Constants Lambda Rec Derivation Total Curvature Gauss Bonnet

The Gauss-Bonnet step

Gauss-Bonnet is a classical result in geometry: for a closed surface, the total curvature is fixed by a topological invariant, the Euler characteristic. For a sphere, this total is always 4π, no matter how the surface is bent or stretched. The framework's declaration total_curvature_gauss_bonnet applies this to the cube, a polyhedral sphere, and proves that the sum of the angular deficits at its eight corners equals 4π. This is not a new physical claim; it is a formal restatement of a known geometric fact, checked by a machine.

The step matters because the framework uses it to derive a cost for curvature. In the framework's ledger, a discrete record of events, the cube Q₃ is the elementary cell of three-dimensional space. The theorem shows that the cube's total curvature is fixed at 4π, which lets the framework define a curvature cost that scales as the square of a length. This cost is then balanced against the cost of posting a single ledger entry, and the balance has a unique positive solution: the recognition length λ_rec equals 1/√2 in cost units. The declaration itself only establishes the geometric fact; the subsequent balance and uniqueness are separate theorems in the same module.

What the declaration does not claim is also precise. It does not assert that physical space is a cube, nor that the curvature cost formula is the only possible one. It only proves the integrated curvature identity for the cube as a topological sphere. The derivation of the recognition length from this identity is a separate chain of reasoning, and the step that connects the cube to physical three-dimensionality is a separate, unformalized bridge. The theorem is a lemma in a larger derivation, not a statement about the universe's geometry on its own.

/-- Total curvature over all 8 vertices = 4π = 2π × χ(S²).
    This is the Gauss-Bonnet theorem for the cube. -/
theorem total_curvature_gauss_bonnet :
    Q3_vertices * angular_deficit_per_vertex = 2 * Real.pi * euler_S2 := by
  simp [Q3_vertices, euler_S2, angular_deficit_value]; ring

/-- Total curvature over all 8 vertices = 4π = 2π × χ(S²).
    This is the Gauss-Bonnet theorem for the cube. -/
theorem total_curvature_gauss_bonnet :
    Q3_vertices * angular_deficit_per_vertex = 2 * Real.pi * euler_S2 := by
  simp [Q3_vertices, euler_S2, angular_deficit_value]; ring

/-- lambda_0 is the unique positive root of the balance residual. -/
theorem balance_unique_positive_root (lambda : ℝ) (hlambda : lambda > 0) :
    balanceResidual lambda = 0 ↔ lambda = lambda_0 := by
  unfold balanceResidual J_curv J_bit_normalized lambda_0
  constructor
  · intro h
    have hsq : lambda ^ 2 = 1 / 2 := by linarith
    have hlam_sqrt : lambda = Real.sqrt (1 / 2) := by
      rw [← Real.sqrt_sq (le_of_lt hlambda), hsq]
    rw [hlam_sqrt, Real.sqrt_div (by norm_num : (0:ℝ) ≤ 1), Real.sqrt_one]
  · intro h
    rw [h, div_pow, Real.sq_sqrt (by norm_num : (0:ℝ) ≤ 2)]
    ring

What this page does not claim

The declaration does not claim that physical space is a cube. The declaration does not by itself derive the recognition length; that requires the separate balance and uniqueness theorems. The declaration does not claim the curvature cost formula is the only possible one.

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/Constants/LambdaRecDerivation.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.

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