Masses Mass Genesis Pattern Readout Equivalence Abstract Equivalence Principle A

The equivalence principle in pattern form

The equivalence principle, in its classical form, states that inertial mass and gravitational mass are the same property of a body. Inertial mass is the resistance an object offers to being accelerated; gravitational mass is the strength with which it responds to a gravitational field. Newton's mechanics treated the two as numerically equal without explaining why. Einstein's general relativity built that equality into the theory's foundations, and experiments have confirmed it to extraordinary precision.

In Recognition Science, the framework generalizes the principle into an abstract theorem about cost, a discrete measure of the effort a recognition event requires. The abstract theorem, proved in the framework's machine-checked library of formal theorems, states: if inertial mass and gravitational mass are both read from the same cost source, then the two readouts are equal. The proof does not define the readouts to be equal. It shows that equality follows necessarily from the single-source assumption.

The declaration abstract_equivalence_principle_available_for_pattern adapts that abstract theorem to the framework's pattern setting, where a pattern is a stable configuration of recognition events. The declaration establishes that the abstract equivalence principle is available as a template for pattern-level mass theories. It constructs, for any pattern theory that reads both masses from one source, the exact evidence object that the abstract theorem requires. It does not itself prove that any particular physical pattern satisfies the single-source condition.

What the declaration does not claim is as important as what it proves. It does not define inertial and gravitational mass to be equal. It does not prove that the two readouts are equal for any real pattern. It only states the conditional: given a pattern theory whose readouts draw from one cost source, the abstract equivalence principle applies. The framework's other theorems, such as pattern_single_source_equivalence and stableLoadReadoutTheory_identity, supply the missing premises for specific pattern theories, but the availability declaration itself carries no such premise.

The declaration's role is architectural. It marks the point where a general principle becomes a reusable tool for a family of specific theories. A reader who wants to know whether inertial and gravitational mass are equal for a given pattern must look to the pattern theory's own proof obligations, not to this declaration. The declaration guarantees that the abstract machinery is in place; it does not guarantee that any particular pattern satisfies the machinery's conditions.

theorem pattern_single_source_equivalence
    (T : PatternSingleSourceMassTheory Λ)
    (ψ : LightPattern Λ)
    (hψ : StableClosedLightPattern ψ) :
    T.readout.inertialMass ψ = T.readout.gravitationalMass ψ := by
  rw [T.inertial_from_source ψ hψ,
      T.gravitational_from_source ψ hψ]

theorem abstract_equivalence_principle_available_for_pattern
    (T : PatternSingleSourceMassTheory Λ)
    (ψ : LightPattern Λ) :
    (shiftedSourceMassTheory T ψ).inertial_mass 1 =
      (shiftedSourceMassTheory T ψ).gravitational_mass 1 :=
  Gravity.EquivalencePrinciple.single_source_equivalence
    (shiftedSourceMassTheory T ψ) 1 zero_lt_one

/-- Pack the existing abstract equivalence principle as the pattern-level
statement at the cost value `sourceMass ψ + 1`. The shift makes the state
positive without changing the pattern readout theorem above. -/
def shiftedSourceMassTheory
    (T : PatternSingleSourceMassTheory Λ)
    (ψ : LightPattern Λ) :
    Gravity.EquivalencePrinciple.SingleSourceMassTheory where
  cost _ := T.sourceMass ψ
  inertial_mass _ := T.sourceMass ψ
  gravitational_mass _ := T.sourceMass ψ
  inertial_from_cost := fun _ _ => rfl
  gravitational_from_cost := fun _ _ => rfl

theorem abstract_equivalence_principle_available_for_pattern
    (T : PatternSingleSourceMassTheory Λ)
    (ψ : LightPattern Λ) :
    (shiftedSourceMassTheory T ψ).inertial_mass 1 =
      (shiftedSourceMassTheory T ψ).gravitational_mass 1 :=
  Gravity.EquivalencePrinciple.single_source_equivalence
    (shiftedSourceMassTheory T ψ) 1 zero_lt_one

What this page does not claim

The declaration does not prove that inertial and gravitational mass are equal for any real physical pattern. The declaration does not define the two mass readouts to be equal by fiat. The declaration does not establish that the framework's restMass value equals any measured particle mass.

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/Masses/MassGenesis/PatternReadoutEquivalence.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.

Derived articles

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