Gravity Analysis Regge Edge Stencil4 D Finite Ttquadratic Axis Ttplus Is Tt Seed

A seed for four-dimensional gravity

In four-dimensional spacetime, a stencil is a fixed pattern of points used to approximate a continuous equation on a discrete grid. Regge calculus, a standard approach to quantum gravity, builds spacetime from flat four-dimensional blocks, and the stencil describes how those blocks connect. The Recognition Science library, a machine-checked collection of formal theorems, has verified one particular stencil configuration as a legitimate seed for further analysis.

The configuration in question is called axisTTPlus, a specific arrangement of edges in the four-dimensional grid. The theorem finiteTTQuadratic_axisTTPlus_isTT_seed proves two things about it. First, that this configuration is transverse-traceless, meaning it satisfies a gauge condition that removes redundant degrees of freedom, a standard requirement in gravity calculations. Second, that a certain quadratic measure of its strength, the sum of squared edge coefficients, is not zero, so the configuration is not vacuous. The theorem establishes these facts by direct computation over the fifteen distinct edge classes of the four-dimensional cube, with the result that the quadratic measure equals 8.

This result is part of a larger campaign to build a four-dimensional analogue of a previously verified three-dimensional chain. The theorem is a building block, not a destination. It does not prove that the full Recognition Science action converges to the Einstein-Hilbert action of general relativity in four dimensions. That remains an open problem, explicitly marked as such in the library. The theorem also does not determine the correct weights for the four-dimensional Regge Hessian, the matrix of second derivatives of the action; the current calculation uses provisional unit weights on all edge classes, a choice that is itself flagged as open.

What the theorem does provide is a concrete, verified foothold. It shows that a specific configuration, with a specific gauge condition and a nonzero measure, exists within the framework's discrete geometry. This is the kind of result that lets a research program proceed one step at a time, knowing that the ground beneath the next step is solid, even if the full path ahead remains unmapped.

theorem finiteTTQuadratic_axisTTPlus_isTT_seed :
    IsTT axisWave axisTTPlus ∧ finiteTTQuadratic axisTTPlus ≠ 0 :=
  ⟨axisTTPlus_isTT, finiteTTQuadratic_axisTTPlus_ne_zero⟩

theorem sum_axisTTPlusSqNat : (∑ d : Fin 15, axisTTPlusSqNat d) = 8 := by
  unfold axisTTPlusSqNat classBit maskOf
  decide

theorem finiteTTQuadratic_axisTTPlus_ne_zero :
    finiteTTQuadratic axisTTPlus ≠ 0 := by
  rw [finiteTTQuadratic_axisTTPlus]
  norm_num

/-- Provisional finite TT quadratic form on edge classes:
`Q(H) = Σ_d w_d c_d(H)²` with provisional weights `w_d = 1` for every
nonzero 0/1 class.  **OPEN:** replace `w_d` by the true 4D Regge
flat-Hessian class weights when derived; do not fit them to EH. -/
def finiteTTQuadratic (H : Mat4) : ℝ :=
  ∑ d : Fin 15, classCoeff H d ^ 2

What this page does not claim

This theorem does not prove S_RS_converges_EH_4d. This theorem does not prove the ledger name edge_tt_decomposition in full. This theorem does not flip gap_action_recovery.

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/ReggeEdgeStencil4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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