argument('--mutant',choices=MUTANTS) + args=parser.parse_args();global _MUTANT;_MUTANT=args.mutant + try:ans=check() + except ValueError as exc: + print(json.dumps({'passed':False,'error':str(exc)},sort_keys=True));raise SystemExit(1) + print(json.dumps(ans,indent=2,sort_keys=True)) + +if __name__=='__main__':main() diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/test_inverse.py b/frontiers/fixed_r_inverse_lifetime_20260930/test_inverse.py new file mode 100644 index 000000000..9695c472e --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/test_inverse.py @@ -0,0 +1,121 @@ +from fractions import Fraction as F +import unittest +import inverse as m + +class InitialTests(unittest.TestCase): + def test_schur_chart(self): + H,e=m.soft_chart([[F(-2)]],[F(1,3)],F(1,10)) + self.assertEqual(H,[[F(-7,45),F(1,3)],[F(1,3),F(-2)]]) + self.assertEqual(e,[F(1),F(1,6)]) + def test_barrier(self): + self.assertEqual(m.barrier(F(4),F(1,10),F(2)),(F(1,20),F(1,12000))) + def test_path_constants(self): + self.assertEqual(m.path_constants(F(1),F(4)),(F(6),F(1,24),F(36),F(9))) + def test_spectral_split(self): + self.assertEqual(m.upper_split(F(1,10)),F(7,500)) + def test_exact_cubic_gap(self): + self.assertEqual(m.cubic_gap(F(1,10),F(2)),F(1,6000)) + def test_tilt_exponent(self): + self.assertEqual(m.tail_exponent(1),F(2,3)) + self.assertEqual(m.tail_exponent(0),F(1,3)) + def test_threshold(self): + self.assertEqual([m.inverse_finite(p) for p in (F(0),F(1,3),F(2,3),F(1))],[True,True,False,False]) + def test_critical_cap(self): + self.assertEqual(m.critical_model(),(F(1),F(-2))) + + +class ExtendedTests(unittest.TestCase): + def test_full_scalar_controls(self): + r=m.check();self.assertEqual(r['path_parameter_cases'],36) + self.assertEqual(r['path_grid_checks'],612) + def test_schur_in_arbitrary_hard_dimensions(self): + count=0 + for n in range(1,5): + B=[[F((i+2*j)%5-2,7) for j in range(n)] for i in range(n)] + C=[[-sum(B[k][i]*B[k][j] for k in range(n))-int(i==j) for j in range(n)] for i in range(n)] + for numerator in (1,2,5): + v=[F(numerator*(i+1),11) for i in range(n)] + for lam in (F(1,100),F(1,10),F(1)): + H,e=m.soft_chart(C,v,lam) + He=[sum(H[i][j]*e[j] for j in range(n+1)) for i in range(n+1)] + self.assertEqual(He,[-lam]+[F(0)]*n) + self.assertEqual(m.determinant(H),-lam*m.determinant(C)) + for x in (F(-1),F(2,3)): + y=[F(i-1,3) for i in range(n)];vec=[x]+y + q=sum(vec[i]*H[i][j]*vec[j] for i in range(n+1) for j in range(n+1)) + cv=m.inverse(C);z=[y[i]+x*sum(cv[i][j]*v[j] for j in range(n)) for i in range(n)] + self.assertEqual(q,-lam*x*x+sum(z[i]*C[i][j]*z[j] for i in range(n) for j in range(n))) + count+=1 + self.assertEqual(count,36) + def test_uniform_scalar_barrier_inequality(self): + for L in (F(1,3),F(1),F(4),F(24)): + for K in (F(1),F(3),F(20)): + for ratio in (F(1,100),F(1,3),F(1)): + lam=K*ratio;R,gap=m.barrier(L,lam,K) + self.assertLess(R,L/2) + self.assertLessEqual(-lam*R**2/2+K*R**3/6,-gap) + for j in range(1,11): + t=R*F(j,10) + self.assertLess(-lam*t*t/2+K*t**3/6,0) + def test_scalar_cubic_stationary_point(self): + for lam in (F(1,100),F(1,7),F(1)): + for g in (F(1,2),F(1),F(7)): + t=2*lam/g + self.assertEqual(-lam*t+g*t*t/2,0) + self.assertGreater(-lam+g*t,0) + self.assertEqual(-lam*t*t/2+g*t**3/6,-m.cubic_gap(lam,g)) + def test_lower_path_at_worst_fourth_derivative(self): + for g0 in (F(1,3),F(1),F(3)): + for M in (F(1),F(10),F(100)): + B,cut,C,gain=m.path_constants(g0,M) + for g in (g0,2*g0): + lam=cut/2 + for h in (-M,F(0),M): + val=lambda t:-lam*t*t/2+g*t**3/6+h*t**4/24 + self.assertGreaterEqual(val(B*lam),gain*lam**3) + self.assertTrue(all(val(B*lam*F(j,20))>=-C*lam**3 for j in range(21))) + def test_weighted_split_by_independent_primitives(self): + for s in (F(1,100),F(1,3),F(1)): + direct=s*s/2+s**3*(1/s-1) + self.assertEqual(direct,m.upper_split(s)) + self.assertLessEqual(direct,F(3,2)*s*s) + self.assertGreaterEqual(direct,s*s/2) + def test_random_norm_factor_can_be_absorbed(self): + for A in (F(1),F(3,2),F(10)): + for x in (F(1,100),F(1,3),F(1),F(5)): + self.assertLessEqual(min(F(1),A*x),A*min(F(1),x)) + def test_capped_model_piecewise_integration(self): + for s in (F(1,20),F(1,3),F(1)): + for n in (0,1,3,4,5): + direct=s**(2-n)+2*(1-s**(2-n))/F(2-n) + self.assertEqual(m.cap_model(s,n),direct) + self.assertGreaterEqual(direct,1) + def test_critical_model_logarithm(self): + # For p=2/3, the lower part is1 and the upper integral is -2log(s). + self.assertEqual(m.critical_model(),(1,-2)) + self.assertEqual(F(2)/3,m.tail_exponent()) + def test_wrong_inputs_rejected(self): + for val in (True,0.5,'1/2'): + with self.assertRaises(TypeError):m.cubic_gap(val,1) + for C,v,l in (([[1]],[0],1),([[-1,1],[0,-1]],[0,0],1),([[-1]],[0],0)): + with self.assertRaises(ValueError):m.soft_chart(C,v,l) + with self.assertRaises(ValueError):m.barrier(0,1,2) + with self.assertRaises(ValueError):m.barrier(1,3,2) + with self.assertRaises(ValueError):m.path_constants(0,1) + with self.assertRaises(ValueError):m.upper_split(0) + def test_cli_both_modes_and_mutants(self): + import subprocess,sys + from pathlib import Path + path=str(Path(m.__file__).resolve());baseline=None + for flags in (['-B','-S'],['-B','-O','-S']): + run=subprocess.run([sys.executable,*flags,path],capture_output=True,timeout=25) + self.assertEqual(run.returncode,0,run.stderr);self.assertEqual(run.stderr,b'') + if baseline is None:baseline=run.stdout + self.assertEqual(run.stdout,baseline) + for mutant in m.MUTANTS: + bad=subprocess.run([sys.executable,*flags,path,'--mutant',mutant],capture_output=True,timeout=25) + self.assertEqual(bad.returncode,1,(mutant,bad.stdout,bad.stderr)) + bad=subprocess.run([sys.executable,*flags,path,'--mutant','unknown'],capture_output=True,timeout=25) + self.assertEqual(bad.returncode,2) + +if __name__=='__main__':unittest.main() diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/test_verify.py b/frontiers/fixed_r_inverse_lifetime_20260930/test_verify.py new file mode 100644 index 000000000..1fccbe702 --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/test_verify.py @@ -0,0 +1,69 @@ +"""Real-Git regressions for the reused, narrowed source-verification driver.""" +import hashlib,json,os,subprocess,tempfile,unittest +from pathlib import Path +import verify as v + + +def entry(path,data,**kw): + return dict(path=path,bytes=len(data),sha256=hashlib.sha256(data).hexdigest(), + git_blob=hashlib.sha1(b'blob '+str(len(data)).encode()+b'\0'+data).hexdigest(),**kw) + +class SourceTests(unittest.TestCase): + def setUp(self): + t=tempfile.TemporaryDirectory();self.addCleanup(t.cleanup);self.repo=Path(t.name) + self.root=self.repo/'frontiers/packet';self.root.mkdir(parents=True) + self.git('init','-q');self.data=b'source fixture\n' + (self.repo/'P.md').write_bytes(self.data);self.git('add','P.md');self.git('commit','-qm','fixture') + self.sha=self.git('rev-parse','HEAD').decode().strip();self.e=entry('P.md',self.data,id='P',commit=self.sha);self.write() + def git(self,*args): + env=os.environ.copy();env.pop('GIT_NO_REPLACE_OBJECTS',None) + return subprocess.run(['git','-c','user.name=Fixture','-c','user.email=fixture@example.invalid',*args],cwd=self.repo,env=env,capture_output=True,check=True,timeout=15).stdout + def write(self,es=None): + (self.root/'SOURCES.json').write_text(json.dumps({'sources':[self.e] if es is None else es})) + def test_original_nested_source(self):self.assertEqual(v.sources(self.root),1) + def test_identity_drift(self): + for k,x in [('bytes',0),('bytes',True),('sha256','0'*64),('git_blob','0'*40)]: + self.write([dict(self.e,**{k:x})]) + with self.subTest(k=k),self.assertRaises(ValueError):v.sources(self.root) + def test_empty_duplicate_sources(self): + for es in ([],[self.e,self.e]): + self.write(es) + with self.assertRaises(ValueError):v.sources(self.root) + def test_noncommit_and_unsafe_path(self): + self.write([dict(self.e,commit=self.git('rev-parse','HEAD^{tree}').decode().strip())]) + with self.assertRaises(ValueError):v.sources(self.root) + self.write([dict(self.e,path='../P.md')]) + with self.assertRaises(ValueError):v.sources(self.root) + def test_replace_cannot_change_history(self): + new=b'forged\n';(self.repo/'P.md').write_bytes(new);self.git('add','P.md');self.git('commit','-qm','change') + self.git('replace',self.sha,self.git('rev-parse','HEAD').decode().strip()) + self.assertEqual(self.git('show',self.sha+':P.md'),new) + self.assertEqual(v.sources(self.root),1) + self.write([entry('P.md',new,id='P',commit=self.sha)]) + with self.assertRaises(ValueError):v.sources(self.root) + def test_symlink_source(self): + (self.repo/'P.md').unlink();(self.repo/'P.md').symlink_to('elsewhere') + self.git('add','P.md');self.git('commit','-qm','link') + self.write([dict(self.e,commit=self.git('rev-parse','HEAD').decode().strip())]) + with self.assertRaises(ValueError):v.sources(self.root) + +class PacketTests(unittest.TestCase): + def setUp(self): + t=tempfile.TemporaryDirectory();self.addCleanup(t.cleanup);self.root=Path(t.name) + self.data=b'packet\n';(self.root/'PROOF.md').write_bytes(self.data) + (self.root/'MANIFEST.json').write_text(json.dumps({'files':[entry('PROOF.md',self.data)]})) + def test_original(self):self.assertEqual(v.inventory(self.root),1) + def test_changed_payload(self): + (self.root/'PROOF.md').write_bytes(b'changed') + with self.assertRaises(ValueError):v.inventory(self.root) + def test_extra_file(self): + (self.root/'extra').write_bytes(b'x') + with self.assertRaises(ValueError):v.inventory(self.root) + def test_duplicate_json(self): + (self.root/'MANIFEST.json').write_text('{"files":[],"files":[]}') + with self.assertRaises(ValueError):v.inventory(self.root) + def test_symlink_payload(self): + (self.root/'PROOF.md').unlink();(self.root/'PROOF.md').symlink_to('MANIFEST.json') + with self.assertRaises(ValueError):v.inventory(self.root) + +if __name__=='__main__':unittest.main() diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/verify.py b/frontiers/fixed_r_inverse_lifetime_20260930/verify.py new file mode 100644 index 000000000..22db229cf --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/verify.py @@ -0,0 +1,110 @@ +"""Replay the fixed-r inverse-lifetime packet; --local-only excludes historical sources.""" +import argparse +import hashlib +import json +import os +from pathlib import Path, PurePosixPath +import re +import subprocess +import sys + +IDS={'P'} +MUTANTS=('lose-determinant-weight','wrong-cubic-power','wrong-soft-direction', + 'admit-critical-inverse','drop-endpoint-margin','unconditional-tail-shortcut') + + +def read_json(path): + if path.is_symlink() or not path.is_file():raise ValueError('regular JSON required') + def unique(pairs): + out={} + for k,v in pairs: + if k in out:raise ValueError('duplicate JSON key') + out[k]=v + return out + data=json.loads(path.read_text(encoding='utf-8'),object_pairs_hook=unique) + if not isinstance(data,dict):raise ValueError('JSON object required') + return data + + +def parts(path): + if not isinstance(path,str) or not path:raise ValueError('nonempty path required') + p=PurePosixPath(path) + if p.is_absolute() or str(p)!=path or any(x in ('.','..') or not re.fullmatch('[A-Za-z0-9_.-]+',x) for x in p.parts): + raise ValueError('unsafe or noncanonical path') + return p.parts + + +def identity(data,e): + blob=hashlib.sha1(b'blob '+str(len(data)).encode()+b'\0'+data).hexdigest() + if type(e['bytes']) is not int or len(data)!=e['bytes']: + raise ValueError('size mismatch') + if hashlib.sha256(data).hexdigest()!=e['sha256'] or blob!=e['git_blob']: + raise ValueError('hash mismatch: '+e['path']) + + +def git(root,*args): + run=subprocess.run(['git','--no-replace-objects',*args],cwd=root, + env=dict(os.environ,GIT_NO_REPLACE_OBJECTS='1'), + capture_output=True,timeout=45) + if run.returncode:raise ValueError('Git read failed: '+run.stderr.decode(errors='replace')[:160]) + return run.stdout + + +def sources(root): + es=read_json(root/'SOURCES.json')['sources'] + if not isinstance(es,list) or len(es)!=len(IDS) or {e['id'] for e in es}!=IDS: + raise ValueError('source inventory mismatch') + seen=set() + for e in es: + path=e['path'];ps=parts(path);commit=e['commit'] + if not isinstance(commit,str) or not re.fullmatch('[0-9a-f]{40}',commit): + raise ValueError('full commit identity required') + if (commit,path) in seen:raise ValueError('duplicate source binding') + seen.add((commit,path)) + if git(root,'cat-file','-t',commit).strip()!=b'commit':raise ValueError('not a commit object') + for j in range(1,len(ps)+1): + prefix='/'.join(ps[:j]) + rows=[s for s in git(root,'ls-tree','--full-tree','-z',commit,'--',prefix).split(b'\0') if s] + if len(rows)!=1:raise ValueError('missing or ambiguous source path') + meta,name=rows[0].split(b'\t',1);mode,kind,blob=meta.decode().split() + if name.decode()!=prefix:raise ValueError('source path mismatch') + if j0 is taken first. No whole-cluster maximum or finite-r +rate is to be inferred. + +Design: reciprocal radius z=1/Z makes the physical cutoff an interval with +explicit endpoints. Map it to [-1,1], prove a smooth even integrated function of +1/t using Gaussian derivative domination, and compute its first two coefficients. +Deduce radial and fixed latent-mark total-variation rates, keeping the sign- +retaining law separate. Preserve the complete original anisotropic covariance, +Haar factors and full normalizer. No original source or peer branch is edited. + +Execution: isolated new runtime directory and new additive GitHub branch. +Dylan's repeated autonomy instruction authorizes foreground work; no timers or +background agents are restarted. No subagent tool is available here. + +1. Read source R11-R13, exact Gaussian/spectral interface and current peer claims. +2. Write and observe failing tests for exact shape integrals and endpoint series. +3. Implement standard-library rational controls; test both signs and parity. +4. Write complete domination and coefficient proof, plus all scope exclusions. +5. Verify exact historical source identities, packet inventory and both Python + modes; include hostile Git fixtures and semantic mutants. +6. Publish a new source-bound candidate and request nonauthor analytic review. + +Separate supporting task: bounded nonauthor review of PR173 Route C; accept the +conditional count implication and repair its inadmissible fixed-R substitution +by a diagonal argument. No graph execution or parent revalidation. diff --git a/frontiers/radial_rate_20260930/PROOF.md b/frontiers/radial_rate_20260930/PROOF.md new file mode 100644 index 000000000..65f24b1f9 --- /dev/null +++ b/frontiers/radial_rate_20260930/PROOF.md @@ -0,0 +1,334 @@ +# Smooth inverse-radius expansion and sharp rates for microscopic extreme marks + +Object: OA-RADIAL-RATE-20260930-v1. Author: OpenAI / GPT-6 Astra Pro. +Disposition: AUTHOR-SIDE CONDITIONAL PROOF CANDIDATE; NONAUTHOR REVIEW OPEN. +Scientific effect NONE. No parent proof, register, Boolean, prize, or earlier +review is changed. This conversation authored SC/P and contributed to the +related cluster line; no independent revalidation of those premises is implied. + +## 1. Exact object and scope + +Fix d>=2, L>0, b real, k>0 and an orthonormal frame, under the original model +and full endpoint normalizer. This paper concerns the ALREADY-FORMED limiting +microscopic POINT intensity of [R]. First the original field radius r tends to +zero; only then does the microscopic distance threshold t tend to infinity. +No finite-r convergence rate is proved. Whole-cluster maximum selection and +its moving tie boundaries are not consumed or treated here. + +Exact source identities are in SOURCES.json. [R] is PR166 RADIAL_TAIL.md at +80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337, blob0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4: +R7-R13 supply the root-resolved integral, Gaussian domination and positive leading +coefficient. [SC] is PR162 PROOF.md at382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf, +blob16c56821b52fd76b0be791622b9c3809eafde75a: the nondegenerate raw Gaussian contact +jet, orthogonal spectral convention and finite nonempty microscopic measure. +[P] is imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md at +7a1cb09a9d58d179393e6d29146252f714b9c9bf, blobdfed3b8d318a3ab1950957f393307733a4bef3f2: +only the exact periodized covariance and contact conditioning, used in Section7. +Neither PR169's whole-cluster candidate, its companion-ratio addendum, nor any +numerical coefficient note is a premise. Source reviews are not reviews here. + +Use [R]'s variables lambda=(a,h,O,xi), with ordered positive hard curvatures h, +normalized Haar O, and remaining cubic entries xi. Put + + A=a/(12k), gamma=sqrt(1+A^2), + beta0=a^2/(12k), c0=a^3/(144k^2), + A0=O diag(0,-h) O^T, + H(h)=product h_j^3 product_(i0 on S. + +For y=1/Z define + + g_lambda,u,v(y)=G_lambda(beta0+k b0 y^2, + c0+(a b0/4)y^2+2k(v-b0u)y^3). (2) + +[R, R11] says exactly that the tail F(t)=integral sum_i 1{|z_i|>t}dM is + + F(t)=K0 integral_(S,lambda) Q H + integral_R |Z|^-12 g(1/Z) + 1{(u-AZ)^2+Z^2>t^2}dZ du dv d lambda. (3) + +The multiplicity of roots is already correct in this formula. No extra factor +for a double cluster, pin density, or endpoint normalizer is introduced. + +## 2. A fixed-domain smooth even representation + +**Theorem 1.** There is a smooth even function C(delta), defined for +|delta|<2/3 and with every fixed derivative bounded on |delta|<=1/3, such that + + F(t)=t^-11 C(1/t), t>=3. (4) + +Smoothness also holds in L1 for the underlying UNSIGNED shape/parameter measures. +The claim is C-infinity, not analyticity or convergence of an infinite Taylor +series. Consequently for every fixed N>=0 there are finite coefficients C_(2j) +with + + F(t)=sum_(j=0..N) C_(2j) t^(-11-2j)+O_N(t^(-13-2N)), + -F'(t)=sum_(j=0..N)(11+2j)C_(2j)t^(-12-2j) + +O_N(t^(-14-2N)). (5) + +C_(0)>0 is [R]'s C_star. Constants are at the fixed source parameters; no growing +dimension, volume, vanishing-gap, or parameter-uniform claim is made. + +**Proof.** The reciprocal substitution in (3) gives |Z|^-12 dZ=|y|^10 dy on +both signs. For t>|u| the physical radius condition is + + (t^2-u^2)y^2+2uAy-gamma^2<0. + +Set delta=1/t and y=delta w. Its two endpoints in w are + + w_plus/minus(delta)=m(delta) +/- d(delta), + m(delta)=-uA delta/(1-u^2 delta^2), + d(delta)=sqrt(gamma^2-u^2 delta^2)/(1-u^2 delta^2). (6) + +Thus (4) holds with + + J_delta(u,v,lambda)=d(delta) integral_(-1)^1 + [m(delta)+d(delta)x]^10 + g(delta[m(delta)+d(delta)x]) dx, + C(delta)=K0 integral Q H J_delta du dv d lambda. (7) + +There is no moving integration boundary left. On |delta|<=delta0<2/3 the +denominator is uniformly positive and the square-root radicand is bounded +below by a positive constant, since |u|<=3/2 and gamma>=1. Each fixed derivative +of m,d is bounded by a polynomial in1+|a| with constants depending on delta0,k. + +For every fixed j the nondegenerate Gaussian contact density satisfies + + sup_(beta,c) |partial_(beta,c)^j G_lambda(beta,c)| + <= C_j exp[-c_j(|h|^2+a^2+|xi|^2)] (8) + +uniformly in O. Indeed each derivative is a polynomial times the same Gaussian; +absorb that polynomial into a weaker Gaussian exponent and then discard beta,c +from its quadratic lower bound. A bounded mean changes the constants only. +Uniform tensor norm equivalence under O supplies the displayed retained variables. +No inverse hard eigenvalue is used. This is an elementary derivative extension +of [R]'s raw Gaussian bound, not an extra stochastic assumption. + +The chain rule in (7), for any fixed number of delta derivatives, now gives an +integrable majorant of the form + + C Q H (1+|a|)^M exp[-c(|h|^2+a^2+|xi|^2)], (9) + +on the bounded shape domain and x in[-1,1]. The actual shifted beta,c values +may be large: (8) is a supremum over them, not an invalid compactness assumption. +This proves differentiation under all integrals and smoothness in L1 as well +as for the total mass. The exact identities m(-delta)=-m(delta), +d(-delta)=d(delta), followed by x->-x, show J_(-delta)=J_delta. This cancellation +holds even when g(y) is not even; its cubic y^3 term is retained. + +Taylor's theorem with the integrated derivative bounds proves (5), including +its differentiated remainder. It is NOT obtained by formally differentiating +an unspecified big-O estimate. C(0) is the positive source coefficient. QED. + +## 3. Explicit second coefficient + +Write C_0=C_(0), C_2=C_(2), to distinguish coefficients from their tail powers11,13. +Set the differential operator acting on G_lambda at fixed a,h,O,xi + + D_a=k partial_beta+(a/4)partial_c. + +The pointwise density has expansion g(y)=G0+g2 y^2+g3 y^3+O(y^4), with + + g2=b0 D_a G0. (10) + +There is no linear y term. Expanding the exact endpoints, or differentiating +(7), gives + + j0=(2/11)gamma^11 G0, + j2=u^2 gamma^9(1+12A^2)G0+(2/13)b0 gamma^13 D_a G0, + J_delta=j0+j2 delta^2+O_L1(delta^4). (11) + +For the geometric part, write w_+=gamma-uA delta+ +[u^2 gamma-u^2/(2gamma)]delta^2+O(delta^3), with the corresponding reflected +negative endpoint. Then (w_+^11-w_-^11)/11 equals +2gamma^11/11+u^2 gamma^9(1+12A^2)delta^2+O(delta^4). +The g2 term contributes2g2 gamma^13 delta^2/13. The odd g3 term first contributes +at order delta^4 after the two physical sides are summed; it is not dropped +from the all-order argument. + +The necessary exact shape integrals are + + I=integral_S Q=246528/35, + U2=integral_S u^2 Q=240192/35, + B=integral_S b0 Q=-428544/7=3I-12U2, + Uabs=integral_S |u|Q=5898627/880. (12) + +They follow by integrating the polynomial Q in v and then on the strips cut +at u=-1/2 and u=0. The companion computes them in exact Fraction arithmetic. +In particular + + C_0=K0*(2I/11) integral H gamma^11 G0 d lambda, + C_2=K0 integral H [U2 gamma^9(1+12A^2)G0 + +(2B/13)gamma^13 D_a G0]d lambda. (13) + +Both integrals are finite by (8). C_0>0. No general sign or nonzero assertion +for C_2 is made in arbitrary dimension/frame; Section7 supplies an exact +noncancellation result at a narrower source scope. No numerical evaluation of +either Gaussian integral is claimed. + +## 4. Rates for the radial ratio and unsigned marks + +Let P_t be point-INTENSITY sampling conditioned on physical radius R>t. Define +q=R/t>=1 and retain Y=(u,v,lambda), but FORGET sign Z. These are the root-resolved +latent coordinates of [R], not a configuration-uniform selection. Define the +finite measures with densities + + dmu0=K0 Q H j0 dY, dmu2=K0 Q H j2 dY, + c=C_2/C_0. (14) + +mu0 has mass C_0 and mu2 mass C_2. The joint law of (q,Y) has the L1 expansion + + dP_t(q,Y)=11q^-12 dq dmu0(Y)/C_0 + +t^-2 [13q^-14 dq dmu2(Y)/C_0 + -11c q^-12 dq dmu0(Y)/C_0]+O_L1(t^-4). (15) + +In particular its total-variation distance to the stated independent limiting +product is O(t^-2). Projection to Y, or to the downward height +eta=u+1/2+v/6, retains this rate. This supplies an O(t^-2) TV rate for the +intensity-selected height law H_eta/I in [R, H1-H2]. No spatial-configuration +TV claim is made: the map to actual positions depends on t. + +Proof: for fixed Y, the radius tail density is t^-11 K0 QH J_(1/t). +Differentiate the smooth expression already justified in Section2; at tq its +radial density times t equals + +t^-11 K0 QH q^-12[11J_(1/(tq))+(1/(tq))J'_(1/(tq))]. + +Divide by F(t)=t^-11 C(1/t) and use the L1 Taylor estimates. The remainder is +bounded after integration by a constant times t^-4 q^-12, integrable on q>=1. +This proves (15) without differentiating a merely pointwise asymptotic. QED. + +Let Pareto_11 have density11q^-12 on q>=1. For the scalar radial ratio, (15) gives + + p_t(q)=11q^-12[1+c t^-2((13/11)q^-2-1)]+O_L1(t^-4). + +**Sharp scalar coefficient:** + + dTV(Law_(P_t)(R/t),Pareto_11) + =kappa |c| t^-2+O(t^-4), + kappa=(2/13)(11/13)^(11/2). (16) + +The same leading coefficient holds for the supremum distance of their CDFs. +The correction changes sign only at q0=sqrt(13/11). Its positive integral is +q0^-11-q0^-13=kappa; its total integral is zero. Half its L1 norm is therefore +kappa, proving (16). The uniform CDF remainder follows from the L1 remainder; +the maximum of q^-11(1-q^-2) is also kappa. If C_2=0 the stated error is O(t^-4), +not an unjustified claim of a nonzero second-order term. + +As another quantitative consequence, uniformly over q>=1, + + F(tq)/F(t)=q^-11[1+c t^-2(q^-2-1)]+O(t^-4 q^-11). (17) + +This is a source-specific second-order regular-variation statement. Generic +second-order tail theory is not claimed novel. + +## 5. Retaining the sign has a different, strictly slower sharp rate + +The cancellation in Section2 used both signs. Keep epsilon=sign Z and the +same Y. The two half integrals are exactly + + J_delta,epsilon=integral_0^(d+epsilon m) w^10 g(epsilon delta w)dw, + J_delta,epsilon=j0/2+epsilon h1 delta+O_L1(delta^2), + h1=-uA gamma^10 G0. (18) + +The linear term comes from the physical cutoff shift m. g'(0)=0, so no density +term cancels it. Applying the differentiated density argument to each half yields + + dTV(Law_(P_t)(epsilon,R/t,Y), + UniformSign x Pareto_11 x mu0/C_0) + =B_sign t^-1+O(t^-2), + B_sign=(K0/C_0) Uabs integral H |A|gamma^10 G0 d lambda>0. (19) + +Indeed the first signed correction has density +12 epsilon h1 q^-13 K0 QH/C_0; its half-L1 norm, summed over the two signs and +integrated over q, is the displayed B_sign because integral_1^infinity12q^-13dq=1. +Finiteness follows from (8); strict positivity follows from the full positive +Gaussian density on open sets with a!=0 and positive hard spectrum. The gauge +convention for O is exactly that in [R]; (19) is a result for those retained +latent variables, not a statement that a sign alone has a nonzero marginal bias. +Marginalizing can cancel a correction, and specifically forgetting epsilon +cancels this one. Thus signed and unsigned rates must not be conflated. + +## 6. A finite part for the critical eleventh moment + +Let M11(t)=integral sum_i min(R_i,t)^11 dM. The prior theorem gave its leading +logarithm. There is now a finite constant L11 such that + + M11(t)=11 C_0 log t+L11-(11/2)C_2 t^-2+O(t^-4). (20) + +One exact definition is + + L11=11 integral_0^1 x^10 F(x)dx + +11 integral_1^infinity[x^10 F(x)-C_0/x]dx. + +The first integral is finite by finite total point mass. The second converges +by (5) at N=1. Tonelli gives M11(t)=11 integral_0^t x^10 F(x)dx; subtracting +its limiting finite part and integrating the controlled remainder proves (20). +No numerical value for L11 is asserted. This is again after the r limit only. + +## 7. Exact positive second correction in the aligned planar periodic model + +**Theorem 2.** In d=2, with the axial coordinate aligned with one of the lattice +axes of [P]'s EXACT periodized kernel, C_2>0 for every fixed L,k>0 and b real. +Consequently the radial TV and CDF errors in (16) are genuinely Theta(t^-2), +with the positive coefficient in (16); not merely an upper bound. + +Proof: [P]'s kernel factors as K_L(x,z)=k_L(x)k_L(z). Let m2,m4,m6 be the even +spectral moments of the one-dimensional variance-one factor; m2>0 and the full +positive lattice spectrum gives m4-m2^2>0 and m6-m4^2/m2>0. At contact condition +on (f,fx,fxx,fxxx,fz,fxz)=(b,0,0,12k,0,0). + +Even-total-order jets and odd-total-order jets are independent. For the odd +soft cubic jets (a,beta,c)=(fxxz,fxzz,fzzz), separability and reflection in each +axis give a centered DIAGONAL conditional Gaussian with variances + + va=vbeta=m2(m4-m2^2), vc=m6-m4^2/m2. (21) + +For transparency: Cov(beta,(fx,fxxx))=(-m2^2,m2 m4) is -m2 times the first row +of Cov(fx,fxxx), so its conditional mean is -m2 fx=0 even though fxxx=12k. +a and c correlate only with fz in that odd conditioning block, with covariances +-m2^2 and -m4; their raw covariance m2 m4 is removed by that regression. Their +conditional covariance with beta vanishes by coordinate reflection. The diagonal +variances follow by subtracting these rank-one regressions. The even soft +curvature has its own strictly positive conditional density at zero and merely +multiplies this cubic Gaussian by a positive b-dependent scalar. + +Therefore on the cusp + + D_a G0=-[a^2/(12vbeta)+a^4/(576k^2 vc)]G0<=0. (22) + +In (13), U2>0, B<0 and G0>0. The geometric term is strictly positive and the +second term is nonnegative. Thus C_2>0, without replacing finite L by a continuum +covariance or numerically approximating a lattice sum. The same common even +b-dependent factor cancels from c=C_2/C_0 and B_sign at this planar aligned +scope. We do not extend this diagonal regression or noncancellation assertion +to arbitrary rotations or d>=3 without a separate argument. QED. + +## 8. Verification and review boundary + +A: reciprocal physical cutoff, fixed-domain map, C-infinity L1 Gaussian derivative +domination, exact parity and differentiated all-order asymptotic expansion. +B: j0/j2 coefficients, all four exact shape integrals, signed/unsigned TV and CDF +constants, finite-part moment, and aligned planar regression/noncancellation. + +The stdlib companion checks shape integrals, formal endpoint equations, exact +Taylor coefficients, cancellation and signed bias, independent scalar controls +and deliberately false variants. These are finite checks, not a formal proof +of any Gaussian input or differentiation under an infinite-dimensional integral. +No Lean formalization, global status change, coefficient enclosure or self-merge. + +No rate for r->0, no uniform simultaneous t(r)->infinity statement, no claim +about the remote singleton population, no convergence rate for whole-cluster +maximum selectors across ties, and no spatial independence/elder pairing is +added by this packet. Sign-forgetting is part of (15), not an optional omission. diff --git a/frontiers/radial_rate_20260930/README.md b/frontiers/radial_rate_20260930/README.md new file mode 100644 index 000000000..cc0e2f5a4 --- /dev/null +++ b/frontiers/radial_rate_20260930/README.md @@ -0,0 +1,61 @@ +# Microscopic radial rates and an even inverse-radius expansion + +OpenAI / GPT-6 Astra Pro. Source-bound conditional candidate; nonauthor review +OPEN. Scientific effect NONE. No self-merge or parent/status/graph modification. + +The earlier point-intensity result F(t)~C0 t^-11 is sharpened here to a smooth +even expansion in delta=1/t, with controlled derivatives to every fixed order. +The original field limit r->0 is taken FIRST. This is not a rate for that limit. + +Main manuscript PROOF.md: +- F(t)=C0 t^-11+C2 t^-13+O(t^-15), with an explicit finite Gaussian-derivative + formula for C2 and all-even fixed-order extensions. +- The radial ratio's Pareto(11) TV error has coefficient + (2/13)(11/13)^(11/2)|C2/C0| t^-2+O(t^-4). +- The unsigned shape/parameter/radial joint law has an L1 second-order expansion; + its height marginal consequently converges in TV at O(t^-2). +- RETAINING sign Z gives a strictly positive first-order joint TV coefficient, + Bsign t^-1+O(t^-2). Forgetting sign cancels that term. +- The capped eleventh moment has a finite renormalized limit and an explicit + t^-2 correction, not merely a leading logarithm. +- For the exact aligned PLANAR periodized kernel, Gaussian separability proves + C2>0 at every fixed L,k>0 and b real. Thus the radial error is genuinely + Theta(t^-2) at that narrower scope. No isotropic replacement is used. + +No numerical Gaussian coefficient, finite-r spatial moment/rate, simultaneous +radius limit, whole-cluster maximum rate, spatial TV convergence or independence +between pin pairs is asserted. All-rate statements concern the root-resolved +POINT measure. The parent whole-cluster and elder-geometry lanes are untouched. + +## Replay + +From a full checkout with the three pinned historical source objects: + + python -B -S frontiers/radial_rate_20260930/verify.py + +Standalone download: + + python -B -S verify.py --local-only + +The latter reports source_pins_checked=false. Real local Git fixtures test +custody behavior, not the actual project sources. The full workflow verifies +all source identities and the flat packet before replaying both Python modes, +deterministic results and eight semantic negative controls. No third-party +Python library is required by the shipped scripts. + +Forty local tests passed in each normal/optimized mode at author validation. +They include exact shape integrals, endpoint identities, signed/unsigned formal +series, a high-precision scalar diagnostic, planar Gaussian Schur complements, +and real Git replacement/noncommit/path/inventory attacks. Tests do not prove +the Gaussian domination, differentiation, or the new probability limits. + +## Review requests + +A: Sections2-3, exact physical-cutoff inversion, fixed-domain representation, +C-infinity L1 domination, parity, derivatives and explicit second coefficient. +B: Sections4-7, normalized joint/radial TV rates, signed first-order bias, finite +critical-moment part and exact planar regression/noncancellation. + +The author is exposed to SC/P and related source packets. New reviews must state +that exposure and bind the exact mathematical bytes; existing source verdicts +or green workflows are not automatic acceptance of these new deductions. diff --git a/frontiers/radial_rate_20260930/RECONNAISSANCE.md b/frontiers/radial_rate_20260930/RECONNAISSANCE.md new file mode 100644 index 000000000..3eb935b8d --- /dev/null +++ b/frontiers/radial_rate_20260930/RECONNAISSANCE.md @@ -0,0 +1,33 @@ +# Narrow primary-source reconnaissance — 30 September 2026 UTC + +Exact searches actually executed: +1. second order regular variation Pareto approximation total variation Hall class tail expansion paper +2. site.arxiv.org Palm theory extremes anchored tail processes Planinic 2104.03810 + +Primary records consulted: +- Hrvoje Planinic, Palm theory for extremes of stationary regularly varying time + series and random fields, arXiv:2104.03810v2 (14 March2023). + https://arxiv.org/abs/2104.03810 + Direct abstract/metadata open succeeded. It distinguishes typical exceedances + from typical clusters; no generic sampling/Palm principle is claimed new here. +- Second-order regular variation, convolution and the central limit theorem, + Stochastic Processes and their Applications69(2),139-159(1997), + DOI10.1016/S0304-4149(97)00042-2. + https://www.sciencedirect.com/science/article/pii/S0304414997000422 + The search response supplied the abstract describing second-order tail theory. + Direct open returned403. No full-text/theorem-level audit is claimed. +- L. de Haan and L. Peng, Rates of Convergence for Bivariate Extremes, + Journal of Multivariate Analysis61(2),195-230(1997), DOI10.1006/jmva.1997.1669. + https://doi.org/10.1006/jmva.1997.1669 + Search abstract states rates in total variation and uniform metrics under a + second-order regular-variation condition. Direct open returned403. + +This new packet derives its particular inverse-radius expansion from exact +GitHub source R11, not by importing an unchecked second-order tail theorem. +Generic second-order regular variation and metric convergence rates are prior +frameworks. No exhaustive novelty, priority, or absence claim is made. + +Consensus was not invoked in this pass: the user requested GitHub/Superpowers, +and prior actual repeated attempts had returned the quota error. No changed +quota status is asserted. All external access depth above is limited to the +actual returned abstracts/metadata. No PDF or full paper was inspected. diff --git a/frontiers/radial_rate_20260930/RESULTS.json b/frontiers/radial_rate_20260930/RESULTS.json new file mode 100644 index 000000000..c7494046e --- /dev/null +++ b/frontiers/radial_rate_20260930/RESULTS.json @@ -0,0 +1,21 @@ +{ + "mathematical_acceptance": false, + "passed": true, + "scientific_effect": "NONE", + "scope": "finite rational coefficient and parity controls only", + "series_cases": 12, + "shape_integrals": { + "ABS_U": "5898627/880", + "B": "-428544/7", + "I": "246528/35", + "U2": "240192/35" + }, + "signed_joint_rate": 1, + "tail_exponents": [ + 11, + 13, + 15 + ], + "tv_crossing_squared": "13/11", + "unsigned_radial_rate": 2 +} diff --git a/frontiers/radial_rate_20260930/SOURCES.json b/frontiers/radial_rate_20260930/SOURCES.json new file mode 100644 index 000000000..37f3dc7b5 --- /dev/null +++ b/frontiers/radial_rate_20260930/SOURCES.json @@ -0,0 +1,33 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "sources": [ + { + "id": "R", + "commit": "80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337", + "path": "frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md", + "bytes": 12533, + "sha256": "250897858c8b314c0ff85ec1860a725efacfdae0ff7409fced4cadf7bfb85973", + "git_blob": "0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4", + "use": "R7-R13 exact point-intensity integral/majorant; H1-H2 limiting height law." + }, + { + "id": "SC", + "commit": "382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf", + "path": "frontiers/spectral_cluster_closure_20260929/PROOF.md", + "bytes": 25006, + "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", + "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a", + "use": "Raw nondegenerate contact Gaussian, orthogonal spectral conventions, finite microscopic measure. Author exposure." + }, + { + "id": "P", + "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "use": "Exact product-periodized covariance and contact pin vector; no new parent acceptance." + } + ] +} diff --git a/frontiers/radial_rate_20260930/VALIDATION.md b/frontiers/radial_rate_20260930/VALIDATION.md new file mode 100644 index 000000000..5a4a11b11 --- /dev/null +++ b/frontiers/radial_rate_20260930/VALIDATION.md @@ -0,0 +1,45 @@ +# Author validation and limits + +The initial eight-test mathematics scaffold returned seven failing assertions +(the zero odd-series test already passed on the zero scaffold). After exact +implementation all eight passed. Expanded mathematics tests total21. + +The nineteen-test custody scaffold returned25 failing assertions including +subtests. The implemented verifier passed all19. It rejects changed identities, +missing/duplicate/unexpected source IDs, unsafe or missing paths, noncommit tree +and tag objects, Git symlinks, replacement refs, and altered packet inventories. +The replacement fixture verifies that ordinary Git reads really see the forged +bytes before confirming that the protected verifier still authenticates only +the actual original source. This is a test fixture, not observed source tampering. + +Complete local result:40 tests PASS in each normal and optimized Python mode; +12 exact formal-series parameter cases; eight explicit semantic mutants rejected +with rc1 in each mode; unknown mutant rc2; deterministic baseline equal to +RESULTS.json. The source/packet verifier checks its inventory before and after +execution. No temporary output is placed in the packet during replay. + +Exact controls cover all four shape integrals, formal physical-cutoff equations, +odd-term cancellation, the signed first-order term, the density multiplier13, +the TV crossing at sqrt(13/11), and exact planar Schur regressions. A Decimal +scalar diagnostic checks the constant-density endpoint remainder at three +thresholds; it is a finite diagnostic, not a Gaussian coefficient enclosure. +Exploratory SymPy integration reproduced the shape constants, but SymPy is not +a shipped dependency or a substitute for the analytic derivation. + +Local runtime Python3.13.5, Git2.47.3. The workflow requests Python3.11.16, as in +the actual upstream workflows. All shipped syntax uses standard-library3.11 +features. The local raw.githubusercontent.com request failed with DNS resolution +error, so no complete upstream checkout was obtained. Local-only replay states +source_pins_checked=false. The real three-source authentication remains a +HOSTED requirement, not inferred from local Git fixtures or manifest strings. +No hosted pass is anticipated in this author-time record. + +The full Gaussian derivative domination and L1 smoothness are written analytic +obligations. There is no Lean formalization of this new packet. No numerical +value of C0,C2,Bsign or the critical-moment finite part is evaluated. The retained +sign and the order r->0 before t->infinity are load-bearing qualifications. + +Peer support actually delivered: native review5360751376 accepts PR173 §3 Route C +at its conditional count-estimate scope and requests an explicit diagonal repair +of the alternate fixed-R/moving-cutoff reading. It does not authorize graph +execution or count as an independent review of this conversation's source proofs. diff --git a/frontiers/radial_rate_20260930/rate.py b/frontiers/radial_rate_20260930/rate.py new file mode 100644 index 000000000..d213a2762 --- /dev/null +++ b/frontiers/radial_rate_20260930/rate.py @@ -0,0 +1,170 @@ +"""Exact finite controls for the inverse-radius asymptotic candidate. + +Standard library only. Series calculations are formal algebra, not authentication +of Gaussian premises or a proof of domination/total-variation convergence. +""" +import argparse +from fractions import Fraction as F +import json +from math import comb + +MUTANTS=('omit-shift','omit-density-term','drop-a-shear','odd-term-survives', + 'unsigned-is-signed','wrong-b-integral','wrong-density-factor','wrong-tv-root') + + +def rational(x): + if type(x) not in (int,F): raise TypeError('exact int/Fraction required') + return F(x) + + +def add(a,b): + out=dict(a) + for i,x in b.items():out[i]=out.get(i,F(0))+x + return {i:x for i,x in out.items() if x} + + +def scale(a,c): return {i:x*c for i,x in a.items() if x*c} + + +def mul(a,b): + out={} + for i,x in a.items(): + for j,y in b.items():out[i+j]=out.get(i+j,F(0))+x*y + return {i:x for i,x in out.items() if x} + + +def power(a,n): + out={0:F(1)} + for _ in range(n):out=mul(out,a) + return out + + +def integral(a,lo,hi): + return sum((x*(hi**(i+1)-lo**(i+1))/F(i+1) for i,x in a.items()),F(0)) + + +def shape_integrals(): + b={0:F(3),2:F(-12)};u={1:F(1)};hi={0:F(3),1:F(-6)} + # Q=(4v^2-b^2)(b-4uv); dictionary index is v power. + Q={3:scale(u,-16),2:scale(b,4),1:scale(mul(u,power(b,2)),4),0:scale(power(b,3),-1)} + ans={key:F(0) for key in ('I','U2','B','ABS_U')} + for lo_u,hi_u,sgn_b,sgn_u in ((F(-3,2),F(-1,2),-1,-1),(F(-1,2),F(0),1,-1),(F(0),F(1,2),1,1)): + low=scale(b,F(sgn_b,2));vint={} + for j,c in Q.items(): + difference=add(power(hi,j+1),scale(power(low,j+1),-1)) + vint=add(vint,scale(mul(c,difference),F(1,j+1))) + for key,weight in [('I',{0:F(1)}),('U2',{2:F(1)}),('B',b),('ABS_U',{1:F(sgn_u)})]: + ans[key]+=integral(mul(vint,weight),lo_u,hi_u) + return ans + + +def smul(a,b,n): + out=[F(0)]*(n+1) + for i,x in enumerate(a[:n+1]): + for j,y in enumerate(b[:n+1-i]):out[i+j]+=x*y + return out + + +def spow(a,m,n): + out=[F(1)]+[F(0)]*n + for _ in range(m):out=smul(out,a,n) + return out + + +def endpoints(u,A,g,n): + u,A,g=map(rational,(u,A,g)) + if type(n) is not int or not 0<=n<=12: raise ValueError('series order 0..12 required') + if g<=0 or g*g!=1+A*A: raise ValueError('positive exact gamma^2=1+A^2 required') + inv=[F(0)]*(n+1);root=[F(0)]*(n+1) + binom=F(1) + for j in range(n//2+1): + inv[2*j]=u**(2*j) + if j:binom*= (F(1,2)-(j-1))/j + root[2*j]=g*binom*(-u*u/(g*g))**j + d=smul(inv,root,n) + m=[F(0)]+[-u*A*x for x in inv[:n]] + plus=[x+y for x,y in zip(m,d)];minus=[x-y for x,y in zip(m,d)] + return plus,minus + + +def validate_phi(phi,n): + if not isinstance(phi,dict) or any(type(j) is not int or j<0 for j in phi): + raise ValueError('nonnegative polynomial orders required') + return {j:rational(x) for j,x in phi.items() if j<=n} + + +def tail_series(u,A,g,phi,n): + plus,minus=endpoints(u,A,g,n);phi=validate_phi(phi,n) + out=[F(0)]*(n+1) + for j,c in phi.items(): + p=spow(plus,11+j,n-j);m=spow(minus,11+j,n-j) + for i in range(n-j+1):out[i+j]+=c*(p[i]-m[i])/F(11+j) + return out + + +def half_series(u,A,g,phi,n,sign): + if type(sign) is not int or sign not in (-1,1): raise ValueError('sign must be +/-1') + plus,minus=endpoints(u,A,g,n);phi=validate_phi(phi,n) + endpoint=plus if sign==1 else [-x for x in minus] + out=[F(0)]*(n+1) + for j,c in phi.items(): + p=spow(endpoint,11+j,n-j) + for i in range(n-j+1):out[i+j]+=c*sign**j*p[i]/F(11+j) + return out + + +def density_series(coeffs):return [(11+j)*x for j,x in enumerate(coeffs)] + + +def predicted_second(u,A,g,phi0,phi2): + return u*u*g**9*(1+12*A*A)*phi0+F(2,13)*g**13*phi2 + + +def check(mutant=None): + vals=shape_integrals() + expected={'I':F(246528,35),'U2':F(240192,35),'B':F(-428544,7),'ABS_U':F(5898627,880)} + if mutant=='wrong-b-integral': vals['B']=-vals['B'] + if vals!=expected: raise ValueError('shape integral mismatch') + cases=0 + for u in (F(-5,4),F(-3,4),F(-1,4),F(1,4)): + for p in (F(-1,2),F(0),F(1,3)): + A=2*p/(1-p*p);g=(1+p*p)/(1-p*p) + phi={0:F(2),2:F(-3),3:F(5),4:F(7)} + got=tail_series(u,A,g,phi,6) + if got[0]!=2*g**11*phi[0]/11:raise ValueError('leading coefficient') + guess=predicted_second(u,A,g,phi[0],phi[2]) + if mutant=='omit-shift':guess-=10*u*u*A*A*g**9*phi[0] + if mutant=='omit-density-term':guess-=F(2,13)*g**13*phi[2] + if mutant=='drop-a-shear':guess=predicted_second(u,F(0),F(1),phi[0],phi[2]) + if got[2]!=guess:raise ValueError('second coefficient mismatch') + odd=[got[i] for i in (1,3,5)] + if mutant=='odd-term-survives':odd[0]=1 + if any(odd):raise ValueError('odd inverse powers do not cancel') + hplus=half_series(u,A,g,phi,6,1);hminus=half_series(u,A,g,phi,6,-1) + if [a+b for a,b in zip(hplus,hminus)]!=got:raise ValueError('half sum') + h1=-u*A*g**10*phi[0] + if mutant=='unsigned-is-signed':h1=F(0) + if hplus[1]!=h1:raise ValueError('signed first-order bias') + den=density_series(got) + expected_den=13*got[2] + if mutant=='wrong-density-factor':expected_den=11*got[2] + if den[2]!=expected_den:raise ValueError('radial density coefficient') + cases+=1 + crossing2=F(13,11) if mutant!='wrong-tv-root' else F(11,13) + if F(13,11)/crossing2!=1:raise ValueError('TV crossing root') + return {'passed':True,'mathematical_acceptance':False,'scientific_effect':'NONE', + 'scope':'finite rational coefficient and parity controls only', + 'shape_integrals':{k:str(v) for k,v in expected.items()}, + 'series_cases':cases,'tail_exponents':[11,13,15], + 'unsigned_radial_rate':2,'signed_joint_rate':1,'tv_crossing_squared':str(crossing2)} + + +def main(): + parser=argparse.ArgumentParser(description=__doc__);parser.add_argument('--mutant',choices=MUTANTS) + args=parser.parse_args() + try:result=check(args.mutant) + except ValueError as exc: + print(json.dumps({'passed':False,'error':str(exc)},sort_keys=True));raise SystemExit(1) + print(json.dumps(result,sort_keys=True,indent=2)) + +if __name__=='__main__':main() diff --git a/frontiers/radial_rate_20260930/test_rate.py b/frontiers/radial_rate_20260930/test_rate.py new file mode 100644 index 000000000..4074549e5 --- /dev/null +++ b/frontiers/radial_rate_20260930/test_rate.py @@ -0,0 +1,137 @@ +from fractions import Fraction as F +import unittest +import rate + +class InitialTests(unittest.TestCase): + def test_shape_mass(self): self.assertEqual(rate.shape_integrals()['I'],F(246528,35)) + def test_shape_u2(self): self.assertEqual(rate.shape_integrals()['U2'],F(240192,35)) + def test_shape_b(self): self.assertEqual(rate.shape_integrals()['B'],F(-428544,7)) + def test_shape_abs_u(self): self.assertEqual(rate.shape_integrals()['ABS_U'],F(5898627,880)) + def test_leading_boundary(self): + c=rate.tail_series(F(-3,4),F(3,4),F(5,4),{0:F(1)},4) + self.assertEqual(c[0],2*F(5,4)**11/11) + def test_second_boundary(self): + u,A,g=F(-3,4),F(3,4),F(5,4) + c=rate.tail_series(u,A,g,{0:F(1),2:F(2)},4) + self.assertEqual(c[2],u*u*g**9*(1+12*A*A)+F(4,13)*g**13) + def test_even_series(self): + c=rate.tail_series(F(-3,4),F(3,4),F(5,4),{0:F(1),2:F(2),3:F(7)},6) + self.assertEqual([c[i] for i in (1,3,5)],[0,0,0]) + def test_signed_first_coefficient(self): + u,A,g=F(-3,4),F(3,4),F(5,4) + c=rate.half_series(u,A,g,{0:F(1),2:F(2),3:F(7)},4,1) + self.assertEqual(c[1],-u*A*g**10) + + +class ExtendedTests(unittest.TestCase): + def test_endpoint_equation(self): + for u in (F(-5,4),F(-1,2),F(1,4)): + for p in (F(-1,2),F(0),F(1,3)): + A=2*p/(1-p*p);g=(1+p*p)/(1-p*p);n=8 + for B in rate.endpoints(u,A,g,n): + square=rate.smul(B,B,n) + for j in range(n+1): + value=square[j]-(u*u*square[j-2] if j>=2 else 0) + value+=2*u*A*B[j-1] if j>=1 else 0 + value-=g*g if j==0 else 0 + self.assertEqual(value,0) + def test_general_polynomial_parity(self): + phi={0:F(1),1:F(3),2:F(-5),3:F(7),4:F(2),5:F(-1)} + got=rate.tail_series(F(-4,3),F(3,4),F(5,4),phi,10) + self.assertTrue(all(got[j]==0 for j in (1,3,5,7,9))) + def test_half_reflection(self): + u,A,g=F(-3,4),F(3,4),F(5,4);phi={0:F(2),2:F(-3),3:F(5)} + plus=rate.half_series(u,A,g,phi,8,1);minus=rate.half_series(u,A,g,phi,8,-1) + self.assertEqual(minus,[x*(-1)**j for j,x in enumerate(plus)]) + def test_cubic_density_term_first_enters_at_four(self): + args=F(-3,4),F(3,4),F(5,4) + x=rate.tail_series(*args,{0:F(1)},6);y=rate.tail_series(*args,{0:F(1),3:F(5)},6) + self.assertEqual(x[:4],y[:4]);self.assertNotEqual(x[4],y[4]) + def test_shear_zero_removes_first_signed_bias(self): + got=rate.half_series(F(-3,4),F(0),F(1),{0:F(1),2:F(3),3:F(5)},6,1) + self.assertEqual(got[1],0);self.assertNotEqual(got[3],0) + def test_density_multiplier(self): + coeff=[F(2),0,F(3),0,F(5)] + self.assertEqual(rate.density_series(coeff),[F(22),0,F(39),0,F(75)]) + def test_b_integral_relation(self): + c=rate.shape_integrals();self.assertEqual(c['B'],3*c['I']-12*c['U2']) + def test_constant_density_decimal_remainder(self): + from decimal import Decimal,localcontext + with localcontext() as ctx: + ctx.prec=70;D=Decimal;u=D('-0.75');A=D('0.75');g=D('1.25') + c0=2*g**11/11;c2=u*u*g**9*(1+12*A*A) + vals=[] + for t in (D(40),D(80),D(160)): + delta=1/t;disc=(g*g-u*u*delta*delta).sqrt();den=1-u*u*delta*delta + bp=(-u*A*delta+disc)/den;bm=(-u*A*delta-disc)/den + J=(bp**11-bm**11)/11 + vals.append(abs((J-c0-c2*delta*delta)/delta**4)) + self.assertTrue(all(x0 for x in expected)) + def test_second_correction_positive_in_aligned_case(self): + vals=rate.shape_integrals() + for a in (F(0),F(1),F(-3)): + for k in (F(1,2),F(1),F(2)): + vbeta,vc=F(2),F(6);da=-(a*a/(12*vbeta)+a**4/(576*k*k*vc)) + self.assertGreater(vals['U2'],0) + self.assertGreaterEqual(vals['B']*da,0) + +if __name__=='__main__':unittest.main() diff --git a/frontiers/radial_rate_20260930/test_verify.py b/frontiers/radial_rate_20260930/test_verify.py new file mode 100644 index 000000000..258288f8e --- /dev/null +++ b/frontiers/radial_rate_20260930/test_verify.py @@ -0,0 +1,102 @@ +"""Custody regressions on real local Git objects, never on invented upstream data.""" +import copy +import hashlib +import json +import os +from pathlib import Path +import subprocess +import tempfile +import unittest +import verify as v + + +def entry(path,data,**kw): + return dict(path=path,bytes=len(data),sha256=hashlib.sha256(data).hexdigest(), + git_blob=hashlib.sha1(b'blob '+str(len(data)).encode()+b'\0'+data).hexdigest(),**kw) + +class SourceTests(unittest.TestCase): + def setUp(self): + t=tempfile.TemporaryDirectory();self.addCleanup(t.cleanup);self.repo=Path(t.name) + self.root=self.repo/'frontiers'/'packet';self.root.mkdir(parents=True) + self.git('init','-q');self.data=b'fixture\n' + for name in ('R','SC','P'):(self.repo/(name+'.md')).write_bytes(self.data) + self.git('add','R.md','SC.md','P.md');self.git('commit','-qm','fixture') + self.commit=self.git('rev-parse','HEAD').decode().strip() + self.es=[entry(x+'.md',self.data,id=x,commit=self.commit) for x in ('R','SC','P')];self.write() + def git(self,*args): + env=os.environ.copy();env.pop('GIT_NO_REPLACE_OBJECTS',None) + return subprocess.run(['git','-c','user.name=Fixture','-c','user.email=fixture@example.invalid',*args], + cwd=self.repo,env=env,check=True,capture_output=True,timeout=15).stdout + def write(self,es=None): + (self.root/'SOURCES.json').write_text(json.dumps({'sources':self.es if es is None else es})) + def test_valid_nested(self):self.assertEqual(v.sources(self.root),3) + def test_identity_fields(self): + for k,val in (('bytes',0),('bytes',True),('sha256','0'*64),('git_blob','0'*40)): + es=copy.deepcopy(self.es);es[0][k]=val;self.write(es) + with self.subTest(k=k,val=val),self.assertRaises(ValueError):v.sources(self.root) + def test_missing_and_duplicate_inventory(self): + for es in ([],self.es[:-1],self.es+[self.es[0]]): + self.write(es) + with self.assertRaises(ValueError):v.sources(self.root) + def test_wrong_id(self): + es=copy.deepcopy(self.es);es[0]['id']='X';self.write(es) + with self.assertRaises(ValueError):v.sources(self.root) + def test_unsafe_paths(self): + for p in ('../R.md','/R.md','./R.md','x//R.md','x/../R.md',''): + es=copy.deepcopy(self.es);es[0]['path']=p;self.write(es) + with self.subTest(path=p),self.assertRaises(ValueError):v.sources(self.root) + def test_missing_file(self): + es=copy.deepcopy(self.es);es[0]['path']='missing.md';self.write(es) + with self.assertRaises(ValueError):v.sources(self.root) + def test_noncommit_tree(self): + es=copy.deepcopy(self.es);es[0]['commit']=self.git('rev-parse','HEAD^{tree}').decode().strip();self.write(es) + with self.assertRaises(ValueError):v.sources(self.root) + def test_annotated_tag(self): + self.git('tag','-a','tag','-m','tag');es=copy.deepcopy(self.es) + es[0]['commit']=self.git('rev-parse','tag').decode().strip();self.write(es) + with self.assertRaises(ValueError):v.sources(self.root) + def test_missing_commit(self): + es=copy.deepcopy(self.es);es[0]['commit']='0'*40;self.write(es) + with self.assertRaises(ValueError):v.sources(self.root) + def test_leaf_symlink(self): + (self.repo/'R.md').unlink();(self.repo/'R.md').symlink_to('SC.md');self.git('add','R.md');self.git('commit','-qm','link') + es=copy.deepcopy(self.es);es[0]['commit']=self.git('rev-parse','HEAD').decode().strip();self.write(es) + with self.assertRaises(ValueError):v.sources(self.root) + def test_replacements_disabled(self): + changed=b'forged bytes\n';(self.repo/'R.md').write_bytes(changed);self.git('add','R.md');self.git('commit','-qm','changed') + newer=self.git('rev-parse','HEAD').decode().strip();self.git('replace',self.commit,newer) + self.assertEqual(self.git('show',self.commit+':R.md'),changed) + self.assertEqual(v.sources(self.root),3) + es=copy.deepcopy(self.es);es[0].update(entry('R.md',changed));self.write(es) + with self.assertRaises(ValueError):v.sources(self.root) + +class PacketTests(unittest.TestCase): + def setUp(self): + t=tempfile.TemporaryDirectory();self.addCleanup(t.cleanup);self.root=Path(t.name);self.data=b'payload\n' + (self.root/'proof.md').write_bytes(self.data) + self.entries=[entry('proof.md',self.data)];self.write() + def write(self): (self.root/'MANIFEST.json').write_text(json.dumps({'files':self.entries})) + def test_valid(self):self.assertEqual(v.inventory(self.root),1) + def test_changed(self): + (self.root/'proof.md').write_bytes(b'changed') + with self.assertRaises(ValueError):v.inventory(self.root) + def test_extra(self): + (self.root/'extra').write_bytes(b'x') + with self.assertRaises(ValueError):v.inventory(self.root) + def test_missing(self): + (self.root/'proof.md').unlink() + with self.assertRaises(ValueError):v.inventory(self.root) + def test_duplicate(self): + self.entries+=self.entries;self.write() + with self.assertRaises(ValueError):v.inventory(self.root) + def test_symlink_payload(self): + (self.root/'proof.md').unlink();(self.root/'proof.md').symlink_to('MANIFEST.json') + with self.assertRaises(ValueError):v.inventory(self.root) + def test_symlink_manifest(self): + p=self.root/'MANIFEST.json';p.rename(self.root/'real');p.symlink_to('real') + with self.assertRaises(ValueError):v.inventory(self.root) + def test_json_duplicate(self): + p=self.root/'MANIFEST.json';p.write_text('{"files":[],"files":[]}') + with self.assertRaises(ValueError):v.inventory(self.root) + +if __name__=='__main__':unittest.main() diff --git a/frontiers/radial_rate_20260930/verify.py b/frontiers/radial_rate_20260930/verify.py new file mode 100644 index 000000000..d5789cbe7 --- /dev/null +++ b/frontiers/radial_rate_20260930/verify.py @@ -0,0 +1,104 @@ +"""Exact packet replay; --local-only does not authenticate upstream Git sources.""" +import argparse +import hashlib +import json +import os +from pathlib import Path, PurePosixPath +import re +import subprocess +import sys + +IDS={'R','SC','P'} +MUTANTS=('omit-shift','omit-density-term','drop-a-shear','odd-term-survives', + 'unsigned-is-signed','wrong-b-integral','wrong-density-factor','wrong-tv-root') + + +def read_json(path): + if path.is_symlink() or not path.is_file():raise ValueError('regular JSON file required') + def unique(pairs): + ans={} + for k,v in pairs: + if k in ans:raise ValueError('duplicate JSON key') + ans[k]=v + return ans + ans=json.loads(path.read_text(encoding='utf-8'),object_pairs_hook=unique) + if not isinstance(ans,dict):raise ValueError('JSON object required') + return ans + + +def path_parts(path): + if not isinstance(path,str) or not path:raise ValueError('nonempty path required') + p=PurePosixPath(path) + if p.is_absolute() or str(p)!=path or any(x in ('.','..') or not re.fullmatch('[A-Za-z0-9_.-]+',x) for x in p.parts): + raise ValueError('unsafe path') + return p.parts + + +def identity(data,e): + blob=hashlib.sha1(b'blob '+str(len(data)).encode()+b'\0'+data).hexdigest() + if type(e['bytes']) is not int or len(data)!=e['bytes'] or hashlib.sha256(data).hexdigest()!=e['sha256'] or blob!=e['git_blob']: + raise ValueError('identity mismatch: '+e['path']) + + +def git(root,*args): + run=subprocess.run(['git','--no-replace-objects',*args],cwd=root, + env=dict(os.environ,GIT_NO_REPLACE_OBJECTS='1'),capture_output=True,timeout=45) + if run.returncode:raise ValueError('Git object read failed: '+run.stderr.decode(errors='replace')[:200]) + return run.stdout + + +def sources(root): + es=read_json(root/'SOURCES.json')['sources'] + if not isinstance(es,list) or len(es)!=len(IDS) or {e['id'] for e in es}!=IDS: + raise ValueError('missing, duplicate or unexpected source IDs') + seen=set() + for e in es: + parts=path_parts(e['path']);commit=e['commit'] + if not isinstance(commit,str) or not re.fullmatch('[0-9a-f]{40}',commit):raise ValueError('exact commit required') + pair=(commit,e['path']) + if pair in seen:raise ValueError('duplicate source binding') + seen.add(pair) + if git(root,'cat-file','-t',commit).strip()!=b'commit':raise ValueError('not a commit object') + for j in range(1,len(parts)+1): + prefix='/'.join(parts[:j]);records=git(root,'ls-tree','--full-tree','-z',commit,'--',prefix).split(b'\0') + records=[x for x in records if x] + if len(records)!=1:raise ValueError('missing or ambiguous source path') + meta,name=records[0].split(b'\t',1);mode,kind,blob=meta.decode().split() + if name.decode()!=prefix:raise ValueError('path mismatch') + if j