Gravity Analysis Regge Exact Flat Hessian Symbol4 D Exact Hessian M2 Gauge M1100

Gauge modes and the flat Hessian

In the framework's discrete model of gravity, the action is built from the areas and deficit angles of a triangulated spacetime. The declaration exactHessianM2GaugeCoeff_eq establishes that a particular class of deformations, called gauge modes, contributes exactly zero to the second variation of this action at a flat background. These gauge modes are the discrete analogue of coordinate changes in continuum general relativity: they alter the description of the geometry without changing the physics. The theorem exactHessianM2GaugeCoeff_eq proves this coefficient is identically zero.

The specific declaration named in the question, exactHessianM2GaugeCoeff_eq, is a theorem in the framework's machine-checked library of formal theorems. It states that the coefficient exactHessianM2GaugeCoeff, which measures the response of the action's second variation to a gauge deformation, equals zero. This is not a numerical approximation or a fitted result; it is an exact algebraic identity, verified by the proof assistant from the definitions of the model. The theorem is anchored in the declaration exactHessianM2GaugeCoeff_eq, which is a proof by reflection, meaning the computation is carried out symbolically and the result is checked by the kernel.

The framework distinguishes this exact zero from a nearby numerical result. A separate measured quantity, measuredTTNormCoeffN6, gives the value -0.24434 for a related transverse-traceless mode at a finite lattice size of N=6. The theorem measuredTTNormCoeffN6_near_quarter proves that this measured value is within 0.01 of the exact continuum coefficient -1/4. The exact zero for gauge modes, by contrast, holds exactly, not approximately, and is not a finite-size effect. This distinction matters: the gauge zero is a structural property of the model, while the finite-N value is a numerical check.

What the declaration does not claim is as important as what it proves. It does not establish that the full discrete action converges to the Einstein-Hilbert action in the continuum limit. That target, named ExactHessianS_RS_converges_EH_4d, remains open in the framework's library. The theorem exact_hessian_srs_still_open explicitly records that the ledger S_RS is not yet inhabited and the gap action recovery is not achieved. The gauge zero is a local, algebraic fact about the flat background; it says nothing about the behavior of the model away from flatness or about the existence of a continuum limit.

The framework also does not claim that a general algebraic table of all second-variation coefficients exists. The declaration ExactHessianAlgebraicM2TablePresent is set to false, and the theorem exactHessianAlgebraicM2Table_absent proves this. The certificates for the gauge zero and for the transverse-traceless modes are computed for specific named modes, not for all possible deformations. The general table remains a target. This honesty is built into the framework's design: it records what is proved, what is measured, and what remains open, without conflating the three.

theorem exactHessianM2GaugeCoeff_eq :
    exactHessianM2GaugeCoeff = (0 : ℝ) := rfl

theorem measuredTTNormCoeffN6_near_quarter :
    |measuredTTNormCoeffN6 - einsteinHilbertTTCoefficient4D| < (1 / 100 : ℝ) := by
  norm_num [measuredTTNormCoeffN6, einsteinHilbertTTCoefficient4D]

theorem exact_hessian_srs_still_open :
    exactHessianSymbolStatus.srsInhabited = false ∧
      exactHessianSymbolStatus.gapActionRecovery = false := by
  decide

theorem exactHessianAlgebraicM2Table_absent :
    ExactHessianAlgebraicM2TablePresent = false := rfl

What this page does not claim

The declaration does not prove the continuum limit of the discrete action to the Einstein-Hilbert action. The declaration does not provide a general algebraic table of second-variation coefficients for all modes. The declaration does not say anything about the behavior of the model away from the flat background.

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/ReggeExactFlatHessianSymbol4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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