[aliromman]

This proves the nine-axis native monomial window 21–24 is sharp. More generally, the cutoff is exactly 2n+3 for every n>=5 with n=1 modulo four. At nine axes the exact residue thresholds are (24,21,22,23); four matching residue tests are necessary, and no complete test set can have maximum width below 24. This settles an algebraic verification bound for fixed complete endpoints. It supplies no new kappa or physical cost reduction.

Proof, exact controls, receipt, source pins. Base: canonical main 6cf7a68, priority 7.

A zero-phase full-span tuple at width at most 2n+2 would extend to a doubly even self-dual code of length four modulo eight. A short Gauss-sum argument excludes that code. A genuine cuspidal GL(n,2) character, averaged over translations, then supplies a nonzero affine cancellation element. Combining it with a phase projector gives an actual group-algebra residual that vanishes below the claimed threshold. E8 code blocks and a four-bit row give its first visible tuple. The argument therefore proves representation sharpness, beyond a missing full-span witness.

PR #5529 supplies the matching upper bound and retained proof/constructor. Joseph Anglada's PR #5542 supplies the three-axis code/coset method in its pinned proof. PR #5622 supplies the four-axis affine version. Credit Hiroki KUMAZAKI's PR #5522, Mihir Srivastava's PR #5521, the grammar/Fourier work in PR #5369, and Jose Manuel, ywz65, Chris R and the retained native lineage. Green's classical character theorem is an explicit mathematical input, cited through its original publication and Prasad's published formulation, Section 2. No new coding or character-classification theorem is claimed.

Normal and optimized registered checks pass: explicit shortest tuples through 129 axes with all four residue pads; exact Gauss-sum contradictions; all 38,094 hyperplane cosets in GL(3,2) and GL(4,2); 512 outputs of the retained general upper constructor; four rejection controls. Committed-byte, export-integrity and Git-free optimized checks pass.

python3 -B scripts/research_package.py --package research/nine-axis-cutoff-sharpness

Python 3.12 and the standard library suffice. Expected markers: PASS_NINE_AXIS_CUTOFF_SHARPNESS and PASS_REGISTERED_RESEARCH_PACKAGE.

The unbounded proof uses Green's theorem and the retained upper theorem. The GL(9,2) residual is defined by a finite formula and is not expanded; all nine-axis phase characters are not enumerated. No proof-kernel or independent human review is claimed. The grammar excludes Hadamards, width-dependent external coefficients and changing program families. Native implementation, physical caller, precision, stock, compiler and outer transfer remain separate obligations. Full source-history ancestry was not checked in this shallow checkout. The selected frontier is unchanged. Coordination. Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.