Holography Turn Ratio Carrier Phase Cost Vanishes On Covers

Why phase blindness matters

A phase angle is a way of describing a position on a circle, measured in turns or radians, like the hands of a clock. In the Recognition Science framework, one natural idea was to price the cost of a recognition event by plugging this phase angle into the same cost function J that works for real numbers. The declaration phaseCost_vanishes_on_covers proves that this approach is blind: it assigns a cost of zero to every full turn of the phase, whether that is one turn, two turns, or any whole number of turns. The theorem states that for any integer n, the phase cost at an angle of n times 2π is exactly zero.

This matters because the framework's real cost function, applied to the ratio of a cycle's period to a fixed reference, has a unique zero: it is zero only when the ratio is exactly one, and strictly positive for every other positive ratio. That property lets the framework single out one special period as the only cost-free one. The phase version loses that ability entirely. Since it vanishes on every cover, meaning every multiple of the base period, it can never distinguish the minimal cycle from its repetitions. The theorem phaseCost_nonpos shows the same formula never even produces a positive cost, so it cannot penalize any deviation from the preferred period at all.

In Recognition Science, this result is called a poison lemma: it kills one whole route to deriving a key physical constant. The framework's library of machine-checked theorems uses it to show that the phase branch of pricing is dead, and that the real turn ratio must be used instead. The library also proves that extending the cost function J from real numbers to complex phases is not a forced choice: two different extensions both agree with J on the real line yet differ at the imaginary unit, and one of them even vanishes there. So the phase approach fails twice: its natural cost is lattice-blind, and the extension itself is underdetermined.

What phaseCost_vanishes_on_covers does not claim is just as important as what it proves. It does not say that phase angles are physically meaningless in general, only that this particular cost formula cannot see the difference between covers. It does not prove that the real turn ratio cost is the only possible pricing scheme; that conclusion depends on a separate premise called CensusPricing, which remains a named assumption rather than a derived theorem. And it says nothing about what happens for non-integer multiples of the base period, since the statement covers only whole-number turns.

/-- **Poison lemma (lattice blindness):** the phase-branch cost vanishes on EVERY
`n`-sheeted cover, so it can never single out `n = 1`. Contrast
`turnRatioCost_cover_pos`. -/
theorem phaseCost_vanishes_on_covers (n : ℤ) : phaseCost ((n : ℝ) * (2 * Real.pi)) = 0 := by
  rw [phaseCost_eq, Real.cos_int_mul_two_pi]
  ring

/-- **Poison lemma (sign death):** the phase-branch cost is NEVER strictly positive,
so it cannot penalize any deficit. The phase branch cannot force the period. -/
theorem phaseCost_nonpos (δ : ℝ) : phaseCost δ ≤ 0 := by
  rw [phaseCost_eq]
  linarith [Real.cos_le_one δ]

/-- **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

/-- **Kernel record: the zero set of an agreeing extension is not forced.** `Jsecond`
agrees with `J` on `ℝ₊` yet vanishes at `I ≠ 1`: imposing "zero iff closure" on an
extension ASSUMES B2's conclusion rather than deriving it. -/
theorem u1_extension_zero_set_not_forced :
    ∃ J' : ℂ → ℝ,
      (∀ x : ℝ, 0 < x → J' (x : ℂ) = Cost.Jcost x) ∧
      J' Complex.I = 0 ∧ (Complex.I : ℂ) ≠ 1 := by
  refine ⟨Jsecond, fun x hx => Jsecond_agrees x hx, Jsecond_I, ?_⟩
  intro h
  have := congrArg Complex.im h
  simp at this

/-- **Headline (B2 math half): the deficit-free period is the UNIQUE zero of the
per-cycle cost.** `C(T) = 0 ↔ T = 2π/κ`. Strictly stronger than minimality
(`DeficitFreePeriod.euclideanPeriod_isLeast`): no other positive period, lattice or
not, deficit or excess, has zero cost. Pure T5 (`Cost.Jcost_eq_zero_iff`). -/
theorem turnRatioCost_eq_zero_iff (kappa T : ℝ) (hk : 0 < kappa) (hT : 0 < T) :
    turnRatioCost kappa T = 0 ↔ T = DeficitFreePeriod.euclideanPeriod kappa := by
  unfold turnRatioCost
  rw [Cost.Jcost_eq_zero_iff _ (turnRatio_pos hk hT)]
  exact turnRatio_eq_one_iff kappa T hk

/-- Strict positivity off the deficit-free period: any other positive period, deficit
or excess, costs strictly positive recognition per cycle. -/
theorem turnRatioCost_pos_of_ne_period (kappa T : ℝ) (hk : 0 < kappa) (hT : 0 < T)
    (hne : T ≠ DeficitFreePeriod.euclideanPeriod kappa) :
    0 < turnRatioCost kappa T := by
  rcases lt_or_eq_of_le (turnRatioCost_nonneg hk hT) with hpos | heq
  · exact hpos
  · exact absurd ((turnRatioCost_eq_zero_iff kappa T hk hT).mp heq.symm) hne

What this page does not claim

Phase angles are physically meaningless in general. The real turn ratio cost is the only possible pricing scheme without the CensusPricing premise. The phase cost is zero for non-integer multiples of the base period.

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

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: