Constants Lambda Rec Derivation Lambda Rec Native Voxel Convention

The native voxel convention

The recognition length λ_rec is a quantity in the Recognition Science framework: the unique positive length at which the cost of posting one ledger entry, a discrete record of events, balances the cost of supporting the bounding spherical geometry. The framework's machine-checked library of formal theorems proves this length exists and is unique. In the cost units used for that proof, the length is 1/√2, roughly 0.7071.

The declaration lambda_rec_native_voxel_convention is not a further physical result. It is a unit convention, a definitional choice: it sets the reference length ℓ₀ equal to λ_rec, and then declares that reference length to be exactly 1 in the framework's native units. The theorem states λ_rec = ℓ₀ and ℓ₀ = 1. This is the framework choosing its own yardstick, not discovering a new fact about the world.

Why make this choice? The derivation that produces λ_rec runs from the recognition composition law through dimension forcing and Gauss-Bonnet to a balance condition, with no free parameter and no calibration to measured data. The balance condition has a unique positive solution, and that solution is normalization-dependent: in the bit-cost normalization it is 1/√2. The native voxel convention rescales so that the fundamental length is simply 1, which makes the framework's internal arithmetic cleaner. The normalization-free content of the derivation is the ratio λ_rec/ℓ_P = 1/√π, where ℓ_P is the Planck length.

What the convention does not claim is just as important. It does not claim that the recognition length is physically one meter, one Planck length, or any particular SI value. It does not claim that the choice of units has empirical content. The convention is a bookkeeping step inside the framework's own system of units, and it changes nothing about the derivation's substance. The derivation's content lies in the uniqueness and the ratio, not in the number 1.

/-- 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

/-- 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

/-- 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

What this page does not claim

The convention does not assign the recognition length any particular SI value, such as one meter or one Planck length. The convention does not add empirical content to the derivation; it is a unit choice. The convention does not prove that the recognition length is physically fundamental.

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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