Gravity Analysis Regge Bloch Local Incidence4 D M2 Mean Local Orbit Slot Coeff E

The scaling identity

The declaration m2MeanLocalOrbitSlotCoeff_eq_scaled is a theorem in the framework's machine-checked library of formal theorems. It states a precise algebraic identity: for any 4x4 matrix H, any direction vector dir, and any choice of orbit type, slot, and term, the coefficient called m2MeanLocalOrbitSlotCoeff is equal to the reciprocal of the orbit's star size multiplied by another coefficient called m2TransportedOrbitSlotCoeff. In plain words, the mean-local coefficient is not a new object; it is the transported coefficient scaled down by a factor that depends only on which orbit type is being used.

This identity is part of a larger investigation into how to build local-incidence kernels in four dimensions. The framework models a discrete geometry of hinges and orbits, and it tests whether different ways of assembling these kernels give the same physical results. The theorem is one step in proving that the mean-local construction, which averages over a star of hinges, is equivalent to the distinct-hinge construction, which treats each hinge separately. The scaling factor, the reciprocal of the orbit's star size, is exactly what compensates for the averaging.

The theorem is a formal statement about the algebraic structure of these coefficients. It does not claim that the mean-local construction successfully reproduces the Einstein-Hilbert action or any other physical result. The framework's own status record explicitly marks the path B position-resolved construction as not closing the Einstein-Hilbert action, and the theorem does not change that. The identity is a building block, not a physical conclusion.

What the theorem does establish is a clean relationship between two families of coefficients. It shows that the mean-local slot coefficient is a scaled version of the transported slot coefficient, with the scaling factor being the reciprocal of the orbit's star size. This is a precise, checkable fact that holds for every orbit type, slot, and term. It is a piece of the larger puzzle of understanding how different kernel constructions relate to each other, and it is a necessary step for the equivalence result that the framework does prove.

theorem m2MeanLocalOrbitSlotCoeff_eq_scaled (ty : HingeOrbitType) (H : Mat4)
    (dir : Fin 4 → ℝ) (s : Fin 24) (t : Fin 10) :
    m2MeanLocalOrbitSlotCoeff ty H dir s t =
      (orbitStarSize ty)⁻¹ * m2TransportedOrbitSlotCoeff ty H dir s t := by
  change m2MeanLocalOrbitSlotCoeff ty H dir s t =
    (orbitStarSize ty)⁻¹ * m2TransportedOrbitSlotCoeffTrunc ty H dir s t
  unfold m2MeanLocalOrbitSlotCoeff m2TransportedOrbitSlotCoeffTrunc
  by_cases h : isOrbit ty s t
  · simp only [h, ↓reduceIte]
    have hsum :
        (∑ d : Fin 15,
            slotOrbitMeanLocalKer ty s t d * classCoeff H d *
              (phaseScaleDir dir (hingeBase s t) d) ^ 2) =
          (orbitStarSize ty)⁻¹ *
            ∑ d : Fin 15,
              slotOrbitDeficitKer ty s t d * classCoeff H d *
                (phaseScaleDir dir (hingeBase s t) d) ^ 2 := by
      have hpt : ∀ d : Fin 15,
          slotOrbitMeanLocalKer ty s t d * classCoeff H d *
              (phaseScaleDir dir (hingeBase s t) d) ^ 2 =
            (orbitStarSize ty)⁻¹ *
              (slotOrbitDeficitKer ty s t d * classCoeff H d *
                (phaseScaleDir dir (hingeBase s t) d) ^ 2) := by
        intro d
        rw [slotOrbitMeanLocalKer_eq_scaled]
        ring
      simp_rw [hpt, ← Finset.mul_sum]
    rw [hsum]
    ring
  · simp only [h, ↓reduceIte, mul_zero]

theorem m2MeanLocalAllOrbitMoment_eq_distinctHinge (H : Mat4)
    (dir : Fin 4 → ℝ) :
    m2MeanLocalAllOrbitMoment H dir =
      m2TransportedAllOrbitMomentDistinctHinge H dir := by
  unfold m2MeanLocalAllOrbitMoment m2TransportedAllOrbitMomentDistinctHinge
  refine Finset.sum_congr rfl fun ty _ =>
    m2MeanLocalOrbitMoment_eq_scaled ty H dir

theorem Regge4DPathBPositionResolvedClosesEH_status_open :
    Regge4DPathBPositionResolvedClosesEH = False :=
  rfl

What this page does not claim

The theorem does not imply that the mean-local construction produces the Einstein-Hilbert action, a result explicitly marked as false in the framework's status record. The theorem does not establish any physical prediction or measurement. The theorem does not claim that the mean-local and distinct-hinge constructions are identical in all respects, only that their moments agree.

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/ReggeBlochLocalIncidence4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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