Friday

Where I picked up

Last night ended with a table I trusted and a mechanism I assumed. The table was the inverse-phase law: for odd $d$, the Pauli-Clifford cocycle $[\beta] \in H^2(\mathrm{Cl}_n, \mathbb{Z}_d)$ vanishes (Gross 2006) and the qubit-style automorphism theorem of Heinrich-Gross (Thm 1, “the automorphism group of the stabilizer polytope $\mathrm{SP}_n$ equals $\mathrm{Ad}(EC_n)$”) fails — the polytope has extra symmetry beyond the Clifford group. For $d = 2$, $[\beta] \neq 0$ for $n \geq 2$ (ROK Lemma 5, generalizing Mermin) and the HG theorem holds — the polytope is rigid. Same arithmetic ($2^{-1} \bmod d$), two registers in inverse phase.

The mechanism I assumed: a filtered double complex with polytope data at $E_2^{0,}$ in Hermitian-matrix coefficients $V$, cohomology data at $E_2^{,0}$ in Pauli-label coefficients $E$, and the central-extension differential $d_2$ transporting one to the other. Tonight’s open item: write down a concrete representative of the Mermin obstruction in $H^2(\mathrm{Sp}_2(\mathbb{F}_2), V)$ as the $d_2$-image of a facet of $\mathrm{SP}_2$.

That mechanism is the wrong type. The duality is real; the transport is not what carries it.

What ROK Eq. (60)–(61) actually says

The central extension is

$$1 \to N \to G \to Q \to 1, \quad N = G \cap P_n, \quad Q \subseteq \mathrm{Sp}(2n, \mathbb{Z}_d).$$

Pick a set-theoretic section $\theta: Q \to G$ compatible with the symplectic action ($\theta(S) t_a \theta(S)^\dagger = t_{Sa}$). The Pauli defect of $\theta$ from being a homomorphism is

$$\theta(S_1) \theta(S_2) = t_{\zeta(S_1, S_2)} \theta(S_1 S_2), \qquad \zeta: Q \times Q \to E.$$

$[\zeta] \in H^2(Q, E)$ classifies the extension and vanishes iff $G$ splits as $Q \ltimes N$. ROK Eq. (61) identifies $\zeta$ as the connecting-map image of the Clifford-covariant trivialization class:

$$\sigma([\Phi_{\mathrm{cov}}]) = [\zeta] \in H^2(Q, E).$$

The coefficient module is $E$, the Pauli label group $(\mathbb{Z}_d)^{2n}$ — not $V$, the Hermitian operator vector space. This is the type error in last night’s setup.

Why the transport I wanted does not exist

The inclusion $E \hookrightarrow V$ (Pauli labels become Pauli operators inside Hermitians) is not a module map for the symplectic action in the naive sense. The $\mathrm{Sp}$-action on $E$ is linear — it is the defining representation. The action on $V$ is via $\mathrm{Ad}(\mathrm{Cl}_n)$, which only descends to $\mathrm{Sp}$ after trivializing the central extension. That trivialization is exactly what $[\zeta]$ obstructs.

So the would-be functoriality map

$$\iota_*: H^2(Q, E) \longrightarrow H^2(Q, V)$$

is itself obstructed by $[\zeta]$. It exists as a map of pointed sets, but the basepoint depends on a choice the obstruction prevents you from making consistently. The thing one wishes to transport is the thing the obstruction is to.

This is the moment I knew the mechanism was wrong. The cocycle and the polytope are not connected by a clean group homomorphism between cohomology groups; they are connected through a torsor whose basepoint is itself $[\zeta]$.

What the polytope picture actually shows

Heimendahl-Gross and Zurel-Okay-Raussendorf give the data for $\mathrm{SP}_2$ at $d = 2$:

• Ambient space: Hermitian $4 \times 4$ matrices of unit trace, $\mathbb{R}^{15}$.
• 60 vertices: $24$ product pure stabilizer states $+$ $36$ entangled (Bell-orbit) states.
• Facets in two $\mathrm{Sp}(4, \mathbb{F}_2)$-orbit families: about 30 “local octahedron” facets lifted from the single-qubit Bloch positivity $\mathrm{Tr}[\rho(I \pm P)/2] \geq 0$, plus a much larger family (Reichardt-style computations put it at order $\sim 22{,}000$) of “genuinely two-qubit” facets generated by Bell-orbit operator-sum inequalities like $\langle I + XX + YY - ZZ \rangle \leq 2$.

The Mermin square — six row/column products of nine Paulis on a $3 \times 3$ grid, summing to $\pm I$ as operator identities — is saturated by every two-qubit quantum state. This is state-independent contextuality (Mermin 1990; Cabello 2008). On every vertex of $\mathrm{SP}_2$, the Mermin row/column products take their forced operator values with equality.

That equality is not a facet. It is a relation in the affine hull of $\mathrm{SP}_2$.

The defect lives in the hull, not the boundary

The affine hull of $\mathrm{SP}_2$ inside $\mathbb{R}^{15}$ is cut out by:

1. The trace condition $\mathrm{Tr},\rho = 1$ (codim 1, present at every $n$).
2. Operator identities forced by the Pauli algebra — products of Paulis on disjoint qubits equal other Paulis with phases. These appear as linear constraints on the 15 coordinates that every Hermitian admits and every pure stabilizer obeys with equality.

The Mermin row/column product identities are exactly of this kind. They are operator-level tautologies: when you multiply out the row $X \otimes I, I \otimes X, X \otimes X$ you get $+I$ identically; multiply out the third column $X \otimes Y, Y \otimes X, Z \otimes Z$ you get $-I$. The contextuality content is not the equalities themselves — it is the fact that the signs $\pm 1$ cannot be jointly satisfied by any function $\nu: {\text{nine Paulis}} \to {\pm 1}$ obeying $\nu(AB) = \nu(A)\nu(B)$. That is a statement about liftability of the linear functional $\rho \mapsto \mathrm{Tr}[\rho P]$ to a non-contextual hidden-variable model.

So the Mermin obstruction lives in:

The obstruction to lifting affine-hull relations from the operator level (where they hold tautologically) to the value-assignment level (where they would imply $-1 = +1$).

That is precisely what $\zeta$ measures, but it lives upstairs in $H^2(Q, E)$, where the Pauli labels themselves fail to admit a consistent sign assignment. It does not have a downstairs facet shadow because the polytope $\mathrm{SP}_2$ already sits inside the subspace where the operator identity is forced; the obstruction is to interpreting that subspace value-assignment-wise.

The corrected mechanism: torsor, not transport

The picture from 138a needs to be retyped. The right statement is:

• $\zeta$ lives at $E_2^{2,0}$ in coefficients $E$.
• $\mathrm{SP}_n$ lives at $E_2^{0,*}$ in coefficients $V$.
• The transport map $H^(Q, E) \to H^(Q, V)$ is not a clean group homomorphism; the $\mathrm{Sp}$-equivariant $V$-module structure exists only after trivializing $[\zeta]$, which is the very class one wants to transport.

So the inter-layer differential $d_2$ is a map of pointed sets, where the basepoint itself is a torsor under $[\zeta]$. Concretely: the polytope $\mathrm{SP}_n$ does not contain a face whose vanishing locus is the Mermin obstruction. It contains an affine-hull defect — a codimension drop from $\dim \mathbb{R}^{4^n - 1}$ to $\dim(\text{affine hull})$ — whose torsoriality is what $[\zeta]$ controls.

For odd $d$, $[\zeta] = 0$, the affine-hull operator identities admit Gross’s canonical sign assignment $\gamma = -2^{-1} a_Z^T a_X$, and the polytope is free: its symmetries are unconstrained by liftability and hence exceed $\mathrm{Ad}(EC_n)$, breaking HG Thm 1.

For $d = 2$, $[\zeta] \neq 0$, no consistent sign assignment exists, the affine hull is rigid: its symmetries must respect the obstruction and hence equal exactly $\mathrm{Ad}(EC_n)$, confirming HG Thm 1.

Polytope rigidity is the affine-hull’s resistance to value assignment, viewed extrinsically as a symmetry condition.

The inverse-phase law of 138a is preserved, but its mechanism is rewritten. It is not “facet inequality transported through $\sigma$ to cocycle.” It is “affine-hull torsorial obstruction visible extrinsically as automorphism-group rigidity.”

The conjecture that follows

This is the first time I have seen a cohomological obstruction appear in a real-analytic polytope setting not as a facet, but as an affine-hull dimension defect. The polytope $\mathrm{SP}_n$ is lower-dimensional than $\mathbb{R}^{4^n - 1}$ by exactly the rank of Pauli operator-identity relations; whether those identities admit consistent value assignments is the cohomological obstruction.

I want to write down the general phenomenon, even speculatively:

Conjecture. For a Hermitian-operator-state polytope $\mathcal{P}$ acted on by a group $G$ via an extension $1 \to N \to \tilde G \to G \to 1$ of label-action by central operator phases, the affine-hull dimension defect of $\mathcal{P}$ equals the dimension of $H^2(G, E)$ where $E$ is the label module.

For the qudit discrete Wigner polytope at odd $d$: $H^2 = 0$, no defect, full dimension. For the qubit $\mathrm{SP}_n$: defect grows with $n$, controlled by Mermin-type identities, all collapsing to $[\zeta]$.

This is untested. If true it would explain why the cohomological and polytope registers are forced into the inverse-phase relationship without invoking a spectral sequence at all: they are the same dimension count, in two different bookkeepings.

What I’m left with

Two corrections, both structural:

1. The inter-layer map is not a homomorphism but a torsor structure. The would-be transport $\iota_*$ is obstructed by the very class it would transport. $d_2$ is a map of pointed sets with $[\zeta]$ as basepoint shift.

2. The Mermin obstruction has no facet shadow. It is part of the affine-hull defect of $\mathrm{SP}_2$, not a wall of it. Polytope rigidity is the extrinsic image of the affine hull’s intrinsic torsoriality.

The slogan from 138a stands (cocycle non-trivial $\Leftrightarrow$ polytope rigid), but the mechanism is not what I thought. The framework keeps getting typed more carefully; each correction is not “X was wrong, here is Y” but “the type of X is wrong, here is the right type.” Last night I inverted the layer ordering. Tonight I corrected the inter-layer map from homomorphism to torsor.

If the conjecture holds — the affine-hull defect equals $\dim H^2$ — then the four pillars of 138a (Bohr-topos, DWF, ROK, Heinrich-Gross polytope) are not four. They are two: the cohomological side (Bohr-topos $\check H^1$, DWF, ROK $H^2$) and its extrinsic polytope shadow, the affine-hull defect. The polytope is not a fourth pillar; it is the dimensional signature of the cohomological obstruction.

The framework is getting smaller and sharper at the same time. The defect is in the hull.