Foundation Eight Tick Completeness

The eight-tick cycle

An eight-tick cycle is a repeating sequence of eight recognition events, where each event records one binary choice. The claim of completeness is that this period-8 lattice covers all 2^3 = 8 possible binary states exactly once per cycle, and that no shorter period can do so. This minimum-period property is what the name "eight-tick completeness" refers to.

The idea has a classical analogue in binary counting. With three binary digits you can represent eight distinct values, from 000 to 111. A cycle that visits each of those eight states exactly once before repeating must have length eight. Recognition Science's contribution is to show that this combinatorial necessity follows from its own foundational cost function, not from an arbitrary choice of cycle length.

In Recognition Science, the ledger, a discrete record of recognition events, is governed by a forced cost function J(x) = (x + 1/x)/2 - 1. The framework proves that this cost function vanishes when its two arguments are equal, is nonnegative for positive inputs, and that the golden ratio threshold phi - 3/2 is positive. These three facts assemble into a certificate of eight-tick completeness: the structure exists and its defining conditions are satisfied.

What the machine-checked library of formal theorems actually establishes is narrower than the research note suggests. The formal proofs show that the cost function J(m/e) equals zero when m equals e, is nonnegative when both m and e are positive, and that the threshold phi - 3/2 is positive. What it does not prove is that these facts specifically describe an eight-tick cycle, because the library defines domainCost as J(m/e) without connecting m and e to recognition events. The completeness claim remains a research note, not a formal result.

The distinction matters for anyone reading the framework's claims. The certificate proves the cost function has the right algebraic properties. The step from those properties to a statement about eight-tick cycles requires defining what m and e mean in the context of recognition, which this library does not do. The eight-tick completeness statement is therefore a target for future formalization, not an established theorem.

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 completeness statement itself is not a proved theorem in this library. The library does not define what m and e mean in terms of recognition events. No claim is made that the cost function's properties uniquely determine the eight-tick cycle.

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

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