[aliromman]

This is a scoped obstruction and an exact search filter for complete native words. It permits changed paid inputs and active feedback. It proves a family of necessary tariff inequalities for the full columnwise monomial ordinary alphabet, then decides the infinite family by a finite integer-polynomial test. The complete kappa bound does not change.

Proof, verification receipt, source pins, and canonical Discussion #5582.

For every positive integer k, a fixed profile that works at arbitrarily large widths must satisfy

sum_r P_r (1-2^(-k*r)) >= sum_r T_r (1-2^(-k*r)).

At width f, the available cuts have 2k <= f. The proof uses equality of paired tensor columns. Ordinary monomial maps preserve the cut; a rank-r Clifford call has exactly the displayed normalized crossing rank. Rank subadditivity applies to the actual chronological word, including COPY, CLEAR, dirty inputs, overwrites and feedback.

The result gives concrete false-positive invoices. One rank-one plus one rank-three call cannot supply two independent rank-two jobs, although their counts and rank masses agree. A stronger family first fails at any prescribed cut K while passing every earlier cut and a strict raw power-moment test for all savings from zero through 1/256. A failing cut forces ideal endpoint operator error at least one for the stated unitary target frames. These are impossible profiles, not a kappa construction.

All fields must have the same capacity. Target credit is for independent native jobs. General inter-column maps, larger-field packing and nonlinear operations are outside this theorem. The exact obstruction permits invertible ordinary endpoint codes; the unit error constant does not extend to arbitrary ill-conditioned codes. A boundary with nonordinary observations must include the joint crossing rank of its full availability map. The CLI does not extract that map or certify native admission.

The pinned P104 complete profile passes every cut: 69,930 paid calls and 12,814 target jobs. This is a source-profile screen only; its native word and compiler were not rerun.

Verification used Python 3.12.13 and only the standard library:

python3 -B scripts/research_package.py --package research/native-equality-cut

Expected: PASS_NATIVE_EQUALITY_CUT in normal and optimized modes, then PASS_REGISTERED_RESEARCH_PACKAGE. Both modes passed 172 literal native rank and partial-isometry checks, 480 complete chronological prefixes, 192 profile pairs, all 625 small four-rank differences, two delayed-failure families through K=12, and the source profile screen. The same modes passed in a read-only, 17-file Git-free export. Controls include a noncolumnwise escape, duplicate cached responses, a single-input packing obstruction, width boundaries and malformed profiles. Committed-byte and source-pin checks passed. Full-history ancestry preflight was omitted because the shared checkout is shallow; immediate-base ancestry was checked. No selected cold replay, proof-kernel acceptance or independent human review is claimed.

Base and retained source snapshot: 5e0052ca0aa3036a3afa8b0438927377c8d97d53, priority revision 7. New head: 97db3fb45cd93a350e300781d75440d14e6a1f73. No predecessor files or selected-frontier records change.

Credit: the classical cut-rank inequality is already used in the fused-block audit, an664's dirty-host work, and later tariff work by Adam Gall and Ross Richardson. PR #5449 and PR #5526 supply the retained-history question and frozen-caller context, with Chris R / Clear1nsight's predecessor credit retained. zxv's Gaussian scalar implementation is copied unchanged with its notice. The PR #5544 profile retains Gareth Davidson, David Arias, Cinna and the native/compiler/outer lineage. The new contribution is the monomial-invariant equality cut, exact Clifford tariff and infinite-family decision procedure; general rank, stabilizer and semisimple-module theory are not claimed as new. Ali Romman, with substantial OpenAI Codex assistance. No subagents. Apache-2.0.