Constants Lambda Rec Derivation G Derivation Chain Complete

The derivation certificate

The declaration G_derivation_chain_complete is a machine-checked certificate, a single formal object that bundles five theorems from the Recognition Science library into one chain. The chain starts from a cost function, the price paid for a discrete record of events, which the library proves must take the form J(x) = ½(x + x⁻¹) − 1. From that cost, the chain forces a three-dimensional cube as the elementary recognition cell, then uses polyhedral Gauss-Bonnet to fix the cube's total curvature at 4π, which yields a curvature cost of 2λ². The balance condition, equating this curvature cost to the bit cost of posting one ledger entry, has a unique positive solution: λ_rec = 1/√2 in cost units, or λ_rec = 1 in the framework's native voxel convention.

The certificate's final step is the Planck gate identity, πℏG = c³λ_rec², which defines G as an output of the chain, never an input. The derivation is non-circular: steps one through four involve no G, no ℏ, no c, and no λ_rec. The certificate is a theorem in the framework's library, meaning it is proved in the Lean 4 proof assistant with no axioms beyond the standard three, and it is a claim about the framework's internal structure, not about measured physics.

What the certificate does not claim is equally important. It does not claim to derive the fine-structure constant α; the framework's expression for α lands within about 5.6 parts per million of the measured value with nothing fitted, but its seed, 44π, is an identification, not a derived coupling. It does not claim to prove the Riemann Hypothesis; every statement about RH in the library is an equivalence, not a proof. It does not claim that G is a free parameter or that the chain rests on nothing; the audit to [propext, Classical.choice, Quot.sound] is a claim about postulates, not about the ambient type theory.

The certificate's value is that it makes the derivation auditable: every step, from the cost function to the balance condition to the definition of G, is checked by a machine against the framework's axioms. This is not a claim that the framework is true, but that its internal logic is sound. The certificate is a target for verification, not a proof of physical reality.

/-- 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

/-- J_curv = 2λ² is the curvature cost per recognition token.
    Derivation: |κ_normalized| × (4πλ²) / (2π × χ(S²))
    = 1 × (4πλ²) / (2π × 2) = 2λ² / 2 ... wait, let's be precise:
    J_curv = (|κ|/(2χ)) × (A/(2π)) where |κ| = 4, χ = 2, A = 4πλ²
    = (4/4) × (4πλ²/(2π)) = 1 × 2λ² = 2λ². -/
theorem J_curv_derivation (lambda : ℝ) :
    J_curv lambda = 2 * lambda ^ 2 := rfl

/-- 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

/-- The RS-native convention sets the voxel length to one recognition length:
`lambda_rec = ell0 = 1`.  The derived content is `lambda0_forced_in_cost_units`;
this theorem records the subsequent native-unit gauge choice. -/
theorem lambda_rec_native_voxel_convention :
    lambda_rec = ell0 ∧ ell0 = 1 := by
  constructor
  · rfl
  · rfl

theorem G_derivation_chain_complete : GDerivationChain where
  step1_Q3_vertices := rfl
  step2_gauss_bonnet := total_curvature_gauss_bonnet
  step3_J_curv_formula := J_curv_derivation
  step4_balance_unique := balance_determines_lambda
  step5_G_formula := rfl
  step6_kappa := Constants.kappa_einstein_eq

theorem G_derivation_chain_complete : GDerivationChain where
  step1_Q3_vertices := rfl
  step2_gauss_bonnet := total_curvature_gauss_bonnet
  step3_J_curv_formula := J_curv_derivation
  step4_balance_unique := balance_determines_lambda
  step5_G_formula := rfl
  step6_kappa := Constants.kappa_einstein_eq

What this page does not claim

The certificate does not derive the fine-structure constant α. The certificate does not prove the Riemann Hypothesis. The certificate does not claim G is a free parameter or that the chain rests on nothing.

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.

Derived articles

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