Gravity Analysis Regge Bloch Transported All Orbit4 D M2 Transported Orbit Slot

The transported orbit slot

In the Recognition Science framework's gravity analysis, a central object is the ledger, a discrete record of geometric events that the framework uses to build physical quantities. The declaration m2TransportedOrbitSlotCoeffFull_eq_trunc_of_ker0 concerns a specific coefficient in this ledger: the m² transported orbit slot coefficient. This coefficient measures how a second-order moment, a quantity akin to a variance, is distributed across the 24 slots and 10 time steps of a 4D orbit. The declaration establishes that under a particular condition, the full version of this coefficient equals a simplified, truncated version.

The condition is named ker0, which refers to the kernel, or core data, being zero in a specific sense. When this holds, the full coefficient, which includes all contributions, reduces to the truncated form. This is a structural identity: it shows that the extra terms in the full version vanish under the kernel-zero condition, leaving only the simplified expression. The equality is a theorem in the machine-checked library of formal theorems, meaning it is proved from the definitions without additional assumptions.

The declaration does not claim that the kernel is always zero, nor that the full and truncated coefficients are equal in general. It only establishes the equality under the stated condition. It also does not claim that the m² moment itself is zero, only that the slot coefficient takes a particular simplified form. The declaration does not address the behavior of the moment in the continuum limit, which remains an open target in the framework.

/-- Full cosine two-jet of the phased product `A(μ)K(μ)`:
`A0*K2 + A2*K0` with `A2 = -½ Aph²` and `K2 = -½ Kph²`.
Equals the truncated form whenever `slotOrbitKerDot = 0`. -/
def m2TransportedOrbitSlotCoeffFull (ty : HingeOrbitType) (H : Mat4)
    (dir : Fin 4 → ℝ) (s : Fin 24) (t : Fin 10) : ℝ :=
  if isOrbit ty s t then
    (∑ d : Fin 15, slotOrbitAreaCov ty s t d * classCoeff H d) *
        (-(1 / 2 : ℝ) *
          ∑ d : Fin 15,
            slotOrbitDeficitKer ty s t d * classCoeff H d *
              (phaseScaleDir dir (hingeBase s t) d) ^ 2) +
      (-(1 / 2 : ℝ) *
          ∑ d : Fin 15,
            slotOrbitAreaCov ty s t d * classCoeff H d *
              (phaseScaleDir dir (hingeBase s t) d) ^ 2) *
        slotOrbitKerDot ty H s t
  else 0

/-- Full cosine two-jet of the phased product `A(μ)K(μ)`:
`A0*K2 + A2*K0` with `A2 = -½ Aph²` and `K2 = -½ Kph²`.
Equals the truncated form whenever `slotOrbitKerDot = 0`. -/
def m2TransportedOrbitSlotCoeffFull (ty : HingeOrbitType) (H : Mat4)
    (dir : Fin 4 → ℝ) (s : Fin 24) (t : Fin 10) : ℝ :=
  if isOrbit ty s t then
    (∑ d : Fin 15, slotOrbitAreaCov ty s t d * classCoeff H d) *
        (-(1 / 2 : ℝ) *
          ∑ d : Fin 15,
            slotOrbitDeficitKer ty s t d * classCoeff H d *
              (phaseScaleDir dir (hingeBase s t) d) ^ 2) +
      (-(1 / 2 : ℝ) *
          ∑ d : Fin 15,
            slotOrbitAreaCov ty s t d * classCoeff H d *
              (phaseScaleDir dir (hingeBase s t) d) ^ 2) *
        slotOrbitKerDot ty H s t
  else 0

What this page does not claim

The kernel is always zero in physical situations. The full and truncated coefficients are equal without the kernel-zero condition. The m² moment itself evaluates to zero under the kernel-zero condition.

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/ReggeBlochTransportedAllOrbit4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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