Foundation Ledger Canonicality Zero Parameter Comparison Ledger

The zero-parameter ledger

A ledger, in Recognition Science, is a discrete record of events. The declaration ZeroParameterComparisonLedger defines the minimal such record that the framework's theorems consume. It packages five ingredients: a countable set of possible states, a way to compare any two states with a symmetric cost, a conserved scalar quantity called log-charge, no external adjustable settings, and closure under composition so that compound comparisons are always well-defined. The name "zero-parameter" is the load-bearing part: the parameter record is trivial, meaning the object carries no free knobs that could be tuned to fit a result.

The cost function in this ledger is not arbitrary. It is an AdmissibleCost: a function on positive real numbers satisfying reciprocal symmetry, unit normalization, strict convexity, continuity, and calibration. In plain terms, the cost of comparing x to y must equal the cost of comparing y to x, comparing something to itself costs nothing, and the function bends upward smoothly. The calibration condition fixes the second derivative at a specific point. These are the minimal axioms from which the framework's central theorem, the uniqueness of the cost function J(x) = (x + 1/x)/2 - 1, is proved in the machine-checked library.

The ledger also defines a neutral sector: the set of states with zero charge. A class HasNeutralStates asserts this sector is nonempty in the generic case, and HasMultilevelComposition adds that events compose at more than one scale, inducing a discrete hierarchy of level sizes. The local composition class then constrains adjacent levels to interact, with positive integer coefficients. The docstring points to a separate proof that zero-parameter minimality forces those coefficients to (1,1), which yields the golden ratio φ. That derivation lives in HierarchyDynamics, not in this declaration itself.

What this declaration does not claim is as important as what it does. It does not assert that any particular physical system realizes this ledger; it only defines the interface that downstream theorems consume. It does not prove conservation of charge under dynamics; that requires a separate transition map, as the docstring explicitly notes after deleting a former field that asserted nothing. And it does not itself derive the golden ratio or any emergence theorem. It is the foundation stone, not the building.

In Recognition Science, this object is the entry point for the unconditional inevitability theorem: the claim that any system satisfying these minimal conditions must exhibit the structure the framework derives. The payoff is that a reader can see exactly what is assumed, and exactly what is not. The ledger names its own limits, which is what makes the framework's larger claims checkable.

/-- A zero-parameter local conserved comparison ledger packages all the
primitive structure needed for the unconditional theorem. -/
structure ZeroParameterComparisonLedger where
  Carrier : Type
  carrier_nonempty : Nonempty Carrier
  carrier_countable : ∃ (f : ℕ → Carrier), Function.Surjective f
  cost : AdmissibleCost
  charge : ConservedCharge Carrier
  no_free_knobs : ∀ (embed : ℝ → Carrier), ¬ Function.Injective embed
  cost_sufficient : ∀ (x₁ x₂ y : ℝ), 0 < x₁ → 0 < x₂ →
    cost.J x₁ = cost.J x₂ → 0 < y →
    cost.J (x₁ * y) + cost.J (x₁ / y) = cost.J (x₂ * y) + cost.J (x₂ / y)
  has_composition : ∀ x y : ℝ, 0 < x → 0 < y →
    ∃ P : ℝ → ℝ → ℝ,
      cost.J (x * y) + cost.J (x / y) = P (cost.J x) (cost.J y)
  composition_continuous : ∀ x y : ℝ, 0 < x → 0 < y →
    ∃ P : ℝ → ℝ → ℝ, Continuous (Function.uncurry P) ∧
      cost.J (x * y) + cost.J (x / y) = P (cost.J x) (cost.J y)

/-- A comparison cost on positive reals satisfying the minimal ledger
axioms: reciprocal symmetry, unit normalization, strict convexity,
continuity, and calibration. -/
structure AdmissibleCost where
  J : ℝ → ℝ
  reciprocal_sym : ∀ x : ℝ, 0 < x → J x = J (x⁻¹)
  unit_norm : J 1 = 0
  strict_convex : StrictConvexOn ℝ (Set.Ioi 0) J
  continuous : ContinuousOn J (Set.Ioi 0)
  calibration : (deriv (deriv (fun t => J (Real.exp t)))) 0 = 1

/-- A ledger scalar quantity on a type `α` with values in `ℝ`.

Conservation under dynamics is not a property of a bare charge function: it
requires a transition map (see `ClosedFramework.ClosedObservableFramework`,
field `charge_conserved : ∀ s, charge (T s) = charge s`). Former field
`charge_conserved : ∀ s₁ s₂, charge s₁ = charge s₂ → True` deleted (Move A):
equal charges implying `True` excluded nothing. -/
structure ConservedCharge (α : Type) where
  charge : α → ℝ

/-- The neutral sector of a ledger is the set of states with zero charge. -/
def neutralSector (L : ZeroParameterComparisonLedger) : Set L.Carrier :=
  { s | L.charge.charge s = 0 }

What this page does not claim

This declaration does not prove conservation of charge under dynamics; that requires a separate transition map. It does not itself derive the golden ratio or any emergence theorem; those live in HierarchyDynamics. It does not assert that any particular physical system realizes this ledger.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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