Foundation Neutral Sector Parameter Free Ratios Are Unity

The zero-parameter rule

A ledger, in this framework, is a discrete record of events and their relative sizes. The declaration parameter_free_ratios_are_unity states a constraint on any such ledger that is meant to be fully self-contained: if no external data is allowed to pick out a special size, then every observable ratio between states must be exactly 1.

The reasoning is direct. An observable state must be repeatable and internally generated, requiring no outside input to specify. A nonzero conserved-sector label, written Q, would encode a real number, an extra free parameter. Under a zero-parameter posture, the only admissible label is Q = 0, the additive identity. Since the log-charge of a ratio is just its natural logarithm, a zero log-charge forces the ratio itself to be 1. The formal theorem chains these steps: parameter-free observables are neutral, neutral ratios equal one, and therefore parameter-free ratios are unity.

This is a conditional result, not a claim about any particular physical system. It applies only to ledgers that adopt the zero-parameter posture, and only to ratios that are observable in the sense of being repeatable and internally generated. It does not say that all ratios in nature equal 1; it says that if you forbid free parameters, you cannot have a ratio different from 1.

In Recognition Science, this result is a bridge, labeled B4, that supports later structure by showing where parameters may and may not enter. It is a theorem in the framework's machine-checked library, proved from the definitions of an observable ratio model and a free sector label. The proof is short and does not depend on the deeper cost-function chain.

The practical consequence is a sharp boundary: any theory that wants a ratio other than 1 must introduce a parameter explicitly. The zero-parameter ledger cannot hide a nontrivial scale; it must either be neutral or admit a knob.

/-- **Theorem (Parameter-free observables are neutral)**:
If every observable state must be specifiable without free parameters,
and specifying a nonzero log-charge requires a free real knob, then
all observable states have zero log-charge. -/
theorem parameter_free_observables_are_neutral
    {α : Type}
    (model : ObservableRatioModel α)
    (h_no_knob : ∀ Q : ℝ, Q ≠ 0 → ¬ sectorLabelIsFreeKnob model Q)
    (s : α) :
    model.log_charge s = 0 := by
  by_contra h
  exact h_no_knob (model.log_charge s) h ⟨s, rfl⟩

/-- Neutral log-charge forces the ratio to equal 1. -/
theorem neutral_ratio_eq_one
    {α : Type}
    (model : ObservableRatioModel α)
    (s : α)
    (h_neutral : model.log_charge s = 0) :
    model.ratio s = 1 := by
  have hlog : Real.log (model.ratio s) = 0 := by
    rw [← model.log_charge_eq s]; exact h_neutral
  exact Real.log_injOn_pos (Set.mem_Ioi.mpr (model.ratio_pos s))
    (Set.mem_Ioi.mpr one_pos) (by rw [hlog, Real.log_one])

/-- **Bridge B4 core (unconditional)**: parameter-free observable
ratios in a zero-parameter ledger are all equal to 1. -/
theorem parameter_free_ratios_are_unity
    {α : Type}
    (model : ObservableRatioModel α)
    (h_no_knob : ∀ Q : ℝ, Q ≠ 0 → ¬ sectorLabelIsFreeKnob model Q)
    (s : α) :
    model.ratio s = 1 :=
  neutral_ratio_eq_one model s (parameter_free_observables_are_neutral model h_no_knob s)

What this page does not claim

Not a claim that all physical ratios are 1, only those in a zero-parameter ledger. Not a claim that the zero-parameter posture is mandatory; it is a definitional choice. Not a claim about ledgers that allow free parameters.

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

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