[aliromman]

Bind the complete narrow periodic Gaussian Reference family to a direct cyclic-band solve. The previous direct consumer covers one 385-row fixture. This result supports every original Reference parameter tuple that passes its explicit positive-gap and analytic-error gates, including multiple holes and arbitrarily small funded gaps. It retains every periodic and cross-cell coefficient, returns every target coordinate, and copies holes exactly.

The written proof, consumer, and exact receipt define the result. Analytic source error, coefficient rounding, the direct solve and final grid rounding receive separate allowances. The gap is rounded down adaptively; there is no fixed 16-bit gap floor. Preparation rejects an unfunded source.

For source size n, retained halfwidth w and target period N, the retained solver uses at most 4nw² factor products and 8nw products per complex solve. Full input/output scans, gather/scatter, final truncations, buffer clearing, source formation and retained caches/factors remain paid. This is an all-parameter conditional source/consumer interface, not production parameter funding or a physical tape tariff. The analytic Reference and interval contracts remain inherited. It does not improve complete kappa or change the selected frontier, whose conditional value remains 60403/20000000.

Base and all live source inputs are pinned to ff3cc58e370d0f678e529246f8c73e8094e18fcf. Jose Manuel (@jmbarrancoml) supplies the unchanged Reference. JoyboyBrian supplies the original source/error interface in PR #5159 and the weighted recurrence in PR #5216. Ali Romman's PR #5260, at 65cdd6b36e002ce9d13bd96dbc504da90f77f2c9, supplies the fixed-source direct predecessor. His PR #5284, consumed at 72269696a9c5ae5b1c0b469afdd3db08080635cf, supplies the unchanged cyclic solver and precision theorem, retained in this branch's ancestry. Dugongue's PR #4554 and Ryanson Jonathan's audit supply the earlier band/Woodbury scope. Source pins retain the Gaussian, interval and classical elimination credit. Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.

Python 3.12 and its standard library:

python3.12 -B scripts/research_package.py --package research/periodic-gaussian-uniform-solve

Both registered isolated, warning-strict normal and optimized commands pass with PASS_PERIODIC_GAUSSIAN_UNIFORM_SOLVE. Eight actual sources, 103,717 original entries, 24 signed complex RHS cases, six separate dense rational solves, eleven rejection controls and seven gap cases pass. Four code mutations fail in both modes. The 258-row fixture uses 492 factor products and 1,314 solve products, with source setup recorded separately. Original-matrix internal residuals plus final rounding certify every finite target. The final-grid residual alone is too loose in two fixtures; the receipt records that limitation explicitly.

Committed bytes, all source-pin ancestry, official archive validation and both read-only Git-free export replays pass, with all 22 exported files unchanged. No production Tensor/native execution, selected cold admission, Lean, Docker, independent human proof review, interval-backend acceptance or full multiplication calculation was run. The full-history wrapper is unpassed in this shallow checkout. The next task is to fund one actual production policy and price source bit work and physical delivery together.

Hosted run 39995095815 passed planning and the registered Linux replay for published head 234142af90b5e7afe0b33c4b78c3c0bd622d5fc3. Hosted execution is separate from independent scientific review.