Gravity Analysis Regge Edge Ttattachment4 D Decoy Tt Not Gauge Discrete Lie Axis

The decoy test

In the framework's work on gravity, a key question is whether a small disturbance to the geometry can be explained away as a mere coordinate choice, a gauge artifact, rather than a real physical change. The declaration decoyTT_not_gaugeDiscreteLie_axis2 answers this for one specific, deliberately constructed case. It proves, in the machine-checked library of formal theorems, that a particular matrix, called decoyTT, cannot be written as a pure gauge effect along the second axis of a four-dimensional lattice. This is a negative result: it establishes that this particular decoy is not a disguise.

The proof works by examining the edge load, a number that measures how much a matrix stretches a specific lattice edge. For the decoy matrix on the second axis, this load equals 1. The definition of a pure gauge effect on that axis requires the load to equal 2 times the wave momentum on that axis times some vector component. The wave chosen for the test has zero momentum on the second axis. Therefore, a pure gauge effect would have to produce a load of zero, but the decoy produces a load of one. Since one does not equal zero, the decoy fails the test. The proof is a direct contradiction, verified by the computer.

This result is a small but concrete step in a larger campaign. The framework does not claim this proves anything about the full four-dimensional Einstein equations, nor that it recovers general relativity from the lattice model. It does not prove that the decoy matrix is physically meaningful or that it represents a real gravitational wave. The declaration is a precise, limited statement: this specific matrix, under this specific wave, on this specific axis, is not a pure gauge effect. It is a building block, not a conclusion.

The value of this result is in what it rules out. It shows that the framework's decomposition of a matrix into transverse, gauge, and residual parts is not vacuous. There exist matrices that cannot be absorbed into a gauge choice. This is a necessary check for the framework's program of building gravity from discrete lattice principles. The proof confirms that the distinction between real and apparent geometric change is meaningful within the model, at least in this one carefully constructed case.

theorem decoyTT_not_gaugeDiscreteLie_axis2 :
    ¬ IsGaugeDiscreteLieOnAxis axisWave decoyTT 2 := by
  rintro ⟨v, hv⟩
  have hm : axisWave 2 = 0 := by simp [axisWave]
  rw [decoyTT_edgeLoad_axis2, hm] at hv
  norm_num at hv

theorem decoyTT_edgeLoad_axis2 :
    edgeLoad decoyTT (axisDisp 2) = 1 := by
  simp [decoyTT, edgeLoad_axis, axisTTPlus]

theorem decoyTT_not_gaugeDiscreteLie_axis2 :
    ¬ IsGaugeDiscreteLieOnAxis axisWave decoyTT 2 := by
  rintro ⟨v, hv⟩
  have hm : axisWave 2 = 0 := by simp [axisWave]
  rw [decoyTT_edgeLoad_axis2, hm] at hv
  norm_num at hv

What this page does not claim

This does not prove the decoy matrix represents a physical gravitational wave. This does not prove the full four-dimensional Einstein equations are recovered from the lattice model. This does not prove that all matrices can be decomposed into transverse, gauge, and residual parts.

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/ReggeEdgeTTAttachment4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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