d6g8k5htny-coder · Oct 4, 2026 · Sep 30, 2026 · Sep 30, 2026 · Sep 30, 2026 · Sep 30, 2026
@@ -0,0 +1,69 @@
+name: C6 microscopic tail numerics (replay)
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/c6_microscopic_tail_numerics_20260930/**'
+      - '.github/workflows/c6-microscopic-tail-numerics.yml'
+      - 'frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md'
+      - 'frontiers/spectral_cluster_closure_20260929/PROOF.md'
+      - 'frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json'
+      - 'frontiers/planar_cubic_cluster_20260929/PROOF.md'
+      - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md'
+      - 'frontiers/extreme_cluster_sampling_20260930/PROOF.md'
+      - 'frontiers/radial_rate_20260930/PROOF.md'
+  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_microscopic_tail_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 / 'tail_constants.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 ['shape-integral', 'gamma-power', 'cusp-shift']:
+                  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
@@ -0,0 +1,177 @@
+# Numerical values of the planar microscopic radius-tail constants for the parent kernel
+
+**Object:** `CL-C6-MICRO-TAIL-NUMERICS-20260930-v1`. **Author lane:** Anthropic / Claude. **Kind:** numerical note.
+**Scientific effect:** NONE. **Certified:** no (deterministic Gauss quadrature with reported convergence; not an
+enclosure). Third of the planar numerics notes after Math-#168 (near coefficients `alpha_1, alpha_2`) and
+Math-#174 (remote coefficient `k integral_X Lambda`).
+
+## 1. What is evaluated
+
+[R] (Math-#166, merged) proves for the already-formed limiting microscopic measure `M` (the `r -> 0` limit taken
+first) the point-intensity radius tail
+
+    F(t) = integral sum_i 1{R_i > t} dM ~ C_* t^(-11),
+    C_* = (216/11) k^7 (c_m/z_0) I J_cusp,    I = 246528/35,                                        (R13)
+
+with `J_cusp = integral H(h) gamma(a)^11 h_0(A_0, Rot_O tau_cusp) da dxi dh dO` (R12) and
+`gamma(a) = sqrt(1 + (a/(12k))^2)`, and lists "no numerical value of its Gaussian integral" as a non-claim. The
+companions Math-#169 ([E]) and Math-#176 ([Q]), both merged but still conditional (§5), express further constants of the same measure through
+the same cusp integrals: the whole-cluster split of the tail (E6) and the second-order coefficient `C_2`, the
+total-variation coefficient `kappa |C_2/C_0|` of the radial ratio and the signed constant `B_sign` ((13), (16),
+(19)). This note evaluates all of them in the plane for the parent kernel `K(z) = exp(-|z|^2/2)` ([LP] §2).
+
+**Planar reduction.** In `d = 2` the spectral measure (17) of [SC] has `m = d - 1 = 1`, `c_1 = 1`, no hard
+curvatures, and the Haar variable `O in O(1) = {±1}` averages trivially, because `p_odd` is invariant under
+`(a, beta, c) -> (-a, beta, -c)`; `H(h) = 1`. Its density `h_0(A_0, tau)` at `A_0 = 0` is the raw contact density of
+`(f_zz; f_xxz, f_xzz, f_zzz)` at `(0; a, beta, c)` given the contact pins `(f, f_x, f_z, f_xx, f_xz, f_xxx) =
+(b, 0, 0, 0, 0, 12k)`, which by the exact regression of Math-#168 factors as `p_b(0) . p_odd(a, beta, c)` with
+`p_b(0) = phi(b/sqrt2)/sqrt2` (the density of `f_zz | (f, f_xx, f_xz) = (b, 0, 0) ~ N(-b, 2)` at `0`) and
+`p_odd = N(0, diag(2, 2, 6))`, for every `k, b`. These are the continuum jet covariances. For the periodized parent
+kernel at side `L` they hold up to the lattice-moment deviation recorded by Math-#168 (`lattice_vs_continuum`: at
+most `1.4e-13` at `L = 12, 24`; `L = 6` is not covered), so the values below are for the continuum covariance and
+apply at `L >= 12`. Hence
+
+    c_1 h_0(0, tau)/z_0 = [p_b(0)/z_0] p_odd(a, beta, c),   p_b(0)/z_0 = phi(b/sqrt2) / (sqrt2 . 36 k^2 m_(2,b)),
+
+the prefactor of Math-#168, and `J_cusp(k) = integral gamma(a)^11 p_odd(a, a^2/(12k), a^3/(144k^2)) da` is a
+one-dimensional Gaussian integral, `b`-free; all `b`-dependence of the constants sits in the prefactor.
+
+**Near-mass identity (control).** The same root-resolved integral (R11) without the radius indicator is the near
+point mass `integral n dM = a_1 + 2 a_2`, `a_j = integral 1{n = j} dM` ([SC] (18), (20); `n <= 2` by the classifier of
+[SC] §6), which at `d = 2` are Math-#168's `alpha_j` (the near part of `Lambda_1` in [CL] (1.6), Math-#159, not
+pinned here):
+
+    alpha_1 + 2 alpha_2 = 108 k^7 [p_b(0)/z_0] integral_S Q(u, v) integral_(Z != 0) |Z|^(-12)
+                          integral p_odd(a, beta(u, v, Z, a), c(u, v, Z, a)) da dZ du dv,                 (N)
+
+with `beta = a^2/(12k) + k b_0(u)/Z^2`, `c = a^3/(144k^2) + a b_0(u)/(4Z^2) + 2k(v - b_0(u) u)/Z^3`. This
+four-dimensional integral is evaluated independently here and compared with the merged Math-#168 values; agreement
+is a consistency check between the [SC]/[R] parametrization (jet Jacobian `2`, root Jacobian `6`, weight `9`, multiplicity by the map) and the
+[CL]/[CUB] parametrization of Math-#168, which integrates the classifier over the jets directly.
+
+## 2. Exact structure
+
+- Shape integrals over `S = {-3/2 < u < 1/2, |b_0|/2 < v < 3 - 6u}` with `Q = (4v^2 - b_0^2)(b_0 - 4uv)`, in rational
+  arithmetic: `I = 246528/35`, `U_2 = integral u^2 Q = 240192/35`, `B = integral b_0 Q = -428544/7 = 3I - 12 U_2`,
+  `U_abs = integral |u| Q = 5898627/880` ([Q] (12)); `E eta = 5771/7062` under `Q/I` ([R] H3); the outer doublet
+  integral `J = 1083417/280` ([E] (E5), derived from the primitive (E20)); `11 U_2/(2I) = 4587/856`, `11 U_abs/(2I) = 4587821/876544` (author-side
+  corollary in Math-#176 comment 5903080058, not part of the pinned [Q] bytes). The inner doublet integral `D = 27066286003/223205220 - (79298560/4782969) log 2 = 109.7699503...`
+  ([E] E21) is quoted, not recomputed exactly (it was confirmed to `1e-13` by quadrature in the Slice C read).
+- `kappa = (2/13)(11/13)^(11/2) = q_0^(-11) - q_0^(-13)` at `q_0 = sqrt(13/11)` ([Q] (16)).
+- With the aligned planar cusp density ([Q] (21)–(22), `v_a = v_beta = 2`, `v_c = 6` for the continuum kernel):
+  `D_a G_0 = -[a^2/24 + a^4/(3456 k^2)] G_0 <= 0`, so `C_2 > 0`.
+- Assembly (`K_0 = 108 k^7 p_b(0)/z_0`; `C_0` below is [E]'s per-shape amplitude, `C_* = C_0 I` is [Q]'s `C_0`):
+
+      C_0 = (216/11) k^7 [p_b(0)/z_0] J_cusp,           C_* = C_0 I,
+      G_max ~ C_0 (I - D),  G_2 ~ C_0 J,  B_2 ~ C_0 D,  singleton ~ C_0 (I - D - J),
+      C_2 = K_0 [ U_2 T_1 + (2B/13) T_2 ],   T_1 = integral gamma^9 (1 + 12A^2) G_0 da,  T_2 = integral gamma^13 D_a G_0 da,
+      c = C_2/C_*,   d_TV(R/t, Pareto_11) = kappa c t^(-2) + O(t^(-4)),
+      B_sign = (K_0/C_*) U_abs integral |A| gamma^10 G_0 da,   A = a/(12k).
+
+  Every integrand in `a` is even, so the integrals are taken on `[0, 14]` and doubled (this also removes the kink of
+  `|A|` at `0`); 96-point Gauss–Legendre, checked against 48 points.
+
+## 3. Values
+
+**Cusp integrals** (`b`-free):
+
+| `k` | `J_cusp` | `T_1` | `T_2` | `integral |A| gamma^10 G_0` | max rel. dev. 48 vs 96 points |
+|---|---|---|---|---|---|
+| `0.5` | 0.0569758 | 0.0933384 | -0.0077782 | 0.0115205 | `1e-14` |
+| `1.0` | 0.0485522 | 0.0561351 | -0.00467793 | 0.00468021 | `1e-15` |
+| `2.0` | 0.0465858 | 0.0483768 | -0.0040314 | 0.0022051 | `2e-15` |
+
+**Tail constants** (`F(t) ~ C_* t^(-11)` is the `r^(-3)`-scaled expected number of near points at scaled radius
+`R > t`; the split rows are values of [E] (E6); `c`, `kappa c`, `B_sign` of [Q] (13), (16), (19) (conditional on [E]/[Q]; see §5)):
+
+| `k` | `b` | `C_*` | `G_max` | `G_2` | `B_2` | singleton | `c = C_2/C_*` | `kappa c` | `B_sign` |
+|---|---|---|---|---|---|---|---|---|---|
+| `0.5` | `0.0` | 1.9297 | 1.8996 | 1.0601 | 0.030073 | 0.83957 | 9.7826 | 0.6 | 1.058 |
+| `1.0` | `0.0` | 52.621 | 51.801 | 28.907 | 0.82006 | 22.894 | 6.9042 | 0.424 | 0.5045 |
+| `2.0` | `0.0` | 1615.7 | 1590.5 | 887.55 | 25.179 | 702.95 | 6.2011 | 0.381 | 0.2477 |
+| `0.5` | `1.0` | 0.55249 | 0.54388 | 0.30351 | 0.0086102 | 0.24038 | 9.7826 | 0.6 | 1.058 |
+| `1.0` | `1.0` | 15.066 | 14.831 | 8.2763 | 0.23479 | 6.5549 | 6.9042 | 0.424 | 0.5045 |
+| `2.0` | `1.0` | 462.58 | 455.37 | 254.11 | 7.209 | 201.26 | 6.2011 | 0.381 | 0.2477 |
+
+`c` and `B_sign` are `b`-free (the prefactor cancels); the ratios `G_max/C_* = (I-D)/I = 0.98442`, `G_2/C_* = J/I
+= 0.54934`, `B_2/C_* = D/I = 0.015584` are universal. In `RESULTS.json`, `C2_over_C0` uses [Q]'s `C_0 = C_*`;
+`C0_per_shape` is [E]'s `C_0`. The numbers satisfy the bounds of the Math-#176 author-side corollary (comment 5903080058; not in the
+pinned [Q] blob): `c > 4587/856 = 5.3586`
+and `0 < B_sign < 4587821/876544 = 5.2340`. Reading: at `k = 1`, `b = 0` the near point mass is `1.380` and `C_* = 52.6`, so the asymptotic law puts `C_* 2^(-11) = 0.026` (about `1.9%` of it) beyond twice the pin distance and `3e-4` beyond
+three; the radial ratio's total-variation distance from Pareto(11) at threshold `t` is `0.42/t^2` to leading order
+(`4.7%` at `t = 3`), and the signed-mark distance decays only like `0.50/t`.
+
+**Near-mass identity** (`alpha_1 + 2 alpha_2` from (N) at three shape-grid levels, against Math-#168):
+
+| `k` | (N) at 32/64/96 shape points (`ns = 160`) | (N), `b = 0` | Math-#168 GH60 | Math-#168 MC (s.e. of `alpha_1`) | rel. dev. vs GH60 / MC |
+|---|---|---|---|---|---|
+| `0.5` | 37.9952 / 37.871 / 37.8825 | 1.1874 | 1.1877 | 1.1844 (0.0027) | -2.67e-04 / +2.48e-03 |
+| `1.0` | 175.923 / 176.071 / 176.082 | 1.3798 | 1.3801 | 1.3786 (0.0022) | -2.22e-04 / +8.33e-04 |
+| `2.0` | 1119.62 / 1120.53 / 1120.7 | 2.1954 | 2.196 | 2.1948 (0.0033) | -2.34e-04 / +2.87e-04 |
+
+The shape-grid refinement at fixed `ns = 160` moves the integral by at most `3e-04` relative between the 64- and
+96-point levels (the sequence is monotone with observed order about 2 at `k = 2` and about 3.5 at `k = 1`, and is not
+monotone at `k = 1/2`, so no order is claimed; the integrand has an integrable singularity at the corner
+`(u, v) = (-1/2, 0)` of `S`, where the `|Z|`-scale of the Gaussian cutoff diverges), and the log-`Z` order is converged: at `32 x 32` shape points,
+`ns = 64, 96, 160, 256` give `175.6152`, `175.9278`, `175.9235`, `175.9235` at `k = 1` (an under-resolved `ns = 64`, the order of the
+first version of this note, biased the identity low by `1.8e-3`, exactly the deviation then reported; the `a`-order
+`32` and `48` agree to all digits). At the finest level the identity lies within `1.1`, `0.5` and `0.2` Monte Carlo
+standard errors of `alpha_1` of the Math-#168 values at `k = 1/2, 1, 2`, and within `2.7e-04`, `2.2e-04`, `2.3e-04` relative of the GH60
+values. This is the check that the two parametrizations agree; it is not a certification of either.
+The deviations from GH60 are all negative and of the size of the remaining shape-grid movement of (N), which still
+increases between the two finest levels; at `k = 2` the GH60 reference is itself not settled at this level (Math-#168
+§4: `J_2(2)` GH60 lies 3.2 MC s.e. below MC and the third digit of `alpha_2(2)` is not established, about `4e-4`
+relative in `alpha_1 + 2 alpha_2`).
+
+## 4. Precision (empirical, not bounds)
+
+- The one-dimensional cusp integrals agree between 48 and 96 Gauss–Legendre points to the relative deviations in
+  the table (all at most `1.3e-14`); the constants inherit that precision and the exact prefactor.
+- Three exact identities stated in Math-#190 (elementary: `G_0 = exp(-12 k^2 (gamma^6 - 1)) / sqrt(192 pi^3)` on the
+  cusp, and `(a gamma^11)' = 12 gamma^11 - 11 gamma^9`) hold in this quadrature at machine precision and are
+  `--check` rules since v3: `T_2 = -T_1/12` (`< 3e-18` absolute at every `k`), `T_b sqrt(192 pi^3) = (12k^2 + 1)/(36k^3)`
+  (`8/9`, `13/36`, `49/288`; `< 6e-16`), and the rational factor `(11/2)(U_2 - B/78)/I = 66451/11128` of
+  `c = C_2/C_* = (66451/11128) T_1/J_cusp` (exact). Since `T_1/J_cusp = 12 - 11 I_9/I_11 in (1, 12)`, Math-#190
+  reports (information only, not consumed here) the bounds `5.9715 < c(k) < 71.658`, which would sharpen the corollary
+  `c > 4587/856`; the floating-point values here (`9.78`, `6.90`, `6.20`) lie inside.
+- Math-#190 (reviews/c6_tail_constant_certified_20260930, other Claude session; merged 2026-10-01 at `77b854c` as a review
+  record filed as certified-arithmetic information, not a promotion; not consumed here)
+  reports certified interval enclosures of every constant of this table and finds each value of `RESULTS.json`
+  (blob `0b5ce76f`) within `9e-16` relative of its enclosure. That record is information for the reader; it is not a
+  certification of this note, which stays floating point.
+- The near-mass integral moves by at most `3e-04` relative between the 64- and 96-point shape levels at `ns = 160`,
+  and by `4e-09` between `ns = 160` and `256`; its remaining error is of the former order, below the Math-#168 Monte
+  Carlo standard error.
+
+## 5. What the numbers are not
+
+Not a proof and not an enclosure. `C_*` is a value of the reviewed formula (R13) of the merged [R]; the split and the
+second-order/signed constants are values of formulas of [E] (Math-#169) and [Q] (Math-#176). Both are merged on main
+after scoped nonauthor review, but their pinned bytes keep the disposition 'author-side conditional proof candidate'
+and their source hypotheses remain. These values are therefore conditional on [E] and [Q] and their sources,
+evaluated at their stated scope (planar, `r -> 0` first), with continuum spectral moments and no finite-`L`
+periodization correction. Planar only (`d = 2`), parent
+kernel only, aligned frame; no rate for `r -> 0`, no finite-`r` statement, no statement about the remote singleton
+population. No register, catalog or STATUS change. Same GitHub account as every lane; zero
+organizational-independence credit. The author will not merge.
+
+## 6. Provenance
+
+`tail_constants.py` (standard library; `--check` replays the exact controls, the cusp integrals and the complete
+constants table for each `k` and both `b`, the low-level near-mass integral for each `k` with its log-`Z` study
+entry, the assembly of the identity from the stored numbers, the identity tolerance against `RESULTS.json`, and the
+three exact identities of §4; mutants `shape-integral`,
+`gamma-power`, `cusp-shift` exit 1), `RESULTS.json` (full run, about fifteen minutes), `SOURCE_MAP.json`
+(pins [R], [SC], [NUM], [CUB], [LP], [E], [Q] on `main` `3e0a91b`, all workflow-checked; [E] and [Q] were unmerged
+companions in v1/v2 and were promoted to pins in v3 with their blobs unchanged),
+`SOURCE_FILES.json`, workflow `c6-microscopic-tail-numerics.yml`.
+
+Revision v3.1 (Slice C review 5974193134 on Math-#178, findings F1–F8, wording only): [E]/[Q] values are stated as
+conditional although both are merged; the corollary bounds are attributed to Math-#176 comment 5903080058, not to the
+pinned [Q]; Math-#190 is recorded as merged at `77b854c` and its bounds as information only; precision `1.3e-14`;
+near point mass `1.380` (`C_* 2^(-11)` about `1.9%` of it); the two `C_0` conventions in `RESULTS.json` named;
+continuum spectral moments stated. With the Slice A review 5974218952 (F1–F3): the planar regression is stated as the
+continuum covariance, valid at `L >= 12` by the Math-#168 lattice control; the trivial `O(1)` average is named; `J`
+is cited at [E] (E5). With the Slice B review 5974449197 (F1–F4): no convergence order is claimed for the shape grid;
+the sign and size of the GH60 deviations are explained; the `±` is the MC standard error of `alpha_1`; the identity is
+derived from the pinned [SC], with [CL] (1.6) cited as context only. `RESULTS.json` and `tail_constants.py` unchanged.
@@ -0,0 +1,22 @@
+# Planar microscopic radius-tail constants (parent kernel, d = 2)
+
+**Object:** `CL-C6-MICRO-TAIL-NUMERICS-20260930-v1`. **Author lane:** Anthropic / Claude. **Kind:** numerical note
+(deterministic Gauss quadrature with reported convergence; no proof, no enclosure). **Scientific effect:** NONE.
+
+Evaluates, for the parent kernel `exp(-|z|^2/2)` in the plane, the point-intensity tail constant `C_*` of the
+already-formed microscopic measure ([R] (R13); `F(t) ~ C_* t^-11`), the whole-cluster split of that tail
+(Math-#169, merged; values conditional on it), the second-order coefficient `C_2/C_0`, the TV coefficient `kappa |C_2/C_0|` and
+the signed constant `B_sign` (Math-#176, merged; values conditional on it), for `k in {1/2, 1, 2}` and `b in {0, 1}`. In `d = 2`
+the [SC] measure reduces to the exact contact regression of Math-#168, so every constant is a one-dimensional
+Gaussian integral; the four-dimensional near-mass identity `alpha_1 + 2 alpha_2 = K_0 int Q |Z|^-12 p_odd` is
+evaluated as a control against the merged Math-#168 values.
+
+- `NOTE.md` — what is evaluated, exact structure, values, precision, non-claims.
+- `tail_constants.py` — standard-library script (Python `>= 3.11`): `--check` (exact controls and replay against
+  `RESULTS.json`, including the exact cusp identities `T_2 = -T_1/12`, `T_b sqrt(192 pi^3) = (12k^2+1)/(36k^3)`), full run regenerates `RESULTS.json` (about fifteen minutes); `--mutant {shape-integral, gamma-power,
+  cusp-shift}` must exit 1.
+- `RESULTS.json`, `SOURCE_MAP.json` (seven main-resident pins: [R], [SC], [NUM], [CUB], [LP], [E], [Q]), `SOURCE_FILES.json`.
+- Workflow `.github/workflows/c6-microscopic-tail-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.