Masses Mass Genesis T10 Posting Rate Verdict Canonical Octave Parity Occupation

The posting rate verdict

The declaration canonicalOctave_parityOccupation_satisfies_onePostingPerTick is a formal theorem in the Recognition Science framework's machine-checked library of formal theorems. It proves that a particular readout of the framework's ledger, a discrete record of events, changes by at most one unit from one tick to the next across the canonical eight-tick cycle. The readout in question is the parity-phase occupation, which is a two-valued indicator: at any tick it reports either 0 or 1, depending on whether the ledger's parity state matches a chosen phase.

The proof is trivial in a precise sense. Because the readout is two-valued, its change between any two consecutive ticks can only be -1, 0, or +1. The theorem proves this bound for every possible run of the ledger, settled or not, legal or not, including the wrap-around from tick seven back to tick zero. It requires no assumption about the ledger's legality. The declaration is therefore a derived form of a definitional fact: the bound excludes nothing that the two-valued nature of the readout did not already exclude.

The deeper content of the theorem is what it does not claim. It does not say the ledger's total activity is one post per tick. A faithful count that registers every post, such as the total count of debits and credits, rises by eight over the eight ticks and thus violates the bound at the wrap tick. The theorem no_faithfulCountReadout_onePostingPerTick proves that no such faithful count can satisfy the bound on any run of the eight-cycle. The canonical balance readout violates it by computation, moving four units in the wrap step alone. The theorem's reach is therefore narrow: it certifies the posting rate only for the two-valued occupation readout, not for any count of actual posts.

This verdict settles a question left open in an earlier audit. The framework's mass line needs a readout that changes by at most one per tick, and the theorem shows which reading of "standing count" can supply it. The two-valued parity-phase occupation satisfies the bound automatically. A faithful count cannot. The missing piece is not a rate law but an identification: the framework must choose which readout its matter windows use. For photon and settlement windows that identification is made; for matter windows it remains an open gap, recorded as such in the library. The theorem closes the rate question while leaving that identification explicitly open.

/-- The parity-phase occupation of the same canonical octave does satisfy the
bound: non-vacuity for the positive half on the identical object. The two
readouts of one run decide the verdict. -/
theorem canonicalOctave_parityOccupation_satisfies_onePostingPerTick (phase : Fin 8) :
    OnePostingPerTick
      (parityPhaseOccupation q3ZeroLedger q3GrayPostingSchedule phase) :=
  parityPhaseOccupation_onePostingPerTick _ _ _

/-- **The committed occupation satisfies the rate bound on every run, with no
premise.** Legality is never used: an indicator is two-valued, so its change
across any tick, including the cyclic wrap, is at most one. The theorem is
DERIVED in form and DEFINITIONAL in content. -/
theorem parityPhaseOccupation_onePostingPerTick (initial : LedgerState 3)
    (schedule : ℕ → PostInstr 3) (phase : Fin 8) :
    OnePostingPerTick (parityPhaseOccupation initial schedule phase) :=
  onePostingPerTick_of_twoValued (a := 0) (b := 1) (by decide)
    (parityPhaseOccupation_twoValued initial schedule phase)

/-- **No faithful standing count satisfies the cyclic rate bound.** Over seven
steps the count rises by seven, so the wrap tick alone moves seven units. This
is the sharp half of the verdict: the bound is not merely underived for
counts, it is false. -/
theorem no_faithfulCountReadout_onePostingPerTick (r : LedgerState 3 → ℤ)
    (hf : FaithfulCountReadout r) (initial : LedgerState 3) (sched : ℕ → PostInstr 3) :
    ¬ OnePostingPerTick (fun t : Fin 8 => r (run initial sched t.val)) := by
  intro hrate
  have hstep : (r (run initial sched (7 : Fin 8).val) -
      r (run initial sched ((7 : Fin 8) + 1).val)).natAbs ≤ 1 := hrate (7 : Fin 8)
  have hwrapval : ((7 : Fin 8) + 1).val = 0 := by decide
  have hv7 : (7 : Fin 8).val = 7 := by decide
  rw [hwrapval, hv7] at hstep
  rw [faithfulCountReadout_run r hf initial sched 7,
    faithfulCountReadout_run r hf initial sched 0] at hstep
  have hdiff : r initial + ((7 : ℕ) : ℤ) - (r initial + ((0 : ℕ) : ℤ)) = 7 := by
    push_cast
    ring
  rw [hdiff] at hstep
  exact (by decide : ¬ ((7 : ℤ).natAbs ≤ 1)) hstep

/-- **The countermodel.** The library's canonical settled legal octave, read
by the standing balance of axis zero, violates the cyclic posting-rate bound:
the wrap tick compares a balance of `4` against a balance of `0`. So the bound
is false, not merely underived, for the integer "standing count" reading,
already on the one settled octave the library exhibits. -/
theorem canonicalOctave_axisBalance_violates_onePostingPerTick :
    ¬ OnePostingPerTick (axisBalanceOccupation q3ZeroLedger q3GrayPostingSchedule 0) := by
  intro hrate
  have hstep : (phiVec (run q3ZeroLedger q3GrayPostingSchedule (7 : Fin 8).val) 0 -
      phiVec (run q3ZeroLedger q3GrayPostingSchedule ((7 : Fin 8) + 1).val) 0).natAbs ≤ 1 :=
    hrate (7 : Fin 8)
  have hwrapval : ((7 : Fin 8) + 1).val = 0 := by decide
  have hv7 : (7 : Fin 8).val = 7 := by decide
  rw [hwrapval, hv7, canonicalOctave_axisBalance_seven,
    canonicalOctave_axisBalance_zero] at hstep
  exact (by decide : ¬ ((4 : ℤ) - 0).natAbs ≤ 1) hstep

What this page does not claim

The theorem does not claim that the ledger's total activity is one post per tick. The theorem does not claim that any faithful count readout satisfies the posting rate bound. The theorem does not close the open identification of a two-valued occupancy readout for matter windows.

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/T10PostingRateVerdict.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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