Masses Mass Genesis T10 Pattern Cyclic Shift Eight Tick Window Equivariant Simul

The shift invariance

The declaration eightTickWindowEquivariant_simultaneousCyclicShift is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It concerns light patterns, which are discrete records of events across eight time positions, or ticks. A pattern is eight-tick window equivariant when every one of its eight overlapping eight-tick windows has the same internal structure. The theorem states that if you take such a pattern and apply the same one-tick shift to every window, the resulting pattern is also eight-tick window equivariant. In other words, the property of having uniform windows is preserved under a global shift of the clock.

The proof is direct. It relies on the fact that shifting every window by the same amount commutes with the shift operation itself: shifting a window and then moving it is the same as moving it and then shifting. The theorem is a formal statement, verified by the Lean 4 proof assistant, and it is part of a larger effort to show that matter, as described by masses and charges, emerges from the recognition process. The declaration is one of several that establish the invariance of various quantities, such as predicted mass and energy, under this simultaneous cyclic shift.

What the theorem does not claim is equally important. It does not prove that any particular pattern is eight-tick window equivariant; it only shows that the property is preserved by the shift. It does not derive the physical settled readout of a pattern, nor does it show that a pattern's sigma value is zero, nor does it establish the uniqueness of matter. These are separate, later goals in the framework's program. The theorem is a piece of quotient infrastructure, a way of grouping patterns that are equivalent under shifting, not a statement about which patterns actually occur in nature.

In Recognition Science, this shift invariance is a step toward the claim that the origin of the clock, the choice of which tick is labeled zero, is a coordinate choice that does not affect the underlying physics. The theorem supports the idea that the recognition process is independent of this choice, a necessary condition for the framework's account of mass to be well-defined. The framework models the universe as maintaining a ledger of recognition events, and the cost of recognition is forced by a proved functional equation. This theorem is a small but necessary piece of that larger structure.

/-- Simultaneous clock shift preserves the Q3 window-action law. -/
theorem eightTickWindowEquivariant_simultaneousCyclicShift
    {ψ : LightPattern (Fin 8)}
    (h : EightTickWindowEquivariant ψ)
    (k : Fin 8) :
    EightTickWindowEquivariant (simultaneousCyclicShift k ψ) := by
  intro x
  rw [simultaneousCyclicShift_window, simultaneousCyclicShift_window,
    h x]
  exact cyclicShift_iterate_comm k.val (ψ.window x)

structure T10PatternCyclicShiftCert : Prop where
  q3_preserved :
    ∀ (ψ : LightPattern (Fin 8)) (k : Fin 8),
      Q3MatterCarrier ψ →
        Q3MatterCarrier (simultaneousCyclicShift k ψ)
  species_label_preserved :
    ∀ (ψ : LightPattern (Fin 8)) (row : ChargedMassRow) (k : Fin 8),
      HasChargedSpeciesLabel ψ row →
        HasChargedSpeciesLabel (simultaneousCyclicShift k ψ) row
  load_cost_preserved :
    ∀ (ψ : LightPattern (Fin 8)) (k : Fin 8),
      loadRecognitionCost (simultaneousCyclicShift k ψ) =
        loadRecognitionCost ψ
  full_pattern_quotient :
    Equivalence PatternRHatOrbitEquivalent
  raw_origin_changes :
    simultaneousCyclicShift 1
        (worldlinePattern gapOneTwoPhaseMode) ≠
      worldlinePattern gapOneTwoPhaseMode
  gap_classes_distinct :
    ¬ PatternRHatOrbitEquivalent
      (worldlinePattern gapOneTwoPhaseMode)
      (worldlinePattern gapTwoTwoPhaseMode)

What this page does not claim

This answer does not claim that any specific light pattern is eight-tick window equivariant. This answer does not claim that the theorem derives the physical settled readout or matter uniqueness. This answer does not claim that the framework's account of mass is complete.

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/T10PatternCyclicShift.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: