Gravity Analysis Regge Exact Flat Hessian Bloch Data4 D

The exact coupling table

In the Recognition Science framework, gravity is studied not as a smooth field but as a discrete lattice of points, and this data file is a piece of that program. It is a generated data file, not a theorem about physics. The file defines a table of 1,208 entries, each one a rational coupling between two edges of a unit cell in a four-dimensional lattice. The word "exact" in the name means the numbers are rational fractions, not floating-point approximations. The table is chunked into 16 arrays, each with 80 entries except the last with 8, and the machine-checked library proves the total size is exactly 1,208.

Each entry in the table is a coupling, a number that says how strongly two lattice edges influence each other. The structure stores a rational number s = p/q, two edge displacement vectors De and Dep in {0,1}^4, and a doubled midpoint offset delta2. These feed a Bloch symbol, a standard tool in solid-state physics for studying waves in periodic media. The continuum symbol is written as (1/2) Σ s · (D_eᵀ H D_e)(D_epᵀ H D_ep) · cos(k·Δ), where H is the Hessian matrix and Δ is half the midpoint offset. This is the exact midpoint Bloch symbol, meaning it uses the true midpoint between edges rather than an approximation.

The purpose of the table is to make a specific calculation tractable. In lattice gravity, one wants to know how curvature or flatness emerges from the discrete structure. The Hessian matrix H encodes second derivatives of an action, and the couplings describe how those derivatives interact across the lattice. By fixing these couplings as exact rationals, the framework can compute the flat-space behavior without rounding error. This data file establishes, in plain language, that the exact flat-space Hessian for a four-dimensional lattice can be written down as a finite, explicit list of rational numbers, ready for further analysis.

This is a definitional step, not a physical claim. The table defines the data; it does not prove that gravity behaves this way. The framework's larger chain of theorems, which forces three spatial dimensions and the constants hbar = phi^-5 and G = phi^5/pi, lives elsewhere. Here, the contribution is narrower: a precise, machine-checked object that later analysis can rely on. The table is generated by a script and marked "do not hand-edit", because its size and structure are meant to be trusted as given, not altered by human error.

/-- Generated coupling table (`n = 1208`). -/
def couplingTable : Array Coupling :=
  couplingChunk0 ++ couplingChunk1 ++ couplingChunk2 ++ couplingChunk3 ++ couplingChunk4 ++ couplingChunk5 ++ couplingChunk6 ++ couplingChunk7 ++ couplingChunk8 ++ couplingChunk9 ++ couplingChunk10 ++ couplingChunk11 ++ couplingChunk12 ++ couplingChunk13 ++ couplingChunk14 ++ couplingChunk15

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]

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

This module does not prove that gravity is a lattice theory. The table does not derive the fine-structure constant or any measured physical constant. The module does not establish the physical recognition-to-linking bridge for gravity.

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

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