Foundation Eight Tick Derivation V3

The eight-tick cycle

An eight-tick cycle is a complete traversal of the eight binary states of a recognition lattice: three positions, each on or off, give 2³ = 8 combinations. The number eight is not chosen; it is forced once the framework establishes that the recognition lattice has three spatial dimensions. In Recognition Science, a ledger, a discrete record of recognition events, organizes these states, and the cycle is one full pass through them.

The framework's machine-checked library of formal theorems proves three general facts about its cost function J(x) = (x + 1/x)/2 - 1. First, the cost vanishes when the two inputs are equal: J(1) = 0. Second, the cost is never negative for positive inputs. Third, the golden ratio φ ≈ 1.618 satisfies φ - 3/2 > 0, a threshold fact used elsewhere. These are general properties of the cost function, not specific to any one subject.

The module named Eight_Tick_Derivation_v3 defines its cost as J(m/e) for two real numbers m and e, but it never defines what m and e mean. The docstring records the intended research direction: that m and e should be masses and energies of a particle, and that the eight-tick cycle should follow from the three dimensions. The Lean code proves the three general facts above, but it does not prove the eight-tick derivation, because the definitions of m and e are missing.

What the module does establish is a certificate: a packaged set of the three cost facts, ready to be reused. The certificate is inhabited, meaning the facts are consistent and available. This is a small but real step. The general cost facts are proved once in a shared template and cited here. The eight-tick conclusion remains a target, not a result, until someone defines m and e in a subject's own terms.

In plain language: the framework has a cost function with three clean properties, and a module that packages them. The eight-tick cycle is the intended payoff, but the module itself does not deliver it. The reader can see exactly what is proved and what is not: the cost facts are solid, the cycle is a goal.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The eight-tick cycle is not proved in this module; it is a documented research goal. The cost function J is not derived here; it is assumed from the shared template. No claim is made about what m and e physically represent.

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/Foundation/Eight_Tick_Derivation_v3.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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