Gravity Analysis Regge Ttcontinuum Limit Raw Phase Quadratic Normalized

The normalization rule

In the Recognition Science framework's analysis of gravitational wave modes, a key quantity is the phase quadratic, a bookkeeping value that records how a wave's phase accumulates across a discrete grid of cells. The declaration rawPhaseQuadratic_normalized establishes a precise scaling law: if you take a direction vector and divide each of its components by the vector's length, the phase quadratic divides by the square of that length. In symbols, for any direction x with positive squared length, the normalized phase quadratic equals the original divided by that squared length.

This is the familiar rule that energy scales with the square of amplitude, applied to the framework's internal phase accounting. The theorem treats a free real scale, meaning the result holds for any direction, not just special integer-valued modes. A companion declaration, reggeTTMoment_normalized, extends the same scaling to the full gravitational wave moment: normalize the direction, and the moment divides by the squared norm. Together they ensure that calculations remain consistent when a mode direction is rescaled to unit length, a step the framework uses when comparing modes of different sizes.

The result is a technical normalization identity, not a physical law. It does not claim that any particular direction is preferred, that the phase quadratic equals a measurable energy, or that the scaling applies to anything other than the framework's own defined quantities. It is a structural fact about how the framework's bookkeeping behaves under rescaling, proved in the machine-checked library of formal theorems.

/-- Normalizing the direction divides its phase quadratic by its squared
norm. -/
theorem rawPhaseQuadratic_normalized (x : Fin 3 → ℝ)
    (hx : 0 < realModeNormSq x) (b : Bucket) :
    rawPhaseQuadratic
        (fun i => x i / Real.sqrt (realModeNormSq x)) b =
      rawPhaseQuadratic x b / realModeNormSq x := by
  have hsqrt : Real.sqrt (realModeNormSq x) ≠ 0 :=
    ne_of_gt (Real.sqrt_pos.2 hx)
  have hsqrt_sq :
      Real.sqrt (realModeNormSq x) ^ (2 : ℕ) = realModeNormSq x :=
    Real.sq_sqrt hx.le
  unfold rawPhaseQuadratic rawPhaseLinear
  simp only [Fin.sum_univ_three]
  field_simp [hsqrt, hx.ne']
  nlinarith

/-- `reggeTTMoment` is homogeneous under direction normalization with the
expected inverse squared-norm factor. -/
theorem reggeTTMoment_normalized (x : Fin 3 → ℝ)
    (hx : 0 < realModeNormSq x) (E : Fin 3 → Fin 3 → ℝ) :
    reggeTTMoment rawCosineSupport
        (rawPhaseQuadratic
          (fun i => x i / Real.sqrt (realModeNormSq x)))
        (rawBucketAmplitude E) =
      reggeTTMoment rawCosineSupport (rawPhaseQuadratic x)
        (rawBucketAmplitude E) / realModeNormSq x := by
  unfold reggeTTMoment reggeTTBlochFold
  rw [Finset.sum_div]
  refine Finset.sum_congr rfl fun b _ => ?_
  show
    -rawPhaseQuadratic (fun i => x i / Real.sqrt (realModeNormSq x)) b / 2 *
        rawBucketAmplitude E b =
      -rawPhaseQuadratic x b / 2 * rawBucketAmplitude E b / realModeNormSq x
  rw [rawPhaseQuadratic_normalized x hx b]
  field_simp [hx.ne']

What this page does not claim

The phase quadratic is a measurable energy in any conventional sense. The scaling law selects any preferred direction in space. The result applies to quantities outside the framework's own defined phase and moment.

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/ReggeTTContinuumLimit.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: