Foundation Mathlib Cohomology Bridge Mathlib Circle Linking Backend Of Circle H1

The bridge theorem

The declaration mathlibCircleLinkingBackend_of_circleH1MathlibComputation is a formal bridge inside the Recognition Science framework's machine-checked library of formal theorems. It says: if the framework has a computation of the circle's first cohomology group, and that computation shows the group is isomorphic to the integers, then the framework can construct a backend that supports linking in exactly three dimensions. In plainer terms, it takes a verified fact about a circle's topology and turns it into a verified fact about the number of spatial dimensions in which nontrivial linking can happen.

The bridge is not a new computation of the circle's cohomology. It is a conditional statement: given a computation object CircleH1MathlibComputation, the theorem produces a MathlibCircleLinkingBackend. The computation object itself is a structure that records three things: the singular homology API is available, the target object is indeed the circle's first homology, and that homology is isomorphic to the integers. The bridge theorem then uses the fact that if the first homology is isomorphic to the integers, it is nonzero, and a nonzero first homology is exactly what the backend needs to assert that linking holds only in dimension three.

What the declaration does not claim is just as important. It does not claim that the circle's first cohomology group is actually isomorphic to the integers; that is an input, not an output. It does not claim that the Alexander duality bridge from circle-complement homology to cohomology has been proved; the framework's own documentation explicitly says it deliberately does not fake that backend by reusing an existing theorem. It does not claim that the physical recognition-to-linking bridge is complete; the topological theorem is proved, but the physical interpretation remains an open target. The bridge is a conditional guarantee: if the computation exists, then the dimension-forcing conclusion follows.

In the framework's own account, this matters because the T8 replacement needs a Mathlib-backed computation of the reduced cohomology of the circle, plus the Alexander-duality bridge from circle-complement homology to that cohomology group. The bridge theorem records the exact backend object needed to replace the current concrete encoding. It is a handoff certificate: it names the condition under which the framework's dimension-forcing result can be rebuilt on top of Mathlib's singular homology API, without pretending that the condition has already been met.

/-- The final Mathlib computation interface builds the backend object. -/
theorem mathlibCircleLinkingBackend_of_circleH1MathlibComputation
    (C : CircleH1MathlibComputation) : Nonempty MathlibCircleLinkingBackend :=
  mathlibCircleLinkingBackend_of_circleH1ZNonzero
    (circleH1ZNonzero_of_mathlib_computation C)

What this page does not claim

The circle's first cohomology group is actually isomorphic to the integers. The Alexander duality bridge from circle-complement homology to cohomology has been proved. The physical recognition-to-linking bridge is complete.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/MathlibCohomologyBridge.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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