[Matarener]

Requested by Mark

Change and mathematical scope

Audit with a positive scoped result; no new κ. The selected κ is unchanged.

The #4311 coverage map lists COMPLEX.md's complete-cover execution in #4100 with no audit. This package checks the finite and algebraic statements of COMPLEX.md's common cover, at #4100 head 337d049:

• Cover size. Gamma = 2^1296 ∏_{i=1}^{36}(2^(2i)−1) = |O(73,2)| holds. A written proof shows that for odd n, O(n,2) ≅ Sp(n−1,2) (the all-one vector is fixed, and its complement is symplectic). Brute-force counts for n = 1, 3, 5, 7 match.
• Terminal defect correction. P_g C_all P_g^−1 = C_all X_delta with delta_i = (wt(row_i g)−1)/2 mod 2 holds for orthogonal g, with or without a translation. The written proof is a weight expansion mod 4 in the Hadamard picture. The check runs on all of O(3,2) and O(5,2) and on 400 random elements of O(73,2). A corollary, observed on every sample: wt(delta) ≡ 0 mod 4.
• H0 = ((1−i)/2) D C D. This holds exactly, together with the tensor i^wt diagonals and the parity split of ((1−i)/2)^h.
• Depth and stock. D is monotone, every child rank ≤ 48 drops D, and two levels halve the width. The stock passed to children stays divisible, and the parked volumes are geometric.
• Presentational correction. The padded row range R = ceil(2^q/M)·M is strictly larger than 2^q whenever D ≥ 1, because M has odd factors. So it needs q+1 reserved bits (or a radix-R field), not q. Only a constant changes.

Lift. The formal free-word lift is #4184's (ToikY), and it is not re-audited. As a supplementary cross-check, all four literal PR4088 words have only real dyadic scalars. So i^a can only sit inside warp generators, as COMPLEX.md's i^a → i^(fa) sentence assumes. The words also execute correctly with random i^a monomial generators raised to ⊗f, for f = 1, 2, 3, forward and dual, with relative error ≤ 1.7·10⁻¹⁴.

Update: native warps are i^a monomials. This answers the native-realization question that #4184 left open. native_warps.py binds each canonical word (evidence/{NATIVE,THREE}-CANONICAL-*) to its literal input word: B positions, G columns and frame spans. It then checks every warp edge class T_E K_G^(±1) T_U^-1: all 720 classes over the four words are exact Z4-monomial matrices, global phase included. This independently re-derives PR4088's own warp-recipe receipt. Controls: without its quadratic phase, P_G is not monomial, and swapped frames fail on 86 classes.

Not covered: the moment and root (#4171), the rank-r child reading (#4375), the composition of local recipes into the global emitted program (open question), and the physical gather/execute schedule.

Task and overlap

• Card. uniform-transfer-audit, priority revision 7, refined to COMPLEX.md's common-cover algebra.
• Base. 12fa8c5c844be21bfefb38a19503fbd867a11244.
• Overlap. Checked against Map independent audit coverage of #4100's obligations (navigation; no new κ) #4311 and against open PRs mentioning Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100 up to Answer #4371's open question 1: one coefficient's error path through #4100's transforms carries one factor of T (m = 1), so C = 5 suffices for b >= 13 (no new κ) #4415. Audit PR4100’s free-word tensor lift on all four actual programs #4184 covers the formal lift, Independent numerical cross-check of PR4100's exact-root and paid-power ledger (finite layer; no new kappa) #4171 the moment, Bind mixed-provider precision to the rounded Gaussian tensor caller #4178 precision, Separate the tensor-frame child from factorwise repetition. #4375 the rank-r child reading, and Complete source-bound four-provider finite admission for PR4088 #4180 finite admission. No PR checks the cover size, the X_delta correction or the stock arithmetic.

Validation

python3.12 -B research/complex-cover-lift-audit/verify.py
python3.12 -B research/complex-cover-lift-audit/test_verify.py
python3.12 -B research/complex-cover-lift-audit/native_warps.py
python3.12 scripts/research_package.py --package research/complex-cover-lift-audit
python3.12 scripts/check_research_commit.py --package research/complex-cover-lift-audit --check-source-history

All pass on CPython 3.12 (Windows). verify.py and test_verify.py use the standard library only; native_warps.py needs numpy (pinned numpy==2.2.6) and takes about 2.5 min. verify.py takes about 25 s; with --full it adds the n = 7 count (about 8 min).

Rejecting controls in test_verify.py:

• non-orthogonal g, and delta = 0, fail the defect identity;
• the padded range exceeds q bits;
• reading scalar signs as phase symbols breaks the lift;
• non-inverse G^−1 breaks the lift.

The finite checks support the written proofs in the README; they do not replace them.

Credit

The construction is by Cinna Davis (#4100, PR4088). The free-word format read here is from Sagar Thakrar's interpreter in PR4088's vendor. #4184 (ToikY), #4171 (petalas), #4311 (Ryanson Jonathan) and #4375 (unpaidmoon) are cited above. Prepared for Mark (Matarener) with substantial Claude Code assistance. Apache-2.0.

Open next questions

Does each PR4088 word's global emitted program compose its local frame, centre and warp recipes into exactly the free-word shadow? PR4088's PHASES receipts list this as a separate obligation. See TASK.json.

🤖 Generated with Claude Code