Foundation Primitive Recognition Calculus Delta Real Calibration Calibration Is

One continuum act

Recognition Science studies systems that keep a discrete record of events, a ledger. The cost of recognizing an event is forced by five plain conditions, but one freedom remains: the unit in which cost is measured. You can rescale the cost function and still satisfy every condition. The discrete ledger alone cannot tell you which scale is right.

The framework's machine-checked library of formal theorems shows exactly what the discrete data can and cannot do. Distinct units always give distinct costs, so the freedom is a genuine one-real torsor: a line of possible units with no preferred point. No discrete datum fixes the unit. The gap is real and must be closed from outside the ledger.

In Recognition Science, the closure comes from a single continuum-interface quantity: the one-act curvature. For a cost member with unit c, the one-act curvature is the second derivative at the limit ratio t = 0 of the function cosh(c·t) − 1. The theorem proves this curvature equals c². The continuum interface reads the residual gauge parameter as a second derivative.

The key normalization is "one-act curvature equals 1." For positive c, this condition holds if and only if c = 1. The theorem calibration_is_one_continuum_act assembles the full classification: the curvature is c², the normalization forces c = 1, and distinct units remain distinct. A single continuum datum, no more and no less, selects the canonical cost unit.

The framework also defines a minimal normalized interface: a positive unit together with the assertion that its one-act curvature is 1. Any such interface forces the unit to be 1, and the canonical unit 1 itself carries such an interface. The calibration gap is closed exactly by this minimal second-order recognition interface. The honest conditional is that the unit is not discrete-forced; it is fixed by one named continuum act.

What this does not claim: the continuum act is not a full continuum or a field completion, only the single curvature datum. The theorem does not derive the existence of a continuum from the discrete ledger; it assumes the interface. And the theorem does not say the unit is forced by the discrete laws alone, which the proof explicitly denies.

/-- The one-act curvature is `c²`: the residual gauge parameter read as a
second derivative. -/
theorem oneActCurvature_eq (c : ℝ) : oneActCurvature c = c ^ 2 :=
  Calibration.logCurvature c

/-- **One continuum datum fixes the unit.** At the continuum interface, the single
normalization "one-act curvature equals 1" forces `c = 1`, i.e. selects the
canonical J. -/
theorem unit_forced_by_one_act {c : ℝ} (hc : 0 < c) :
    oneActCurvature c = 1 ↔ c = 1 := by
  unfold oneActCurvature
  exact Calibration.curvature_one_iff_J hc

/-- **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

/-- 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

/-- 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 continuum act is not a full continuum or field completion, only the single curvature datum. The theorem does not derive the existence of a continuum from the discrete ledger. The unit is not forced by the discrete laws alone.

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:

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

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