[Clear1nsight]

Change and mathematical scope

PR3068's period-domain handoff leaves phase-compatible state exactness open. This contribution constructs an explicit Gaussian Newton domain for the complete literal P104 collector, with a self-contained all-rank proof and a separate implementation.

For pi=1+i, let B_r contain the arrays whose multilinear coefficient a_S is divisible by pi^|S|. The encoder is E_r = [[1,0],[1,pi]]^tensor-r; the butterfly becomes the integral coefficient update (c0,c1) -> (c0+c1,i*c1).

• Boolean Möbius inversion proves that B_r is exactly the integral entrance on which every partial coordinate product C_T remains integral.
• Explicit integral intertwiners prove closure under the actual affine-address/quadratic-Gaussian-phase wrapper alphabet. The credited signed-floor pair is ideal at every admitted prefix.
• The original PR2586 events 2025–2039, including both separately archived U and V recipes and all four wrappers, implement (D,Q,C=0) -> (D,Q+U sum(D),C=0) exactly for five borrowed sources and an arbitrarily dirty query in 2^-f B_9.

This is a different phase-compatible domain, with no global containment claim about the earlier Gamma/Lambda lattices. Classical Clifford/Barnes–Wall constructions retain their authorship. The added value is the quantified domain, its actual complete collector binding and its explicit codec/consumer boundary.

Local bill and remaining obligations

The actual wrappers have cycles [64,128,160,64]: 416 cycles and 2,464 swaps. A rejecting control prevents substitution of PR3025's different paired-inverse invoice, 384 cycles/2,432 swaps. The direct collector retains eleven array ADDs, 3,584 pair applications, 39,936 real additions/subtractions, 14,336 floor halves, component traversal and explicit private workspace.

An optional source-preserving cold coefficient caller pays for five owned copies, six encodes, six decodes and ten copy/cleanup ADDs. It adds 120,832 real additions/subtractions, 27,648 exact halvings and 2,560 owned complex words. Its logical target is E^-1 U E. The proof and receipt include sufficient direct and composed finite headroom.

Earlier-producer admission, later quarter-source membership in 4 B_r, an actual compatible consumer, physical owners/routing/control/precision, native recursive tariffs and a favorable complete supplier bill remain open. The package claims no new precision threshold, physical saving, supplier, kappa or selected-frontier change. Human/proof-kernel review, native compiler and whole-word/selected-baseline replays are unrun.

Validation

• Primary: 2,450 signed pair cases; 1,234 generator intertwiners; four dense signed collectors with every prefix checked; the full cold codec schedule; source/frame/ownership/cleanup/domain/invoice controls.
• Separate reconstruction: all 512 ideal complex columns, all 1,024 signed module columns, 3,670,016 pair-level exact comparisons, all wrapper addresses/phases and three additional dense collectors. It was implemented before reading the primary model/proof and imports no predecessor executable. This is separate automated checking, not independent human review.
• Registered CPython 3.12.14 normal and optimized replays and metadata/source-pin validation PASS. A 23-file source-complete Git-free export passes the optimized full replay with all bytes unchanged; missing/corrupted scientific inputs fail before execution.
• Complete receipt SHA256: 7aa3d2274fdc0cd3f93da9c89f0b783991af657de8fa683627f0d7bbc8365a52. The public commit's tree is byte-identical to the reviewed local tree.

python3.12 -B scripts/research_package.py --package research/p104-clifford-exactness

The package README, proof, complete receipt and independent review provide the exact contracts. Hosted CI is separate from the local PASS evidence.

Task and overlap

Ten additive package files, based on canonical main 6f100d4bb5083f8c35d96fd95a539811c62f245c, priority revision 6. Task p104-side-correction, changed scope, original scope key c5367b1a1b7bed20db53. The original card, seed, family gate and checkpoints are retained in TASK.json.

Advance claim in Discussion2750. Canonical main and guidance were unchanged at the 23:43 UTC checkpoint. The fresh check covered all 287 open PR records and 181 visible Discussion records across five threads; Discussion pagination and sparse local coverage remain incomplete. New P104 relay/channel coefficient work in PR #3363/PR #3365/PR #3371/PR #3372 has a different obligation. No exhaustive novelty claim is made.

Credit and open next questions

Credit ywz65/PR2586 (7953c9a8f788a8de5cf2ced63d9a61b67c22c6f1), Tekalign/PR3025 (a544ef068a46f4de3bc47d1d02149cbdc3ba547f), Ross Richardson/PR3068 (7d2c519a1fdcc6bf66f6f08757eed7c48762d3f2), Arham Tahir's retained collector/numerical work, Nick Petalas's period/precision work, Alejandro Zarzuelo Urdiales's Gaussian parity work and the retained native/geometry/compiler lineage. Jose Manuel's PR3087 physical monomial work remains related context with its own assumptions. Nebe, Rains and Sloane retain the classical Clifford/Barnes–Wall literature credit. Exact consumed source pins, original headers and notices are retained.

The smallest useful next step is to bind one actual producer or downstream consumer, prove its Newton divisibility/intertwiner and charge the whole boundary. The Open next questions record the blockers and stopping decision. This result is retained as one local contract; further implementation expansion requires that evidence.

Prepared by Clear1nsight with substantial OpenAI assistance, including three concurrent agents for algebra, independent reconstruction and source/cost review. Apache-2.0.
p104-clifford-exactness-source.zip

Source ZIP: 180,889 bytes, SHA256 f096ec3f4723bb84c58090c89df50bdd4ace62f622d9cff6e3424b40d4d12aac; ten package files, twelve declared inputs and the standard export inventory at commit 2834523b1cd4ea83ba7a043e2e2f5f3332cf9909.