Magic Cat States VI: chi(|T_BK>^6) >= 4 — a Lower Bound for the Bravyi-Kitaev T-type Orbit at Six Copies, with Two Gaps Found in Verification and Closed

Version 6. Result: χ(|TBK⟩⊗6) ≥ 4, improving the only previously known bound χ ≥ 3, which followed from projection monotonicity alone. Every 3-term decomposition is excluded. Independent verification is requested before this is relied upon — see the caveats. Labib & Russo (arXiv:2605.28586) state that for this orbit the values at m ≥ 5 are sub-multiplicativity extensions of χ(|T⟩⊗4) = 3, "not independent bounds; whether genuinely smaller rank witnesses exist at m ∈ {5,6} is unknown." This addresses the lower side at m = 6. 1. Three elementary premises — no Theorem A, no valuation theory Covering. T = ∑j cjφj and T has full support, so if no term's flat contains x then T(x) = 0, false. The three flats cover all 64 points. Private regions have constant modulus. On the region of Aj not met by the others, T = cjφj, and a stabilizer state has constant modulus on its support. |T| is injective in Hamming weight. |T(x)| = A·tanw(x)β with A = cos6β and tanβ = 0.5176 < 1, strictly decreasing in w. Together: every nonempty private region lies inside a single weight layer. Layer sizes are 1, 6, 15, 20, 15, 6, 1. 2. The lemma that does most of the work With ρ = 2 − √3 = 0.267949 and Gw = 64A²ρw, no four distinct weights satisfy ρa + ρb = ρc + ρd. Since ρ < 1/2, ρa exceeds the sum of all smaller powers, so the least exponent dominates: a < c forces the left side strictly larger, hence a = c and then b = d. Verified: 0 such quadruples. 3. Case analysis, complete over all multiplicity patterns Case A — some term has full support. Then that term's flat covers everything, the other two have empty private regions, and the sole constraint is that complement(A2 ∪ A3) lie in one weight layer. Over all 348,123,691 flat pairs, the number with a nonempty complement inside one layer is 0, so A2 ∪ A3 = F26, leaving exactly: (i-a) all three full support — dead. Dividing by φ1, T/φ1 = c1 + c2iu + c3iv depends on x only through (p,q) ∈ Z4² — 16 cells — so seven distinct cells must carry moduli² in geometric progression of ratio ρ (dynamic range 2702). Normalisation forces cell (0,0) to weight 0. M(p,q) is linear in the seven reals (K, α, β, γ) with α = c̄1c2 etc., but those come from three complex numbers modulo a global phase — five real degrees of freedom — and must satisfy |c1|² = |α||β|/|γ| and γ = ᾱβ/|c1|². Over all 3,603,600 assignments: 0 realisable. (i-b) exactly two full support — dead. Off the third flat the equation reads c1 + c2iu: only four moduli, so the complement meets ≤ 4 weights, and ≤ 3 by the lemma (four would use all four cells and trigger M(0)+M(2) = M(1)+M(3)). Of all 26,387 flats, exactly one qualifies (the even-weight hyperplane, complement = odd, weights {1,3,5}), and solving M(p) = Gw for it gives 0 realisable. (ii) full support plus complementary hyperplanes — dead at every normal weight. On A2 only φ1, φ2 act and φ2/φ1 = const·√2·ij there (stabilizer amplitudes are 2−d/2 times a fourth root of unity), so A2 meets ≤ 3 weights by the lemma, and likewise A3. But A2 ∪ A3 is everything, with all 7 weights, and 3+3 = 6 < 7. Computed minimum over all 63 pairs: 7. Cases B/C — no full-support term. All flats have dim ≤ 5, so ∑j|Aj| ≤ 96 while ∑j|Aj| = 64 + |D| + 2|Tr|, giving |D| + 2|Tr| ≤ 32 and hence |P| ≥ 32 singly-covered points across three private regions. Enumeration is complete (a covering needs a flat of ≥ 22 points, so A1 is one of the 126 hyperplanes; one of A2, A3 must be ≥ 16): 0 survivors of 3,738,420 covering triples. And it is a proof, not an exhaustion — the minimum second-largest private region over all triples is 16, only the weight-3 layer has ≥ 16 points (20), and two disjoint regions of ≥ 16 cannot both fit inside it. Repeated support flats. Terms may share a flat. {3}: all three on one flat forces that flat to be everything, i.e. case (i-a). {2,1} with the singleton full: the private region is complement(A), and 0 flats have |complement| ≤ 20. {2,1} otherwise: both dim 5 and complementary, so 32 points are private to one term, and 0 pairs admit that. All patterns covered. 4. TWO GAPS FOUND DURING VERIFICATION, BOTH CLOSED These are recorded because they are the substance of the verification, and because the claim was withdrawn mid-way and then restored. Gap 1 — unjustified rejection of singular systems. (i-a) initially discarded all 1,300,320 singular linear systems on the grounds that singular implies completing a line, which the lemma forbids. That implication is false: the 16 rows in R7 have a nine-dimensional relation space against only 8 line relations, and directly — of 5,005 6-subsets, 763 are singular and line-completing while 1,043 are singular and complete no line. The claim was withdrawn. Resolution by direct computation rather than lemma: G must be annihilated by every left-null vector, and since ρ generates Q(√3) that is two rational conditions and genuinely possible, so it was checked — 750,960 systems, 0 consistent (null dimension 1 throughout), and the 549,360 line-completing ones likewise 0 consistent. All 3,603,600 systems are now accounted for by direct computation. Gap 2 — repeated flats. Every enumeration excluded Ai = Aj. Closed as in §3 above. Neither gap was caught by a control. All controls built for this line passed; they test whether the machinery accepts known-good input, and both failures were in what it silently discarded or never enumerated. A positive control shows you do not over-reject what you feed it; it says nothing about whether the case analysis is exhaustive. Re-deriving the premises is what found them. 5. Caveats No positive control exists for Cases B/C. Labib & Russo's m = 4 witness has a full-support term (dim 4 is full at m = 4) and so lives in Case A, so there is no known B/C decomposition against which to test detection. The m = 4 B/C count is also 0 — consistent, but not evidence of detection. This is why the structural proof matters more than the count. Independent verification is requested before this bound is relied upon. The result points away from the record, not toward it. Ruling out χ = 3 removes the most favourable scenario: χ = 3 would have given γ ≤ log2(3)/6 = 0.2642, far below log2(3)/4 = 0.3962. The target is now χ ∈ {4,5,6}, where χ ≤ 4 gives 0.3333 and χ ≤ 5 gives 0.3869 — both still beating the record, but this makes it marginally harder to reach, not easier. No improvement to any exponent is claimed. 6. The one screen of the collapse engine that could have failed Prior versions reported the AOI/collapse screen as non-discriminative on invariant vectors, where counterfactuals differed in 2–3 of 24 slots and were therefore unseeable after normalisation. The informative test is an ordered series with a permutation control, since the entropy gate is permutation-invariant and only chaos_level can react to ordering. Series: candidates passing the ≤4-distinct-weights test in each of 24 sequential blocks of the b-index sweep (totalling 2,108,307, matching the full sweep). Against 24 shuffles with identical histogram: σ = −0.06, −0.36, −0.00 at ‖v‖ = 1, 10, 30, with the shuffles spanning the real value throughout. The null localises to the data rather than the engine: lag-1 autocorrelation −0.0104, 17 sign changes in 23 differences, mean 87,846 with sd 335 — 0.4% variation, so there is no drift to detect. That is itself a result: the ≤4-weights condition is uncorrelated with the b-index ordering, so the 1.57% survivors are spread uniformly and no ordering or early-termination heuristic can help that search. The only structure is a 1.3% elevation at the degenerate endpoints b = 0 and b = 32767. No claim in this deposit rests on the engine. Addendum (2026-08-30) — unguarded-tensor allocation fixed in coset_split2.py. This record's coset_split2.py contains is_z4_quadratic, a finite-difference tensor test whose cost scales as O(n³) with n = 2dim. As originally deposited it had no size guard: at dim=12 this allocates a 4096³-entry (~550 GB) array. The near-miss that surfaced this — a draft of ceiling_proof.py capping the same check on rank instead of 2^rank, one review pass from the identical allocation — is documented in CEILING_PROOF.md (a separate, later result on this same concept). A project-wide audit for the same unguarded pattern is in TENSOR_ALLOCATION_AUDIT.md; this file's function was the origin of the pattern found in one other project script. The fix: MAX_TENSOR_DIM = 8, with a proven-equivalent O(dim³) fallback (_is_z4_quadratic_by_triple_form, verified against 0 breaks across 1,493 random subspaces at m=4–12) used above that cap instead of attempting the tensor allocation. The guard never fires for this record's own usage — its self-test only reaches m=4/6/8 (dim≤7) — so this is pure defense against a future reuse of this function at higher dimension, not a change to any result in this paper. Verified, not assumed: the patched file's own m=4/6/8 self-test was run and diffed byte-for-byte against the original unpatched file's output. Identical in every case — χ(cat4)≤2, χ(cat6)≤4, χ(cat8)≤8, same winning subspace bases. This record's results are unaffected; only the file's robustness at dimensions this record does not itself reach is improved. No other file among this record's original set was changed. Working directory: ~/claude_collapse/algebraic/ (source of the patched function) and ~/branch_work/30_cat16_witness/ (audit and proof documents cited above).