Gravity Analysis Regge4 Dflat Second Variation Schlaefli Candidate Vanishes On A

A vanishing test direction

In the study of discrete gravity, one often examines how a geometric action changes when the shape of a simplex is perturbed. The declaration schlaefliCandidate_vanishes_on_axisTTPlus concerns a particular candidate expression, a quadratic form built from a four-by-four matrix, evaluated on a specific direction called axisTTPlus. The theorem proves that this candidate expression equals zero when the matrix is that special direction: schlaefliCandidateZeroMom axisTTPlus = 0. This is a concrete, machine-checked fact about a particular input, not a general statement about all directions.

The candidate expression is part of a larger program to connect a discrete, piecewise-linear model of gravity, known as Regge calculus, to the smooth Einstein-Hilbert action. In three dimensions, a known identity (Schläfli's differential identity) allows the second variation of the action to be reduced to a simpler form. The framework's library has proved this reduction in three dimensions. In four dimensions, the same reduction is partially complete: certain flat-space pieces are proved, but the full off-flat version remains open. The vanishing result on axisTTPlus is one of the proved flat-space pieces; it shows the candidate expression behaves correctly on that particular test direction.

What the declaration does not claim is equally important. It does not prove that the candidate expression vanishes on all directions, nor does it establish the full four-dimensional Schläfli reduction. The library explicitly marks the full off-flat pathwise Schläfli identity as absent, and the elevation of the nonlinear action to the candidate as open. The vanishing on axisTTPlus is a necessary check, not a sufficient one. It is a single data point in a larger verification effort, not the completion of that effort.

The practical consequence for a reader is a precise accounting of what is known. The result is a proved theorem about a specific test case. It does not close the gap between the discrete model and the smooth theory in four dimensions. That gap remains, and the library's status flags record it as such. This vanishing result is one step on a path, and the path is not finished.

theorem schlaefliCandidate_vanishes_on_axisTTPlus :
    schlaefliCandidateZeroMom axisTTPlus = 0 :=
  trueWeightZeroMomQuadratic_axisTTPlus

theorem regge4DFlatSecondVariationStatus_flags :
    regge4DFlatSecondVariationStatus.candidateIdentified = true ∧
      regge4DFlatSecondVariationStatus.candidateBlochFaceEvaluated = true ∧
        regge4DFlatSecondVariationStatus.freudenthal4FlatSchlaefliPresent =
          true ∧
          regge4DFlatSecondVariationStatus.freudenthal4FlatDirectionalPresent =
            true ∧
            regge4DFlatSecondVariationStatus.freudenthal4PathwiseSchlaefliPresent =
              false ∧
              regge4DFlatSecondVariationStatus.schlafliElevationOpen = true ∧
                regge4DFlatSecondVariationStatus.gapActionRecovery = false := by
  decide

What this page does not claim

The declaration does not prove the candidate vanishes on all directions. The declaration does not establish the full four-dimensional Schläfli reduction. The declaration does not close the gap between the discrete model and the smooth theory in four dimensions.

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/Analysis/Regge4DFlatSecondVariation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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