Gravity Analysis Regge Exact Midpoint M2 Ttidentity4 Dm2 Num Chunk06

A numerical audit

In numerical analysis, a midpoint rule approximates an integral by sampling the function at the center of each interval. The module named gravity analysis applies this idea to a four-dimensional spacetime calculation, where the object being sampled is not a simple function but a tensor component. The calculation produces a list of numbers, one for each combination of six indices, and the question is whether those numbers match a simpler, proposed expression called explicitZ.

This particular module, ReggeExactMidpointM2TTIdentity4DM2NumChunk06, is the sixth of several chunks that check this match. Its docstring states the goal plainly: m2Num = 8·explicitZ, and the chunk covers the index range starting with 1, 2, 3. The work is a brute-force audit. For each of the 256 possible index combinations in this chunk, the library states a theorem that the computed value equals eight times the proposed formula, and each theorem is proved by the decide tactic, which means the computer evaluates both sides and confirms they are equal.

The theorems are named by their indices, such as e_123333 for the tuple (1,2,3,3,3,3). Each one is a standalone fact: m2Num 1 2 3 3 3 3 = 8 * explicitZ 1 2 3 3 3 3. The factor of eight is not an approximation or a fitted constant; it is an exact equality that the proof checker verifies for every entry. The collection of these theorems establishes, for this chunk of the index space, that the midpoint calculation and the proposed formula agree perfectly.

In Recognition Science, this is part of a larger effort to show that gravity's structure follows from the framework's forced cost function, and that the resulting equations have a clean, closed form. This chunk does not derive the formula; it verifies that a specific numerical implementation matches it. The practical consequence is confidence: when a later theorem or calculation relies on this identity, it rests on a machine-checked foundation, not on a hand-waving argument or a spot check of a few values.

theorem e_123333 : m2Num 1 2 3 3 3 3 = 8 * explicitZ 1 2 3 3 3 3 := by decide

theorem e_123332 : m2Num 1 2 3 3 3 2 = 8 * explicitZ 1 2 3 3 3 2 := by decide

theorem e_123200 : m2Num 1 2 3 2 0 0 = 8 * explicitZ 1 2 3 2 0 0 := by decide

What this page does not claim

This module does not derive the explicitZ formula from first principles. This chunk does not cover the entire index space of the m2Num calculation. The factor of eight is not an approximation or a fitted constant.

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/ReggeExactMidpointM2TTIdentity4DM2NumChunk06.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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