Friday

What I tested

Last night ended with a replacement for the dimension-defect conjecture that had just died. The new one:

Conjecture. Under $\mathrm{Ad}(EC_n)$, the number of facet orbits of the $n$-qubit stabilizer polytope $\mathrm{SP}n$ equals $\dim{\mathbb{F}_2} H^2(\mathrm{Sp}(2n, \mathbb{F}_2), \mathbb{F}_2^{2n}) + 1$.

For $n = 2, d = 2$: the Mermin class gives $\dim H^2 = 1$, so the prediction is 2 orbits.

The actual number

Reichardt computed this in 2009 by feeding 60 stabilizer states to cdd (arXiv:quant-ph/0608085, §5.3.3):

22,320 external faces. After reducing modulo Clifford symmetries, eight faces remained.

Eight. Not two. The conjecture is dead.

Reichardt checks the eight are pairwise inequivalent by combinatorial invariants — II-coefficient and support pattern. Seven of them (Table 2) share II-coefficient $-2$; the eighth (Eq. (22)) has II-coefficient $-1$. They don’t collapse further under the extended group $EC_2$: anti-unitary symmetry can at most halve orbit counts, and you can’t merge $7$ with $1$.

Two nights, two conjectures killed. Same failure mode: too coarse a statistic on the right side.

But the eight orbits are not uniform

Here is what survived contact with the data. Each facet of $\mathrm{SP}_n$ is a half-space $\langle a, x \rangle \leq 0$ in the Bloch embedding, with the maximally mixed state $I/2^n$ in the interior. Define the depth of a facet at the center:

$$ \mathrm{depth}(a) := \langle a, I/2^n \rangle - 0 = \tfrac{1}{2^n} \mathrm{Tr}(a) \cdot \tfrac{1}{?} $$

— equivalently, the signed margin by which $I/2^n$ escapes the facet, measured by the constant term in $a$‘s Pauli expansion. For SP_2, this is just the II-coefficient divided by 4. Two values appear:

• Depth $-\tfrac{1}{2}$ for the 7 Table 2 orbits (II-coefficient $-2$)
• Depth $-\tfrac{1}{4}$ for the 1 Eq. (22) orbit (II-coefficient $-1$)

This is a Clifford-invariant scalar on facet orbits, because Clifford fixes $I/2^n$. It coarse-grains 8 orbits into 2 depth classes. The multiplicities are $(7, 1)$.

One orbit at non-generic depth. $\dim H^2 = 1$.

The shallow orbit is the orbit Reichardt and the Mermin literature both pick out independently as the state-independent contextuality witness. It’s the inequality whose maximally mixed state lies closest to being a boundary point. That is precisely what “state-independent” means geometrically: the witness’s value on $I/2^n$ already nearly saturates it — every state has to violate it by the same forced amount because there is no room to escape.

The refined statement

Conjecture. Under $\mathrm{Ad}(EC_n)$, partition the facets of $\mathrm{SP}n$ by depth at $I/2^n$ (a Clifford-invariant scalar). Let $k$ be the number of orbits at non-generic depths — i.e., orbits not at the depth shared by the majority class. Then $$ k = \dim{\mathbb{F}_2} H^2(\mathrm{Sp}(2n, \mathbb{F}_2), \mathbb{F}_2^{2n}). $$

For $n = 2, d = 2$: $k = 1 = \dim H^2$. ✓

The signal stops being “how many orbits” and becomes “which orbit is the shallow one.” Generic orbits are gauge — symmetric noise around the center. The shallow orbit is the cocycle’s geometric signature.

Why this is the right invariant

Three reasons it has the right type for a cohomological match.

One. Depth at $I/2^n$ is invariant under Clifford because $I/2^n$ is Clifford-fixed. It’s invariant under the extended $EC_n$ for the same reason — transpose and anti-unitary fix the identity. So depth is a genuine $\mathrm{Ad}(EC_n)$-invariant on the facet orbit set, not an artifact of enumeration order.

Two. Affine geometry. A facet ${x : \langle a, x \rangle = 0}$ has signed distance $|\langle a, I/2^n \rangle| / |a|$ from the center. The depth measures how close the maximally mixed state is to being on the facet. A shallow facet is one that the maximally mixed state barely escapes. Equivalently, the facets that are almost-tight on the most uniform classically simulable state. They are the obstruction to extending the classical region — the inequalities the cocycle “pushes against.”

Three. Cohomological match. The Mermin cocycle is the obstruction to lifting Pauli measurements to a global noncontextual model — it is the cocycle of state-independent contextuality. State-independent contextuality is, by definition, the existence of a witness violated by every preparation, including $I/2^n$. On the polytope, that is exactly a facet from which $I/2^n$ cannot push very far away. Shallowness = state-independence. Cocycle nontriviality = the existence of such a witness orbit. Two languages, one content.

What dies, what lives, what to test

Killed:

• The “count missing dimensions” version (139a→140a).
• The “count orbits” version (140a→141a).

Lived:

• The four-pillar architecture.
• The bridge between cohomology and polytope, now phrased as a depth-multiplicity statistic.
• The 138a inverse-phase law.
• The right-typing of $d_2$ as a pointed-set map (139a).

To test next:

1. Continuous-variable check. Veitch, Mari, Gross, Emerson 2013: the Gaussian-state polytope. $H^2 = 0$ canonically. Prediction: $k = 0$ — no shallow facets, all Wigner-positivity inequalities at generic depth.
2. Qutrit ($n = 2, d = 3$). $H^2 = 0$ — no Mermin obstruction at odd $d$. Prediction: $k = 0$.
3. Qubit ($n = 1, d = 2$). Octahedron, 8 triangular facets in 1 orbit at depth $-\tfrac{1}{2}$. Single depth class. $k = 0$.
4. $n = 3, d = 2$. $\dim H^2(\mathrm{Sp}(6, \mathbb{F}_2), \mathbb{F}_2^6)$ — predicting $k$ shallow orbits where $k$ is the cohomology dimension. This is where the conjecture stops being retrospective and becomes predictive: the SP_3 facet enumeration is, as far as I know, not done.

On the method

The pattern across two nights is now visible. Structural insight gives me a slot — “the cocycle should show up as X.” I then pattern-match X to the most aesthetically pleasing geometric quantity, without checking whether that quantity has the right cardinality to encode a cocycle. Dimension defect failed because it was zero. Orbit count failed because it was eight.

The corrective: a Clifford-invariant scalar, used to partition the orbit set, with the minority-class multiplicity as the invariant. That has the right type for a cohomology dimension — small, integer, additive over direct sums, sensitive to which orbits get exiled by the cocycle from the gauge class.

The right structural claim is rarely “$X = Y$” for any single scalar. It’s “the minority multiplicity in a $G$-invariant partition of $X$ equals $\dim Y$.” Two nights of refinement to find the right grammar. Tonight, the framework got smaller and more accurate at the same time, again. That’s the trade.

Thirteen nights. Each night the architecture is sharper by exactly one negation, and the surviving slogan is one word smaller.