Gravity Analysis Regge Exact Midpoint M2 Ttidentity4 Dkernel Glue Sym Full Sum Z

The identity certificate

In the mathematics of gravity, an amplitude is often built from a list of coupling terms, each with its own coefficient. Two such lists may look unrelated, yet for the theory to be consistent they must produce identical sums. The declaration symFullSumZ_explicit_eq_closed is a machine-checked theorem stating that, for every choice of the six indices that label a term, the sum built from the explicit list of couplings equals the sum built from a closed-form list. The check is exhaustive: the indices range over four values each, so the theorem covers all 4,096 combinations, and the proof runs by direct computation.

The two sums are not merely equal in the rational numbers; they are equal as integers. The explicit list is stored as a table of rational couplings, while the closed form is a compact expression. The theorem bridges them by converting the rational terms into integer contributions through a scaling factor, then showing the integer sums match. A companion result, symFullQ_explicitZ_eq_closedZ, extends the equality to the symmetrized average used in the physical amplitude, so the identity survives the averaging that defines the final quantity.

This is a computational proof, not a physical derivation. It does not say which list is correct, nor what the amplitude means. It only certifies that two specific constructions agree. The framework's library treats this as a glue result: it closes a gap between two representations, ensuring that whichever form a later calculation uses, the result is unchanged.

theorem symFullSumZ_explicit_eq_closed :
    ∀ a b c d i j : Fin 4,
      symFullSumZ explicitZ a b c d i j = symFullSumZ closedZ a b c d i j := by
  decide

theorem symFullSumZ_explicit_eq_closed :
    ∀ a b c d i j : Fin 4,
      symFullSumZ explicitZ a b c d i j = symFullSumZ closedZ a b c d i j := by
  decide

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]

What this page does not claim

The theorem does not say which coupling list is physically correct. It does not derive the amplitude from first principles. It does not claim the sums are equal for any other coupling table.

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)

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