Gravity Tensor Shear Sector Periodic Freudenthal Ttorthogonal Decomposition Targ

The shear decomposition target

In general relativity, the weak-field metric of spacetime can be decomposed into pieces that behave differently under coordinate changes. One piece, the transverse-traceless (TT) part, describes gravitational waves: ripples in spacetime that travel at the speed of light and have no longitudinal component. The Recognition Science framework studies a discrete analogue of this decomposition on a periodic lattice, a three-dimensional grid with 5 vertices per side. The declaration periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_longitudinalCoefficientSolutionData is a theorem in the framework's machine-checked library of formal theorems that establishes the existence of such a decomposition for a specific class of edge-length perturbations.

The theorem states that if one has a solution to a certain coefficient equation, then the space of all edge-length perturbations on the periodic lattice can be split into three orthogonal subspaces: the conformal part, which preserves angles; the gauge part, which corresponds to coordinate changes; and the TT part, which is the gravitational-wave-like component. This is a formal statement, proved in the Lean proof assistant, that the decomposition exists. It is a target, not a physical derivation: the theorem establishes the mathematical possibility of the decomposition, not that the framework's physical model of gravity produces it.

The theorem is a step toward a full tensor/shear sector in the framework's model of gravity. The existing scalar ansatz, which assigns a single potential to each vertex, cannot represent pure shear, and thus cannot cover gravitational-wave modes. This theorem shows that a richer description, one with a degree of freedom per edge, can be decomposed into the three parts needed to describe all weak-field modes. It is a scaffolding result, a proof that the mathematical structure exists, not a claim about the physical content of the framework's gravity model.

/-- Residual-defined coefficient-solution data closes the concrete longitudinal
TT decomposition target. -/
theorem periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_longitudinalCoefficientSolutionData
    (D : PeriodicTTLongitudinalCoefficientSolutionData5) :
    PeriodicFreudenthalTTOrthogonalDecompositionTargetAtN5
      (PeriodicLongitudinalGaugeIdx5 → ℝ) periodicLongitudinalGaugeMap5 :=
  periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_longitudinalCoefficientData
    (PeriodicTTLongitudinalCoefficientProjectorData5.ofSolutionData D)

/-- Residual-defined coefficient-solution data closes the concrete longitudinal
TT decomposition target. -/
theorem periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_longitudinalCoefficientSolutionData
    (D : PeriodicTTLongitudinalCoefficientSolutionData5) :
    PeriodicFreudenthalTTOrthogonalDecompositionTargetAtN5
      (PeriodicLongitudinalGaugeIdx5 → ℝ) periodicLongitudinalGaugeMap5 :=
  periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_longitudinalCoefficientData
    (PeriodicTTLongitudinalCoefficientProjectorData5.ofSolutionData D)

What this page does not claim

The theorem does not claim that the framework's physical model of gravity produces this decomposition. It does not claim that the decomposition is unique. It does not claim that the TT part corresponds to observable gravitational waves.

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

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