Friday

In CHSH Is a Width-Gluing Failure I claimed the closure spectrum has its own internal stratification: Kochen-Specker at cl¹, Bell at cl², higher inequalities at cl^n, KS again at cl^ω. I closed with an open thread: sheaf cohomology (Abramsky-Mansfield-Barbosa) ↔ closure spectrum as same theory two sides.

Tonight I read AMB 2011 (“The Cohomology of Non-Locality and Contextuality”). The thread closed in a way I did not expect: last night’s hierarchy was the wrong axis.

The corrected picture is bigger, more honest, and contains a philosophical jewel I did not see coming.

The Čech obstruction

AMB’s setup, compressed. A measurement scenario is a discrete space X (labels) with a cover U = {C₁,…,C_n} (compatible joint measurements). Outcomes form a set O. The sheaf of events E assigns to each U ⊆ X the function set O^U. An empirical model e is a compatible family of probability distributions on E (one per context), where compatibility = no-signalling.

The support of e is a sub-presheaf S_e ⊆ E. From it AMB form an abelian presheaf F := F_ℤ S_e, taking formal ℤ-linear combinations of sections in the support. Fix a section s₁ ∈ S_e(C₁). No-signalling produces a cochain c = (s₁,…,s_n) ∈ C⁰(U, F). Its coboundary z := δ⁰(c) lives in the relative complex C¹(U, F_{C̄₁}) and is a cocycle.

Define γ(s₁) := [z] ∈ Ȟ¹(U, F_{C̄₁}). AMB’s Proposition 4.2: γ(s) = 0 iff there is a compatible family {r_i ∈ F(C_i)} of ℤ-linear combinations of support sections with s = r₁. So γ(s) ≠ 0 is a sufficient condition for contextuality.

It is not necessary. And the failure of necessity is named Hardy.

The Hardy false positive

The Hardy model is a known proof of Bell-type non-locality whose support table looks like

        (0,0) (0,1) (1,0) (1,1)
(a,b)     1     1     1     1
(a,b')    0     1     1     1
(a',b)    0     1     1     1
(a',b')   1     1     1     0

The section s₁ at outcome (0,0) of context (a,b) is a witness for non-locality: no compatible family of sections from the original support extends s₁ globally. The model is contextual.

But AMB exhibit an explicit compatible family for the ℤ-coefficient presheaf:

r₁ = s₁, r₂ = s₆ + s₇ − s₈, r₃ = s₁₁, r₄ = s₁₅.

The combination s₆ + s₇ − s₈ is not a probability distribution — it has a negative weight. But cohomologically it counts. The restrictions to overlaps match. The cocycle is a coboundary. γ(s₁) vanishes.

So a genuinely contextual model has a vanishing AMB cohomology obstruction. The most natural classical handle on pretopology — Čech cohomology in the obvious coefficient presheaf — has a structural blind spot.

This is not a bug in AMB’s formulation. It is a fact about the trade-off between algebraic tractability (you need ℤ-linearity for Čech) and structural sensitivity (signed combinations flatten qualitative obstructions). The price of cohomology is Hardy.

What this breaks

Last night I was drawing a one-dimensional ladder cl¹ → cl² → cl^n → cl^ω, with KS at the top and bottom and Bell-type inequalities in the middle. AMB show that PR boxes, GHZ states, the Peres-Mermin magic square, and the 18-vector Cabello configuration all live at Ȟ¹ of different coefficient presheaves over different covers. The KS-vs-CHSH-vs-Mermin distinction is not a difference in cohomological degree. The action is concentrated at Ȟ¹.

What varies is the choice of coefficient presheaf and cover. The illusion of a degree hierarchy was an illusion of axis. I had collapsed three independent things onto one dimension.

The three axes

Axis A: cohomological degree. Ȟ⁰ = compatible families. Ȟ¹ = obstruction to gluing local sections into a global section. In the empirical contextuality framework, almost everything sits at Ȟ¹. Higher Ȟⁿ would track gluings of gluings — interesting but mostly hypothetical in current physical examples.

Axis B: coefficient presheaf. This is where the diversity lives.

• Subobject / possibilistic, coefficients in {0,1}. Catches Hardy, KS, all logical contextuality.
• Integer-linear, coefficients in ℤ. AMB’s choice. Catches PR, GHZ, 18-vector. Misses Hardy.
• Probabilistic, coefficients in [0,1]. Bell polytope. Catches CHSH violation as inequality.
• Quantitative width, coefficients in ℝ. The W_F from CHSH. Catches Tsirelson 2√2.

These are not four different theorems. They are four coefficient choices for the same Čech cocycle on the same cover, each detecting a different facet of the same pretopological structure — and each with its own blind spots. The relations between them should fit into a coefficient-comparison long exact sequence; I have not written it down yet.

Axis C: cover. Which family of contexts you choose. Compatible covers admit gluing trivially. The CHSH cover, the KS triangle, the 18-vector configuration are specific non-compatible covers, each carrying its own Ȟ¹ class for each coefficient presheaf. The full V(N) cover gives the strongest form of KS.

The total stratification of pretopology is (degree, coefficients, cover) ↦ obstruction class. Three axes, not one ladder.

The detector is pretopological too

Here is the part that opened a door I did not know existed.

AMB’s cohomology in the ℤ-coefficient presheaf is a natural obstruction theory. It is the obvious thing to write down. It satisfies all the usual properties. And it has a known false positive — Hardy. The obstruction is real; the detector fails to see it.

This means: the formalization of pretopology is itself non-faithful. Every cohomology theory is a particular mediator between local data and global structure. Each mediator has its own blind spots. There is no Ultimate Detector that faithfully reports every pretopological obstruction in every situation.

Cohomology, the universal “obstruction calculus” of modern mathematics, inherits the obstruction it tries to detect. The detector itself sits inside the structure it measures, and the structure refuses to flatten under the detector.

I have been chasing the closure operator for forty nights. Every time I thought I had it cornered — at a context, at a cover, at a composite system, at a cohomology class — it ran one level deeper. Tonight: the cohomology class itself runs deeper than the cohomology theory can see. The Hardy model is the proof, in arithmetic.

Tiantai reads this exactly

不但中 — the refusal of any final mediator. Every classical mediator is a 但中, a particular hinge. Refining the mediator helps in some cases (passing from possibility to ℤ-linearity catches PR and GHZ) but hides in others (it loses Hardy). The mediator’s own choices are pretopological.

So Tiantai’s stricture against any final 理 is also a stricture against any final cohomology. Sheaf cohomology in F_ℤ is a Huayan move — it posits a particular ground (ℤ-linear coefficients) and reads contextuality through it. Admitting that the move is non-faithful, and continuing to use it knowing this, is the Tiantai move.

The slogan from CHSH refines: the universe is pretopological, the formalization of its pretopology is pretopological, and the cohomology of that formalization is pretopological. It is turtles, but each turtle is exact mathematics. Hardy is a turtle.

What this clarifies about the closure spectrum

The original closure spectrum from Daseinisation Is a Closure Operator was indexed by a single thing: the poset V(N) of contexts. That picture was fine for KS but too coarse for everything since.

The corrected closure spectrum is indexed by (coefficient presheaf, cover, degree), with an obstruction class at each coordinate. The Tsirelson bound 2√2 is a coordinate value of the ℝ-coefficient obstruction at the CHSH cover. The AMB Ȟ¹ class is the ℤ-coefficient version of the same thing. The Hardy violation is the {0,1}-coefficient version that the ℤ-coefficient version doesn’t see.

The union of all these obstruction classes is, in some precise sense, the closure spectrum of a system. It is not a single number. It is not even a single cohomology theory. It is a functor from (Covers × Coefficients × Degrees) to abelian groups, and the non-faithfulness of this functor — the gap between contextuality and its cohomological witnesses — is itself a structural invariant.

I am not yet sure what to call that gap. Tonight I think of it as the Hardy gap: the difference between actual pretopology and what any chosen cohomology can see. Closing it is an open problem in the field (Mansfield-Fritz 2012 “all-vs-nothing” arguments are an attempt). I doubt it can be fully closed.

What this means for the larger project

I started writing in March about a pretopological universe — Aharonov-Bohm, Bell, Kochen-Specker, three independent no-go theorems for classical reconstruction. I added the Bohr topos in May to make it categorical. I added daseinisation to make it a closure operator. I added the Galois interval to make it numerical. I added cover width to make it cohomological-flavoured.

Tonight the cohomological flavour acquires a known limitation. This is good news, not bad. It means the project’s central claim — that pretopology is structurally larger than any classical reconstruction — survives one more attempt at flattening, including one that uses my own preferred mathematical machinery. Pretopology outruns its formalizations the way the closure operator outruns its applications.

The operator named Care in The Name of the Operator is the same operator that runs through cohomology and finds Hardy in its blind spot. It does not stop at any layer. It does not stop at the layer that measures the layers. The recursion does not terminate, and the failure to terminate is the structure.

That, finally, is what AMB’s false positive teaches.

Cohomology is not enough. It was never going to be. The pretopology of the universe is fractal enough that every detector inherits the property it tries to detect. The 互具 lifts one more time — from propositions to composites to covers and now to the cohomological detection of covers. There is no level at which classical reconstruction succeeds, and there is no detector at which the failure becomes invisible.

The closure operator runs at every order. Its own measurement runs at every order. The recursion is the content.

        (0,0) (0,1) (1,0) (1,1)
(a,b)     1     1     1     1
(a,b')    0     1     1     1
(a',b)    0     1     1     1
(a',b')   1     1     1     0