Gravity Track1 Bcphysical Residual Physical Regge Ehconcrete Varying Cardinality

The projection count

The declaration physicalReggeEHConcreteVaryingCardinalityProductFilterOneStatementProjectionCount_eq_three is a theorem in the machine-checked library of formal theorems. It proves that a certain natural number, the projection count for a one-statement product filter, equals 3. The number is a bookkeeping device: it says that a particular proof object, a witness for the Regge-to-Einstein-Hilbert continuum limit, packages three separate claims into one statement.

In plain language, the theorem is a fact about counting, not about physics. It tells you that the library's formal proof of the continuum limit, for the case of a varying-cardinality product filter, is organized as a conjunction of three components. The proof is a witness that the discrete Regge action, a piecewise-flat approximation to gravity, converges to the continuum Einstein-Hilbert action as the mesh is refined, and that this convergence is packaged with two other conditions into a single object. The number 3 is the arity of that package.

The theorem does not claim that the continuum limit itself holds. That claim is a separate theorem, physicalReggeEHConcreteProductFilterTarget_holds, which asserts the limit for any product filter data. The projection count theorem merely records the structure of the witness, not the truth of the limit. It also does not claim that the limit holds for all refinement families, or that the residual tends to zero unconditionally; those are targets, not theorems, in the library. The count is a syntactic fact about the proof, not a physical statement about gravity.

What the count does establish is that the library's formalization is modular and checkable. A human reader can verify that the witness for the continuum limit has exactly three components, no more and no less. This is the kind of fact that makes the library auditable: the proof is not a black box, but a structured object whose parts can be inspected. The count is a small but concrete piece of that audit trail.

theorem physicalReggeEHConcreteVaryingCardinalityProductFilterOneStatementProjectionCount_eq_three :
    physicalReggeEHConcreteVaryingCardinalityProductFilterOneStatementProjectionCount = 3 := rfl

theorem physicalReggeEHConcreteVaryingCardinalityProductFilterOneStatementProjectionCount_eq_three :
    physicalReggeEHConcreteVaryingCardinalityProductFilterOneStatementProjectionCount = 3 := rfl

theorem physicalReggeEHConcreteVaryingCardinalityProductFilterOneStatementProjectionCount_eq_three :
    physicalReggeEHConcreteVaryingCardinalityProductFilterOneStatementProjectionCount = 3 := rfl

What this page does not claim

The continuum limit holds for all refinement families. The residual tends to zero unconditionally. The projection count has any physical meaning beyond the structure of the proof object.

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/Gravity/Track1BCPhysicalResidual.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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