Gravity Analysis Regge Exact Midpoint M2 Ttidentity4 Dkernel Glue M2 Num Div256

The numerical bridge

In the study of gravity, physicists often need to verify that two very different-looking expressions actually compute the same number. The declaration m2Num_div256_eq_explicitZ_div32 is a machine-checked proof of exactly such an equality. It shows that a quantity called m2Num, divided by 256, equals another quantity called explicitZ, divided by 32. Both are evaluated at six indices that label the components of a gravitational interaction, and the equality holds for every possible choice of those indices.

The two sides of the identity come from different construction methods. The left side, m2Num, is built by summing contributions from a list of coupling constants, each term carrying a factor of one quarter and various delta functions. The right side, explicitZ, is a closed-form expression written directly in terms of integer arrays. The theorem bridges these two approaches, confirming that the laborious sum over the coupling table produces exactly the same result as the compact formula.

This equality is not a numerical approximation. It is an exact statement about rational numbers, verified by a computer proof assistant that checks every step of the reasoning. The proof works by first relating m2Num to eight times explicitZ, then dividing both sides by the appropriate factors to reach the final form. The declaration also supports a further result: the symmetrized version of the explicit expression matches the symmetrized version of another closed form, called closedZ.

What this declaration does not claim is any physical interpretation of the quantities involved. It does not say what m2Num represents in terms of forces or particles, nor does it assert that the equality has experimental consequences. The theorem is purely computational: it establishes that two symbolic expressions are identical. Any physical meaning must come from the surrounding framework that defines these quantities, not from this identity itself.

theorem m2Num_div256_eq_explicitZ_div32 (a b c d i j : Fin 4) :
    (m2Num a b c d i j : ℚ) / 256 = (explicitZ a b c d i j : ℚ) / 32 := by
  have h := m2Num_eq_eight_explicitZ a b c d i j
  rw [h]
  push_cast
  ring

theorem m2Num_div256_eq_explicitZ_div32 (a b c d i j : Fin 4) :
    (m2Num a b c d i j : ℚ) / 256 = (explicitZ a b c d i j : ℚ) / 32 := by
  have h := m2Num_eq_eight_explicitZ a b c d i j
  rw [h]
  push_cast
  ring

What this page does not claim

The declaration assigns no physical meaning to the quantities it equates. The identity does not claim any experimental or observational consequence. The theorem does not state that the coupling table itself is derived from first principles.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: