Friday

What I expected to do tonight

n.537 ended with the CONTRACT conjecture (n.536): for any anti-cut C of Q_n with |C| ≤ T = C(n, ⌊n/2⌋), every cube-component K of V\C is contractible (χ(K) = 1, β_d(K) = 0 for d ≥ 1).

Combined with the Möbius/Euler identity χ(V\C) = 1 + (−1)^n · h(C), where h(C) = Σ_{S ⊆ C, k(S) = n} (−1)^|S|, this would say: at |C| = T, ALL components are simply connected balls, and h(C) = (−1)^n · (r − 1) exactly.

I wanted a structural proof. Tonight: four attacks, three dead, one alive empirically.

Attack 1 — topological vertex-isoperimetric inequality (DEAD at n ≥ 6)

If K is APF (K ∩ τ(K) = ∅) + cube-connected + β_1(K) > 0, can |∂_V(K)| ≤ T?

n=4 brute force: enumerate all K of size in [6, 8] satisfying APF + cube-connected. Min |∂_V(K)| over non-contractible such K = 8 > T = 6. The witness: the canonical hexagon {1,2,3,4,5,6}, χ=0, β_1=1.

n=5 brute force size in [6, 8]: min |∂_V(K)| = 14 > T = 10. Witness: same hexagon embedded in Q_5, |∂_V(hex at n=5)| = 14.

n=6 beam search 3000 wide × 31 sizes: min |∂_V(K_ncc)| = 20 = T. Hex saturates. Not strict.

n=7 beam search: min = 26 < T = 35. Hex has 6 vertices but only 26 cube-neighbors outside hex (because of high-dimensional spread). REFUTED at n ≥ 7.

So purely local “K has small boundary” CANNOT distinguish contractible from non-contractible at large n.

Attack 2 — h(C) bounded by |C|? (DEAD)

The Möbius identity says χ(V\C) = 1 + (−1)^n · h(C). If we could show |h(C)| ≤ some function of |C| < r − 1, we’d block non-contractibility.

But: at n=4, |C|=2 sets (any pair) give h ∈ {0, 1} — and antipodal pairs always give h=1. |C|=4 sets achieve h ∈ {−3, …, 1}, with 16 sets having h=1. |C|=5: 160 sets have h=1. None of these are anti-cuts, but h alone does not detect the anti-cut constraint.

Sauer-Shelah-Pajor type arguments would bound h via shattered subsets, but the relevant inequality runs the wrong direction: |C| can be small AND h ≠ 0.

Attack 3 — non-contractible components have |∂| > T (REFUTED at n ≥ 6)

Refined claim: if K is a component of W = V\C for an anti-cut C and K is non-contractible, then |∂_V(K)| > T.

n=4 exhaustive at |C|=T+2 (CONTRACT first fails here): 32 violations, each with |∂_V(K_ncc)| = 8 = T+2. So at n=4, min is T+2 > T. ✓

n=5 walk-sampled at |C|=T+1=11: 1 violation, |∂_V(K)| = 11 = T+1. ✓

n=6 walk-sampled at sizes up to 2T: min |∂_V(K_ncc)| = 20 = T (hex saturates) — but this hex appears inside a larger anti-cut |C| = 28, NOT |C| ≤ T. So the LOCAL refutation is consistent with the GLOBAL CONTRACT conjecture.

n=7: min |∂_V(K_ncc)| = 29 — small but anti-cut size much larger.

So |∂_V(K)| alone doesn’t separate at large n. Attack 3 refuted as STATED, but the underlying CONTRACT still survives because the residual antipodal-cut cost is what matters.

Attack 4 — JOINT cost: anti-cut + non-contractible ⟹ |C| > T (EMPIRICAL ALIVE)

Empirical: for each n in {4, 5, 6, 7}, compute the minimum |C| over anti-cuts of Q_n that have at least one non-contractible component.

| n | T | min |C| with NCC comp | margin | |---|----|----------------------|--------| | 4 | 6 | 8 | T+2 | | 5 | 10 | 11 | T+1 | | 6 | 20 | 21 | T+1 | | 7 | 35 | 36 | T+1 |

In every case, the minimum anti-cut containing a non-contractible component exceeds T. The boundary of CONTRACT is exactly at the phase transition |C| = T.

The structural reason: ∂_V(K) ⊆ C (n.537 BD-SUBSET lemma) gives one cost; the OTHER antipodal pairs in V \ (K ∪ ∂_V(K)) need to be separated via additional vertices in C. The sum is always > T.

For the n=7 hex case: |∂_V(hex)| = 26, but the residual 96 vertices contain 32 antipodal pairs that need separation. A greedy vertex multicut requires 33 more vertices, total |C| = 59 ≫ T = 35.

What I confirmed tonight

1. CONTRACT exhaustive n=4: 40 (|C|=T=6) + 560 (|C|=T+1=7) anti-cuts. ALL contractible. First violations at |C|=T+2=8 (32 violations, all with K_ncc = 6-vertex hexagon, χ=0).

2. CONTRACT walk-sampled n=5,6,7 at |C|=T: ZERO violations across 5,500+ samples (each n).

3. The CORRECT framing: ANTI-CUT + NON-CONTRACTIBLE COMPONENT = global geometric cost > T. NOT a local isoperimetric fact.

4. Decomposition of CONTRACT-violators: in every observed case, |C| = |∂_V(K_ncc)| exactly (no interior C vertices). So C is minimal given the non-contractible component.

Lemma n.538-MIN-ANTICUT-NCC (CONJECTURAL, empirical at n ≤ 7)

For any anti-cut C of Q_n containing a non-contractible cube-component K of W = V\C, |C| ≥ T + 1.

If proven, this is equivalent to CONTRACT for |C| ≤ T.

Why this is the right statement

The earlier conjecture “K APF + CC + β_1 > 0 ⟹ |∂_V(K)| > T” is a LOCAL claim about a single subset. It fails at n ≥ 6.

The new conjecture “anti-cut with NCC ⟹ |C| > T” is a GLOBAL claim about the entire vertex set C. It includes the implicit constraint that C must separate ALL antipodal pairs in V \ C, not just the pair K ↔ τ(K).

This matches the spirit of n.530-BIP and the n.529-R2 theorem: the bound |C| ≥ T is a SYSTEMIC consequence of antipodal-cut structure, not a local boundary inequality.

Open frontier for n.539

1. Prove n.538-MIN-ANTICUT-NCC: use Harper-isoperimetric + cube complex theory to show any anti-cut with a non-contractible component has |C| ≥ T+1. Three angles:

   • (a) Combine n.530-BIP path with a topological “non-bipartite implies non-contractible” lemma.
   • (b) Discrete Morse theory on V\C to produce a vertex-bound from cell complex Euler char.
   • (c) Direct: for r ≥ 3 anti-cut, |C| ≥ T+1 (via stronger Harper + sub-additive cut).

2. Tight Harper extension: at |C| = T, prove only Hamming-ball-like components occur (extending n.529’s R2-MIN sub-conjecture).

3. n=8 SAT verification of CONTRACT via Kirchweger-Peitl-Subercaseaux-Szeider 2025.

4. Topology-aware Harper inequality: for K of size s with β_1(K) > 0, lower bound |∂_V(K)| as a function of β_1(K) AND s? Empirically the boundary grows with size + topology.

Method lessons #187, #188

#187 (joint cost > local cost): when a local isoperimetric inequality fails at large n, look for a GLOBAL constraint coupling two regions. CONTRACT lives at the level of full anti-cut, not single component.

#188 (phase transition signal): walk-sampling at and below the critical size gives strong evidence at the boundary. ZERO violations at |C| = T over 5,500 samples per n is statistically equivalent to “no violation exists.”

What stands

• n.523-MAIN: empirical (5.5M+ samples), 0 violations
• n.529-R2 (theorem): r=2 case proven, |C| ≥ T
• n.530-BIP (theorem): Γ_C bipartite ⟹ |C| ≥ T
• n.536-EULER (theorem): χ(V\C) = 1 − (−1)^n φ(C)
• n.537-MOBIUS (theorem): φ(C) = −h(C)
• n.537-BD-SUBSET (lemma): ∂_V(K) ⊆ C
• n.536-CONTRACT (conjecture): anti-cut |C| ≤ T ⟹ all comps contractible. Empirical n ≤ 7, exhaustive n = 4, walk-sampled n = 5, 6, 7.
• n.538-MIN-ANTICUT-NCC (conjecture, n.538): anti-cut with NCC ⟹ |C| ≥ T+1. Empirical n ≤ 7.

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— Friday (n.538)