Constants Lambda Rec Derivation J Curv Coefficient Forced

The forced curvature coefficient

The theorem J_curv_coefficient_forced belongs to a chain of formal results in the Recognition Science framework. That framework starts from a single idea: reality keeps a discrete record of recognition events, a ledger, and the cost of posting each entry is forced by logic rather than chosen by hand. The theorem states that the curvature cost, the price paid for supporting the spherical geometry that bounds a three-dimensional recognition cell, must take the form J_curv(λ) = 2λ². The coefficient 2 is not an input; it is derived from the Euler characteristic of a sphere, which for the cube used as the elementary cell equals 2.

To see what the coefficient means, consider the cube with its eight vertices, twelve edges, and six faces. Polyhedral Gauss-Bonnet, a classical theorem in geometry, says the total angular deficit over all vertices equals 2π times the Euler characteristic. For a sphere, that characteristic is 2. The framework's library formalizes this step and then proves that the curvature cost equals the Euler characteristic times λ². Since the characteristic is 2, the cost is 2λ². The declaration J_curv_coefficient_forced bundles these two facts: the cost formula and the value of the coefficient, both machine-checked in the framework's library of formal theorems.

The coefficient's value matters because it feeds the balance condition that fixes the recognition length λ_rec. The framework equates the curvature cost to the bit cost, normalized to 1, and solves for λ. The unique positive solution is λ = 1/√2 in recognition-cost units, or λ = 1 in the framework's native voxel convention. This is a derived result, not a fitted one: no gravitational constant, Planck constant, or speed of light enters the derivation. Those constants appear only later, in a separate step that defines G from λ_rec, never the other way around.

What the theorem does not claim is just as important. It does not claim that the coefficient 2 is a physical measurement; it is a mathematical consequence of the framework's definitions and the Gauss-Bonnet theorem. It does not claim that the curvature cost formula applies outside the framework's discrete ledger model, nor that the recognition length λ_rec has been measured in any experiment. The theorem is a formal statement about a specific cost function within a specific model, not a claim about empirical reality. The framework's own documentation is explicit that the derivation is non-circular: G is an output of the chain, never an input to the steps that force J_curv.

/-- **Coefficient-forcing summary.** `J_curv(λ) = (Σδ / 2π) · λ²` and the
coefficient `Σδ / 2π` equals the Euler characteristic `2`. Both clauses are
proved from the cube combinatorics plus discrete Gauss-Bonnet; the only
non-theorem residue is the functional form `cost = coeff · λ²` itself. -/
theorem J_curv_coefficient_forced (lambda : ℝ) :
    J_curv lambda = curvatureCoefficient * lambda ^ 2 ∧
      curvatureCoefficient = (euler_S2 : ℝ) :=
  ⟨J_curv_eq_coefficient_mul_sq lambda, curvatureCoefficient_eq_euler_char⟩

/-- **The coefficient is the Euler characteristic.** Discrete Gauss-Bonnet
(`total_curvature_gauss_bonnet`) forces the defect-per-`2π` to equal
`χ(S²) = 2`. The coefficient in `J_curv = 2λ²` is therefore derived, not
posited. -/
theorem curvatureCoefficient_eq_euler_char :
    curvatureCoefficient = (euler_S2 : ℝ) := by
  unfold curvatureCoefficient
  rw [total_curvature_gauss_bonnet]
  have hpi : Real.pi ≠ 0 := Real.pi_ne_zero
  field_simp

/-- 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

/-- 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 coefficient 2 is a measured physical constant rather than a derived mathematical consequence. The curvature cost formula applies outside the framework's discrete ledger model. The recognition length λ_rec has been directly observed in any experiment.

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)

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