Conditional κ = 7.09424410568126e-4: fixed prime, seven finite levels and η = 10⁻²⁴ on #240's five-stage banks

.github/workflows/five-stage-493-fixed-prime.yml

@@ -0,0 +1,69 @@
+name: Fixed prime, seven levels and eta 1e-24 on the 493-source five-stage banks
+on:
+  pull_request:
+    paths:
+      - 'research/five-stage-493-fixed-prime/**'
+      - 'research/five-stage-banks-493/**'
+      - '.github/workflows/five-stage-493-fixed-prime.yml'
+  workflow_dispatch:
+permissions:
+  contents: read
+jobs:
+  full:
+    runs-on: ubuntu-24.04
+    timeout-minutes: 90
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
+        with:
+          persist-credentials: false
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
+        with:
+          repository: CrocSwap/integer-mult-bounds
+          ref: af3fe331ca60c936229e060681ffccbc1c208678
+          path: .dependencies/pr234
+          persist-credentials: false
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
+        with:
+          repository: eumemic/integer-mult-bounds
+          ref: 13311491eb74a3a9a443ba4e318c68c263f064f0
+          path: .dependencies/pr210
+          persist-credentials: false
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
+        with:
+          python-version: '3.13'
+      - run: python -m pip install sympy==1.14.0
+      - run: python -B research/five-stage-493-fixed-prime/verify.py --pr210-root .dependencies/pr210 --pr234-root .dependencies/pr234 --full
+      - run: git diff --exit-code
+  prime-kernel:
+    runs-on: ubuntu-latest
+    timeout-minutes: 30
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
+        with:
+          python-version: '3.12'
+      - name: Install the fixed Lean release in runner temporary storage
+        run: |
+          python -m pip install zstandard==0.25.0
+          curl -fsSL https://github.com/leanprover/lean4/releases/download/v4.24.0/lean-4.24.0-linux.tar.zst -o "$RUNNER_TEMP/lean.tar.zst"
+          python - <<'PY'
+          import hashlib,os,tarfile,zstandard
+          from pathlib import Path
+          root=Path(os.environ['RUNNER_TEMP'])
+          expected='b14f5e5159219dd1a1956c3b806813319f5e94ccd5bdfd56f54520609a5bb5ec'
+          assert hashlib.sha256((root/'lean.tar.zst').read_bytes()).hexdigest()==expected
+          with (root/'lean.tar.zst').open('rb') as f:
+              with zstandard.ZstdDecompressor().stream_reader(f) as reader:
+                  with tarfile.open(fileobj=reader,mode='r|') as archive:
+                      archive.extractall(root,filter='data')
+          PY
+          echo "$RUNNER_TEMP/lean-4.24.0-linux/bin" >> "$GITHUB_PATH"
+      - name: Kernel-check the prime in a separate project copy
+        env:
+          MATHLIB_CACHE_DIR: ${{ runner.temp }}/mathlib-cache
+        run: |
+          cp -R research/five-stage-493-fixed-prime "$RUNNER_TEMP/prime-proof"
+          cd "$RUNNER_TEMP/prime-proof"
+          lake update
+          lake exe cache get Mathlib.NumberTheory.LucasLehmer
+          lake env lean Prime.lean

.github/workflows/five-stage-banks-493.yml

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+name: Verify five-stage banks with the 493-source helper
+on:
+  pull_request:
+    paths:
+      - 'research/five-stage-banks-493/**'
+      - '.github/workflows/five-stage-banks-493.yml'
+  workflow_dispatch:
+permissions:
+  contents: read
+jobs:
+  full:
+    runs-on: ubuntu-24.04
+    timeout-minutes: 90
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
+        with:
+          persist-credentials: false
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
+        with:
+          repository: CrocSwap/integer-mult-bounds
+          ref: af3fe331ca60c936229e060681ffccbc1c208678
+          path: .dependencies/pr234
+          persist-credentials: false
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
+        with:
+          repository: eumemic/integer-mult-bounds
+          ref: 13311491eb74a3a9a443ba4e318c68c263f064f0
+          path: .dependencies/pr210
+          persist-credentials: false
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
+        with:
+          python-version: '3.13'
+      - run: python -m pip install sympy==1.14.0
+      - run: python -B research/five-stage-banks-493/verify.py --pr210-root .dependencies/pr210 --pr234-root .dependencies/pr234 --full
+      - run: git diff --exit-code

.github/workflows/five-stage-banks.yml

@@ -0,0 +1,29 @@
+name: Verify five-stage banks
+on:
+  pull_request:
+    paths:
+      - 'research/five-stage-banks/**'
+      - '.github/workflows/five-stage-banks.yml'
+  workflow_dispatch:
+permissions:
+  contents: read
+jobs:
+  full:
+    runs-on: ubuntu-24.04
+    timeout-minutes: 60
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
+        with:
+          persist-credentials: false
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
+        with:
+          repository: CrocSwap/integer-mult-bounds
+          ref: af3fe331ca60c936229e060681ffccbc1c208678
+          path: .dependencies/pr234
+          persist-credentials: false
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
+        with:
+          python-version: '3.13'
+      - run: python -m pip install sympy==1.14.0
+      - run: python -B research/five-stage-banks/verify.py --pr234-root .dependencies/pr234 --full
+      - run: git diff --exit-code

research/five-stage-493-fixed-prime/NOTICE.md

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+# Notice
+
+Fixed prime, seven finite levels and eta = 10^-24 on PR240's banked five-stage bit supplier. Apache-2.0.
+Prepared by Rohan Arun (rohanarun) with Anthropic Claude assistance.
+
+Reused, with credit:
+- Gabriele Nespoli, PR235 (OpenAI Codex assistance): the fixed-prime rare-density argument, the seven-level
+  finite bootstrap, the paid q-power overcharge, `Prime.lean`, `lakefile.toml`, `lean-toolchain` and
+  `base_two_moment.py` (copied unchanged), applied there to PR237.
+- Rohan Arun, PR237/PR240: completed width-120 banks and the 493-source five-stage word (unchanged here).
+- hcg890 and Jacob Sussman, PR234: five-stage cover, moment engine, outer-47 assembly and finite bill.
+- eumemic, PR210 (with Dugongue, PR216, and sennemmi, PR230): the 493-source helper.
+- Evan McKinney (PR197), Chafik Boukhalfa (PR200), Avi Eisenberg (PR193), and every contributor credited by
+  those packages. Original notices remain in the pinned sources.

research/five-stage-493-fixed-prime/Prime.lean

@@ -0,0 +1,22 @@
+/- Copyright 2026 Gabriele Nespoli. Apache-2.0. Developed with OpenAI Codex.
+   Lucas-Lehmer criterion and norm_num extension: mathlib contributors.
+   This verifies prime eligibility only, not the integer multiplication theorem. -/
+import Mathlib.NumberTheory.LucasLehmer
+
+theorem fixed_prime_prime : (mersenne 127).Prime :=
+  lucas_lehmer_sufficiency _ (by simp) (by norm_num)
+
+theorem fixed_prime_value : mersenne 127 = 170141183460469231731687303715884105727 := by
+  norm_num [mersenne]
+
+theorem fixed_prime_large : 2 ^ 80 < mersenne 127 := by
+  norm_num [mersenne]
+
+theorem fixed_prime_density :
+    (0 : ℚ) < 3456000 / 170141183460469231731687303715884105727 ∧
+    (3456000 / 170141183460469231731687303715884105727 : ℚ) < 1 / 10 ^ 16 := by
+  norm_num
+
+#print axioms fixed_prime_prime
+#print axioms fixed_prime_large
+#print axioms fixed_prime_density

research/five-stage-493-fixed-prime/README.md

@@ -0,0 +1,38 @@
+# Fixed prime and seven finite levels on PR240's five-stage banks
+
+Conditional κ = **709424410568125629699 / 10^24 = 7.09424410568125629699 × 10^-4**.
+
+That is +3.91 × 10^-12 over PR240 (`28376976266399/(4·10^16)` = 7.09424406659975 × 10^-4). PR240's banked five-stage word, its 493-source helper and all of its checks are unchanged. Only three fixed arithmetic parameters change:
+
+| Parameter | PR240 | This |
+|---|---|---|
+| Rare-class density in the paid moment | envelope 10^-16 | proved 2m^3/q with the fixed prime q = 2^127 − 1, full 32m^2 fallback still paid |
+| Finite ordinary levels | 3 | 7 |
+| Outer-47 backoff η (β = 10^-9 unchanged) | 10^-12 | 10^-24 |
+
+The first two are PR235's refinement of PR237 (Gabriele Nespoli), applied here to PR240. The third is new: all 47 strict inequalities and 7 margins of PR234's unchanged assembly still hold at η = 10^-24; it adds about 2.1 × 10^-15 on its own.
+
+## Reproduce
+
+```sh
+git worktree add --detach ../pr210 13311491eb74a3a9a443ba4e318c68c263f064f0
+git worktree add --detach ../pr234 af3fe331ca60c936229e060681ffccbc1c208678
+python3 -m pip install sympy==1.14.0
+python3 -B research/five-stage-493-fixed-prime/verify.py --pr210-root ../pr210 --pr234-root ../pr234 [--full]
+```
+
+The verifier first runs PR240's own verifier from the same checkout (all F2 and integer columns, telescope, five-stage lowering, geometry, primes, finite bill, 231 charts, 176,915 banks and the 60-colouring) and requires PR240's certificate to be reproduced. Then it:
+
+- bisects the coarse saving `177482012714287685545103/(2.5·10^26)` on the 10^-27 grid with PR234's engine, checks both sides with an independent atanh engine, and checks that the old 10^-16 density fails at this point;
+- runs seven acyclic levels from the retained completed saving 384599/10^10, with PR234's cutoff checks;
+- runs the outer-47 assembly on the 10^-24 κ grid and rejects the next grid point;
+- reproduces PR240's published κ with η = 10^-12 and three levels;
+- recomputes the fully paid q-power constant with PR240's fresh finite-bill values (coefficient 62,150,081,742,953,667,360 < 2^80 < q).
+
+About 5 minutes; `--full` also runs PR234's fresh verifier (about 10). `certificate.json` holds the key values and the SHA-256 of the complete recomputed record; `--dump PATH` writes the full record. `Prime.lean` kernel-checks q with mathlib's Lucas–Lehmer theorem (CI job `prime-kernel`).
+
+## Scope
+
+This is a conditional finite certificate with exactly PR234/PR240's hypotheses. The large fixed prime makes initialization and table constants enormous but finite and paid, as argued in PR235's PROOF.md. No construction changes.
+
+Prepared by Rohan Arun with Anthropic Claude assistance. Apache-2.0.

research/five-stage-493-fixed-prime/base_two_moment.py

@@ -0,0 +1,63 @@
+#!/usr/bin/env python3
+"""Maintainer arithmetic cross-check of the pinned PR39 certificate.
+
+Uses no producer/checker imports. Counts are inputs, not proved by this script.
+80-term rational logarithms and 12-term exponentials independently enclose
+the two moments. The accompanying written review addresses their realization.
+"""
+import argparse
+from fractions import Fraction as Q
+from hashlib import sha256
+import json
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[1]
+GRID = 10**30
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def log_unit(x):
+    require(1 <= x <= 2, "logarithm range")
+    z = (x-1)/(x+1)
+    lo = 2*sum((z**(2*j+1)/(2*j+1) for j in range(80)), Q(0))
+    return lo, lo+2*z**161/(161*(1-z*z))
+
+
+def log_bounds(x):
+    k = 0
+    while x > 2:
+        x /= 2
+        k += 1
+    lo, hi = log_unit(x)
+    l2, u2 = log_unit(Q(2))
+    lo, hi = lo+k*l2, hi+k*u2
+    # Outward rational rounding keeps later certificates compact.
+    return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID)
+
+
+def exp_bounds(x):
+    require(0 <= x < 1, "exponential range")
+    term = total = Q(1)
+    for j in range(1, 13):
+        term *= x/j
+        total += term
+    tail = term*x/13/(1-x/14)
+    return total, total+tail
+
+
+def moment(m, W, entries, saving):
+    lo = hi = Q(0)
+    for t, count in entries:
+        t, count = int(t), int(count)
+        require(0 < t < m and count > 0, "improper child")
+        l, u = log_bounds(Q(m, t))
+        lower = exp_bounds(saving*l)[0]
+        upper = exp_bounds(saving*u)[1]
+        lo += Q(t*count, m*W)*lower
+        hi += Q(t*count, m*W)*upper
+    return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID)
+

research/five-stage-493-fixed-prime/certificate.json

@@ -0,0 +1,52 @@
+{
+  "beta": "1/1000000000",
+  "coarse": "177482012714287685545103/250000000000000000000000000",
+  "coarse_grid": 1000000000000000000000000000,
+  "complex_saving": "29898197786071/40000000000000000",
+  "controls": [
+    "cyclic supplier",
+    "unattained coarse ordinary leaf",
+    "corrupt Lucas-Lehmer residue",
+    "old-density paid moment at new coarse point",
+    "adjacent coarse and kappa grid points"
+  ],
+  "depth": 7,
+  "eta": "1/1000000000000000000000000",
+  "excluded_coarse": "709928050857150742180413/1000000000000000000000000000",
+  "full_record_note": "SHA-256 of the compact sorted JSON of the complete recomputed record; write it out with --dump PATH.",
+  "full_record_sha256": "de8771e266553c196243a2c979de68d42d5423102c69ccca2eef0b2044d7fb6d",
+  "gain_over_pr240": "3908150629699/1000000000000000000000000",
+  "kappa": "709424410568125629699/1000000000000000000000000",
+  "kappa_decimal": "0.000709424410568125629699",
+  "kappa_grid": 1000000000000000000000000,
+  "minimum_strict_constraint": "10832642377581035494695296821178533499673945779813408255286918911514209978022657405185936620440519269216604609649967868394833616688605989605177135955517175213216408326174719786469010781230802601418458882803839/15258789062500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+  "parent_eta": "1/1000000000000",
+  "parent_eta_seven_level_kappa": "354712205282999181517/500000000000000000000000",
+  "pr240_certificate_sha256": "9309284d492d5631e794cef134e183c266e96c48715994bdbed7113ce7233362",
+  "pr240_kappa": "28376976266399/40000000000000000",
+  "prime": {
+    "criterion": "Lucas-Lehmer sufficiency, independently kernel-checked in Prime.lean",
+    "fixed_before_all_widths": true,
+    "iterations": 125,
+    "p": 127,
+    "q": 170141183460469231731687303715884105727
+  },
+  "prime_residues_sha256": "6cf3fcaaa2a1b05097bab0a570cfb0f9c1b56efb83ec49aa3dff9a08281dfc5a",
+  "q_power_bound": {
+    "coefficient": 62150081742953667360,
+    "enlarged_role_charge": 654573856200,
+    "low_matrix_power": 14400,
+    "original_coefficient": 1567472751689902,
+    "original_copies": 12,
+    "per_selector_coefficient": 5088069708,
+    "route_families": 208450,
+    "selector_calls": 12211167492,
+    "strict_upper": "q^14401"
+  },
+  "rare_density": "3456000/170141183460469231731687303715884105727",
+  "relative_gain_over_pr240": "3908150629699/709424406659975000000",
+  "schema": "five-stage-493-fixed-prime/1",
+  "six_level_kappa": "354712205284062814807/500000000000000000000000",
+  "strict_constraints": 47,
+  "three_level_kappa": "354712205164106803993/500000000000000000000000"
+}

research/five-stage-493-fixed-prime/lakefile.toml

@@ -0,0 +1,6 @@
+name = "fixed_prime_certificate"
+version = "0.1.0"
+[[require]]
+name = "mathlib"
+git = "https://github.com/leanprover-community/mathlib4.git"
+rev = "f897ebcf72cd16f89ab4577d0c826cd14afaafc7"

research/five-stage-493-fixed-prime/lean-toolchain

@@ -0,0 +1 @@
+leanprover/lean4:v4.24.0

research/five-stage-493-fixed-prime/requirements.txt

@@ -0,0 +1 @@
+sympy==1.14.0