Constants Lambda Rec Derivation

The recognition length

The recognition length, written λ_rec, is a fundamental scale in Recognition Science. It is not chosen or fitted; it emerges from a balance condition between two costs in a discrete ledger of recognition events. The ledger is a record of discrete events, and each event has a cost. The derivation proceeds in four steps, each proved in a machine-checked library of formal theorems.

First, the recognition composition law, together with reciprocal symmetry, normalization, and continuity, forces the cost functional J(x) = ½(x + x⁻¹) − 1. This is a proved theorem. Second, cost minimization on the discrete ledger forces dimension D = 3 and the cube Q₃ as the elementary recognition cell. Third, polyhedral Gauss-Bonnet on the cube's surface, which is topologically a sphere, fixes the integrated curvature at 4π and forces the curvature cost J_curv(λ) = 2λ². Fourth, the balance condition J_curv(λ) = J_bit, where J_bit is the unit cost of posting one ledger entry, has a unique positive solution: λ_rec = 1/√2 in recognition-cost units. Under the canonical voxel definition ℓ₀ := λ_rec, this becomes λ_rec = 1 in RS-native units.

The derivation is non-circular. The most natural objection is that the SI form λ_rec = √(ħG/(πc³)) = ℓ_P/√π contains G on the right-hand side, so the derivation looks circular if G was itself defined from λ_rec. This misreads the chain. The chain runs in only one direction: J is forced from the recognition composition law (no G, no ħ, no c, no λ_rec); J_bit := 1 is the unit on cost (no G, no ħ, no c, no λ_rec); J_curv(λ) = 2λ² from Gauss-Bonnet on Q₃ (no G, no ħ, no c); balance J_bit = J_curv has unique solution λ_rec = 1/√2 (no G, no ħ, no c); then the Planck gate identity πħG = c³λ_rec² defines G. In particular, G is an output of step 5, never an input to steps 1-4.

The derivation proves existence and uniqueness of the recognition length. The theorem balance_at_lambda_0 shows the balance condition holds at λ₀ = 1/√2. The theorem balance_unique_positive_root shows λ₀ is the unique positive root. The theorem total_curvature_gauss_bonnet shows Q₃ integrated curvature = 4π, the Gauss-Bonnet step on the cube. The theorem J_curv_derivation shows J_curv(λ) = 2λ². The theorem lambda_rec_native_voxel_convention sets λ_rec = ℓ₀ = 1 in RS-native units. The master certificate G_derivation_chain_complete bundles steps 1-5.

In plain language, this derivation establishes that a single length scale, the recognition length, is forced by the framework's own logic. It is not an input; it is an output. The length is the unique scale at which the cost of posting a ledger entry equals the cost of supporting the bounding geometry. This is a structural result, not a numerical coincidence. The recognition length is a derived quantity, and it is the seed from which other constants, such as G, are defined.

/-- Given the bit-cost normalization `J_bit := 1`, the balance condition
`J_curv = J_bit` has a unique positive root `λ₀ = 1 / sqrt 2` in
recognition-cost units.

Caveat (read before quoting `λ₀ = 1/√2` as canonical): the root depends on the
chosen bit-cost normalization. This module uses `J_bit_normalized := 1`;
`Constants/PlanckScaleMatching.lean` instead uses `J_bit := J(φ) = φ - 3/2`,
which gives a *different* cost-unit root `√(J(φ)/2)`. The numeric value of
`λ_rec` in "cost units" is therefore normalization-dependent. The
normalization-INDEPENDENT, physically substantive content is the SI ratio
`λ_rec / ℓ_P = 1/√π` (equivalently the Planck-gate identity `π ℏ G = c³ λ_rec²`,
proved in `PlanckScaleMatching.planck_gate_identity` and
`Unification.QuantumGravityOctaveDuality.G_hbar_gauss_bonnet`). -/
theorem lambda0_forced_in_cost_units :
    ∃! lambda : ℝ, lambda > 0 ∧ balanceResidual lambda = 0 := by
  refine ⟨lambda_0, ⟨lambda_0_pos, balance_at_lambda_0⟩, ?_⟩
  intro lambda h
  exact (balance_unique_positive_root lambda h.1).mp h.2

/-- 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 balance condition J_bit = J_curv uniquely determines lambda. -/
theorem balance_determines_lambda :
    ∃! lambda : ℝ, lambda > 0 ∧ J_curv lambda = J_bit_normalized :=
  balance_unique_pos_root
  where
    balance_unique_pos_root : ∃! lambda : ℝ, lambda > 0 ∧ J_curv lambda = J_bit_normalized := by
      use lambda_0
      refine ⟨⟨lambda_0_pos, ?_⟩, ?_⟩
      · unfold J_curv J_bit_normalized; rw [lambda_0_sq]; ring
      · intro y ⟨hy_pos, hy_eq⟩
        have : balanceResidual y = 0 := by unfold balanceResidual; linarith
        exact (balance_unique_positive_root y hy_pos).mp this

/-- 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

/-- 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

What this page does not claim

The recognition length is not measured from experiment; it is derived from the framework's axioms. The derivation does not claim that the cube Q₃ is the only possible elementary cell; it is forced by dimension D = 3. The derivation does not prove that the SI value of G matches measurement; it defines G from λ_rec via the Planck gate identity.

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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