[aliromman]

The new verifier passes the complete selected PR4610 local provider in both directions. It also proves a scoped obstruction: all 30,328 early/late fusions of two paid climbs on one persistent role change the complete endpoint at every positive tensor width. The pairs need not be adjacent, have rank one, or have the same middle frame. This closes the fixed-pair route on this provider and gives a saved contact certificate for each rejected pair. There is no improvement to the complete kappa calculation.

Start with the provider proof, result receipt, source pins and research record. This is an enabling proof, verification algorithm and scoped negative result. It changes no selected record or predecessor package.

The provider importer keeps all 632 persistent inputs, including 408 arbitrary helpers, and the arbitrary owned input. Every output is checked; the owned return must be zero. Each direction contains 21,449 source events. The provider replay checks 7,000 phase-bearing native recipe classes with the quadratic-sum engine: all 5,710 paid classes, 360 warp classes, 34 helper boundary classes and 896 main adapter classes. These are labelled classes, not distinct matrices. Exact normalized Gram value one proves equality of each unitary pair, including absolute phase, and thus equality of all its tensor powers.

The complete physical programs have 24,827 forward and 24,819 backward instructions. Both pass compact.certify and a separate sparse all-input interpreter against an explicit full reference. Peak support is 22,331 and 12,930 terms respectively. The source reduces exactly under signed-word cancellation; this is not a general bound on support or a claim that this baseline requires all eight width tests. No dense tensor matrices are made.

For every pair A,B on one persistent role, let T be the complete intervening physical program. The two tested replacements change BTA to TBA or BAT, with all other operations fixed. The 15,164 contact certificates give an exact off-diagonal interval block q U^tensor f touching that role. A separate role-list enumeration checks the whole pair census; the earlier independent-input engine rechecks every coefficient. At each paid cut, owned storage is unused or zero. The complete persistent baseline is invertible, so a nonzero interval defect survives its prefix and suffix.

Each relevant exact trace satisfies Re(tau) <= 1/2 and |tau|^2 <= 1/2. The affected block therefore has normalized squared defect q^2 (2 - 2 Re(tau^f)) >= q^2 > 0 for every positive f. The smallest saved q^2 is 1/256. This bounds the selected cut blocks, not the full-provider norm. The twelve full-program basis witnesses also give literal width-one failures. A separate frozen source amplitude evaluator checks their values without Gram traces or quadratic sums.

The obstruction covers one fixed pair at a time. It excludes neither changed contacts, simultaneous rewrites, interior placement, cross-role pairs, owned-center pairs, new operators nor later providers. The initial H1+H1→H2 opportunity comes from the PR #5243 handoff. The required physical pair remains absent. The local paid profile stays {1:2201,2:412,3:510,4:127,5:40,8:8}: 3,298 jobs and rank mass 5,327 per direction. No hypothetical saving is charged to the controller.

The reusable results from the same PR remain:

• The original proof and extensions give the eight-width test 8n-3,...,8n+4 for the full dyadic Clifford group on n >= 2 binary axes. One axis needs widths 5,6,7,8; maximum cutoff eight is sharp. Zero axes use widths 1,2,3,4.
• Some width in the full range 1,...,8n+4 has normalized squared residual at least 75M/76, where M is the squared coefficient norm after exact operator grouping. This constant is not claimed for only the eight selected widths.
• The compact producer uses exact quadratic sums and bitwise witness search. Its work is polynomial in expanded support, word length, axis count and tested width. Expanded support can still be exponential in program length.

Programs and scalar coefficients are fixed as width varies. Native operators are unitary dyadic Clifford matrices; their phases stay inside tensor powers. The proof permits complex coefficients; the implementation uses Gaussian rationals. Width-dependent programs, non-Clifford or nonunitary native gates, clean-input restrictions, rectangular types and rounded or nonlinear leaves are outside this contract.

Validation used CPython 3.12.13 on macOS and the standard library:

python3.12 -B scripts/research_package.py --package research/clifford-tensor-gram
python3.12 -I -W error -B research/clifford-tensor-gram/verify_provider.py
python3.12 -I -O -W error -B research/clifford-tensor-gram/verify_provider.py

The registered runner passes all six original, extension and provider replays in normal and optimized modes. The provider's final line is:

PASS_COMPLETE_PR4610_PROVIDER_AND_30328_FUSION_OBSTRUCTIONS

The runner ends with PASS_REGISTERED_RESEARCH_PACKAGE research/clifford-tensor-gram outcome=completed. Provider controls reject an omitted paid class, wrong absolute phase, borrowed erasure, a live owned cut and a missing contact. An identity interval is a positive control. The importer also checks that each nonidentity main adapter surrounds all uses of its role; a separate moved-adapter negative check rejects the altered word. These are author checks with separate computational paths, not independent human review or proof-kernel verification.

Committed-byte checks and official 50-file export integrity pass at 848cdce9003262c4b6e02ec4558f302e4fae56ad. All six replays pass from a fresh read-only export with Git absent from PATH. All 51 regular files, including its manifest, remain unchanged. Export SHA-256: a40f20e4f2e6d22748919ce19954847fe6b949e1395d9d6ed6ca32b712a802c0. The earlier original mutation receipt and extension mutation receipt remain unchanged; their mutation scripts were run in the preceding revision and were not rerun for this provider-only addition.

The first hosted provider replay passed the native classes and both complete endpoints, then hit its 240-second limit. The manifest now assigns 90 seconds to each earlier check and 420 seconds to each provider check, within the unchanged 1,200-second package budget. The scientific code and certificates did not change.

No new complete controller pricing, outer admission, selected cold replay, Lean or Docker run is claimed. The selected five framework assumptions remain. Physical ownership, stock, precision, storage, tape execution and analytic recovery remain separate obligations. The optional full-history ancestry preflight was omitted in this shallow checkout; exact source Git objects and bytes were checked.

The original verifier's scientific base remains 7c472563474e60185e6ca101a17622133d047e56, priority revision 7. This continuation refreshed main to 1ec1119972c9aefe42525172d07b332fcff81cb4 and consumes the selected provider author head 20bb93cd990d2ff683e4d0fc56b87a42cd5d3296 and independent admission head 13efb5fc5f3b7c14ec2cfa181dd9eb6480972cf2. Their data are copied from that main snapshot with exact object and SHA-256 pins. The selected conditional kappa remains 15108879/5000000000 = 0.0030217758. The final overlap refresh covered 350 open PRs and 4,684 Discussion records with complete remote pagination; sparse local notes and historical diffs remain coverage limits. The canonical claim and handoff records this scope.

Credit and source distinctions:

• Cao Yifan (@intqwq) supplies the selected construction in PR #4610 and its source proof.
• Chris R (@Clear1nsight) supplies PR #5241's paid catalogue, independent admission and earlier quantified free-group lift. That lift is prior work; this PR applies the new engine and adds the pair obstruction.
• Cinna Davis (@1337isnot1337) supplies the helper, warp and retained controller chain. The provider notice links the source credit for zxv, Jacob Alan Sussman, Dugongue, Rohan Arun, Jeremy Kothe, icekylinx, DreamingOfClouds and chafreaky. Tekalign (@tekuuu) retains the native kernel; its full lineage notice is unchanged.
• @zxv supplies the scalar layer and independent-operator interface in PR #5235. @krzysztofwojt supplies the inverse-interface handoff in PR #5262. lm1909's projection note, following Jacob Alan Sussman, already uses positive-width separation. Chris R supplies related finite-alphabet work in PR #4928 and PR #4978, with wider coefficient quantifiers.
• Related PR #5369 gives a four-width affine-quadratic monomial criterion. Related PR #5393 studies a frozen eight-relation provider. Neither is an input to this pair proof, and neither is subsumed by it. The coupled-helper proposal in PR #5402 changes the helper boundary and is also outside this fixed-endpoint pair class.
• The code invariant is classical work of Runge, as credited in Nebe, Rains and Sloane. Clifford quadratic representations are classical work of Dehaene and De Moor. No general stabilizer-method or literature priority is claimed.

New proof, importer, verifier application, census and receipts by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0. Next useful targets are a fully priced changed contact or an explicit joint rewrite. Individual rejection does not prove joint rejection.