Masses Mass Genesis Integrated Load Nontrivial Neutral Load Iff Exists Site Load

The load certificate

A pattern in the Recognition Science framework is a finite arrangement of activity over a substrate. Each active site, one location in that arrangement, carries a number called its site load, which measures how much meaningful structure is concentrated there. The total load of the whole pattern is simply the sum of these site loads over all its active sites. This is a definitional choice, not a derived result: the framework models a pattern's total meaningful content as the finite sum of its localized parts.

The declaration nontrivialNeutralLoad_iff_exists_site_load_pos states an equivalence: a pattern has a nontrivial neutral load, meaning its total meaningful content is not zero, if and only if at least one of its active sites has a strictly positive site load. In plainer terms, the whole is positive exactly when some part is positive. This is a proved theorem in the machine-checked library of formal theorems, and it follows directly from the definition of total load as a sum of nonnegative site loads. The proof is a routine verification that a sum of nonnegative terms is zero exactly when every term is zero, and positive exactly when at least one term is positive.

The theorem also comes with a certificate, a packaged collection of related facts. Together they state that the total load is always nonnegative, that it is zero exactly when no site has positive load, that it is positive exactly when some site does, and that the total load does not change when the pattern evolves through the framework's cyclic shifts. This last point means the total meaningful content of a pattern is invariant under the framework's time evolution, a property the certificate records as a theorem.

What the declaration does not claim is just as important. It does not identify the load with the phi-ladder mass law; the docstring for the module says so explicitly. The load layer is algebraic and closed on its own terms, and the connection to physical masses remains a separate, later step. The theorem also does not say anything about which specific patterns have positive load, only that if a pattern has one, then some site does. It is a structural fact about how the load is defined, not a statement about any particular pattern's content.

/-- The existing `NontrivialNeutralLoad` predicate is exactly positive
site load on the finite support. -/
theorem nontrivialNeutralLoad_iff_exists_site_load_pos
    (ψ : LightPattern Λ) :
    NontrivialNeutralLoad ψ ↔
      ∃ x ∈ ψ.support, 0 < siteMeaningLoad ψ x := by
  rfl

theorem integratedMeaningLoad_eq_support_sum (ψ : LightPattern Λ) :
    integratedMeaningLoad ψ =
      ψ.support.sum (fun x => siteMeaningLoad ψ x) := by
  rfl

theorem integratedMeaningLoad_eq_of_evolvePattern
    (ψ : LightPattern Λ) (k : ℕ) :
    integratedMeaningLoad (evolvePattern k ψ) = integratedMeaningLoad ψ :=
  integratedMeaningLoad_eq_of_sameLoadOnSupport
    (sameLoadOnSupport_evolvePattern ψ k)

What this page does not claim

The theorem does not identify the load with the phi-ladder mass law. The theorem does not state which specific patterns have positive load. The theorem does not say anything about the magnitude of the load, only its positivity.

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

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