[ToikY]

For every INTEGER width f>=10 in the full three-qubit Gaussian-dyadic Clifford alphabet, prove the sufficient normalized tensor-Gram interval

(405/16384) I <= H_f <= (1334151/131072) I,
condition(H_f) <= 16471/40 = 411.775 < 807.

This answers Nick Petalas #6794's second-class trace handoff. The new actual lift is g=(hS) tensor I4, h=(1+i)/2 [[1,1],[1,-1]], S=diag(1,i): trace4, cube -I, quotient order three on one symplectic plane. Exactly6/10 quadratic forms are fixed in the36/28 Arf orbits. The real-part fixed-space bound and exact phase-one character cancellation give T3/d in [-3/36,6/36] or [-5/28,10/28]. Individual fixed-space traces need not be real.

Three exact all-thirty-row inequalities plus feasible primal mixtures yield multiplicity-ratio intervals [2/7,2353/20] for N28 and [1,2621/36] for N36. Every displayed dual has zero transvection coefficient: the new order-three upper constraint alone supplies these bounds, while the matching primals also satisfy the old transvection slab. The extrema are sharp for the declared convex relaxation; actual irreducible decompositions, spectral endpoints and exact condition number are unclaimed. Threshold ten is unchanged.

Only research/clifford-order3-conditioning/ is added (15files). Scope CHANGED/partial from formal-interface-lemma to this written native-algebra consequence; no Lean proposition or original selected-interface gate closed. Canonical main refreshed7c49fe32c55de02c2c873655c804b16e4c3fef0a/revision8, selected6030. Explicit handoff, no picker draw. Fresh769 full open bodies/heads and bounded all-state searches found no exact competing extension within inspected coverage. Sparse/history/unpublished gaps remain unknown. Canonical claim and exposure: #6794 (comment) . Short hub pointer: #1166 (comment) .

Inherited inputs: alephnan #6714 at f7f32a1 supplies group/table identification, width-nine multiplicities, even-sector reciprocity/Pauli orbits, Gram normalization and Schur propagation. Petalas #6794 at c6252da owns the exact one-transvection807 theorem. Own excluded #6718 review is not recounted. Credit Lee–Yu–Peng–Lai's published table (https://arxiv.org/html/2309.14850v2#S6.SS3), Ali Romman, MatShep and retained lineage/classical methods. No published-table reconstruction or predecessor scientific engine ran. Exact5-source prose/numerical copies, pins, LICENSE and NOTICE are retained; source-tree associations were checked separately from canonical generic ancestry.

Validation, CPython3.12.14, standard library:

python3.12 -B scripts/research_package.py --package research/clifford-order3-conditioning
python3.12 -B scripts/check_research_commit.py --package research/clifford-order3-conditioning --check-source-history --check-modes
python3.12 -B scripts/export_research_package.py export --package research/clifford-order3-conditioning --output /tmp/order3.tar
python3.12 -B scripts/export_research_package.py check --check-inputs /tmp/order3.tar

• Actual8x8 Gaussian lift,64 signed Pauli images,64 quadratic forms/4,096distinct form-vector cases;252distinct feasible vertices (84/168), four30-row dual certificates and6 witnessed controls. Two controls show lost-constraint primal admission; they are not rejecting valid representations.
• First run0.702s/24.3125MiB,14then-present files unchanged. Afterward one receipt key was renamed form_vector_evaluations to distinct_form_vector_pairs:4,096 counts cases, each evaluated twice. Entire typed packet otherwise identical; first code/packet and pre-correction replays preserved. Mathematical arithmetic/source/controls unchanged.
• Final direct normal/optimized, both registered commands and read-only Git-free empty-PATH exported replays agree on the complete13,000-byte packet, SHA25638380ddd9be74d0bf0beb6bd2c21a4dddbc7e78e94ffc3607c08bf421118771a. Official16-member export/input integrity, committed bytes/modes/history and14 exact-reason source/typed-result rejections pass, including False changed to integer0.
• Source-aware proof/code/metadata challenges found no blocker; related AI roles are not blind independent, human or proof-kernel acceptance. Original hosted run42321028552, attempt1, completed successfully: https://github.com/CrocSwap/integer-mult-bounds/actions/runs/42321028552 . Both complete13,000-byte scientific packets equal frozen local. It completed711 TOTAL helper tests, including24skips, on CPython3.12.16. Raw logs and the exact synthetic checkout tree/parents are retained; a synthetic test ref is not a repository merge. No native/Lean/Docker build, selected cold replay, full group census or full even-sector table was run.

Next: bind actual quadratic-form stabilizer decompositions to test spectral realization. Physical use separately needs one exact caller and complete preparation/recovery/precision/routing/ownership/restoration/stock/recursive invoice; these remain unknown. All five selected assumptions remain; no supplier admission, kappa/frontier change, merge or follow-on reservation.

Prepared for ToikY with substantial source-aware OpenAI Codex proposal, implementation, proof/code and overlap/metadata assistance. Preclaim count/lift/phase guesses and source exposure are disclosed; no priority over classical methods or predecessor results. Maintainer edits and auto-merge disabled.