Gravity Analysis Regge Exact Flat Hessian Bloch Data4 D Coupling Chunk1 Size

A table of gravity couplings

In numerical work on gravity, a calculation often needs a long list of coefficients. The Recognition Science framework's machine-checked library of formal theorems stores such a list as an array, a data structure that holds a fixed sequence of entries. To keep the formal proof manageable, the library splits the list into chunks. The declaration couplingChunk1_size is a theorem stating that the second chunk, named couplingChunk1, contains exactly 80 entries.

Each entry in this chunk is a coupling, a rational number that weights a pair of edge displacements in a discrete approximation to the Hessian, the matrix of second derivatives of a function. The full table, assembled from sixteen chunks, holds 1,208 such couplings. The theorem about the chunk's size is proved by direct computation: the definition of the chunk is written out, and the size follows immediately from that definition. It is a small, exact fact, not an approximation.

What the theorem does not claim is any physical content. It says nothing about what the couplings mean, how they were derived, or whether the underlying gravity model is correct. It does not assert that the table is complete, that the couplings are accurate, or that the calculation using them converges. The declaration only certifies a bookkeeping fact: the chunk has 80 entries, no more and no fewer.

This kind of exact size statement matters because the library's proofs depend on the table being precisely the right length. If a chunk had 79 or 81 entries, any later theorem that indexes into it could fail. The size theorem gives later formal work a solid footing, a guarantee that the data structure is exactly as large as the rest of the proof expects.

theorem couplingChunk1_size : couplingChunk1.size = 80 := rfl

theorem couplingTable_size : couplingTable.size = 1208 := by
  simp only [couplingTable, Array.size_append, couplingChunk0_size, couplingChunk1_size, couplingChunk2_size, couplingChunk3_size, couplingChunk4_size, couplingChunk5_size, couplingChunk6_size, couplingChunk7_size, couplingChunk8_size, couplingChunk9_size, couplingChunk10_size, couplingChunk11_size, couplingChunk12_size, couplingChunk13_size, couplingChunk14_size, couplingChunk15_size]

/-- One unit-cell edge-pair coupling for the exact midpoint Bloch symbol. -/
structure Coupling where
  num : ℤ
  den : ℕ
  De : Fin 4 → ℤ
  Dep : Fin 4 → ℤ
  /-- Doubled midpoint offset `2(mid_e - mid_ep)`. -/
  delta2 : Fin 4 → ℤ

set_option maxRecDepth 4096

What this page does not claim

The theorem says nothing about the physical correctness of the gravity model. The theorem does not assert the table is complete or that its entries are accurate. The theorem does not claim the calculation using the table converges.

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/ReggeExactFlatHessianBlochData4D.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: