Masses Mass Genesis Load Normalization Integrated Meaning Load Eq Support Card O

The counting theorem

The theorem integratedMeaningLoad_eq_supportCard_of_neutralUnitLoad states a precise counting fact. In the framework's model, a light pattern is a discrete collection of occupied sites, and each site carries a numeric value called its site meaning load, a measure of how much recognition work that site contributes. The integrated meaning load is the sum of these site values across the whole pattern. The theorem says: if every occupied site carries exactly one unit of load, a condition the framework calls neutral unit load, then the integrated load equals the number of occupied sites, the support cardinality.

The proof is a short chain of equalities. The integrated load is first rewritten as the sum over the support. Since each term in that sum is 1 by hypothesis, the sum collapses to the count of terms. The machine-checked library of formal theorems records this argument as a proved lemma, not as an assumption. The same file then derives a corollary: under neutral unit load, the mass law is realized exactly when the support cardinality equals the predicted mass.

This theorem does not say that neutral unit load is the actual physics of any real pattern. The docstring is explicit: CP6-style unit normalization captures a projective-direction normalization, not the amplitude scale needed for mass. The theorem is a conditional bridge. It shows what would follow if unit load held, and it sharpens the open problem: the framework still needs a topology-forced load density, a site load law whose sum equals the predicted mass, derived from deeper constraints rather than assumed.

What the theorem changes is the shape of the remaining work. It isolates the missing piece as a density law, not as a counting problem. The counting side is settled; the amplitude side is the next true bottom-up target.

theorem integratedMeaningLoad_eq_supportCard_of_neutralUnitLoad
    (ψ : LightPattern Λ) (hunit : NeutralUnitLoadOnSupport ψ) :
    integratedMeaningLoad ψ = (ψ.support.card : ℝ) := by
  rw [integratedMeaningLoad_eq_support_sum ψ]
  calc
    ψ.support.sum (fun x => siteMeaningLoad ψ x)
        = ψ.support.sum (fun _x => (1 : ℝ)) := by
          apply Finset.sum_congr rfl
          intro x hx
          exact hunit x hx
    _ = (ψ.support.card : ℝ) := by
          simp

theorem neutralUnitLoad_realized_iff_predictedMass_eq_supportCard
    (ψ : LightPattern Λ) (hunit : NeutralUnitLoadOnSupport ψ) :
    MassLawLoadRealized ψ ↔
      (ψ.support.card : ℝ) = predictedMass ψ := by
  unfold MassLawLoadRealized
  rw [integratedMeaningLoad_eq_supportCard_of_neutralUnitLoad ψ hunit]

/-- CP6-style unit neutral load at every occupied site. This captures the
projective-direction normalization, not the amplitude scale needed for mass. -/
def NeutralUnitLoadOnSupport (ψ : LightPattern Λ) : Prop :=
  ∀ x ∈ ψ.support, siteMeaningLoad ψ x = 1

What this page does not claim

Neutral unit load is not asserted to hold for any actual physical pattern. The theorem does not prove that the integrated load equals the predicted mass. The framework does not yet derive the load density from first principles.

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/LoadNormalization.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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