Foundation Active Edge Budget Active Edge Budget One Statement

The one-edge budget

The active edge budget is a count inside Recognition Science's model of discrete change. The framework pictures reality as a ledger, a discrete record of events, where each tick of time posts a change at exactly one node of a structure. The question this theorem answers is simple: how many edges of that structure does one tick traverse? The answer, proved rather than assumed, is exactly one.

The proof works by looking at the eight ticks that make up one full cycle. The framework's earlier results establish that a complete cycle visits eight vertices, the corners of a three-dimensional cube. The key fact is that consecutive visits in the canonical cycle differ in exactly one bit of their three-bit address, meaning each step moves along exactly one edge of the cube. Since every one of the eight steps traverses precisely one edge, the total edge count for the cycle is eight, and the per-tick average is forced to be one.

The theorem active_edge_budget_one_statement packages this into a single machine-checked statement. It asserts that every consecutive pair in the cycle has Hamming distance one, the total octave edge count is eight, any uniform per-tick count must be one, and both the framework's constants active_edges_per_tick and A equal one. The integer one is not selected from options; it is the only cardinality of a one-bit difference between two binary strings of length three.

In Recognition Science, this derivation matters because a larger budget was previously a postulate. The matter-consciousness duality, written as η_B · Θ_crit = φ, rests on this equality, so replacing an assumption with a proof strengthens the chain. The theorem shows the budget is forced by the combination of atomic ticks, eight-tick periodicity, and Gray-cycle adjacency, with no free parameter left to tune.

The declaration does not claim that the physical universe actually runs on this ledger, nor that the Gray cycle is the only possible sequence of postings. It proves a conditional statement: if the framework's axioms hold, then the active edge budget is one. The empirical question of whether nature obeys those axioms is a separate matter, not settled by this theorem.

/-- **ACTIVE-EDGE BUDGET: ONE-STATEMENT THEOREM.**

(1) For consecutive postings `p, q` in the canonical 8-tick cycle on
    the 3-cube, the Hamming distance equals 1.
(2) The total edge traversal in one octave equals 8 (one per tick).
(3) The downstream constants `active_edges_per_tick` and
    `Foundation.GapDerivation.A` are both forced equal to 1.
(4) Every tick has a unique witnessing edge of the 3-cube. -/
theorem active_edge_budget_one_statement :
    (∀ i : Fin 8, hammingDist (grayCycle3Path i) (grayCycle3Path (i + 1)) = 1) ∧
    octaveEdgeCount = 8 ∧
    (∀ n : ℕ, octaveEdgeCount = 8 * n → n = 1) ∧
    Constants.AlphaDerivation.active_edges_per_tick = 1 ∧
    Foundation.GapDerivation.A = 1 ∧
    (∀ i : Fin 8,
      ∃! k : Fin 3, grayCycle3Path i k ≠ grayCycle3Path (i + 1) k) := by
  refine ⟨?_, ?_, ?_, ?_, ?_, ?_⟩
  · intro i
    exact grayCycle3_per_tick_edge_count i
  · exact octaveEdgeCount_eq_eight
  · exact per_tick_count_from_octave
  · rfl
  · exact gapDerivation_A_eq
  · exact unique_edge_per_tick

/-- 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

/-- **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 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

What this page does not claim

This theorem does not claim that physical reality actually operates on a discrete ledger. It does not claim that the Gray cycle is the only possible sequence of postings within the framework. It does not derive the fine-structure constant or any measured physical constant from this budget alone.

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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