Masses Mass Genesis T10 Pattern Cyclic Shift

The cyclic shift test

A cyclic shift moves the starting point of a repeating sequence. Think of a clock face: shifting the origin from 12 to 3 does not change the hour hand's motion, only where you begin reading it. In the Recognition Science framework, a light pattern (a discrete record of eight recognition events) can be shifted the same way. The module called T10 Pattern Cyclic Shift asks whether this coordinate choice matters for physics. The answer, proved in a machine-checked library of formal theorems, is that it does not: shifting the clock origin leaves the pattern's predicted mass, rest mass, and recognition cost exactly unchanged.

The framework models matter as stable patterns of recognition events. A recognition cost (a forced expense associated with each event) is assigned to each pattern. The core question is whether this cost depends on where you start counting. The theorem loadRecognitionCost_simultaneousCyclicShift proves that applying the same shift to every site window leaves the cost identical. The proof is short and direct: it unfolds the definition of cost and rewrites using the fact that shifting preserves the topology ratio. The same invariance holds for predicted mass (predictedMass_simultaneousCyclicShift) and for rest mass when the pattern is stable (restMass_simultaneousCyclicShift_of_stable).

This invariance is not trivial. The framework's eight-tick cycle means a shift by one tick changes which events land in which window. Yet the mass and cost do not change. The module also builds quotient infrastructure: it defines an equivalence relation identifying patterns that evolve into each other, and proves this relation is an equivalence. Two specific patterns, gapOne and gapTwo, are shown to be genuinely different even under this equivalence (gapOne_patternRHatQuotient_ne_gapTwo). So the shift invariance is not collapsing distinct matter states into one; it only removes the arbitrary choice of where the clock starts.

What this establishes in plain language: the framework's mass predictions are coordinate-invariant under time translation. If you relabel the ticks, you get the same physics. This is a necessary consistency check for any theory that claims to derive particle masses from a discrete ledger. The module does not yet derive the actual settled mass values, nor does it prove matter uniqueness. It certifies that the foundation is stable under a basic symmetry, which is the first step before any mass prediction can be trusted.

/-- Pattern load-recognition cost is independent of clock origin. -/
theorem loadRecognitionCost_simultaneousCyclicShift
    (ψ : LightPattern (Fin 8))
    (k : Fin 8) :
    loadRecognitionCost (simultaneousCyclicShift k ψ) =
      loadRecognitionCost ψ := by
  unfold loadRecognitionCost
  rw [loadTopologyRatio_simultaneousCyclicShift]

/-- The topology-predicted mass is independent of clock origin. -/
theorem predictedMass_simultaneousCyclicShift
    (ψ : LightPattern (Fin 8))
    (k : Fin 8) :
    predictedMass (simultaneousCyclicShift k ψ) = predictedMass ψ :=
  rfl

/-- Stable rest mass is independent of clock origin. -/
theorem restMass_simultaneousCyclicShift_of_stable
    (ψ : LightPattern (Fin 8))
    (k : Fin 8)
    (h : StableClosedLightPattern ψ) :
    restMass (simultaneousCyclicShift k ψ) = restMass ψ :=
  restMass_evolvePattern_of_stable ψ k.val h

/-- Gap one and gap two define different full-pattern quotient points. -/
theorem gapOne_patternRHatQuotient_ne_gapTwo :
    Quotient.mk patternRHatOrbitSetoid
        (worldlinePattern gapOneTwoPhaseMode) ≠
      Quotient.mk patternRHatOrbitSetoid
        (worldlinePattern gapTwoTwoPhaseMode) := by
  intro h
  exact gapOne_not_patternRHatOrbitEquivalent_gapTwo
    (Quotient.exact h)

What this page does not claim

The module does not derive the physical settled mass readout or matter uniqueness. The shift invariance does not imply that all patterns with the same mass are physically identical. The framework's cyclic shift is a coordinate choice, not a physical rotation in space.

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)

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