Constants Lambda Rec Derivation Balance Unique Positive Root

The unique balance point

The declaration balance_unique_positive_root is a machine-checked theorem inside the Recognition Science framework. It states that a certain equation, the balance condition, has exactly one positive solution. The equation equates two costs: the bit cost, which is the price of posting one entry in the framework's discrete record of events, and the curvature cost, which is the price of supporting the geometry that bounds a cube. The unique positive solution is the number one divided by the square root of two, or approximately 0.7071.

This result is the fourth step in a longer chain. The chain begins with a cost functional J(x) = (x + 1/x)/2 - 1, forced by five plain conditions. From that functional, the framework derives a three-dimensional cube as the elementary cell. Polyhedral Gauss-Bonnet, a classical theorem about total curvature, fixes the integrated curvature of the cube's surface at 4π. That curvature gives rise to the curvature cost J_curv(λ) = 2λ². The balance condition sets this equal to the bit cost, normalized to 1. The theorem proves that this condition holds at λ = 1/√2 and nowhere else among positive numbers.

The proof is short and algebraic. It uses the fact that J_curv(λ) = 2λ² and J_bit = 1. Setting them equal gives 2λ² = 1, so λ² = 1/2. Since λ is positive, λ = 1/√2. The theorem's uniqueness part rules out any other positive root. The declaration also carries a normalization note: the value 1/√2 depends on the choice of bit cost as 1. Under a later convention that sets the voxel length to 1, the same balance point becomes λ = 1 in the framework's native units.

The theorem does not claim that the balance point is a measured physical constant. It does not claim that the number 1/√2 appears in any experiment. It does not claim that the balance condition itself is forced by the five plain conditions; that forcing is a separate theorem. The declaration only establishes uniqueness of the positive root given the balance condition. The physical interpretation of the balance point, as a recognition length, is a separate claim in the framework's larger derivation.

/-- 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

/-- Step 4: balance residual.

The balance condition `J_curv(λ) = J_bit` is equivalent to
`balanceResidual(λ) = 0`.  The residual vanishes at exactly one positive
scale, namely `λ_0 = 1/√2`. -/
noncomputable def balanceResidual (lambda : ℝ) : ℝ :=
  J_curv lambda - J_bit_normalized

/-- 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

What this page does not claim

The theorem does not claim that the balance point is a measured physical constant. The theorem does not claim that the balance condition itself is forced by the five plain conditions. The theorem does not claim that the value 1/√2 appears 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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Derived articles

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