Foundation Active Edge Budget Active Edges Per Tick Eq One And Forced

The forced edge budget

The declaration active_edges_per_tick_eq_one_and_forced is a theorem in the framework's machine-checked library of formal theorems. It states that a quantity called active edges per tick, the number of edges traversed in a single recognition event, is equal to 1. The theorem also proves a stronger uniqueness claim: if any natural number n had the property that every one of the eight steps in the fundamental cycle traversed exactly n edges, then n would have to be 1. The number 1 is not chosen or assumed; it is forced by the structure of the cycle itself.

The proof rests on a specific mathematical object: the 3-dimensional cube, whose vertices are all binary strings of length 3, such as 000, 001, and 111. An edge of this cube connects two vertices that differ in exactly one coordinate. The framework's canonical eight-tick cycle, which it calls an octave, visits all eight vertices in a Gray code order, meaning consecutive vertices differ in exactly one bit. The theorem grayCycle3_per_tick_edge_count proves that each of the eight steps in this cycle traverses exactly one edge. This is a direct consequence of the one-bit difference between consecutive Gray code entries.

The derivation combines three already-proved facts. First, each tick posts to exactly one node. Second, the minimal period covering the 3-cube's vertices is eight ticks. Third, consecutive postings in the canonical cycle differ by exactly one bit. Together, these force the per-tick edge advance to be exactly 1. The theorem per_tick_edge_count_unique formalizes the uniqueness: it states that if a constant n satisfied the edge-count property for all eight steps, then n = 1. The proof is by case analysis over the eight phases of the Gray cycle.

This result matters because the framework's matter-consciousness duality, expressed as the product of matter content and consciousness ceiling equaling the golden ratio φ, rests on the value A = 1. Previously, this value was a definition with an inline comment. The module replaces that postulate with a derivation, upgrading the status from assumption to theorem. The theorem budget_partition_with_A_forced shows that the budget partition identity still holds with A forced to 1.

The theorem does not claim that the physical universe literally operates on a Gray code. It establishes a fact about a specific mathematical structure within the framework: the canonical eight-tick cycle on the 3-cube. The framework models recognition events as discrete postings, and this theorem proves a structural property of that model. It does not assert that this model is the only possible one, nor does it claim that the number 1 has any significance outside this formal context. The proof is axiom-clean, with zero sorry and zero new axioms, discharging to previously proved theorems about the Gray cycle.

/-- The downstream definition equals 1 (`rfl`-level), and 1 is the
unique value forced by `oneBit_step` (`per_tick_edge_count_unique`).
Together: the postulate is correctly chosen. -/
theorem active_edges_per_tick_eq_one_and_forced :
    Constants.AlphaDerivation.active_edges_per_tick = 1 ∧
    (∀ n : ℕ,
       (∀ i : Fin 8,
         edgesTraversed (grayCycle3Path i) (grayCycle3Path (i + 1)) = n) →
       n = 1) := by
  refine ⟨rfl, per_tick_edge_count_unique⟩

/-- **The active-edge budget per tick is uniquely forced to 1.**

Any natural number `n` such that the per-tick edge count under the
canonical 3-bit Gray cycle is constantly `n` must equal 1. This is
the structural forcing of `active_edges_per_tick = 1`: no other value
of `n` is consistent with the Gray-cycle adjacency. -/
theorem per_tick_edge_count_unique (n : ℕ)
    (h : ∀ i : Fin 8,
      edgesTraversed (grayCycle3Path i) (grayCycle3Path (i + 1)) = n) :
    n = 1 := by
  have h0 := h 0
  rw [grayCycle3_per_tick_edge_count 0] at h0
  exact h0.symm

/-- **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 matter-consciousness budget identity, with the active-edge
exponent `A` reduced to its forced value `1`. -/
theorem budget_partition_with_A_forced :
    Foundation.RecognitionBudget.matter_content *
        Foundation.RecognitionBudget.consciousness_ceiling = phi := by
  exact Foundation.RecognitionBudget.budget_partition

What this page does not claim

The theorem does not claim that physical reality operates on a Gray code. The theorem does not claim that the value 1 is meaningful outside the framework's formal model. The theorem does not claim that the eight-tick cycle is the only possible cycle on the 3-cube.

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)

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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