Holography Turn Ratio Carrier Turn Ratio Cost Pos Of Ne Period

The unique zero-cost period

A cost function that measures the price of a repeated cycle often has a special input where the price is zero. In the Recognition Science framework, the cost function is cost, a measure of the price of a recognition event, and it takes the form J(x) = (x + 1/x)/2 − 1. For a cycle with a given turn ratio, the cost is J of that ratio. The theorem turnRatioCost_pos_of_ne_period states that this cost is strictly positive for every cycle length except one: the unique period where the turn ratio is exactly 1. At that special length, the cost is zero; at every other length, it is positive.

This is not a statement about the physical world. It is a statement about a mathematical object. The theorem is proved in the machine-checked library of formal theorems, and it relies on the definition of the turn ratio and the properties of the cost function J. The proof is a direct consequence of two earlier results: the cost is always nonnegative, and the cost is zero if and only if the turn ratio is 1. The theorem simply combines these two facts. It does not say that the universe has a preferred period, nor that any physical system must exhibit this period. It only says that within this mathematical model, the cost function has exactly one zero.

The practical content is that the framework's cost function is sharply discriminating. It does not allow a continuum of zero-cost cycles. There is exactly one period that costs nothing, and all others cost something. This sharpness is what allows the framework to derive unique periods and to rule out alternatives. The theorem does not, however, say anything about what the period is in physical units, nor does it say that the cost function is the only possible one. It is a statement about a specific function, not about the world.

/-- 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

/-- **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

/-- The cost is nonnegative for positive rate and period (T5 AM-GM,
`Cost.Jcost_nonneg`). -/
theorem turnRatioCost_nonneg {kappa T : ℝ} (hk : 0 < kappa) (hT : 0 < T) :
    0 ≤ turnRatioCost kappa T :=
  Cost.Jcost_nonneg (turnRatio_pos hk hT)

What this page does not claim

The theorem does not claim that any physical system has a preferred period. The theorem does not claim that the cost function is the only possible cost function. The theorem does not claim that the period of zero cost is the golden ratio or any other specific constant.

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.

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