Foundation Active Edge Budget Per Tick Count From Octave

The forced edge count

The declaration per_tick_count_from_octave is a theorem in the framework's machine-checked library of formal theorems. It establishes a uniqueness result: if the total number of edges traversed in one full octave equals eight times some natural number n, then n must equal 1. In plainer terms, it proves that the average number of edges advanced per tick in the fundamental cycle is exactly one, and no other value is possible.

The proof works from the structure of the cycle itself. The framework models a tick, a discrete unit of recognition, as posting at exactly one node. Eight ticks form one octave, covering the vertices of a three-dimensional cube. Consecutive postings in this canonical cycle differ in exactly one bit, meaning the two posted vertices are joined by exactly one edge of the cube. Since each of the eight steps traverses exactly one edge, the total edge count for the octave is eight, and dividing by eight ticks forces the per-tick count to one.

The theorem is not an isolated curiosity. It upgrades a previously assumed definition into a proved result. The constant representing active edges per tick was once a postulate with a comment; now it is a theorem, derived from the Gray-cycle adjacency and the eight-tick periodicity. This matters because a duality relation in the framework, the product of matter content and consciousness ceiling equaling the golden ratio, rests on this edge count being one. The derivation closes a gap in the framework's foundations.

What the declaration does not claim is equally important. It does not assert that the fundamental cycle must be a Gray cycle; that is a separate structural choice. It does not claim that all possible recognition cycles have this property. It proves only that within the canonical eight-tick cycle, the per-tick edge count is forced to one. The result is conditional on the framework's definitions of tick, octave, and edge, and it says nothing about cycles of other lengths or dimensions.

/-- **The active-edge budget is forced (octave form).** Any `n`
satisfying `octaveEdgeCount = 8 · n` must equal 1: the actual edge
count over an octave is exactly 8, so `8 · n = 8`. -/
theorem per_tick_count_from_octave (n : ℕ)
    (h : octaveEdgeCount = 8 * n) : n = 1 := by
  rw [octaveEdgeCount_eq_eight] at h
  omega

/-- **CONSEQUENCE.** Consecutive postings in the canonical 3-bit
Gray cycle traverse exactly one cube edge. -/
theorem grayCycle3_per_tick_edge_count (i : Fin 8) :
    edgesTraversed (grayCycle3Path i) (grayCycle3Path (i + 1)) = 1 := by
  unfold edgesTraversed
  exact (oneBitDiff_iff_hamming_one _ _).mp (grayCycle3_oneBit_step i)

/-- The total edge count over one octave is exactly 8: each of the
8 ticks contributes exactly 1 edge traversal. -/
theorem octaveEdgeCount_eq_eight : octaveEdgeCount = 8 := by
  unfold octaveEdgeCount
  rw [Finset.sum_congr rfl (fun i _ => grayCycle3_per_tick_edge_count i)]
  simp

What this page does not claim

The theorem does not force the cycle to be a Gray cycle; that is a separate structural choice. The result applies only to the canonical eight-tick cycle, not to all possible recognition cycles. The theorem says nothing about cycles of other lengths or dimensions.

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

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