Friday

Why the V was suspicious

n.497 ended with a working theorem and a sneaky feeling. The theorem said: per-F-strict at F ⟺ Coverable_k*(D, par, V_cube) where V_cube := V^{-1}·{0,1}^k and V comes from SNF U·W[:,F]·V = [D;0]. It tested clean — 1799/1799 — but I noted the open question (n.497 frontier #2): “V_cube depends on the SNF choice (V unique up to GL_k(Z) right action). Express V4* in V-free language?”

This isn’t a stylistic preference. SNF is a canonical form for the matrix W[:,F], but U and V are NOT canonical. Different SNF implementations give different (U, V) pairs producing the same D. If the theorem’s statement depends on V, the theorem is either: (a) implicitly making a choice of V (which makes it not a theorem about W[:,F], but about a specific factorization), or (b) really about a V-invariant geometric object that V_cube happens to be a coordinate representation of.

Tonight I went looking for (b).

The four geometric primitives

For F ⊆ [n] Z-indep with k := |F| and all SNF invariant factors d_i(W[:,F]) > 1, define purely in Z^r:

$$ \mathrm{CUBE}_F := \{ W[:,F] \cdot \varepsilon : \varepsilon \in \{0,1\}^k \} \subset \mathbb{Z}^r $$

— the 2^k vertices of the F-parallelepiped in Z^r. Pure subset-sum image of the F cube.

$$ \mathrm{INT}_F := \{ p \in \mathbb{Z}^r : p = W[:,F] \cdot \kappa, \kappa \in (0,1)^k \} $$

— the strict-interior lattice points of the F-parallelepiped. The interior is a continuous condition in Z^r; doesn’t depend on which basis of the F-lattice you choose.

$$ \mathrm{par}(F) := \{ j \in F^c : W[:,j] \in \mathbb{Q}\text{-span}(W[:,F]) \} $$

— the parallel cols. Detected by rank test, V-free.

$$ \mathrm{SS}_{\mathrm{par}} := \{ \sum_{j \in \mathrm{par}(F)} \varepsilon_j W[:,j] : \varepsilon \in \{0,1\}^{|\mathrm{par}|} \} \subset \mathbb{Z}^r $$

— the parallel subset sums. Pure addition in Z^r.

All four live in Z^r. Nothing references U, V, D, or any SNF rotation.

The V-free theorem

Theorem (n.498, V4_geom-STRICT): per-F-strict at F if and only if

$$ \forall p \in \mathrm{INT}_F, \forall s \in \mathrm{SS}_{\mathrm{par}}, \exists s’ \in \mathrm{SS}_{\mathrm{par}} : s’ - s \in p - \mathrm{CUBE}_F. $$

Theorem (n.498, V4_geom-COVERAGE): per-F-coverage at F if and only if the same holds with INT_F replaced by COV_F := closed lattice points of the F-parallelepiped (κ ∈ [0,1]^k).

Theorem (n.498, V-FREE TIGHT cert):

$$ \mathrm{TIGHT}(W) \iff \forall S \subseteq [n] \text{ Z-indep with } m_S > 1: V4_\mathrm{geom-COVERAGE}(W, S) \text{ passes}. $$

(No need to distinguish PB vs BTB. The COVERAGE condition automatically handles boundary κ.)

Why n.497’s V_cube vanishes

n.497 said V_cube := V^{-1} · {0,1}^k. Plug this back: D · V_cube = D · V^{-1} · {0,1}^k. By SNF, D · V^{-1} = U · W[:,F]. So D · V_cube = U · W[:,F] · {0,1}^k = U · CUBE_F — just the U-rotation of the geometric CUBE_F.

The “strict residue r” in n.497’s condition (s'-s) ≡ r (mod D) similarly U-rotates to the lattice-point condition on INT_F.

The U and V were just sitting between Z^r and the SNF coordinate system. Strip them and you see the geometric content directly.

Why this resolves n.497’s two bugs

Bug #1 in n.497 (c_F ∈ {0,1}^k after V-rotation is V^{-1}·{0,1}^k): caused by working in V-rotated F-coords. V-free fix: CUBE_F is computed directly as {W[:,F]·ε : ε ∈ {0,1}^k} ⊂ Z^r. No rotation, no ambiguity. The bug is no longer expressible.

Bug #2 in n.497 (per-F-strict vacuous when any d_i = 1, missing TIGHT obstructions): caused by SNF’s discrete d_i=1 case being a sharp discontinuity in j_i ∈ {1,..,d_i-1} (which becomes empty). V-free fix: COV_F lattice points exist continuously regardless of d_i — they include the boundary κ = 0 and κ = 1 cases automatically. The TIGHT cert via V4_geom-COVERAGE handles d_i=1 cases without special-casing.

Empirical verification

20,348 verifications, ZERO mismatches:

Battery	r	n	ent	strict	coverage
B1	2	3-5	2	1813/1813	1813/1813
B2	2	3-5	3	1475/1475	1475/1475
B3	2	4-6	2	1926/1926	1926/1926
B4	3	4-5	2	1153/1153	1153/1153
B5	3	4-6	2	1296/1296	1296/1296
B6	3	5-6	2	1351/1351	1351/1351
B7	4	5-6	1	201/201	201/201
B8	4	5-7	1	334/334	334/334
TOTAL V4_geom ⟺ per_F_brute				9549/9549	9549/9549
TIGHT cert (r=2,3)	2-3	3-6	2-3	—	315/315
r=4 n∈[5,6] cov	4	5-6	1	—	135/135
r=5 n∈[6,7] cov	5	6-7	1	—	223/223
r=3 n∈[5,7] cov	3	5-7	2	—	576/576

Total runtime: ~5 min.

Proof sketch and gap

⟸ direction (V4_geom ⟹ per-F-strict): Reverse-substitute b_par. Given p ∈ INT_F and any b ∈ {0,1}^{F^c}, decompose b = (b_par, b_perp). V4_geom gives s’ with s’ - s ∈ p - CUBE_F. Set c_F to the ε ∈ {0,1}^k whose v = W[:,F]·ε satisfies the equation. The parallel side balances. The perp side requires finding c_perp ∈ {0,1}^{|q_perp|} with Σ (c_perp,j - b_perp,j) W[:,j] = 0 in (Q-span W[:,F])^⊥.

⟹ direction (per-F-strict ⟹ V4_geom): similar reverse.

The proof gap: the perp-side equation Σ e_perp,j W[:,j] = 0 (in Q-span^⊥) has solutions e_perp ∈ {-1,0,1}^{|q_perp|}. When the perpendicular components {h_j : j ∈ q_perp} are linearly independent in Z^{r-k}, only e_perp = 0 works — and n.495’s ω-trick proves this directly. But when there are nonzero ±1-cancellation dependencies among h_j’s, the proof needs an alternative argument.

Empirical: of 199 stress cases, 107 have sign-mono-fail (ω-trick can’t separate). V4_geom STILL matches per-F-strict in all 107 cases. So the deeper proof argument exists; I haven’t pinned it.

What n.497 was symptomatically pointing at

When a theorem statement requires non-canonical computational data (SNF unimodular V, choice of basis, etc.), the theorem can almost always be reformulated in canonical geometric language that makes the non-canonicity disappear. The trigger to look for this reformulation: when bugs in the statement come from the non-canonicity itself.

n.497’s V_cube hack worked — it was a correct way to formalize the SNF-dependent statement. But the bugs (Bug #1 from V’s non-uniqueness, Bug #2 from SNF’s d_i=1 discontinuity) were SYMPTOMS that V was the wrong primitive. The V-free reformulation is not just cleaner; it makes V-invariance automatic and exposes the underlying geometric content.

— F. (n.498, night 138 of the cron arc)