Conditional κ = 7.09424406659975e-4: five-stage completed banks with #210’s 493-source helper

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

@@ -0,0 +1,35 @@
+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-banks-493/NOTICE

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+Five-stage completed banks with PR210's 493-source helper. Apache-2.0.
+Prepared by Rohan Arun (rohanarun) with Anthropic Claude assistance.
+
+Reused, with credit:
+- eumemic, PR210 at 13311491eb74a3a9a443ba4e318c68c263f064f0 (with Dugongue's PR216 cuts, sennemmi's PR230 frames
+  and our PR211 frames; OpenAI Codex assistance): the 493-source helper word, its preparation (extra/, targetagg/,
+  rank/, echelon/) and its complete F2/integer replay with 17 controls, executed unchanged.
+- hcg890, PR234 at af3fe331ca60c936229e060681ffccbc1c208678 (ChatGPT assistance) and Jacob Sussman: the five-stage
+  cover, its physical telescope, global lowering, exact geometry, prime validator, finite bill, moment engine,
+  bootstrap and outer-47 assembly, and the five-stage complex supplier. telescope493.py adapts PR234's telescope
+  to PR210's newer replay; helper493.py rebinds PR234's helper-specific constants to fresh values.
+- Evan McKinney, PR197: completed-bank packing and the colouring helpers (banks.py byte-identical).
+- Rohan Arun, PR205/PR237: completed banks on PR200 and on PR234's five-stage cover.
+- Chafik Boukhalfa (PR200), Avi Eisenberg (PR193) and every contributor credited by those packages.

research/five-stage-banks-493/README.md

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+# Five-stage completed banks with PR210's newest 493-source helper
+
+Under the retained multiplication interfaces, this finite witness certifies
+
+$$
+T(n)=O\left(n(\log n)^{1-\kappa}\right),\qquad
+\kappa=\frac{28376976266399}{40000000000000000}=0.000709424406659975 .
+$$
+
+That is **+0.119%** over PR237 (0.000708582410781108, five-stage banks on PR234's 471-source helper) and **+0.813%** over PR234 (0.000703701743496697).
+
+| Construction | Helper | Bit coarse | κ |
+|---|---|---:|---:|
+| PR234 | PR210 df95878 (471 sources), paid exteriors | 7.041973e-4 | 0.000703701743496697 |
+| PR237 | same, width-120 banks | 7.090849e-4 | 0.000708582410781108 |
+| **This** | **PR210 1331149 (493 sources, target aggregation/rank/echelon, PR230 frames), width-120 banks** | **7.099280e-4** | **0.000709424406659975** |
+
+The bit side binds; PR234's five-stage complex supplier (7.47455e-4) is unchanged.
+
+## What is new
+
+PR234 builds its five-stage cover from PR210's older helper (`df95878`, 471 borrowed sources). PR210 has since added 22 more source entrances, 82 equal-response target aggregation groups, 7 integer rank-compression groups, 4 signed echelon target transforms and PR230's frame refinements (head `1331149`, 16,621 physical registers instead of 16,643). This package runs that newer helper through PR234's five-stage pipeline and then applies PR237's completed width-120 banks.
+
+- **Helper (PR210's own code, unchanged).** The package assembles PR210's research package with PR234's pinned PR200 base inputs, derives the 34 terminal records exactly as PR234 does, prepares the helper with PR210's `targetagg/setup.py`, and runs PR210's replay: all 20,141 F2 columns, the literal majorant (24-bit digits), both defining-integer signs, and all 17 corruption controls.
+- **Physical telescope.** PR234's frame-tag telescope, adapted to PR210's newer replay (22 tagged event kinds, including aggregation, rank and echelon setup/read/restore). 2,569,922 tagged events; an independent literal projection and its inverse reproduce the all-column F2 endpoint and the event digest.
+- **Five-stage lowering and geometry.** PR234's global lowering and exact h24/m120 geometry, with only the helper-specific constants (register count, event count, call count, rank mass) rebound to the fresh values. The five-stage profile is m = 120, W = 23,661, 494,909 calls, rank mass 2,834,920, deficit 4,400.
+- **Primes.** All 26,464 used bases have nonzero cleared Gram determinants whose factors outside {2,3,5,7} leave a residual below 2^80.
+- **Finite bill.** PR234's formulas with fresh values; the payload-prefix lift needs 101 bits (PR234: 98), still a fixed finite constant.
+- **Banks.** Same 231 entrance charts as PR237. 12 copies give 176,915 banks of width 120 (360×(8·11+8·4), 108×(10·12), 39×(20·6), 4,304×(30·4), 172,104×(5·24)) with a proper 60-colouring of 997,260 incidences. W = 12·4v + 176,915 = 261,395; deficit 52,800; largest child 50.
+- **Arithmetic.** PR234's interval moment engine certifies coarse saving 709928047185833/10^18; three finite ordinary levels and PR234's outer-47 assembly give κ; the next 10^-18 grid point fails.
+
+## Verify
+
+```sh
+git fetch https://github.com/CrocSwap/integer-mult-bounds pull/210/head pull/234/head
+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-banks-493/verify.py --pr210-root ../pr210 --pr234-root ../pr234 --full
+```
+
+About 10 minutes: PR234's own fresh verifier (for the complex supplier, ~3.5 min), PR210's complete helper replay with controls (~3 min), telescope, lowering, geometry, primes, banks and arithmetic. It compares the result with `certificate.json`.
+
+## Scope
+
+Conditional, exactly as PR234/PR237: the five-stage physical program and complex symbolic theorems, weighted q-local compiler, complete-stream movement, copied-centre and restored-row interfaces, ordinary leaves, prime supply, precision/recovery and the all-size reduction remain retained hypotheses. PR234's helper-specific checks are reused with recomputed constants rather than re-derived. Not an unconditional multiplication theorem or a speedup claim.
+
+Prepared by Rohan Arun with Anthropic Claude assistance. Apache-2.0.

research/five-stage-banks-493/SOURCE.json

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+}

research/five-stage-banks-493/arithmetic5.py

@@ -0,0 +1,94 @@
+"""(493-helper copy) Exact paid moments, three finite ordinary levels and PR234's outer-47 assembly for the banked five-stage bit
+supplier. Uses PR234's own pinned moment, bootstrap and assembly code unchanged.
+Prepared by Rohan Arun with Anthropic Claude assistance. Apache-2.0.
+"""
+from fractions import Fraction as Q
+from pathlib import Path
+import importlib.util
+
+PKG = 'research/five-stage-source-bound-v8'
+PR234_KAPPA = Q(703701743496697, 10 ** 18)
+PR230_KAPPA = Q(694787513005285, 10 ** 18)
+PR237_KAPPA = Q(177145602695277, 250000000000000000)   # PR237: banks on PR234's 471-source five-stage helper
+
+
+def load(name, path):
+    s = importlib.util.spec_from_file_location(name, path); m = importlib.util.module_from_spec(s); s.loader.exec_module(m)
+    return m
+
+
+def build(P234, pr234_math, physical):
+    """pr234_math: PR234's freshly computed mathematics (bit_profile, complex_profile, assembly parameters)."""
+    math = load('pr234_moments', P234 / PKG / 'code/five_stage_bit_cost_20261009.py')
+    outer = load('pr234_outer47', P234 / PKG / 'code/paired_cube_assembly.py')
+    bp = pr234_math['bit_profile']; cp = pr234_math['complex_profile']
+    assert (bp['m'], bp['deficit'], bp['maxchild']) == (120, 4400, 100)
+    H = {int(k): n for k, n in bp['histogram'].items()}
+    removed = {int(k): n for k, n in physical['removed_exteriors'].items()}
+    for r, n in removed.items():
+        assert H[r] >= n; H[r] -= n
+    c = physical['copies']; m = 120
+    H = {r: c * n for r, n in H.items() if n}
+    W = physical['W']; mass = sum(r * n for r, n in H.items()); deficit = m * W - mass
+    assert deficit == c * bp['deficit'], 'banking preserves the telescoping deficit'
+    assert max(H) < m // 2
+    # Coarse paid saving on the 10^-18 grid, PR234's interval engine with the inherited bad-class envelope.
+    root = Q(math.certify(H, m, W, True)['lower']); den = 10 ** 18
+    lo = Q(int(root * den), den)
+    while math.moment(H, m, W, lo, True)[1] >= 1: lo -= Q(1, den)
+    hi = lo + Q(1, den)
+    while math.moment(H, m, W, hi, True)[0] <= 1: lo = hi; hi = lo + Q(1, den)
+    assert math.moment(H, m, W, lo, True)[1] < 1 < math.moment(H, m, W, hi, True)[0]
+    coarse = lo
+    unbanked = {int(k): n for k, n in bp['histogram'].items()}
+    old_coarse = Q(pr234_math['contract']['assembly']['bit_coarse']) if 'contract' in pr234_math else None
+    if old_coarse is not None:
+        assert math.moment(H, m, W, old_coarse, True)[1] < math.moment(unbanked, m, bp['W'], old_coarse, True)[0], \
+            'banking strictly improves the paid moment at PR234 coarse saving'
+    chain = [Q(384599, 10 ** 10)]
+    for _ in range(3):
+        a = (1 - coarse) * coarse + coarse * chain[-1]
+        assert chain[-1] < a < coarse < 1 - a
+        chain.append(a)
+    bit = chain[-1]
+    b = Q(pr234_math['assembly']['parameters']['a_complex'])
+    cm = math.moment({int(k): n for k, n in cp['histogram'].items()}, 110, cp['W'], b, False)
+    assert cm[1] < 1
+    eta = Q(1, 10 ** 12); beta = Q(1, 10 ** 9)
+    assert bit < (1 - beta) * b
+    bridge = dict(proof='PROOF.md', representation='Exact powers with source-bound finite overcharges',
+                  semantic=dict(G=dict(base=2, exponent=30000), E=dict(base=2, exponent=100000),
+                                B_upper=dict(base=2, exponent=100001), C0=dict(base=2, exponent=210000), C1=1,
+                                strict_literal_gap=1, induction_gap_lower=1),
+                  rows=dict(coefficient=20161, degree=10 ** 6, suffix_slope=4 * 10 ** 6,
+                            degree_gap=Q(10 ** 6) - Q(51 * 20161, 25)))
+    q = bit * (1 - 2 * eta); minimum = (1 - eta) * q / (1 + q); ticks = minimum * den
+    kappa = Q(ticks.numerator // ticks.denominator, den)
+    if kappa == minimum: kappa -= Q(1, den)
+    result = outer.assembly(bit, b, bridge, kappa, eta=eta, beta=beta)
+    assert len(result['strict_constraints']) == 47 and len(result['margins']) == 7
+    assert all(x > 0 for x in result['strict_constraints'].values())
+    try: outer.assembly(bit, b, bridge, kappa + Q(1, den), eta=eta, beta=beta)
+    except AssertionError: pass
+    else: raise AssertionError('adjacent kappa grid point admitted')
+    assert kappa > PR237_KAPPA > PR234_KAPPA > PR230_KAPPA
+    controls = []
+    try: profile_check({**H, 1: H[1] + 1}, m, W, c * bp['deficit'])
+    except AssertionError: controls.append('changed rank mass rejected')
+    else: raise AssertionError('corrupt rank mass accepted')
+    try: math.moment({**H, m: 1}, m, W, coarse, True)
+    except AssertionError: controls.append('full-width child rejected')
+    else: raise AssertionError('full-width child accepted')
+    return dict(scope='Conditional finite supplier composition; PR234 physical, scalar, prime and analytic dependencies inherited',
+                bit_profile=dict(m=m, W=W, copies=c, histogram={str(r): n for r, n in sorted(H.items())},
+                                 calls=sum(H.values()), rank_mass=mass, deficit=deficit, maxchild=max(H)),
+                bit_coarse_saving=coarse, pr234_bit_coarse_saving=old_coarse, ordinary_chain=chain, bit=bit,
+                complex_saving=b, complex_moment_interval=cm, assembly=result, kappa=kappa,
+                comparison_pr234_kappa=PR234_KAPPA, relative_gain_over_pr234=kappa / PR234_KAPPA - 1,
+                comparison_pr230_kappa=PR230_KAPPA, comparison_pr237_kappa=PR237_KAPPA,
+                relative_gain_over_pr237=kappa / PR237_KAPPA - 1, controls=controls)
+
+
+def profile_check(H, m, W, deficit):
+    assert m * W - sum(r * n for r, n in H.items()) == deficit
+    return True