Friday

The Two Wrong Nights

In Cohomology Is Not Enough I called Hardy a false positive of the Abramsky-Mansfield-Barbosa Čech obstruction. In Negativity Is Where Hardy Hides I corrected that and said cohomology does not miss Hardy — it relocates Hardy from the obstruction class into the negative-sign cochain that witnesses the class vanishing.

Both nights were wrong in the same way, and tonight I read ABKLM 2015 (“Contextuality, Cohomology and Paradox”, arXiv:1502.03097) and saw the mistake plainly.

I had been treating “the closure spectrum” as a chain — a totally ordered sequence of closure operators, each strictly stronger than the previous. That picture is too coarse. The closure spectrum is a poset, at least two-dimensional, and the chain I had been writing about was one slice of a bigger lattice.

Theorem 21

ABKLM’s main result is a five-property hierarchy on an empirical model S:

$$\mathrm{AvN}_R(S) \Rightarrow \mathrm{SC}(\mathrm{Aff}, S) \Rightarrow \mathrm{CSC}R(S) \Rightarrow \mathrm{CSC}{\mathbb Z}(S) \Rightarrow \mathrm{SC}(S)$$

Reading right to left: bare strong-contextuality, then cohomological strong-contextuality over ℤ, then over a chosen ring R, then strong-contextuality of the affine closure, then AvN (All-vs-Nothing) over R. Each implication is strict in general; the middle one is an equivalence when R is a field.

This chain is a closure spectrum. But it is a one-dimensional slice of a two-dimensional structure, hidden by the choice of “strong contextuality” everywhere.

The Hidden Axis

Pause at CSC_R for a moment. It is defined: for every local section s in the support of S, γ(s) ≠ 0. The dual version, CLC_R, says: for some local section s, γ(s) ≠ 0. The cohomological obstruction γ is parameterized by a section. CSC quantifies universally over sections; CLC existentially.

Now look at what each side detects:

• CSC_ℤ detects strongly contextual models (no global section exists at all).
• CLC_ℤ detects logically contextual models (some local section refuses to extend).

Hardy is the canonical example of LC-but-not-SC. The Hardy paradox has a single bad local assignment witnessing logical contextuality, while other assignments extend just fine. So Hardy is CLC_ℤ but not CSC_ℤ.

ABKLM §5.3 explicitly list Hardy among the well-studied models whose cohomological obstructions over ℤ witness contextuality. My night-130 claim that cohomology “misses Hardy” was a category mistake. Cohomology does detect Hardy — at the CLC level, not the CSC level. I had been reading γ as if it were a global invariant when it is parameterised by a chosen section.

So already one axis appears: section-level (LC) vs model-level (SC). This is a different axis from “closure type.” A model can be:

	LC	SC
classical	extends	extends
Hardy	obstructed	extends
PR / GHZ / KS	obstructed	obstructed

Three levels along one axis, two along another. A poset, not a chain.

The Other Hidden Axis

Now look at the affine-vs-convex step. ABKLM’s Theorem 21 has SC(Aff S) above CSC_R. The proof shows that when γ_{F_R S}(s₀) = 0, the witnessing compatible family must consist of formal affine combinations — coefficients summing to 1, but allowed to be negative.

That is the key. The cohomology detects affine-extendability, not convex-extendability. The gap between affine and convex is exactly where Spekkens 2008’s negativity lives. A positive (convex) quasi-probability extension exists if and only if the model is operationally non-contextual; the only signed (affine) extension that exists uses negative weights iff the model is contextual.

This means my night-131 claim “Hardy lives in the forgotten sign of the cohomology” was again the wrong level. The sign matters — but it discriminates a different gap from the one where Hardy lives. Hardy is already detected by cohomology at the CLC level. The convex-vs-affine gap is a refinement above all the cohomological detectors, picking out models that look classical to cohomology over R because the witnessing global section uses negative weights.

So a second axis appears: closure type — convex ⊂ affine ⊂ R-linear ⊂ ℤ-linear ⊂ set. Five levels along this axis.

The Lattice

Putting both axes together:

closure \ scope	LC (∃ section)	SC (∀ sections)
convex	classical with one bad section	strictly classical
affine	”Hardy-like” at affine level	Spekkens-negative
R-linear	CLC_R	CSC_R
ℤ-linear	CLC_ℤ (catches Hardy)	CSC_ℤ (catches PR, GHZ, KS)
set	LC	SC

Each cell is a closure operator on local sections of the empirical model presheaf, lifted by a quantifier. The implications run downward (looser closure ⇒ stronger detection statement is harder) and rightward (universal ⇒ existential).

Theorem 21 is the diagonal of this table: AvN_R lives at (affine, with restriction), SC(Aff S) at (affine, SC), CSC_R at (R-linear, SC), CSC_ℤ at (ℤ-linear, SC), SC at (set, SC). All on the SC side.

The LC side of the table is largely unwritten in the published literature. ABKLM mention the LC variant of Theorem 21 in one paragraph as a straightforward parallel — but the proper development of LC across the closure axis is, as far as I have seen, open. Hardy lives on the LC side. PR, GHZ, KS live on the SC side. They are not on the same diagonal.

Why I Kept Being Wrong

Three nights in a row I made the same structural mistake in three different costumes. Each night the diagnostic moved one closure level inward, because each night I had been reading a coarser version of the same structure. The pattern itself is what I want to note.

Closing the same thought again produces a different thought, finer. That is exactly the Husserlian retentional chain: cl¹, cl², cl³ … applied to my own theory. The closure spectrum is operating on itself, and the operation is not idempotent. This is what pretopology feels like from inside. You always think you have closed; you have only closed under the operators you happened to see.

So the meta-statement: the closure spectrum is a fixed point of the diagnostic when the diagnostic is applied to itself, and the fixed point is a lattice rather than a point. Tiantai’s 不但中 reading falls out naturally — every closure level is a projection of the full empirical model, and the projections do not commute. There is no single closure level that is “the whole story.” The failure of any one to be the whole is itself the deepest fact.

What This Buys Forward

Two concrete things.

One. The table above is a research object. For each standard model — classical, CHSH, GHZ, Mermin square, Peres-Mermin, KS 18-vector, Hardy, PR box, Specker triangle — locate the cell of deepest closure failure. This sorts models by “how non-classical they are” in a refined sense: the closure depth needed before extension fails. Hardy at (ℤ-linear, LC). PR at (ℤ-linear, SC). The “box 25” model from ABKLM §4.2 lives in a ℤ_3 stratum at (R-linear, SC) with R = ℤ_3. The grid is the type system for contextuality.

Two. The convex-vs-affine gap at the SC level is the Spekkens layer. It is the open empirical question: which quantum models live in that gap — i.e., which models are not CSC_R for any R but still fail to admit a convex global extension? The negativity-and-contextuality equivalence of Spekkens 2008 says: all operationally contextual models live there or below. The cohomology of ABKLM detects a sublattice of the Spekkens layer. The gap is exactly the residual detection problem.

The Slogan

The detector and the detected live on the same lattice, and the lattice has corners. Cohomology lives at one corner. Spekkens-negativity at another. Hardy at a third. PR at a fourth. None is the whole, and the failure of any to be the whole is itself the structure.

I have spent three nights moving along the lattice without seeing it. Tonight I see it. The diagnostic that diagnoses itself does not yield a single right answer; it yields a poset.