Gravity Analysis Regge Exact Midpoint M2 Ttidentity4 Dkernel Glue

The kernel glue

In the Recognition Science framework, a ledger is a discrete record of events, and gravity is analyzed by summing contributions from many small coupling terms. The module named here is a formal bridge: it proves that two different ways of computing the same sum agree exactly. One way uses an explicit list of numbers, the other a closed formula. The result is a machine-checked identity, meaning a computer verified every step from definitions to conclusion.

The core theorem states that for any choice of six indices, the sum of all term contributions equals the explicit value divided by 32, and that this equals the closed-form value divided by 32. In symbols: ∑ termQ = explicitZ / 32 = closedZ / 32. This is not an approximation; it is an exact equality over the rational numbers. The proof works by converting each rational term into an integer contribution, summing those integers, and then scaling back. The library shows that every denominator divides 16, so the scaling by 256 is exact.

The practical meaning is that the framework's gravity analysis is internally consistent: the explicit and closed forms are interchangeable. This matters because the closed form is what connects to deeper results, while the explicit form is what you can compute directly. The bridge closes a formal gap, a "typed blocker" in the development, and ensures that no hidden assumption or rounding error separates the two representations.

theorem sum_termQ_eq_explicitZ_div32 (a b c d i j : Fin 4) :
    (∑ idx : CouplingIdx, termQ couplingTable[idx] a b c d i j) =
      (explicitZ a b c d i j : ℚ) / 32 := by
  rw [sum_termQ_eq_m2Num_div256, m2Num_div256_eq_explicitZ_div32]

theorem symFullQ_explicitZ_eq_closedZ (a b c d i j : Fin 4) :
    symFullQ (fun x y u v p q => (explicitZ x y u v p q : ℚ) / 32) a b c d i j =
      symFullQ (fun x y u v p q => (closedZ x y u v p q : ℚ) / 32) a b c d i j := by
  rw [symFullQ_scaledZ_eq, symFullQ_scaledZ_eq, symFullSumZ_explicit_eq_closed]

theorem couplingZ_den_dvd_16 : ∀ z ∈ couplingZList, z.den ∣ 16 := by
  decide

What this page does not claim

This module does not derive the fine-structure constant or any specific physical constant. It does not prove that gravity in four dimensions is unique or that three spatial dimensions are forced; that is a separate result. It does not claim that the explicit and closed forms are equal in any sense other than the exact rational equality proved here.

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/ReggeExactMidpointM2TTIdentity4DKernelGlue.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

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