Masses Mass Genesis Q3 Support Action

The eight-tick support action

In the Recognition Science framework, a particle's identity is carried by a ledger, a discrete record of events, in this case a pattern of eight phases arranged in a cycle. The Q3 support action is the specific rule that advances this pattern one step at a time: from phase zero to phase one, phase one to phase two, and so on around the full eight-tick cycle. The framework proves that this action is transitive, meaning starting from any one of the eight phases you can reach any other by repeating the step enough times. It also proves the mapping from each phase to its corresponding pattern is bijective, so every phase has exactly one pattern and every pattern has exactly one phase.

The action is built on a Gray code cycle, a sequence where each step changes exactly one bit of a three-bit label. This is the Q3 in the name: three bits, eight possible values. The framework shows that this one-bit-step property holds at every phase transition, which is what makes the cycle a coherent carrier for the mass pattern rather than an arbitrary list. The support is full, meaning every one of the eight phases is occupied, and the load, the amount of meaning carried at each site, is uniform across all eight positions.

Beyond the bare action, the framework establishes a chain of equivalences that tie the pattern's geometry to its mass. The key result is that several different ways of defining the load at the anchor phase, the starting point of the cycle, all agree. The norm of the neutralized window at phase zero equals the total load divided by eight, which is the same as the load divided by the support size, and this in turn is equivalent to a factorization involving the golden ratio phi raised to a power determined by the pattern's rung and charge gap. These equivalences are proved as theorems in a machine-checked library of formal theorems.

What this establishes in plain language is that the eight-tick cycle is not just a decorative structure. The uniform load and the equivalent norm conditions are what allow the framework to derive a particle's rest mass from the pattern's topology and its position on the phi-power ladder. The framework provides the concrete, checkable instance of that derivation for the Q3 carrier, showing that the abstract mass-genesis machinery applies to a specific, finite pattern. This is the support action that makes the mass prediction possible.

/-- The concrete eight-tick support move: advance one Rhat phase. -/
def eightTickSupportStep (i : Fin 8) : Fin 8 :=
  i + 1

theorem eightTickSupportStep_transitive
    (x y : Fin 8) :
    ∃ n : ℕ, (eightTickSupportStep^[n]) x = y := by
  fin_cases x <;> fin_cases y <;>
    first
    | exact ⟨0, rfl⟩
    | exact ⟨1, rfl⟩
    | exact ⟨2, rfl⟩
    | exact ⟨3, rfl⟩
    | exact ⟨4, rfl⟩
    | exact ⟨5, rfl⟩
    | exact ⟨6, rfl⟩
    | exact ⟨7, rfl⟩

theorem q3VertexAtPhase_bijective :
    Function.Bijective q3VertexAtPhase := by
  simpa [q3VertexAtPhase] using Patterns.grayCycle3_bijective

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]

theorem anchorPhaseEightTickNorm_of_anchorPhaseFactorizedPhiNorm
    (ψ : LightPattern (Fin 8))
    (hphi : AnchorPhaseFactorizedPhiNorm ψ) :
    AnchorPhaseEightTickNorm ψ :=
  (anchorPhaseEightTickNorm_iff_anchorPhaseFactorizedPhiNorm ψ).2 hphi

What this page does not claim

This framework does not prove that any specific measured particle mass matches the predicted value. The Q3 support action is not claimed to be the only possible carrier for mass patterns. No claim is made that the eight-tick cycle is physically realized in nature, only that the framework models it.

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)

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.

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