[aliromman]

This package gives scoped negative results for frame gauges on one historical scalar word. It proves a total native-rank bound for all 511 fixed nonzero translation lines, stronger cost bounds for specified smaller families, and a sharp local rank-six bound for arbitrary common-line products. It also retains a counterexample to a broader cost claim. It does not improve the complete kappa calculation or supply a new multiplication algorithm.

Read the proof, source pins, decisive replay receipt, and independent publication review.

Allowed gauge class	Result
One nonzero translation line fixed across the whole word; each scalar component chooses I or C_a	All 511 lines have minimum total native-rank change zero. Each even line is also nondecreasing on each native edge. The result does not settle all child moments for odd lines outside the common family.
One of the 16 common odd lines; the binary product C_128 C_256; or the four states I,C_128,C_256,C_128 C_256	Every increasing concave native-rank cost with f(0)=0 is nondecreasing. Every zero-total-rank choice is constant on all 1,971 native components.
Binary I,T_415 in forward coordinates	Minimum total-rank change is zero, but one cut removes a paid child and adds rank 48. Every changed native cut increases sum r^(1-b) by more than 47/9800 for 0 <= b <= 2/5. The backward claim uses F T_415 F^-1.
Arbitrary products of translation lines from the common five-dimensional subspace, at both ends of a centre descent	Each descent has rank at least six, attained by C_128 C_256 at all eight centres. This local theorem does not price the full graph.

The graph keeps the scalar word and every native event position. Its endpoint condition is diag(G_out) P = P diag(G_in). The proof charges all 5,064 native events, including 1,570 old rank-zero wrappers. It does not give a lower bound for arbitrary centre encodings, changed scalar schedules, or all Clifford gauges. The counterexample prevents extension of the concave-cost claim to the full-frame family.

The consumed source is the pinned historical replay archive at bce54fb3f895cbe5cd4c0eb477c3afeed100190f. The package contains its two exact scalar-word and local-program members.

The common-boundary geodesic annex adds a distinct scope to this PR. It allows arbitrary Clifford gauges and exact adjacent fusion, while retaining productive lifetime paths, legal common-frame scalar accesses, and priced native events, including old rank-zero events. It forbids free productive resets or lifetime transfers into private zero workspace. With common I/full-F productive boundaries and identity joint content, the path theorem forces total rank at least n N. This is a different boundary premise from the primary 511-line gauge result; it does not extend that result to all Clifford gauges under the original virtual-Q boundaries.

For the stated one-shot common-frame inverse, the sharp conditional rank invoice is 73*15488=1130624, against 1116288 before normalization. The required increase is 14,336 globally, or 896 per replica on average. The literal input/output normalizers use 896 rank-one children on each replica and attain the bound. The numerical application inherits the exact native mode endpoint, full-F monomial content inverse, and whole-owner backfill invoice; it does not independently assign the shared owner costs to each replica. The rank-48 split profile has moment one at exponent zero and greater than one for every positive exponent. For every nonnegative exponent b, admitted fixed-path fusions with child ranks at most 73 still have moment at least one. No positive supplier exponent or complete kappa improvement follows from this route. New controllers, changed boundary encodings, and productive lifetime transfers remain outside the theorem.

The annex source manifest, validation, and independent annex review record the proof and replay boundaries. The source remains pinned to historical bce54fb3f895cbe5cd4c0eb477c3afeed100190f; no transfer to selected PR #4470 at publication base b58ad9ef4d4e861b0eaa5dba41742bdc072dca51 is proved. The canonical-subspace triangle argument is prior work in Section C.2, Theorem N, credited by its retained source notice to DreamingOfClouds with Claude Opus 5.5 assistance. Its notice and full prior credit and Apache-2.0 license are preserved. The new theorem, checks, and separate agent review used substantial OpenAI Codex assistance; no human review is claimed.

With Python 3.10 or later, its standard library, one CPU, at least 3 GiB of available memory, and at least 3 GiB of free disk space, run from the repository root with a new external output directory:

python3 -I -O -B research/fixed-word-gauge-limits/annex/geodesic-boundary/replay.py --out /tmp/geodesic-boundary-annex

Expected final line:

PASS_GEODESIC_BOUNDARY_ANNEX

The annex runs only two decisive checks. They cover all four mode words, 1,004 source frame spaces, the exact invoice, and the complete 720-frame two-bit symplectic control. Three damage controls reject a changed native rank, a productive-array CLEAR, and an unequal-frame scalar access; a separate matrix-order control rejects the wrong relative order. They are distinct from the primary package's controls. Normal and relocated optimized runs passed. The primary package, its manifest, and its original replay remain unchanged. The annex does not rerun native mode admission, whole-owner hosting, a global emitter, current-selected replay, or scalar/address and recursive pricing.

The publication base is b58ad9ef4d4e861b0eaa5dba41742bdc072dca51; its selected record identifies PR #4470. No transfer to that changed construction is proved. PR #4700 gives a different chart bound on a different scalar word and allowed class. It is related work, not a consumed premise.

The retained T16 notice credits @jeremykothe2 for the PR #799 local word, @chafreaky for the PR #741 frames and PR #685 owner construction, and @FanPUYen for T16 integration. The package notice preserves the source gauge and pipeline credit to @icekylinx, the recurrence correction credit to @rohanarun, and the earlier supplier and controller credits and Apache-2.0 license. The source-domain lane supplied the certificates and product argument; temporal-reuse and encoded-controller supplied separate checks. Substantial OpenAI Codex assistance was used. These are separate agent reviews, not independent human peer review.

Use Python 3.10 or later, the standard library, one process at a time, and at least 3 GiB of free space. From the repository root:

python3 -I -O -B research/fixed-word-gauge-limits/replay.py --out /tmp/fixed-word-gauge-replay

The output directory must not exist. The expected final line is:

PASS_FIXED_WORD_GAUGE_LIMITS

All nine checks passed in an independent relocated replay with a read-only package and an empty PATH. They checked both scalar endpoints on all 712 input columns, the complete graph, ranks by span and symplectic pairing methods, 10,488 integer paths, 3,384 common-line rank flow/cut cases, 800 common-line capped-cost cases, and 13 rejected damage controls. All nine mathematical receipts agree with the packaged evidence; five match byte for byte, and the rest differ only in paths or measured memory use. The review also checked the historical archive, source pins, checker adaptations, and the sharp local bound.

No current selected-construction cold replay, changed native circuit emission, full recursive repricing, hosting construction, or new mathematical search ran. The all-size compiler, physical supplier, precision, hosting, and analytic obligations remain outside this package. Open next questions state the remaining scope boundaries.