Encyclopedia Gravity Gravity Analysis Regge Ttderivative Gate Is Ttpolarization Of Orthonormal Transv

ARTICLE 3 claims 3 theorems

Gravity Analysis Regge Ttderivative Gate Is Ttpolarization Of Orthonormal Transv

A machine-checked theorem in the framework's gravity program guarantees that every wave vector admits a transverse polarization pair, a necessary step toward studying gravitational wave modes on a lattice.

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The transverse pair theorem

In the Recognition Science framework's study of gravity, a ledger (a discrete record of events) is used to model spacetime as a lattice of tetrahedra. The framework's machine-checked library of formal theorems contains a result named isTTPolarization_of_orthonormal_transverse_pair. In plain language, it proves that for any integer wave vector, one can always find two perpendicular directions that are both transverse to the wave's direction of travel. These two directions form a complete basis for the plane perpendicular to the wave vector.

The theorem is a statement about existence, not about the value of any physical quantity. It shows that the set of transverse-traceless polarizations is never empty: for every integer wave vector, there is at least one such polarization tensor. The proof constructs the pair explicitly, using a Gram-Schmidt-style procedure that handles both the case where the wave vector lies in a coordinate plane and the case where it points along a coordinate axis. The construction is not limited to axis-aligned cases; it works for all integer wave vectors, including the zero vector.

This result is a stepping stone in a larger program. The framework's gravity analysis aims to compute the continuum limit of a lattice action for gravitational waves. The theorem guarantees that the necessary polarization degrees of freedom exist at every wave vector, a prerequisite for studying how plane waves behave on the lattice. It is a purely algebraic and geometric fact about three-dimensional space, independent of any particular physical theory.

The theorem does not claim anything about the value of the continuum action, nor does it establish the existence of second derivatives of the action profile. Those remain open targets. It also does not compute any lattice sum or take any continuum limit. Its scope is strictly local: it concerns a single tetrahedron at a flat configuration, not the assembled periodic lattice. The theorem is a necessary but not sufficient condition for the larger goal of deriving the continuum symbol from the lattice action.

What the theorem changes is the feasibility of the next stage. With the existence of transverse pairs guaranteed for every wave vector, the program can proceed to analyze the first and second derivatives of the action along these directions. Without this guarantee, the entire analysis would be blocked at the first step. The theorem is a small but load-bearing piece of the framework's attempt to derive gravitational physics from a discrete ledger.

THEOREM exists_isTTPolarization · IndisputableMonolith/Gravity/Analysis/ReggeTTDerivativeGate.lean
exists_isTTPolarization · IndisputableMonolith/Gravity/Analysis/ReggeTTDerivativeGate.lean:218 
/-- GATE 0b NON-VACUITY (THEOREM), general form: EVERY integer wave vector
(zero included) carries a TT polarization, built from an explicit
orthonormal transverse frame — planar rotation + cross product when
`(m₀, m₁) ≠ 0`, the axial `x/y` frame when `m₀ = m₁ = 0`.  No axis-only
weakening anywhere. -/
theorem exists_isTTPolarization (m : Fin 3 → ℤ) :
    ∃ E : Fin 3 → Fin 3 → ℝ, ReggeTTSymbolPreflight.IsTTPolarization m E := by
  by_cases h01 : (m 0 : ℝ) = 0 ∧ (m 1 : ℝ) = 0
  · refine ⟨_, isTTPolarization_of_orthonormal_transverse_pair m
      axialTransverse1 axialTransverse2 ?_ ?_ ?_ ?_ ?_⟩ <;>
      simp [axialTransverse1, axialTransverse2, Fin.sum_univ_three, h01.1, h01.2]
  · have h01' : (m 0 : ℝ) ≠ 0 ∨ (m 1 : ℝ) ≠ 0 := by
      by_contra h
      push_neg at h
      exact h01 ⟨h.1, h.2⟩
    have hs : 0 < (m 0 : ℝ) ^ 2 + (m 1 : ℝ) ^ 2 := by
      rcases h01' with h | h
      · have h0 : 0 < (m 0 : ℝ) ^ 2 := by positivity
        nlinarith [sq_nonneg ((m 1 : ℝ))]
      · have h1 : 0 < (m 1 : ℝ) ^ 2 := by positivity
        nlinarith [sq_nonneg ((m 0 : ℝ))]
    have hn : 0 < ((m 0 : ℝ) ^ 2 + (m 1 : ℝ) ^ 2) + (m 2 : ℝ) ^ 2 := by
      nlinarith [sq_nonneg ((m 2 : ℝ))]
    have hsn : 0 < ((m 0 : ℝ) ^ 2 + (m 1 : ℝ) ^ 2) *
        (((m 0 : ℝ) ^ 2 + (m 1 : ℝ) ^ 2) + (m 2 : ℝ) ^ 2) := mul_pos hs hn
    refine ⟨_, isTTPolarization_of_orthonormal_transverse_pair m
      (planarTransverse1 m) (planarTransverse2 m) ?_ ?_ ?_ ?_ ?_⟩
    · simp only [planarTransverse1, Fin.sum_univ_three]
      exact sum3_div_sq _ _ _ _ hs (by ring)
    · simp only [planarTransverse2, Fin.sum_univ_three]
      exact sum3_div_sq _ _ _ _ hsn (by ring)
    · simp only [planarTransverse1, planarTransverse2, Fin.sum_univ_three]
      exact sum3_div_orth _ _ _ _ _ _ _ _ (by ring)
    · simp only [planarTransverse1, Fin.sum_univ_three]
      exact sum3_dot_div _ _ _ _ _ _ _ (by ring)
    · simp only [planarTransverse2, Fin.sum_univ_three]
      exact sum3_dot_div _ _ _ _ _ _ _ (by ring)
THEOREM exists_isTTPolarization_of_ne_zero · IndisputableMonolith/Gravity/Analysis/ReggeTTDerivativeGate.lean
exists_isTTPolarization_of_ne_zero · IndisputableMonolith/Gravity/Analysis/ReggeTTDerivativeGate.lean:256 
/-- GATE 0b NON-VACUITY, panel-locked form: every NONZERO integer wave
vector carries a TT polarization. -/
theorem exists_isTTPolarization_of_ne_zero (m : Fin 3 → ℤ) (_hm : m ≠ 0) :
    ∃ E : Fin 3 → Fin 3 → ℝ, ReggeTTSymbolPreflight.IsTTPolarization m E :=
  exists_isTTPolarization m
THEOREM exists_isTTPolarization · IndisputableMonolith/Gravity/Analysis/ReggeTTDerivativeGate.lean
exists_isTTPolarization · IndisputableMonolith/Gravity/Analysis/ReggeTTDerivativeGate.lean:218 
/-- GATE 0b NON-VACUITY (THEOREM), general form: EVERY integer wave vector
(zero included) carries a TT polarization, built from an explicit
orthonormal transverse frame — planar rotation + cross product when
`(m₀, m₁) ≠ 0`, the axial `x/y` frame when `m₀ = m₁ = 0`.  No axis-only
weakening anywhere. -/
theorem exists_isTTPolarization (m : Fin 3 → ℤ) :
    ∃ E : Fin 3 → Fin 3 → ℝ, ReggeTTSymbolPreflight.IsTTPolarization m E := by
  by_cases h01 : (m 0 : ℝ) = 0 ∧ (m 1 : ℝ) = 0
  · refine ⟨_, isTTPolarization_of_orthonormal_transverse_pair m
      axialTransverse1 axialTransverse2 ?_ ?_ ?_ ?_ ?_⟩ <;>
      simp [axialTransverse1, axialTransverse2, Fin.sum_univ_three, h01.1, h01.2]
  · have h01' : (m 0 : ℝ) ≠ 0 ∨ (m 1 : ℝ) ≠ 0 := by
      by_contra h
      push_neg at h
      exact h01 ⟨h.1, h.2⟩
    have hs : 0 < (m 0 : ℝ) ^ 2 + (m 1 : ℝ) ^ 2 := by
      rcases h01' with h | h
      · have h0 : 0 < (m 0 : ℝ) ^ 2 := by positivity
        nlinarith [sq_nonneg ((m 1 : ℝ))]
      · have h1 : 0 < (m 1 : ℝ) ^ 2 := by positivity
        nlinarith [sq_nonneg ((m 0 : ℝ))]
    have hn : 0 < ((m 0 : ℝ) ^ 2 + (m 1 : ℝ) ^ 2) + (m 2 : ℝ) ^ 2 := by
      nlinarith [sq_nonneg ((m 2 : ℝ))]
    have hsn : 0 < ((m 0 : ℝ) ^ 2 + (m 1 : ℝ) ^ 2) *
        (((m 0 : ℝ) ^ 2 + (m 1 : ℝ) ^ 2) + (m 2 : ℝ) ^ 2) := mul_pos hs hn
    refine ⟨_, isTTPolarization_of_orthonormal_transverse_pair m
      (planarTransverse1 m) (planarTransverse2 m) ?_ ?_ ?_ ?_ ?_⟩
    · simp only [planarTransverse1, Fin.sum_univ_three]
      exact sum3_div_sq _ _ _ _ hs (by ring)
    · simp only [planarTransverse2, Fin.sum_univ_three]
      exact sum3_div_sq _ _ _ _ hsn (by ring)
    · simp only [planarTransverse1, planarTransverse2, Fin.sum_univ_three]
      exact sum3_div_orth _ _ _ _ _ _ _ _ (by ring)
    · simp only [planarTransverse1, Fin.sum_univ_three]
      exact sum3_dot_div _ _ _ _ _ _ _ (by ring)
    · simp only [planarTransverse2, Fin.sum_univ_three]
      exact sum3_dot_div _ _ _ _ _ _ _ (by ring)

What this page does not claim

The theorem does not compute the continuum value of the TT symbol, which remains an open target. The theorem does not establish the existence of second derivatives of the action profile. The theorem does not perform any lattice sum or continuum limit; it concerns only a single tetrahedron at a flat configuration.

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

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