Foundation Neutral Sector Neutral Ratio Eq One

The neutral sector

A ratio is a comparison of two quantities. In physics, ratios appear constantly: the mass of one particle over another, the strength of one force over another, the size of one orbit over another. The declaration neutral_ratio_eq_one concerns a specific kind of ratio, one produced by a model that must work without any free parameters. A free parameter is a number the modeler gets to choose, like a particle mass or a coupling constant, that the model itself does not determine. The claim is that in such a model, any observable ratio must equal 1.

The argument proceeds in two steps, and the declaration formalizes the second one. The first step, covered by the theorem parameter_free_observables_are_neutral, says that if a model must specify its own states with no external input, then every observable state must carry a conserved charge of zero. The reasoning is direct: assigning a nonzero charge would require encoding a real number, which is exactly the kind of free parameter the model is forbidden to use. The second step is the declaration itself. A model assigns to each state a positive real ratio and records the natural logarithm of that ratio as its log-charge. If the log-charge is zero, then the ratio itself must be 1, because the logarithm function is one-to-one on positive numbers and the logarithm of 1 is 0.

The declaration is a theorem in the machine-checked library of formal theorems that the Recognition Science framework maintains. It is proved from the definitions of an observable ratio model and the neutrality condition, with no additional assumptions. The proof uses the injectivity of the real logarithm on positive arguments, a standard fact. The theorem is unconditional in the sense that it requires only the neutrality premise: if a state has log-charge zero, its ratio is 1. The broader statement that parameter-free observable ratios are all unity, called parameter_free_ratios_are_unity, combines this declaration with the earlier neutrality theorem.

In Recognition Science, this result anchors a bridge called B4: in a zero-parameter ledger, a discrete record of events with no adjustable inputs, observable ratios cannot vary. They are all pinned to the value 1. This is a structural constraint, not an empirical one. It does not say that measured ratios in the actual universe are all 1; it says that a model with no free parameters has no room for any other ratio. The practical consequence is a sharp boundary: if a model produces a ratio different from 1, that model must be carrying at least one free parameter, whether acknowledged or hidden.

What the declaration does not claim is equally important. It does not claim that all physical ratios are 1, nor that the universe has no free parameters. It does not claim that a model with free parameters is wrong; it only says such a model is not zero-parameter. The theorem is a logical consequence of a definitional choice, not a statement about measured reality. It also does not claim that the neutrality condition is always satisfied; it merely states what follows when it is.

/-- 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])

/-- 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)

/-- **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

All physical ratios in the actual universe are equal to 1. The universe has no free parameters. A model with free parameters is invalid.

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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Derived articles

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