Independent verification: weighted-stride closures and bilinear dual (no kappa)

[Ambassators]

Result and scope

Independent verification / research note; no new kappa. This adds an independent finite-geometric-sum oracle for (W-rho I)^(-1), where (Wx)_i=tau_i*x_(i-s mod L), and checks the bilinear inverse transpose. It compares exact results with the neighboring rounded single-factor primitive from Chafik Boukhalfa / chafreaky's PR628, after cycle-local gathering and scattering.

For a stride cycle C of length m and phase product h_C, the written identity is

(W_C-rho I)^(-1) = sum_{j=0}^{m-1} rho^(m-1-j) W_C^j / (h_C-rho^m).

The oracle uses exact rational complex arithmetic, not the predecessor's affine two-pass scan or Gaussian elimination. It checks both inverse products on every matrix entry. The dual independently reverses the stride and shifts phases by s, without conjugation, then checks the result against the actual inverse transpose.

A concrete length-6 / stride-2 witness rejects using one global phase product or the global length in place of each cycle's own closure. These are counterexamples to deliberately incorrect extensions, not a defect in PR628's stated neighboring scope. No such defect was found in the tested domain.

Predecessor credit: PR628 already provides constant-gauge arbitrary-stride sorting and the direct-sum infinity-norm argument. Neither is claimed new. The additional evidence covers independently varying phase products on multiple cycles and the bilinear dual through a different exact algorithm. Gather/scatter routing, phase reads, storage and restoration are not declared free or newly improved.

Pins and package

• Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60.
• Contribution: c5b53a5e80372a1fae9830d8dcf21aaabce3f729.
• PR628 source: 0768f97d06c623b8211e651543c66135796b62bf; eight source/proof/layout/notice SHA-256 pins.
• Only research/cyclic-stride-independent/ is added. Existing tracked files, upstream/ and the selected Conditional κ = 0.001635527380375392: coordinated owner repair on PR623 #626/Conditional κ = 0.001587467473762886: an E8 bit word with 420 helpers, c = 0.001653826988 (+4.18% over #510) #543 frontier are unchanged.

The adapter executes the unchanged scan and apply function ASTs from pinned weighted.py, using its original scan.py arithmetic. Unrelated kernel setup is not executed. Setting the initial scalar to one isolates a single factor; it is not a radix-1 kernel certificate or a full cascade replay.

Reproduce

Python 3.9+ and Git; standard library only. Fetch the exact pending source object once if absent:

git fetch --no-tags https://github.com/CrocSwap/integer-mult-bounds.git 0768f97d06c623b8211e651543c66135796b62bf
python3 -B -I research/cyclic-stride-independent/verify.py --check research/cyclic-stride-independent/EXPECTED.json

Expected: PASS_STRIDE_RESOLVENT_AND_DUAL.

Completed fixed domain: lengths 1-12, 16, 18 and 21; every stride residue; two separated inner/outer roots; seed-derived rational unit phases and RHS values. No adaptive search or early-success stopping rule.

• 266 exact cases; 252,730 matrix-entry obligations, including forward and dual; 122 multi-cycle cases and 30 zero-stride cases; signed-stride and zero-root edge checks.
• 4,256 rounded cases / 53,472 coordinates, covering forward/dual, both neighboring enumeration directions, 12/40-bit grids, dense and basis RHSs.
• 16 rejecting controls and eight focused tests. Actual corrupted-source and missing-Git-object paths fail closed. A separate full CLI test rejects a tampered receipt field.

Full normal and optimized isolated replays pass on macOS arm64 / CPython 3.9.6. The independent oracle alone also passes on Linux x86_64 / CPython 3.13.5, with identical complete exact-evidence projection; no full Linux predecessor replay is claimed. Staged whitespace, selected-certificate/source integrity and patch applicability against a separate base index pass.

Finite rounded RHS tests are not an all-input proof. The written elementary identity is not machine-formalized. Full-kernel preparation, cascades, paid tape/routing, dirty restoration, recursive hosting, H1-H5 and inherited all-size/analytic multiplication obligations remain outside scope. Neither root make verify, selected cold replay nor remote CI success is claimed; no existing generator or selected construction changed.

Coordination and attribution

OS-random assignment seed 79, independent-verification track. Three candidate tasks and the completed local claim are recorded; no pinned sprint coordination issue was found. The final queue refresh through #690 found distinct CRT/resultant (#668), composite lifting (#666), dirty-composition (#663/#673/#689) and packed-Gaussian (#690) work. Those are not duplicated or imported here. PR628's source head was rechecked immediately before submission.

Credit chafreaky for PR628's primitive, rounding estimate and existing layout, Ali Romman's antecedent PR621 and retained source notices. Prepared for Ryan Pinto (@Ambassators) with substantial OpenAI AI assistance in research, mathematics, implementation, verification and writing. Apache-2.0. Independent implementation evidence does not imply independent human review or a full multiplication theorem.

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[Mister-Meeseeks]

Integrated at ee055dc44815497927ef618fc985b93c2dcbda2a in the 20-PR pilot, retaining your original commit, source notices and contributor credit.

Independent cycle-local geometric resolvent and bilinear inverse transpose, with varying phases and rounded predecessor comparisons. Finite domain and controls pass; no whole-kernel/tape/compiler proof.

The scoped package check passed at c5b53a5e80372a1fae9830d8dcf21aaabce3f729. This is research integration; the selected κ and inherited theorem assumptions are unchanged. Review, evidence and attribution. Thank you for the contribution.