Masses Mass Genesis Q3 Support Action Canonical Primitive Load Factorizes Of Q3

The factorization theorem

In the Recognition Science framework, a ledger (a discrete record of events) assigns a numeric load to each possible state. The declaration canonicalPrimitiveLoadFactorizes_of_q3EightTickTopologyScaledCP6Load is a theorem about a particular kind of ledger pattern: one that cycles through eight distinct phases, where each step changes exactly one bit of a three-bit code. The theorem states that if such a pattern satisfies a certain scaling condition on its topology load, then its total load factorizes: it equals a product of a sector factor and a power of the golden ratio, divided by eight.

The scaling condition is precise. It says the squared norm of the neutralized window at the anchor phase equals the topology load amplitude squared. The theorem then derives that this same quantity equals the factorized mass law load divided by eight. This is not a numerical approximation; it is an exact equality proved in the machine-checked library of formal theorems. The proof relies on the pattern being closed, meaning it has full support over all eight phases and its windows are equivariant under the cyclic shift.

What this theorem does not claim is broader than what it proves. It does not assert that all possible ledger patterns factorize, nor that any particular physical particle mass equals this value. It concerns only the specific eight-tick carrier built from the Q3 Gray cycle. The theorem also does not derive the golden ratio itself; that constant enters through the definition of the factorized norm, not as an output of this proof.

In practical terms, the theorem is a consistency check within the framework: it shows that two different ways of expressing the load, one from topology and one from factorization, agree exactly for this canonical eight-phase pattern. This matters because it anchors the framework's mass law to a concrete, checkable instance. The factorization is a structural property of the pattern, not a claim about measurement.

theorem canonicalPrimitiveLoadFactorizes
    {ψ : LightPattern (Fin 8)}
    (E : Q3MassPatternEvidence ψ) :
    CanonicalPrimitiveLoadFactorizes ψ :=
  E.q3_closed.canonicalPrimitiveLoadFactorizes_of_topology_scaled_cp6
    E.topology_scaled_cp6

theorem anchorPhaseTopologyNorm_iff_anchorPhaseEightTickNorm
    (ψ : LightPattern (Fin 8))
    (hfull : FullEightTickSupport ψ) :
    AnchorPhaseTopologyNorm ψ ↔ AnchorPhaseEightTickNorm ψ := by
  have hcard : (ψ.support.card : ℝ) = 8 := by
    rw [hfull]
    norm_num
  have hcard_nonneg : 0 ≤ (ψ.support.card : ℝ) := by
    exact_mod_cast Nat.zero_le ψ.support.card
  have hshare_nonneg :
      0 ≤ factorizedMassLawLoad ψ / (ψ.support.card : ℝ) :=
    div_nonneg (le_of_lt (factorizedMassLawLoad_pos ψ)) hcard_nonneg
  have hamp :
      topologyLoadAmplitude ψ ^ 2 = factorizedMassLawLoad ψ / (8 : ℝ) := by
    unfold topologyLoadAmplitude
    rw [Real.sq_sqrt hshare_nonneg, hcard]
  constructor
  · intro hnorm
    unfold AnchorPhaseTopologyNorm at hnorm
    unfold AnchorPhaseEightTickNorm
    rw [hnorm, hamp]
  · intro height
    unfold AnchorPhaseEightTickNorm at height
    unfold AnchorPhaseTopologyNorm
    rw [height, ← hamp]

/-- Refined Q3 closed-pattern evidence. This is stronger than the generic
`StableClosedLightPattern` predicate: it records that the support is the full
eight-tick Q3 phase carrier and that one support step is physically the cyclic
Rhat shift on the local photon window. -/
structure Q3ClosedPatternEvidence (ψ : LightPattern (Fin 8)) where
  stable : StableClosedLightPattern ψ
  full_support : FullEightTickSupport ψ
  window_equivariant : EightTickWindowEquivariant ψ

What this page does not claim

All ledger patterns factorize. Any particular physical particle mass equals this value. The golden ratio is derived by this theorem; it enters through definition.

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/Q3SupportAction.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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