Gravity Analysis Regge Exact Midpoint M2 Ttidentity4 Dkernel Glue Sym Full Q Exp

Two paths, one number

In the Recognition Science framework's gravity analysis, a key calculation involves adding up contributions from many coupling terms. The calculation can be done in two ways: one uses an explicit, directly written-out list of terms, and the other uses a more compact, closed-form expression. The theorem symFullQ_explicitZ_eq_closedZ proves that after applying a specific symmetrization procedure, which averages over index permutations to respect the underlying physics, these two approaches yield exactly the same rational number for every allowed combination of indices.

The proof is by computation: the machine-checked library of formal theorems verifies the equality by deciding it, meaning it checks all finitely many cases. This is not an approximation or a numerical coincidence; it is an exact identity over the rational numbers. The theorem is part of a larger effort to close a typed blocker, a gap in the formal verification of a midpoint identity related to the TT (transverse-traceless) decomposition in gravity.

What the theorem does not claim is that the explicit and closed forms are identical before symmetrization. They are different expressions; only after the averaging over the eight index permutations do they agree. It also does not claim that this symmetrized sum is zero, or that it equals any particular physical observable. The equality is a statement about the internal consistency of two mathematical representations within the framework's gravity analysis, not a direct prediction about measurable quantities.

In Recognition Science, this kind of check matters because the framework derives physical structure from a forced cost function, and its gravity sector relies on algebraic identities that are detailed and lengthy. A machine-checked equality between two large symbolic sums is a practical guarantee that no sign error or indexing mistake has crept into the derivation. It is the kind of bookkeeping that makes the framework's larger claims about deriving constants and dimensions from first principles credible.

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 explicit and closed forms are identical before symmetrization. The symmetrized sum equals zero or any particular physical observable. The theorem makes a direct prediction about measurable gravitational quantities.

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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