Holography Turn Ratio Carrier Eight Tick Multiple Exclusion

The once-per-closure census record

A ledger, a discrete record of events, must often answer a census question: did every sector of a cycle get posted exactly once, or did some get posted twice? The Recognition Science declaration eight_tick_multiple_exclusion settles this for a specific eight-step walk on four sectors. The walk [0, 1, 2, 3, 0, 2, 1, 3] visits each of the four sectors exactly once per closure. Its retrace, a closed sixteen-step walk that goes around twice, visits every sector twice. The theorem, checked by the decide tactic in the framework's machine-checked library of formal theorems, records both facts as a single census record.

This is a posting-discipline fact, not a census-absence fact. The declaration also records that the weaker condition EightTickSubperiodExclusion, which rules out proper subperiods like 1, 2, or 4 ticks, does not by itself exclude multiples. A cycle can have no proper subperiod and still double-post its sectors on a doubled loop. The multiple side of the exclusion is about how often each sector is posted, not about whether a shorter period exists.

In the framework's account, this discrete once-per-closure content is what the CensusPricing premise prices. That premise states that the per-cycle recognition cost of a continued cycle is the function J of the delivered-to-required closure ratio, posted once per closure. The theorem here is the under-posting half of that premise: it shows what a single, correct posting looks like on the forced substrate. Its over-posting half, the double-posting record, is what this declaration supplies.

What the declaration does not claim is equally precise. It does not derive the CensusPricing premise from a more basic ledger; that derivation remains an open target. It does not say that a doubled cycle is forbidden, only that it double-posts. And it does not, by itself, force the eight-tick cycle or the golden ratio; those are separate results in the framework's forcing chain. This declaration is a census record, not a derivation of the cycle itself.

/-- The turn ratio of the `n`-fold cover of the deficit-free period is exactly `n`. -/
theorem turnRatio_cover (kappa : ℝ) (hk : 0 < kappa) (n : ℕ) :
    turnRatio kappa (n * DeficitFreePeriod.euclideanPeriod kappa) = n := by
  have hpi : Real.pi ≠ 0 := ne_of_gt Real.pi_pos
  have hk0 : kappa ≠ 0 := ne_of_gt hk
  unfold turnRatio DeficitFreePeriod.euclideanPeriod
  field_simp

/-- The turn ratio of the `n`-fold cover of the deficit-free period is exactly `n`. -/
theorem turnRatio_cover (kappa : ℝ) (hk : 0 < kappa) (n : ℕ) :
    turnRatio kappa (n * DeficitFreePeriod.euclideanPeriod kappa) = n := by
  have hpi : Real.pi ≠ 0 := ne_of_gt Real.pi_pos
  have hk0 : kappa ≠ 0 := ne_of_gt hk
  unfold turnRatio DeficitFreePeriod.euclideanPeriod
  field_simp

/-- **Kernel record: the U(1)/ℂ extension of `J` is underdetermined.** Two extensions
agree with `Cost.Jcost` on all of `ℝ₊` yet disagree at `I`. Any argument that "extends
J to the phase and reads off the period" is choosing its conclusion. -/
theorem u1_extension_not_unique :
    ∃ J₁ J₂ : ℂ → ℝ,
      (∀ x : ℝ, 0 < x → J₁ (x : ℂ) = Cost.Jcost x) ∧
      (∀ x : ℝ, 0 < x → J₂ (x : ℂ) = Cost.Jcost x) ∧
      ∃ z : ℂ, J₁ z ≠ J₂ z := by
  refine ⟨Jprime, Jsecond, fun x _ => Jprime_agrees x, fun x hx => Jsecond_agrees x hx,
    Complex.I, ?_⟩
  rw [Jprime_I, Jsecond_I]
  norm_num

What this page does not claim

This declaration does not derive the CensusPricing premise from a more basic ledger. It does not forbid a doubled cycle, only records that it double-posts. It does not, by itself, force the eight-tick cycle or the golden ratio.

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/Holography/TurnRatioCarrier.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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