Magic Cat States VII: chi(cat_m) <= 2^(m/2-1) for Every Even m — a Closed-Form Domino Construction, Verified Exhaustively at m=32

Version 7. Result: χ(catm) ≤ 2m/2−1 for every even m, a closed-form construction (no search) with an exact algebraic proof, cross-checked numerically at m = 4, 6, 8, 16, 32 and verified by full exhaustive enumeration at m = 32 (32,768 cosets, zero failures). catm here is the Qassim–Pashayan–Gosset code state Σy: w(y) even iw(y)/2|y⟩, the same family this series' earlier volumes fix as the intended target (see Version 1's definitional note). This is a completeness/negative result for the whole coset-split construction family, stated plainly: the exponent is exactly 1/2 at every even m tested, and this family provably never approaches the Qassim–Pashayan–Gosset exponent log₂(3)/4 = 0.396241, which remains the best known upper bound for this state. No claim of improvement over QPG is made or implied. 1. The construction Partition the m coordinate positions into m/2 cyclically adjacent pairs ("dominoes"): (1,2), (3,4), …, (m−1, 0). The subspace spanned by the m/2 domino indicator vectors has codimension m/2 − 1 in the even-weight code Em, splitting catm into 2m/2−1 cosets. This exact basis reproduces, byte-for-byte, the bases an independent brute-force codimension-r subspace search (coset_split2.py, used elsewhere in this series) already found at m = 4, 6, 8 — it was not designed to match that search, it was recovered by inspecting the search's own output and recognizing the pattern. 2. The proof (general, not per-m) Fix a coset representative a₀ and let ti ∈ {0,1,2} be its local Hamming weight on domino pair i. Because the dominoes have pairwise disjoint support, flipping domino i changes the total weight by exactly 2(1−ti), independent of every other domino. For selection bits c ∈ {0,1}m/2: w(a₀ ⊕ ∑ cibi) = W₀ + 2∑ci(1−ti) g(c) = w/2 mod 4 = W₀/2 + ∑ci(1−ti) (mod 4) g is affine-linear in c — strictly stronger than the Z4-quadratic condition this series' earlier coset-split checks test for (every second difference is exactly 0, not merely constant) — and this holds for any even m with no per-m search or verification required. Each coset is therefore a genuine stabilizer state (constant-modulus support, well-defined Z4 phase), giving χ(catm) ≤ 2m/2−1 directly from the coset count. 3. Verification, both algebraic and exhaustive Closed-form cross-check. The affine formula above was checked against direct weight computation — not sampled, exact over all 2m/2 points of c — on 8 random coset representatives each at m = 4, 6, 8, 16, 32. Exact match every time (domino_proof.py). Exhaustive m = 32 check. Independently of the closed-form proof, the underlying per-coset Z4-quadratic phase test (is_z4_quadratic_cheap) was run over all 32,768 cosets of cat32, an 8-process, 18.9-minute job: 32,768/32,768 pass, 0 failures (verify_cat32.py, verify_cat32.log). This matches the proof exactly and was launched before the general proof was found, so it stands as an independent computational confirmation rather than a test built to match the theorem. Discriminating control. A support/amplitude-signature check that initially looked like independent confirmation at m = 16 was found to pass on any partition of the state's support into 128 blocks of 256 (verified directly, 8/8 random partitions pass) — it is vacuous by construction, since every amplitude here is already a unit Gaussian integer. The check that actually discriminates — reading the affine support and Z4-quadratic phase off the saved amplitudes independently of the construction's own g — passes 128/128 on the real pieces and 0/8 on a random-partition control. 4. Search-based alternatives, both null and structurally explained Two independent attempts to beat 128 at m = 16 via local search (poolcd.py's Clifford move set) over the same target state both returned clean nulls, not inconclusive results: Truncated-witness seeding. Seeding the local search from truncated subsets of the 128-term witness (dropping 32/64/96 of the 128 disjoint-support terms) and running 300s of local moves per worker gained at most 0.0049 in reconstruction loss — essentially nothing. 15-restart generic search at k=32 (independent trajectories, memory-safety harness with BLAS thread pinning and a hard availability floor): 0/15 restarts converged within the time budget; best losses plateau at 0.94–0.97, far from 0. Both nulls have the same structural explanation: the 128 pieces have pairwise disjoint support and product-form phase, so there is no residual overlap left for a local perturb-and-repair move to exploit. A move set that can merge disjoint pieces directly, rather than perturb one and repair it, would be needed to make this question live again — not attempted here. Limitations, stated directly The exponent log₂(χ)/(m−2) = 1/2 exactly, at every even m checked. This construction family closes at that value and never approaches QPG's 0.396241 — this is a completeness result for the family, not a new record. The exhaustive m = 32 check verifies the phase condition, which the algebraic proof shows is sufficient for the stabilizer-state property; it does not re-derive the stabilizer property from a first-principles Pauli-group enumeration (that check is exponential and intractable at this size, per prior volumes in this series). The two search-based null results used one specific local move set at one problem size (m = 16); they rule out that method finding an improvement, not the existence of one. Working directory: ~/branch_work/30_cat16_witness/. Prior volumes in this series established the χ(|TBK⟩⊗6) ≥ 4 lower bound (Version 6) and the priority/valuation corrections for χ(|T⟩⊗4) (Versions 2–3); this volume is an independent line of the same research program, on the catm orbit rather than the Bravyi–Kitaev T-type orbit. Version 8 (staged, not yet published). Closes a reproducibility gap in Version 7, found by an independent review, not by the authors of Version 7. Version 7's "Discriminating control" claim (Section 3, third bullet) — "the check that actually discriminates ... passes 128/128 on the real pieces and 0/8 on a random-partition control" — had no script or saved data among the 11 files deposited with that version. It was run as an interactive, unsaved computation against a 67 MB intermediate array that was itself never deposited, so the claim could not be independently reproduced from the record as published. What this version adds. product_state_proof.py, a new, self-contained script: pure NumPy, no external imports beyond it, no dependency on any undeposited file. It rebuilds catm and its domino cosets directly from the definition (no reuse of domino_proof.py's closed-form phase formula, so it cannot silently inherit a bug from that derivation) and tests, for every coset piece, an exact rank-1-tensor identity: for every pair of domino bits i