Encyclopedia Gravity Gravity Analysis Regge Ttalgebraic Closer Canonical Finite H Div Momentum Norm S

ARTICLE 3 claims 3 theorems

Gravity Analysis Regge Ttalgebraic Closer Canonical Finite H Div Momentum Norm S

A machine-checked theorem shows that a certain normalized gravitational quantity converges to the same constant value from every direction, a property called isotropy.

Verify this page

The isotropy limit

The declaration canonicalFiniteH_div_momentumNormSq_tendsto_isotropy is a theorem in the machine-checked library of formal theorems that Recognition Science builds on. It concerns a quantity called the finite reduced symbol, which is a ratio: a certain finite Hamiltonian divided by the squared norm of a momentum vector. The theorem states that as the system size grows without bound, this ratio converges to a fixed constant, namely -1/4.

The convergence holds for every nonzero integer mode and every transverse-traceless polarization, which is a symmetric matrix with zero trace and zero divergence. In plain terms, the theorem says that the limit value is the same no matter which direction the momentum points. That direction-independence is what physicists call isotropy: the result does not prefer any axis in three-dimensional space.

The proof is finite algebra. It works by showing that the finite ratio equals a moment, an average over a discrete set of points, and that this moment has a closed form: one half times the quadratic form built from the adjugate matrix. Under the transverse-traceless conditions, that closed form collapses to -1/4 times the squared norm, which is 1 for normalized directions. The chain of equalities is checked by the kernel, with no unproved assumptions beyond the standard three axioms of the ambient type theory.

The theorem does not claim that the finite ratio equals -1/4 at any finite size. It only claims convergence in the limit. It also does not claim that the closed form holds for all matrices: the symmetry conditions are essential, and the difference for non-symmetric matrices is a specific rotational square that does not vanish. The theorem does not address aliased small side lengths, where the finite identity is not identified with the bucket fold and no repair is attempted.

In the broader Recognition Science program, this isotropy result is a step toward showing that the continuum limit of a discrete gravitational assembly is rotationally invariant. The value -1/4 is a coefficient in that continuum theory. The theorem is a piece of the algebraic closer that ties the discrete assembly to its continuum limit, and it is one of the few places where the framework's machinery yields a concrete number.

THEOREM canonicalFiniteH_div_momentumNormSq_tendsto_isotropy · IndisputableMonolith/Gravity/Analysis/ReggeTTAlgebraicCloser.lean
canonicalFiniteH_div_momentumNormSq_tendsto_isotropy · IndisputableMonolith/Gravity/Analysis/ReggeTTAlgebraicCloser.lean:413 
/-- **P1.1a + P1.1b COMPOSED (THEOREM): the normalized finite reduced
Regge TT symbol converges to exactly `-(1/4)` for every nonzero integer
mode and every TT polarization.** -/
theorem canonicalFiniteH_div_momentumNormSq_tendsto_isotropy
    (m : Fin 3 → ℤ) (E : Fin 3 → Fin 3 → ℝ)
    (hm : ∃ i : Fin 3, m i ≠ 0) (hTT : IsTTPolarization m E) :
    Filter.Tendsto
      (fun j : ℕ =>
        @canonicalFiniteH (j + 3) (instNeZeroAddThree j) E m /
          momentumNormSq (j + 3) m)
      Filter.atTop (nhds reggeTTContinuumCoefficient) := by
  have h :=
    ReggeTTContinuumLimit.canonicalFiniteH_div_momentumNormSq_tendsto E m hm
  rwa [reggeTTMoment_tt_value m E hm hTT] at h
THEOREM canonicalFiniteH_div_momentumNormSq_tendsto_isotropy · IndisputableMonolith/Gravity/Analysis/ReggeTTAlgebraicCloser.lean
canonicalFiniteH_div_momentumNormSq_tendsto_isotropy · IndisputableMonolith/Gravity/Analysis/ReggeTTAlgebraicCloser.lean:413 
/-- **P1.1a + P1.1b COMPOSED (THEOREM): the normalized finite reduced
Regge TT symbol converges to exactly `-(1/4)` for every nonzero integer
mode and every TT polarization.** -/
theorem canonicalFiniteH_div_momentumNormSq_tendsto_isotropy
    (m : Fin 3 → ℤ) (E : Fin 3 → Fin 3 → ℝ)
    (hm : ∃ i : Fin 3, m i ≠ 0) (hTT : IsTTPolarization m E) :
    Filter.Tendsto
      (fun j : ℕ =>
        @canonicalFiniteH (j + 3) (instNeZeroAddThree j) E m /
          momentumNormSq (j + 3) m)
      Filter.atTop (nhds reggeTTContinuumCoefficient) := by
  have h :=
    ReggeTTContinuumLimit.canonicalFiniteH_div_momentumNormSq_tendsto E m hm
  rwa [reggeTTMoment_tt_value m E hm hTT] at h
THEOREM adjugateQuadraticForm_tt · IndisputableMonolith/Gravity/Analysis/ReggeTTAlgebraicCloser.lean
adjugateQuadraticForm_tt · IndisputableMonolith/Gravity/Analysis/ReggeTTAlgebraicCloser.lean:302 
/-- **STEP (ii) OF THE C8 CLOSED FORM (THEOREM): on the TT variety the
adjugate quadratic form collapses to `-(1/2)·|x|²·⟨E,E⟩`.**  Kernel form
of the eigenvalue argument (`x` is a null eigenvector of `E`, so
`adj(E)x = λ₁λ₂x` with `λ₁ + λ₂ = 0`), discharged as an explicit cofactor
certificate over the seven TT generators. -/
theorem adjugateQuadraticForm_tt (E : Fin 3 → Fin 3 → ℝ) (x : Fin 3 → ℝ)
    (hsym01 : E 0 1 = E 1 0) (hsym02 : E 0 2 = E 2 0)
    (hsym12 : E 1 2 = E 2 1)
    (htr : E 0 0 + E 1 1 + E 2 2 = 0)
    (htrans0 : x 0 * E 0 0 + x 1 * E 1 0 + x 2 * E 2 0 = 0)
    (htrans1 : x 0 * E 0 1 + x 1 * E 1 1 + x 2 * E 2 1 = 0)
    (htrans2 : x 0 * E 0 2 + x 1 * E 1 2 + x 2 * E 2 2 = 0) :
    adjugateQuadraticForm E x =
      -(1 / 2) * (x 0 ^ 2 + x 1 ^ 2 + x 2 ^ 2) *
        (E 0 0 * E 0 0 + E 0 1 * E 0 1 + E 0 2 * E 0 2 + E 1 0 * E 1 0 +
          E 1 1 * E 1 1 + E 1 2 * E 1 2 + E 2 0 * E 2 0 + E 2 1 * E 2 1 +
          E 2 2 * E 2 2) := by
  rw [adjugateQuadraticForm_explicit]
  linear_combination
    (E 0 0 * x 0 * x 1 - E 0 1 * x 0 ^ 2 / 2 + E 0 1 * x 1 ^ 2 / 2 +
        E 0 1 * x 2 ^ 2 / 2 - E 1 0 * x 0 ^ 2 / 2 - E 1 0 * x 1 ^ 2 / 2 -
        E 1 0 * x 2 ^ 2 / 2 - E 2 1 * x 0 * x 2 + E 2 2 * x 0 * x 1) *
      hsym01 +
    (E 0 0 * x 0 * x 2 - E 0 2 * x 0 ^ 2 / 2 + E 0 2 * x 1 ^ 2 / 2 +
        E 0 2 * x 2 ^ 2 / 2 + E 1 1 * x 0 * x 2 - E 1 2 * x 0 * x 1 -
        E 2 0 * x 0 ^ 2 / 2 - E 2 0 * x 1 ^ 2 / 2 - E 2 0 * x 2 ^ 2 / 2) *
      hsym02 +
    (E 0 0 * x 1 * x 2 - E 0 2 * x 0 * x 1 + E 1 1 * x 1 * x 2 +
        E 1 2 * x 0 ^ 2 / 2 - E 1 2 * x 1 ^ 2 / 2 + E 1 2 * x 2 ^ 2 / 2 -
        E 2 1 * x 0 ^ 2 / 2 - E 2 1 * x 1 ^ 2 / 2 - E 2 1 * x 2 ^ 2 / 2) *
      hsym12 +
    (-(E 0 0 * x 0 ^ 2) / 2 + E 0 0 * x 1 ^ 2 / 2 + E 0 0 * x 2 ^ 2 / 2 -
        2 * E 0 1 * x 0 * x 1 - E 0 2 * x 0 * x 2 + E 1 1 * x 0 ^ 2 / 2 -
        E 1 1 * x 1 ^ 2 / 2 + E 1 1 * x 2 ^ 2 / 2 - E 1 2 * x 1 * x 2 +
        E 2 2 * x 0 ^ 2 / 2 + E 2 2 * x 1 ^ 2 / 2 + E 2 2 * x 2 ^ 2 / 2) *
      htr +
    (E 0 0 * x 0 + E 0 1 * x 1 + E 0 2 * x 2) * htrans0 +
    (E 0 1 * x 0 + E 1 1 * x 1 + E 1 2 * x 2) * htrans1 +
    (-(E 0 0 * x 2) + E 0 2 * x 0 - E 1 1 * x 2 + E 1 2 * x 1) * htrans2

What this page does not claim

The finite ratio equals -1/4 at any finite system size. The closed form holds for non-symmetric matrices. The theorem addresses the aliased small side lengths where the finite identity is not identified with the bucket fold.

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/ReggeTTAlgebraicCloser.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red:

MACHINE LAYER · GROUNDED CLAIM TABLE · CLICK TO EXPAND