Constants Kdisplay Core Lambda Kin From Tau Rec

The kinematic display constant

The declaration lambda_kin_from_tau_rec is a small lemma in the Recognition Science framework's machine-checked library of formal theorems. It states a consistency condition between two display constants: the clock-side display constant τ_rec(display) and the length-side display constant λ_kin(display). In plain terms, the lemma says that multiplying the clock-side display constant by the framework's speed constant c gives exactly the length-side display constant. The proof is a one-line algebraic manipulation using the framework's defining relation c · τ₀ = ℓ₀, where τ₀ is a base time unit and ℓ₀ is a base length unit.

The two display constants share a common structure. Both are defined as (2π · base) / (8 ln φ), where φ is the golden ratio and the base is either τ₀ for the clock side or ℓ₀ for the length side. The lemma lambda_kin_from_tau_rec verifies that these two definitions are consistent with the speed relation c = ℓ₀ / τ₀. It is a bookkeeping check, not a new physical law. The framework's library also proves that each display constant divided by its base unit equals a common ratio K_gate_ratio = π / (4 ln φ), which is a constant that appears in the framework's display conventions.

The lemma does not claim that the speed of light c is derived from first principles. The relation c · τ₀ = ℓ₀ is an input assumption of the framework's unit system, not a consequence of this lemma. The lemma also does not establish any numerical value for c, τ₀, or ℓ₀; those are parameters of the framework's unit choice. What the lemma does establish is internal consistency: given the framework's definitions, the two display constants are linked in exactly the way the speed relation requires. It is a coherence result, confirming that the framework's display conventions do not contradict its unit definitions.

In the broader context of the Recognition Science framework, this lemma is a minor but necessary piece. It supports the framework's claim that its display constants are not arbitrary but follow from a single underlying structure. For a reader outside the framework, the practical takeaway is simple: the declaration checks that two defined quantities agree with each other under the framework's own rules. It does not, by itself, connect those quantities to any measured physical value or experimental prediction.

/-- Kinematic consistency: c · τ_rec(display) = λ_kin(display). -/
lemma lambda_kin_from_tau_rec (U : RSUnits) : U.c * tau_rec_display U = lambda_kin_display U := by
  simp only [tau_rec_display, lambda_kin_display]
  -- Goal: U.c * (2 * π * τ₀ / (8 * log φ)) = 2 * π * ℓ₀ / (8 * log φ)
  have h : U.c * U.tau0 = U.ell0 := U.c_ell0_tau0
  calc U.c * (2 * Real.pi * U.tau0 / (8 * Real.log phi))
      = (2 * Real.pi * (U.c * U.tau0)) / (8 * Real.log phi) := by ring
    _ = (2 * Real.pi * U.ell0) / (8 * Real.log phi) := by rw [h]

/-- Clock-side display definition: τ_rec(display) = (2π·τ₀) / (8 ln φ). -/
@[simp] noncomputable def tau_rec_display (U : RSUnits) : ℝ :=
  (2 * Real.pi * U.tau0) / (8 * Real.log phi)

/-- Length-side (kinematic) display definition: λ_kin(display) = (2π·ℓ₀) / (8 ln φ). -/
@[simp] noncomputable def lambda_kin_display (U : RSUnits) : ℝ :=
  (2 * Real.pi * U.ell0) / (8 * Real.log phi)

/-- Clock-side ratio equals K_gate_ratio. -/
@[simp] lemma tau_rec_display_ratio (U : RSUnits) (hτ : U.tau0 ≠ 0) :
  (tau_rec_display U) / U.tau0 = K_gate_ratio := by
  unfold tau_rec_display K_gate_ratio
  field_simp [hτ]
  ring

/-- Length-side ratio equals K_gate_ratio. -/
@[simp] lemma lambda_kin_display_ratio (U : RSUnits) (hℓ : U.ell0 ≠ 0) :
  (lambda_kin_display U) / U.ell0 = K_gate_ratio := by
  unfold lambda_kin_display K_gate_ratio
  field_simp [hℓ]
  ring

What this page does not claim

The lemma does not derive the numerical value of the speed of light c from first principles. The lemma does not establish any physical prediction or experimental comparison. The lemma does not define what the display constants mean physically; it only checks algebraic consistency.

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/KDisplayCore.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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