Foundation Domain Bootstrap Scale Invariant On

Scale invariance

Scale invariance is a symmetry property of a comparison operator: a rule that takes two positive numbers and returns a third number. The property states that multiplying both inputs by the same positive factor leaves the output unchanged. In symbols, for any positive x, y, and λ, the rule C satisfies C(λx, λy) = C(x, y). This is the classical idea that a comparison should depend on the relative size of the inputs, not on the overall scale, just as a map's scale bar works whether the map shows a city or a continent.

The definition appears in the framework's ledger, a discrete record of events, as one of the conditions a comparison operator must meet. The framework's machine-checked library of formal theorems states this condition for any linearly ordered field, a set with an order and arithmetic where the usual rules hold. The declaration ScaleInvariantOn fixes the exact meaning: for all positive x, y, and λ in the field, the comparison of λx and λy equals the comparison of x and y. This is a definitional choice, a precise statement of what scale invariance means in this setting, not a derived result.

The condition does real work in the framework's bootstrap theorem. That theorem shows that a linearly ordered field on which the comparison operator satisfies the four Aristotelian conditions, plus scale invariance and distinguishability, and which is Archimedean and Dedekind-complete, is canonically isomorphic to the real numbers. Scale invariance is one of the ingredients in this forcing result, but it is not the only one. The theorem's conclusion, that the ambient field must be the reals up to isomorphism, depends on the full package of conditions.

Scale invariance alone does not single out the real numbers. The framework's own documentation notes that the comparison operator could live on the rationals or any incomplete subfield without the completeness hypotheses. Those hypotheses, Archimedean and Dedekind-completeness, are the analytic content that the Law of Logic does not provide on its own. Scale invariance is a symmetry constraint, not a completeness constraint; it rules out absolute-size dependence but does not rule out incomplete fields.

What the declaration does not claim is as important as what it claims. It does not assert that scale invariance is sufficient for any uniqueness result. It does not claim that the comparison operator is unique, or that the reals are the only possible domain. It does not even claim that a scale-invariant comparison operator exists on every field. The declaration is a definition, a precise statement of one condition among several, and its role in the bootstrap theorem is one part of a larger argument.

/-- Scale invariance, generic field version. -/
def ScaleInvariantOn [Zero K] [LT K] [Mul K] (C : ComparisonOperatorOn K) : Prop :=
  ∀ x y lam : K, 0 < x → 0 < y → 0 < lam →
    C (lam * x) (lam * y) = C x y

/-- Scale invariance, generic field version. -/
def ScaleInvariantOn [Zero K] [LT K] [Mul K] (C : ComparisonOperatorOn K) : Prop :=
  ∀ x y lam : K, 0 < x → 0 < y → 0 < lam →
    C (lam * x) (lam * y) = C x y

/-- **Bootstrap theorem (named-hypothesis form)**: a linearly ordered
field on which the Law of Logic is supported and which is Archimedean
and conditionally complete is canonically isomorphic to `ℝ` as an
ordered field. The Archimedean and conditional-completeness
hypotheses are the analytic content the Law of Logic does not on its
own provide; they are named here as inputs.

The conclusion is the existence of an order-preserving ring
isomorphism with `ℝ`. -/
theorem bootstrap_to_real
    (K : Type*) [ConditionallyCompleteLinearOrderedField K]
    (_ : LogicSupported K) :
    Nonempty (K ≃+*o ℝ) :=
  ⟨LinearOrderedField.inducedOrderRingIso K ℝ⟩

/-- Scale invariance, generic field version. -/
def ScaleInvariantOn [Zero K] [LT K] [Mul K] (C : ComparisonOperatorOn K) : Prop :=
  ∀ x y lam : K, 0 < x → 0 < y → 0 < lam →
    C (lam * x) (lam * y) = C x y

What this page does not claim

Scale invariance is not sufficient for any uniqueness result on its own. The declaration does not assert that a scale-invariant comparison operator exists on every field. The reals are not claimed to be the only possible domain for a scale-invariant comparison operator.

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/DomainBootstrap.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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