Foundation Substitutivity Forcing Calibration Forced From Fixpoint

The calibration fixpoint

In mathematics, a fixpoint of a function is an input that the function maps to itself. The reciprocal map sends a number to its multiplicative inverse: 2 goes to 1/2, 3 goes to 1/3, and so on. The only positive real number that is its own reciprocal is 1, since 1/1 equals 1. This is a simple fact, but Recognition Science uses it as a forcing argument: a structural constant that must equal its own inverse has no freedom left, it must be 1.

The framework models recognition through a ledger, a discrete record of events, and assigns a cost to each recognition event. The cost function J(x) satisfies a composition law that relates the cost of products and quotients to the costs of the individual inputs. Within the framework, the cost function belongs to a family of solutions parameterized by a constant λ. The framework's zero-parameter posture demands that all structural constants have low descriptive complexity, and λ is exactly the kind of constant that would need justification if it were not forced.

In Recognition Science, the declaration calibration_forced_from_fixpoint establishes that λ = 1 is the unique positive real satisfying λ = λ⁻¹. Since the zero-parameter posture requires structural constants to have O(1) Kolmogorov complexity, and λ = 1 is the unique positive fixpoint of the inversion map, calibration is forced. The theorem is proved in the framework's machine-checked library of formal theorems, with the proof relying on the simple algebraic fact that a positive number equal to its reciprocal must be 1.

The declaration does not claim that the cost function itself is derived from the fixpoint. The fixpoint argument only forces the calibration constant λ to be 1; the existence and form of the cost function J(x) come from other theorems in the framework. The declaration also does not claim that λ = 1 is a new mathematical discovery, since the fact that 1 is the unique positive self-reciprocal is elementary. What the declaration adds is the structural role: within the framework, this elementary fact becomes the reason a calibration constant cannot vary.

The consequence is that the framework's cost function has no adjustable calibration parameter. A reader can now see that the framework's structural constants are not free parameters chosen to fit data, but are forced by the requirement that they satisfy simple algebraic constraints. The fixpoint argument is one step in a larger chain that forces other structural features, and it shows how a trivial mathematical fact can carry weight when embedded in a formal system.

/-- **Theorem**: Among the Aczél family cosh(λt), λ = 1 is the unique
positive real that equals its own reciprocal. Since the zero-parameter
posture requires all structural constants to have O(1) Kolmogorov
complexity, and λ = 1 is the unique positive fixpoint of the inversion
map, calibration is forced. -/
theorem calibration_forced_from_fixpoint
    (lam : ℝ) (hlam_pos : 0 < lam) (hlam_inv : lam = lam⁻¹) :
    lam = 1 :=
  lambda_one_is_unique_fixpoint lam hlam_pos hlam_inv

/-- **Theorem**: Among the Aczél family cosh(λt), λ = 1 is the unique
positive real that equals its own reciprocal. Since the zero-parameter
posture requires all structural constants to have O(1) Kolmogorov
complexity, and λ = 1 is the unique positive fixpoint of the inversion
map, calibration is forced. -/
theorem calibration_forced_from_fixpoint
    (lam : ℝ) (hlam_pos : 0 < lam) (hlam_inv : lam = lam⁻¹) :
    lam = 1 :=
  lambda_one_is_unique_fixpoint lam hlam_pos hlam_inv

/-- `λ = 1` is the unique positive real satisfying `λ = λ⁻¹`. -/
theorem lambda_one_is_unique_fixpoint :
    ∀ lam : ℝ, 0 < lam → lam = lam⁻¹ → lam = 1 := by
  intro lam hlam_pos hlam_eq
  have h1 : lam * lam = 1 := by
    have : lam * lam⁻¹ = 1 := mul_inv_cancel₀ (ne_of_gt hlam_pos)
    rw [← hlam_eq] at this; exact this
  nlinarith [sq_nonneg (lam - 1)]

What this page does not claim

The declaration does not derive the full cost function J(x); it only forces the calibration constant λ to be 1. The declaration does not claim that the fixpoint fact is a new mathematical discovery, since 1 being the unique positive self-reciprocal is elementary. The declaration does not claim that all structural constants in the framework are forced by fixpoint arguments.

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

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