Encyclopedia Constants Constants Lambda Rec Derivation Lambda0 Forced In Cost Units

ARTICLE 5 claims 5 theorems

Constants Lambda Rec Derivation Lambda0 Forced In Cost Units

A single number, the recognition length, emerges from balancing two costs, and its value depends on the units you choose to measure it in.

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The recognition length

The recognition length is a fundamental scale in the Recognition Science framework, and the declaration lambda0_forced_in_cost_units pins down its value under a specific normalization. The framework models reality as a ledger, a discrete record of events, where each entry carries a cost. The declaration establishes that there is exactly one positive value of this length that balances two competing costs: the cost of posting a single ledger entry, and the cost of supporting the geometry that bounds the system. That unique value, in the units where the entry cost is set to one, is 1 divided by the square root of 2, approximately 0.7071.

The balance condition is not arbitrary. The curvature cost is forced by a classical theorem: the total curvature of a sphere, as measured by the Gauss-Bonnet theorem, is 4π. Applied to the cube that the framework identifies as the elementary recognition cell, this fixes the curvature cost as twice the square of the length. Setting this equal to the unit entry cost gives the equation 2λ² = 1, whose only positive solution is λ = 1/√2. The declaration lambda0_forced_in_cost_units is the formal statement of this existence and uniqueness, and it is a proved theorem in the framework's machine-checked library of formal theorems.

The value 1/√2 is not a universal constant in the way the speed of light is. It depends on the choice of units: it is the value when the cost of a single ledger entry is normalized to one. The framework's own later convention sets the recognition length to exactly 1 in its native units, and the declaration lambda_rec_native_voxel_convention records this. The deeper, normalization-free content is the ratio of the recognition length to the Planck length, which the framework derives as 1/√π. The number 1/√2 is the root of the balance equation in one particular unit system, not an intrinsic property of the length itself.

What the declaration does not claim is equally important. It does not introduce Newton's gravitational constant G, the reduced Planck constant ħ, or the speed of light c as inputs. The derivation runs from the cost balance alone, and G appears only later, as an output of a separate step that connects the framework's units to SI units. The declaration also does not claim that the recognition length is measured or observed; it is derived from the framework's postulates. Finally, it does not claim that the value 1/√2 is meaningful outside the specific normalization where the entry cost is one. The framework's own native convention chooses a different value, and the declaration is silent on which convention is physically preferred.

THEOREM lambda0_forced_in_cost_units · IndisputableMonolith/Constants/LambdaRecDerivation.lean
lambda0_forced_in_cost_units · IndisputableMonolith/Constants/LambdaRecDerivation.lean:208 
/-- 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
THEOREM lambda0_forced_in_cost_units · IndisputableMonolith/Constants/LambdaRecDerivation.lean
lambda0_forced_in_cost_units · IndisputableMonolith/Constants/LambdaRecDerivation.lean:208 
/-- 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
THEOREM J_curv_derivation · IndisputableMonolith/Constants/LambdaRecDerivation.lean
J_curv_derivation · IndisputableMonolith/Constants/LambdaRecDerivation.lean:353 
/-- 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
THEOREM lambda_rec_native_voxel_convention · IndisputableMonolith/Constants/LambdaRecDerivation.lean
lambda_rec_native_voxel_convention · IndisputableMonolith/Constants/LambdaRecDerivation.lean:227 
/-- 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 lambda0_forced_in_cost_units · IndisputableMonolith/Constants/LambdaRecDerivation.lean
lambda0_forced_in_cost_units · IndisputableMonolith/Constants/LambdaRecDerivation.lean:208 
/-- 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 declaration does not derive the value of Newton's gravitational constant G, the reduced Planck constant ħ, or the speed of light c. The declaration does not claim that the recognition length is a measured quantity; it is derived from the framework's postulates. The declaration does not claim that the value 1/√2 is meaningful outside the specific normalization where the entry cost is one.

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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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