Foundation Primitive Recognition Calculus Prccalibration Target Gauge Action Tra

The one remaining freedom

The Recognition Science framework begins with a ledger, a discrete record of recognition events, and asks what a cost function, the price of recording each event, must look like. The framework's central theorem forces a specific form, but it leaves one knob free: a positive real number that rescales the cost. The declaration gauge_action_transitive states that this knob is the only freedom, and that turning it moves any cost in the family to any other.

Concretely, the cost family is written as cosh(c·t) − 1, where c is a positive real parameter and t is the log-coordinate of the event. The theorem says that for any two positive parameters c and d, there exists a positive rescaling μ such that cosh(c·(μ·t)) − 1 = cosh(d·t) − 1. In plain words: rescaling the input by the right positive factor turns one cost into another. The family is a single orbit under rescaling, not a collection of unrelated functions.

This is the sharpened form of a calibration result. The framework proves that the discrete ledger structure does not fix the unit of scale; that unit is a gauge, a free choice. The theorem gauge_action_transitive, together with its companion clog_inj, shows the freedom is exactly one real number. The action is free, meaning different rescalings give different costs, and transitive, meaning any cost reaches any other. Together these make the family a torsor: a space where any two points are connected by a unique translation, here a rescaling. The single free number is fixed only by an external calibration datum, such as setting the curvature at the unit to 1, which the discrete structure alone does not supply.

What the theorem does not claim is as important as what it proves. It does not say which value of c is physically realized; that requires the calibration datum. It does not assert that the cost family is the only possible one, only that within this family the freedom is one-dimensional. And it does not imply that the gauge freedom is observable; the theorem is about the mathematical structure of the cost functions, not about what an experiment can measure. The framework's later results, such as the derivation of the golden ratio and the number of spatial dimensions, depend on choosing the calibrated member c = 1, which is exactly the cost J, but that choice is a separate step.

/-- The gauge action `μ • F := F(μ · )` is transitive on the cost family: any
member reaches any other through a positive rescaling of the log-coordinate. -/
theorem gauge_action_transitive {c d : ℝ} (hc : 0 < c) (hd : 0 < d) :
    ∃ μ : ℝ, 0 < μ ∧
      (fun t => Real.cosh (c * (μ * t)) - 1) = (fun t => Real.cosh (d * t) - 1) := by
  refine ⟨d / c, div_pos hd hc, ?_⟩
  funext t
  have hcne : c ≠ 0 := ne_of_gt hc
  have hkey : c * (d / c * t) = d * t := by field_simp
  rw [hkey]

/-- **Item 2, sharpened: the residual freedom is a torsor, exactly one real.** The
gauge action of the positive reals on the cost family is free (`clog_inj`) and
transitive (`gauge_action_transitive`). A free transitive action exhibits the
family as a principal homogeneous space under `(ℝ_{>0}, ·)`, so the residual
freedom in the cost is exactly one positive real, the unit of scale. It is fixed
by one calibration datum (curvature 1), and that datum is not supplied by the
discrete δ structure. -/
theorem cost_freedom_is_one_real_torsor :
    (∀ 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)) :=
  ⟨fun _ _ hc hd h => clog_inj hc hd h, fun _ _ hc hd => gauge_action_transitive hc hd⟩

/-- **Item 2 headline.** The δ-forced cost form leaves a faithful one-parameter
gauge family; its only invariant is the log-curvature c² at the unit; and
"curvature = 1" is exactly the condition selecting J. The unit c is a free
positive real (a gauge), because curvature is a continuum-side second-derivative
property that the discrete δ carrier does not fix. -/
theorem calibration_unit_is_a_gauge :
    (∀ c : ℝ, deriv (deriv (fun t => Real.cosh (c * t) - 1)) 0 = c ^ 2)
      ∧ (∀ c d : ℝ, 0 < c → 0 < d →
          (fun t => Real.cosh (c * t) - 1) = (fun t => Real.cosh (d * t) - 1) → c = d)
      ∧ (∀ c : ℝ, 0 < c →
          (deriv (deriv (fun t => Real.cosh (c * t) - 1)) 0 = 1 ↔ c = 1)) :=
  ⟨logCurvature, fun _ _ hc hd h => clog_inj hc hd h, fun _ hc => curvature_one_iff_J hc⟩

What this page does not claim

The theorem does not claim which value of the positive parameter c is physically realized. The theorem does not claim the cost family is the only possible family, only that within this family the freedom is one-dimensional. The theorem does not claim the gauge freedom is observable.

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

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