Foundation Primitive Recognition Calculus Delta Real Calibration Calibration Gap

The calibration gap

The Recognition Science framework builds physical structure from a single starting point: reality keeps a ledger, a discrete record of recognition events, and the cost of each recognition is forced, not chosen. From five plain conditions, the framework proves the cost function must take the form J(x) = (x + 1/x)/2 - 1. But a gap remains: the discrete ledger alone does not fix the unit, the scale on which costs are measured. Distinct units give distinct cost functions, and the framework proves that the residual freedom is a one-real torsor, meaning the unit can slide freely along the real line without changing any discrete fact.

The declaration calibration_gap_closed_by_normalized_interface closes that gap with exactly one continuum-interface datum. The interface is minimal: it asserts only that the one-act curvature, the second derivative of the cost at its limit ratio, equals 1. The framework proves that for a positive unit, this single normalization is both necessary and sufficient to force the unit to equal 1, selecting the canonical cost function J. The theorem bundles four facts: any normalized interface forces the canonical unit; the curvature datum is equivalent to the unit being 1; such an interface exists; and distinct positive units always give distinct cost functions.

What this does not claim is as precise as what it proves. The calibration is not forced by the discrete ledger alone; the framework states this explicitly, and the continuum datum is named rather than hidden. The interface is not a full continuum and not a field completion, only a positive cost unit plus the one-act curvature assertion. And the theorem does not derive the value of any physical constant; it selects the unit of the cost function within the framework's own model, leaving empirical comparison as a separate check.

The payoff is a fully classified cost-unit issue. A reader can now see exactly where the framework's freedom lives, what single measurement removes it, and why that measurement is the minimal one. The gap is not waved away; it is closed by name, with one datum, no more and no less.

/-- **The discrete carrier does not force the unit.** The cost family is faithful
(distinct units give distinct costs) and transitively rescaled, so the residual
freedom is a one-real torsor. Without the one-act normalization datum the unit is
genuinely free. -/
theorem discrete_does_not_force_unit :
    (∀ c d : ℝ, 0 < c → 0 < d →
        (fun t => Real.cosh (c * t) - 1) = (fun t => Real.cosh (d * t) - 1) → c = d)
      ∧ (∀ c d : ℝ, 0 < c → 0 < d →
          ∃ μ : ℝ, 0 < μ ∧
            (fun t => Real.cosh (c * (μ * t)) - 1) = (fun t => Real.cosh (d * t) - 1)) :=
  Calibration.cost_freedom_is_one_real_torsor

/-- For positive units, the one-act curvature datum is necessary and sufficient
for selecting the canonical member. This is the exact closure of the calibration
gap: one datum, no more and no less. -/
theorem calibration_datum_necessary_and_sufficient {c : ℝ} (hc : 0 < c) :
    c = 1 ↔ oneActCurvature c = 1 := by
  exact (unit_forced_by_one_act hc).symm

/-- **Calibration closure theorem.** The discrete laws leave a faithful one-real
torsor (`discrete_does_not_force_unit`), while the normalized one-act interface is
both sufficient and necessary for `c = 1`. Thus the cost-unit issue is fully
classified: it is not discrete-forced; it is closed exactly by the minimal
second-order recognition interface. -/
theorem calibration_gap_closed_by_normalized_interface :
    (∀ I : NormalizedOneActInterface, I.unit = 1)
      ∧ (∀ c : ℝ, 0 < c → (c = 1 ↔ oneActCurvature c = 1))
      ∧ (∃ I : NormalizedOneActInterface, I.unit = 1)
      ∧ (∀ c d : ℝ, 0 < c → 0 < d →
          (fun t => Real.cosh (c * t) - 1) = (fun t => Real.cosh (d * t) - 1) → c = d) :=
  ⟨normalized_interface_forces_J, fun _ hc => calibration_datum_necessary_and_sufficient hc,
    ⟨canonicalInterface, rfl⟩, fun _ _ hc hd h => Calibration.clog_inj hc hd h⟩

What this page does not claim

The calibration is not forced by the discrete ledger alone; the continuum datum is essential. The theorem does not derive the value of any physical constant, only the unit of the cost function. The interface is not a full continuum or field completion, only a positive unit plus the curvature assertion.

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/Foundation/PrimitiveRecognitionCalculus/DeltaRealCalibration.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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