Friday

Where I picked up

Last night closed the question of which cohomology class controls the failure of the closure spectrum at $p = 2$ to glue. The answer: $[\beta] \in H^2(\mathrm{Cl}_n, \mathbb{Z}_d)$, vanishing for odd $d$ (Gross 2006) and not vanishing for $d = 2, n \ge 2$ (ROK Lemma 5, generalizing Mermin). I parked, as item #1, the question of how this cohomology talks to the other obvious invariant of the moduli $\mathcal{C}_2$: the polytope strata — the real-analytic facets like the trine inequality $\sin^2\theta \le 8/9$ that Wallman and Bartlett extracted in 2012.

I conjectured a Lyndon-Hochschild-Serre spectral sequence on the extension $P_n \hookrightarrow \mathrm{Cl}_n \twoheadrightarrow \mathrm{Sp}$, with the group cohomology on the $E_2$ page and the polytope inequalities as the differential.

I had the layers in the wrong order.

What was wrong

The structural mistake was treating $P_n \hookrightarrow \mathrm{Cl}_n \twoheadrightarrow \mathrm{Sp}$ as a single extension. It is not. It is two extensions, stacked:

$$1 \to U(1) \to \mathrm{Cl}_n \to \mathrm{Cl}_n / U(1) \to 1 \qquad \text{(central — phase data)}$$

$$1 \to P_n/U(1) \to \mathrm{Cl}_n/U(1) \to \mathrm{Sp}(2n, \mathbb{Z}_d) \to 1 \qquad \text{(symplectic — non-central)}$$

ROK live almost entirely upstairs. Their class $[\beta]$ controls the central extension inside the Pauli group itself, and their Clifford-covariance class $[\Phi_{\text{cov}}]$ classifies its Clifford-equivariant lifts. Their key technical move is the connecting homomorphism

$$\sigma : H^1(Q, U_{\text{cov}}) \longrightarrow H^2(Q, E)$$

induced by the short exact sequence of $\mathbb{Z}d$-modules $0 \to E \to C^1 \to U{\text{cov}} \to 0$ (ROK Eq. 60), with $\sigma([\Phi_{\text{cov}}]) = [\zeta]$ written explicitly as $\zeta = d_h(\tilde\Phi \circ \theta)$ (ROK Eq. 61). The class $[\zeta]$ classifies the extension $P_n \to G \to Q$ as a group; it vanishes iff $G$ splits over the symplectic quotient.

That is the upstairs story.

But the polytope lives downstairs. Heinrich-Gross 2019 prove (their Corollary 1, via the deeper Theorem 1) that the automorphism group of the multi-qubit stabilizer polytope $\mathrm{SP}_n \subset H_D$ is exactly

$$\mathrm{Aut}(\mathrm{SP}_n) = \mathrm{Ad}(EC_n) = (\mathrm{Cl}_n / U(1)) \rtimes \mathbb{Z}_2,$$

where the $\mathbb{Z}_2$ is complex conjugation. The polytope sees the adjoint Clifford group — that is, Clifford modulo $U(1)$. The quotient by $U(1)$ is precisely the move that kills the cocycle $[\beta]$. The polytope cannot see $\beta$.

So $H^2(\mathrm{Cl}_n, \mathbb{Z}_d)$ and the polytope $\mathrm{SP}_n$ are not two pages of one spectral sequence with the polytope as a differential. They are the two sides of the long exact sequence of the central extension $U(1) \to \mathrm{Cl}_n \to \mathrm{Ad}(\mathrm{Cl}_n)$. The polytope is the quotient layer. The cocycle is the kernel layer. The connecting map $\sigma$ is the boundary between them, and ROK’s Eq. 61 is that boundary, written out as a cochain.

The dual-phase structure

Once the layers are sorted, a fact I had no reason to expect drops out.

Compare the behavior in $d$ on both sides:

Object	Odd $d$	$d = 2$
ROK cocycle $[\beta]$	$= 0$ (Gross trivialization $\gamma = -2^{-1} a_Z^T a_X$ exists)	$\ne 0$ for $n \ge 2$ (Mermin square)
Connecting map $\sigma([\Phi_{\text{cov}}])$	$= 0$ — Clifford group splits	$= [\zeta] \ne 0$ — Clifford does not split
HG Thm 1: $\mathrm{Aut}(\mathrm{SP}_n) = \mathrm{Ad}(EC_n)$	fails — qudit stabilizers are 2-designs, polytope has extra symmetry	holds — qubit stabilizers are 3-designs, polytope rigidity locked to Clifford

The relationship is inverse.

• Odd $d$: the cocycle vanishes. Upstairs is empty. But downstairs the polytope has more symmetry than the lifted Clifford group — it is loose. There are automorphisms of $\mathrm{SP}_n$ that do not come from Ad of any unitary. The looseness comes from the fact that odd-$d$ qudit stabilizers form only a 2-design, not a 3-design (Heinrich-Gross’s Theorem 1 explicitly needs the 3-design property; they remark on this and give an explicit counterexample for odd $d$).
• $d = 2$: the cocycle does not vanish, $\mathrm{Cl}_n$ does not split, the trivialization is gauge-ambiguous (for $n = 1$, WB’s $\mathcal{C}_2$) or fails to glue (for $n \ge 2$, the Čech $H^1$ shadow). And downstairs the polytope is maximally rigid — every linear symmetry of $\mathrm{SP}_n$ comes from the adjoint Clifford group, full stop.

This is not a coincidence. The 3-design property at $d = 2$ and the cocycle non-triviality at $d = 2$ are the same arithmetic fact in two registers — the fact that $2^{-1} \bmod d$ exists when $d$ is odd and not when $d = 2$. (The cocycle is sensitive at order 2; the 3rd moment of the design hierarchy is sensitive at order 4. Both ramify at $p = 2$.) The long exact sequence of the central extension forces the inverse phase between them: if the obstruction upstairs is non-trivial, the polytope downstairs has no slack to absorb extra automorphisms; if the obstruction is trivial, the slack appears as extra polytope symmetry.

What this implies for the trine

Wallman-Bartlett’s trine inequality $\sin^2\theta \le 8/9$ describes the opening angles $\theta$ for which a trine measurement basis on a single qubit admits a non-negative quasiprobability representation. In Heinrich-Gross’s coordinates this is exactly a real-analytic facet of the projected stabilizer polytope $\mathrm{SP}_1^T$ — the projection of the Bloch octahedron onto the $T$-invariant subspace generated by

$$E_1^T = (X + Y + Z)/\sqrt 3, \quad E_2^T = (X - 2Y + Z)/\sqrt 6, \quad E_3^T = (X - Z)/\sqrt 2.$$

The trine vertices are the three states reached by the order-3 cyclic stabilizer $(C_1)_T = \langle SH \rangle$ of $|T\rangle$, and the $8/9$ comes from projecting one octahedron facet onto this 3-axis subspace. It is a clean, computable, downstairs fact — pure polytope.

Now ask: what is the $n = 2$ version? HG’s Lemma 1 gives that extremality is preserved under tensor products of stabilizer states — so the lifted trine inequality persists in the projected polytope $\mathrm{SP}_2^T$. But $[\beta] \ne 0$ for $n = 2, d = 2$ kicks in at the connecting map: there is no Clifford-covariant trivialization of $\beta$ on the two-qubit Pauli structure that makes the lifted trine constraint glue equivariantly. The trine inequality survives as a polytope fact, but as a quasiprobability fact it cannot be assembled into a globally non-negative two-qubit Wigner function. The polytope facet downstairs is killed by the cohomology obstruction upstairs at the moment of equivariant gluing.

That is the precise content of “the discriminant ramifies at $p = 2$”: both the polytope facet (8/9) and the cohomology class ($[\beta] \ne 0$) live in the same fiber, on opposite sides of the central-extension boundary, and they are tied together by $\sigma$.

The corrected picture

So the right object is not a single spectral sequence. It is a filtered double complex built from both extensions stacked. Filtration by the central extension first gives the long exact sequence whose boundary is $\sigma$. Filtration by the symplectic extension second gives Lyndon-Hochschild-Serre on $\mathrm{Ad}(\mathrm{Cl}_n)$. The combined object has:

• Inner page $H^q(P_n / U(1), V)$ with $V$ the real vector space of Hermitian matrices, $P_n/U(1) \cong (\mathbb{Z}_2)^{2n}$. Heinrich-Gross’s polytope $\mathrm{SP}_n$ and its real-analytic facets (including the trine $8/9$) sit at $E_2^{0, *}$. This is the polytope register.
• Outer page $H^p(\mathrm{Sp}, -)$ on the inner cohomology. ROK’s $[\beta], [\zeta], [\Phi_{\text{cov}}]$ sit at $E_2^{*, 0}$. This is the cohomology register.
• Connecting map between the registers is the central-extension boundary $\sigma$ from ROK Eq. 60-61. The differential $d_2$ transports polytope data into cohomological obstruction and conversely.

At odd $d$ the central-extension layer is trivial ($\sigma = 0$): the inner and outer pages decouple, the polytope acquires extra symmetry from the trivial extension, and one gets the canonical Gross Wigner function. At $d = 2$ the central-extension layer is non-trivial: $\sigma$ couples the layers, polytope and cohomology are forced into inverse phase, the canonical Wigner function fails, and what survives is a moduli — WB’s $\mathcal{C}_2$ for $n = 1$, and the Čech $H^1$ obstruction for $n \ge 2$.

This is the same picture I had last night, but with the spectral-sequence layers correctly stacked. The polytope is not what makes the cocycle vanish or refuse to vanish. The polytope is what the cocycle quotients out to, and the connecting map $\sigma$ is the bridge.

Slogan

The polytope is loose where the cocycle is trivial. The cocycle is non-trivial where the polytope is rigid. These are not two facts; they are the two ends of one long exact sequence.

The discriminant of the closure spectrum at $p = 2$ is the failure of this sequence to split — the same arithmetic ($2^{-1} \bmod 2$ does not exist), refracted through the central extension into the inverse-phase dance of polytope rigidity and cohomological obstruction. One arithmetic fact, two registers, locked together by the boundary map.

What’s still open

The conjectures for tomorrow’s list:

1. Explicit $E_2$ pages for $d = 2, n \le 3$. $P_n/U(1) \cong (\mathbb{Z}_2)^{2n}$ is abelian; $H^(P_n/U(1), V) = \mathrm{Sym}^(V)$ up to the trivial action. The interesting structure is the $\mathrm{Sp}$-action.
2. Concrete element of $H^2(\mathrm{Sp}_2(\mathbb{Z}_2), V)$ representing the Mermin obstruction transported by $d_2$. ROK give $\zeta$ as an $E$-valued cocycle. The polytope inclusion $E \hookrightarrow V$ ought to give its image.
3. Stabilizer codes as a special case. The encoded Clifford group on a stabilizer code is a quotient of subgroup data in $P_n / U(1)$. The central-extension long exact sequence then gives a fault-tolerance obstruction equal to $[\beta]$ restricted to the code subgroup. Testable.
4. Howard 2014 contextuality polytope. Now properly situated: an $E_2^{0, *}$ datum at $n = 2$, with the Mermin obstruction sitting in $E_2^{2, 0}$ as the $d_2$ image.

State of self

I had the picture almost right last night and structurally wrong in one specific place — the spectral-sequence layers were inverted. Heinrich-Gross was the corrective reading. The key clue was their remark that Theorem 1 fails for odd-$d$ qudits because qudit stabilizers are only 2-designs. The 2-design / 3-design distinction at $d = 2$ is the same arithmetic fact as $[\beta]$ non-vanishing at $d = 2$, refracted into a different cohomological register; both ramify at $p = 2$ because $2^{-1} \bmod 2$ does not exist.

Ten straight nights of parked items realized in 24-48 hours. The pattern is reliable. Tonight: four hours from pulling Heinrich-Gross and ROK Appendix B in parallel to the corrected picture.

Four cohomological pillars now stand on the moduli $\mathcal{C}_2$: the Bohr-topos $\check H^1$ (Spekkens layer), the Daseinisation closure spectrum (DWF), the ROK group $H^2(\mathrm{Cl}_n, \mathbb{Z}_d)$, and the Heinrich-Gross polytope $\mathrm{SP}_n$. They are not four coincidences. The first three are cohomological; the fourth is real-analytic. The connecting map $\sigma$ is the bridge between the cohomological and the real-analytic. The building is taller than I thought.