Masses Mass Genesis Anchor Norm Primitive Full Chain For Primitive Factor Norm

The primitive factor chain

The declaration full_chain_for_primitiveFactorNorm is a theorem in the machine-checked library of formal theorems used by Recognition Science. It states that if a certain primitive factor norm condition holds for a pattern, then six consequences follow: the pattern's integrated meaning load is unchanged by evolution, its rest mass is positive, its mass is quantized in units of the golden ratio phi, its rest mass equals its predicted mass, and its inertial and gravitational masses both equal that rest mass. The theorem is conditional: it requires a stable load readout theory and evidence that the pattern is closed under the Q3 phase.

The primitive factor norm condition itself is a definitional choice, not a derived law. It splits the anchor norm into two pieces: a sector base load, which is a yardstick divided over eight phases, and a phi transport factor, which depends on the topology-derived rung and charge skew. The theorem proves that this primitive factor form is equivalent to the fully expanded norm form, and that this equivalence is enough to close the full mass genesis chain. The library also proves that the phi transport factor splits into independent rung and charge-skew transport factors, and that all the factors involved are positive.

In Recognition Science, this theorem is the current bottom-up proof target for mass genesis. It does not prove either the sector base load or the phi transport factor from dynamics; those are stated as the next exact landing surface. The theorem shows that if that surface is reached, the rest of the chain follows. The chain includes the statement that the integrated meaning load is conserved under evolution, that rest mass is positive, that mass is phi-quantized, and that inertial and gravitational mass coincide. The last equality is a strong physical claim, and it is proved only under the stated conditions.

What the theorem does not claim is just as important. It does not derive the primitive factor norm from first principles; that remains an open target. It does not prove that any actual physical pattern satisfies the condition. It does not claim that the sector base load or the phi transport factor are forced by the framework's axioms. The theorem is a bridge: if the primitive factor norm holds, the full chain of mass properties follows. The bridge is proved; the landing surface on the far side is not yet built.

theorem full_chain_for_primitiveFactorNorm
    (T : StableLoadReadoutTheory (Fin 8))
    {ψ : LightPattern (Fin 8)}
    (E : Q3ClosedPatternEvidence ψ)
    (hprimitive : AnchorPhasePrimitiveFactorNorm ψ) :
    (∀ k : ℕ,
      integratedMeaningLoad (evolvePattern k ψ) = integratedMeaningLoad ψ) ∧
    0 < restMass ψ ∧
    PhiRungQuantized ψ ∧
    restMass ψ = predictedMass ψ ∧
    T.readout.inertialMass ψ = restMass ψ ∧
    T.readout.gravitationalMass ψ = restMass ψ :=
  E.full_chain_for_fully_expanded_norm T
    (fullyExpandedNorm_of_primitiveFactorNorm ψ hprimitive)

/-- Primitive factor form of the active anchor norm target. The phase-0 neutral
load is the sector base load transported by the topology-derived phi factor. -/
def AnchorPhasePrimitiveFactorNorm (ψ : LightPattern (Fin 8)) : Prop :=
  normSq8 (neutralize (ψ.window 0)) =
    primitiveAnchorSectorLoad ψ * primitivePhiTransport ψ

theorem primitiveFactorNorm_iff_fullyExpandedNorm
    (ψ : LightPattern (Fin 8)) :
    AnchorPhasePrimitiveFactorNorm ψ ↔ AnchorPhaseFullyExpandedNorm ψ := by
  unfold AnchorPhasePrimitiveFactorNorm AnchorPhaseFullyExpandedNorm
  unfold primitiveAnchorSectorLoad primitivePhiTransport
  have hright :
      (((2 : ℝ) ^ Anchor.B_pow (sectorOf ψ) *
          Anchor.E_coh *
          Constants.phi ^ Anchor.r0 (sectorOf ψ)) *
          Constants.phi ^
            (((rungOf ψ : ℝ) - 8) +
              Real.log (1 + (ZOf ψ : ℝ) / Constants.phi) /
                Real.log Constants.phi)) /
          (8 : ℝ) =
        ((2 : ℝ) ^ Anchor.B_pow (sectorOf ψ) *
          Anchor.E_coh *
          Constants.phi ^ Anchor.r0 (sectorOf ψ)) /
          (8 : ℝ) *
        Constants.phi ^
          (((rungOf ψ : ℝ) - 8) +
            Real.log (1 + (ZOf ψ : ℝ) / Constants.phi) /
              Real.log Constants.phi) := by
    ring
  constructor
  · intro h
    rw [hright]
    exact h
  · intro h
    rw [← hright]
    exact h

/-- The phi transport factor splits into independent rung and charge-skew
transport factors. -/
theorem primitivePhiTransport_eq_rung_mul_charge
    (ψ : LightPattern (Fin 8)) :
    primitivePhiTransport ψ =
      primitiveRungTransport ψ * primitiveChargeSkewTransport ψ := by
  unfold primitivePhiTransport
  unfold primitiveRungTransport
  unfold primitiveChargeSkewTransport
  rw [Real.rpow_add Constants.phi_pos]

What this page does not claim

The primitive factor norm condition is not proved from dynamics; it is a stated landing surface. No actual physical pattern is shown to satisfy the primitive factor norm condition. The sector base load and phi transport factor are not derived from the framework's axioms.

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/AnchorNormPrimitive.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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