[aliromman]

This scoped negative result proves an exact numerator-width gate for the retained two-FULL caller. Its source has 4,524 paid cuts with independent coarse and fine integer input bands. A literal exact numerator at each marked cut needs leading width 2k, including when a shared exponent is allowed. Under the stated uniform per-occurrence linear width tariff, charging only the 3,163 original rank-one cuts rejects the complete declared invoice. The complete kappa calculation does not improve.

The proof, exact result, complete occurrence certificate, and source pins state the exact scope. The base and consumed source commit are 32890bc5de034ab8fc93ac88ead7279a6a416d04; there is no pending-PR runtime dependency.

The policy fixes a sufficient width at each occurrence for all permitted inputs, funds baseline width k+o(k), and charges linearly in width. Both native rank and fiber/column multiplicity stay fixed as k grows. Tensor denominator offsets can grow with multiplicity, so no growing-multiplicity limit is claimed. Separate worst-case witnesses do not prove the sum for an adaptive per-input policy. Joint symbolic state, different words, changed helper boundaries, other tariffs, and already funded larger widths remain outside the theorem.

For nonzero dyadic coefficients p,q, the values p A/2^k + q B have at least 2^(2k-c) distinct members, where c=max(0,v2(p)-v2(q))<=2 at the recorded cuts. A reduced odd-numerator argument also gives width 2k-O(1) despite an exponent field. All helper targets remain credited. The declared bill is:

targets = {4:56,7:14,8:94,9:652,17:224}, mass 10750
raw paid = {1:4656,2:920,3:1386}, mass 10654
width-charged paid = {1:7819,2:920,3:1386}, mass 13817

Strict rank separation gives T_t/P_t <= 10750/13817 < 1 for every 0<=t<=1. The proof uses only original H1 occurrences; it needs no transfer of width witnesses into higher-rank split factors.

With Python 3.12 and its standard library:

python3 -B scripts/research_package.py --package research/two-full-exact-width-gate
python3 -O -B research/two-full-exact-width-gate/verify.py
python3 -B research/two-full-exact-width-gate/test_mutations.py

The registered command ends with:

PASS_REGISTERED_RESEARCH_PACKAGE research/two-full-exact-width-gate outcome=negative-result

Normal and optimized exact reports match. The retained audit checks all 42,898 events and all 72 warp matrices. The new replay reproduces all 4,524 historical witnesses, checks the 520-column restricted caller input space and every helper return, tests all 75 coefficient pairs, and independently evaluates six source cuts. Four internal controls reject false valuation loss, a zero band, a missing final owned clear, and an integer-to-float receipt change. Isolated normal and optimized processes reject changed source bytes and a changed witness coefficient. The final registered replay, committed-byte check, 12-file export integrity, read-only Git-free export replays, and signed invoice preflight passed.

Cao Yifan supplied the round-three source and profile in the PR #4610 lineage. Yiyang (@Lyy-iiis) supplied the actual caller, boundary substitutions and reused moving-frame audit in PR #4966. Its independent-error proof is a different tariff premise. Dillon (@dillona) retains the border and recovery credit from PR #4623. The additive compatibility result in PR #4983, coherent rank gauge in PR #5106, and rounded inverse history frontier in PR #5444 have distinct scopes, recorded in the overlap note.

No full physical Gaussian execution, recursive child compatibility, compiler or outer multiplication replay was run. The optional full-history preflight was not run in the shallow checkout. Hosted CI has not run at publication. The proof received author-side parallel agent review, not independent human acceptance. No selected frontier file changes. New work by Ali Romman (@aliromman), with substantial OpenAI Codex assistance; Apache-2.0, with predecessor credit retained.