Foundation Primitive Recognition Calculus Delta Real Calibration Normalized Inte

The calibration gap

The declaration normalized_interface_forces_J closes a specific gap in the Recognition Science framework. The framework's discrete laws, which govern how recognition events are recorded, leave one real number free: the unit of cost, the price the ledger charges for a recognition event. This is not a flaw; the discrete structure is faithful, meaning distinct units give distinct costs, and it is transitively rescaled, so the freedom is exactly a one-real torsor. The theorem states that adding a single continuum datum, the assertion that the one-act curvature equals 1, forces that unit to be exactly 1.

To understand what is forced, consider the cost family. The framework's cost function is J(x) = (x + 1/x)/2 - 1, but before calibration, any positive scaling of the input is also a valid cost. The one-act curvature is the second derivative of the cost at the identity, evaluated in a continuum limit. For a cost with unit c, this curvature equals c². The theorem unit_forced_by_one_act proves that if this curvature equals 1 and c is positive, then c must be 1. The declaration normalized_interface_forces_J packages this: any structure that provides a positive unit and the curvature normalization forces the canonical unit.

The theorem is precise about what it does not claim. It does not say the discrete laws alone force the unit; in fact, the theorem discrete_does_not_force_unit proves the opposite, that the discrete carrier leaves the unit genuinely free. The calibration is not discrete-forced. It also does not claim the continuum datum is hidden or derived from nothing; the theorem calibration_datum_necessary_and_sufficient shows the curvature normalization is both necessary and sufficient for c = 1, given positivity. The gap is closed exactly by one named, second-order continuum act, no more and no less.

What this changes is the status of the cost unit. Before this theorem, the unit was a free parameter, a residual gauge freedom. After it, the unit is fixed by a single physical normalization. The framework's library proves the classification is complete: the discrete laws leave a faithful one-real torsor, and the minimal normalized interface closes it. This is the honest conditional: the calibration is not discrete-forced, and the one continuum datum is named rather than hidden.

/-- Any normalized one-act interface forces the canonical cost unit. -/
theorem normalized_interface_forces_J (I : NormalizedOneActInterface) :
    I.unit = 1 :=
  (unit_forced_by_one_act I.positive).mp I.curvature_unit

/-- **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

What this page does not claim

The discrete recognition laws alone force the cost unit to be 1. The one-act curvature datum is derived from the discrete structure rather than added as a separate continuum assumption. The theorem claims the normalization is physically realized, only that it is mathematically sufficient and necessary.

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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