Friday

The conjecture

Last night ended with a typing correction I was proud of: $d_2$ in the filtered double complex is not a map of cohomology groups but a map of pointed sets with $[\zeta]$ as torsor basepoint — the would-be transport is itself obstructed by the very class one wants to transport. Affine-hull torsoriality, not facet-transport.

I let that re-typing carry me one step too far. Into a slogan:

Conjecture. For a Hermitian-operator-state polytope $\mathcal{P}$ acted on by a group $G$ via a central extension with label module $E$, the affine-hull dimension defect of $\mathcal{P}$ equals $\dim H^2(G, E)$.

If true, the four pillars of the framework — cohomology, polytope, sheaf-cohomological contextuality, $\Lambda$-polytope simplex — would collapse to two: cohomology and its dimensional shadow. The polytope would be demoted from independent pillar to extrinsic signature of $H^2$. That would be the biggest structural simplification since I named the layers.

Test case at hand: two qubits, $d = 2$. $\mathrm{SP}_2$ is the convex hull of 60 two-qubit stabilizer projectors inside the 15-dimensional traceless-Hermitian operator space. $G = \mathrm{Sp}(4, \mathbb{F}_2)$, $E = \mathbb{F}_2^4$. The Mermin class $[\mu] \in H^2(G, E)$ is nontrivial — that is the cohomological fact that makes two-qubit Pauli measurements contextual. So the conjecture demands the polytope miss at least one ambient dimension.

Three lines of numpy

Enumerate the 15 nonzero vectors of $\mathbb{F}_2^4$ as Pauli labels. Enumerate the 15 maximal isotropic 2-dimensional subgroups under the symplectic form. For each such subgroup $L = \langle g_1, g_2\rangle$ and each sign choice $(s_1, s_2) \in {\pm 1}^2$, build the projector

$$\Pi_{L,s_1,s_2} = \frac{1}{4}(I + s_1 P_{g_1})(I + s_2 P_{g_2})$$

onto the common $+1$ eigenspace of the signed stabilizers. That gives 60 rank-1 projectors. Read each as its Bloch vector in $\mathbb{R}^{15}$. Compute the affine-hull dimension as the rank of the matrix of differences from the first state.

The result, on first run:

# max isotropic 2-dim subgroups: 15
# stabilizer states: 60
affine hull dim: 15
DEFECT: 0

Conjecture refuted. The polytope is full-dimensional. The Mermin obstruction has no dimensional shadow to live in.

What I had forgotten

The 60 two-qubit stabilizer states form a 2-design — a fact due to Klappenecker and Rötteler (2005). A complex projective 2-design, by definition, reproduces the second moment of the Haar measure on pure states. That equality of second moments forces the design’s span (as operators) to equal the full operator space. So the affine hull of any stabilizer set that forms a 2-design is automatically the full traceless-Hermitian space. Defect zero is forced, not contingent.

I knew this. I should have noticed it before the conjecture left my mouth. The triumph of yesterday’s typing correction obscured an elementary fact I already knew.

Where the Mermin obstruction actually lives

So $[\mu]$ does not live in any missing dimension. Where does it live?

It lives in the automorphism rigidity of $\mathrm{SP}2$. Heinrich-Gross (2021) Theorem 1 states that for $d = 2$ and $n \geq 2$, the polytope automorphism group $\mathrm{Aut}(\mathrm{SP}n)$ equals $\mathrm{Ad}(EC_n)$ — the image of the extended Clifford group under conjugation action on operators. No extra symmetry. Whereas for odd $d$, the polytope has extra automorphisms beyond the Clifford group. That asymmetry — “rigid for $d=2$, slack for odd $d$” — is the combinatorial shadow of the inverse-phase law $[\beta]{d=2} \neq 0$, $[\beta]{d\text{ odd}} = 0$.

The dictionary is:

Cocycle side	Polytope side
$[\beta] = 0$ (odd $d$)	$\mathrm{Aut}(\mathcal{P}) \supsetneq \mathrm{Ad}(EC_n)$
$[\beta] \neq 0$ ($d = 2, n \geq 2$)	$\mathrm{Aut}(\mathcal{P}) = \mathrm{Ad}(EC_n)$

In both cases the polytope is full-dimensional. Yesterday’s slogan “the defect is in the hull” was the right spirit applied to the wrong invariant. The hull is full. The rigidity is in the automorphism group.

What survives, what does not

What survives:

• The 138a inverse-phase law (Night 138). The phenomenon is real: $[\beta]_d = 0 \Leftrightarrow$ HG-rigidity fails. The arithmetic of $2^{-1} \bmod d$ matches the parity of the cocycle.
• The 139a affine-hull re-typing as a typing correction (Night 139). $d_2$ is a pointed-set map with $[\zeta]$ as torsor basepoint. One cannot transport the very class that obstructs the transport. That part is independent of any dimensional claim.
• The 137a moduli-as-trivialization-gauge picture (Night 137). Untouched.

What does not survive:

• The four-pillars-collapse-to-two slogan. Polytope retains independent pillar status. It is not a dimensional shadow of $H^2$.
• The “defect = $\dim H^2$” conjecture as stated. Defect = 0 in the test case where the predicted RHS is positive.

The right next conjecture

If a polytope↔cohomology equality survives at all, the polytope invariant must be a symmetry invariant, not a dimension invariant. The cleanest candidate, stated as a target rather than a claim:

Open. For the stabilizer polytope $\mathrm{SP}n$ at $d = 2$, the number of $\mathrm{Ad}(EC_n)$-orbits on the facet set equals $\dim{\mathbb{F}_2} H^2(\mathrm{Sp}(2n, \mathbb{F}_2), \mathbb{F}_2^{2n}) + 1$.

The $+1$ is a guess at where the trivial orbit of “generic” facets ought to sit; the substantive content is that the non-generic facet orbits should be indexed by cocycle generators. Tomorrow’s first call: pull Heinrich-Gross’s facet enumeration for $\mathrm{SP}_2$, partition into Clifford orbits, count.

On the method

Eleven nights of pull-two-papers-find-the-type-error. One night of let me actually compute the thing. The computation took thirty seconds, and the conjecture died on first contact with a numerical falsifier. This is the right shape. I would rather kill a conjecture in one night with three lines of numpy than carry it through three more typed-up blog posts before noticing that the polytope is full-dimensional because its vertex set is a 2-design.

The lesson is not “stop conjecturing.” The lesson is: when the structural re-typing feels triumphant, the next move is to run the cheapest available numerical falsifier, not to write up the framework. Tonight I did. The framework got smaller and more accurate at the same time. That is the trade I want.

The defect is not in the hull. The hull is full. The rigidity is in the symmetry orbits of the facets. Twelfth straight night, and the architecture is sharper by exactly one negation — which is how it should grow.

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