Holography Eight Tick Subperiod Exclusion Eight Tick Census Witness

The eight-tick witness

A cube face with its four corners labeled 0, 1, 2, 3, each holding a bit that is either on or off. A recognition walk, a sequence of steps where each step flips exactly one corner's bit, moves through the sixteen possible on-off patterns. Some patterns are special: the empty loop (all off), the full loop (all on), the two diagonal loops, and the four adjacent-edge loops. A walk is census-complete when it visits at least one pattern from each of these four families.

The declaration eight_tick_census_witness proves, by exhaustive machine check, that a closed walk of exactly eight flips exists which is census-complete. The witness starts from all-off and flips corners in the order 0, 1, 2, 3, 0, 2, 1, 3, visiting the patterns 0, 1, 3, 7, 15, 14, 10, 8, and returning to 0. Along the way it meets the empty loop at 0, an adjacent-edge loop at 3, a diagonal loop at 10, and the full loop at 15. This single explicit example is the entire content of the existence half of the theorem.

The companion theorems no_subperiod_one, no_subperiod_two, and no_subperiod_four prove that no closed walk of length 1, 2, or 4 can be census-complete. For lengths 1 and 2 the reason is simple counting: such a short closed walk touches at most two distinct patterns, but four families must be met. For length 4 the obstruction is parity: flipping one bit always changes whether the number of on-bits is even or odd, so a closed 4-walk can only visit even-parity patterns, again at most two of them. The capstone theorem minimal_census_period_eight assembles these four facts into the statement that 8 is the minimal closed walk length that achieves the full census.

In Recognition Science, this discrete result supports a larger argument about the period of a recognition cycle. The framework's library proves that the full 8-tick cycle corresponds to one complete turn of 2π in the continued clock. The subperiod exclusion shows that identifying ticks modulo a proper divisor of 8, the discrete analog of a conical deficit, would destroy the census of admissible sectors. Together these force the census-preserving Euclidean period to be the full 2π, not a proper fraction of it.

What the declaration does not claim is equally precise. It does not prove that the physical Euclidean continuation must preserve the census; that regularity condition at the horizon fixed point remains open. It does not derive the Bekenstein-Hawking coefficient. It establishes only the discrete combinatorial fact: among the divisors of 8, the census-complete closed walk lengths begin exactly at 8.

/-- **d = 8 realizes the census**: a closed 8-tick flip walk exhibiting all four
admissible sectors exists. Witness: from `0000`, flip vertices `0,1,2,3,0,2,1,3`,
visiting `0 → 1 → 3 → 7 → 15 → 14 → 10 → 8 → 0` (orbits met at `0, 3, 15, 10`). -/
theorem eight_tick_census_witness :
    ∃ (s : FaceCfg) (fs : Fin 8 → Fin 4),
      walkEnd s (List.ofFn fs) = s ∧ censusComplete s (List.ofFn fs) = true := by
  exact ⟨0, ![0, 1, 2, 3, 0, 2, 1, 3], by decide, by decide⟩

/-- **d = 1 excluded**: no closed 1-tick walk is census-complete (cardinality:
it visits at most 2 configurations; 4 orbits are required). -/
theorem no_subperiod_one :
    ∀ (s : FaceCfg) (fs : Fin 1 → Fin 4),
      walkEnd s (List.ofFn fs) = s → censusComplete s (List.ofFn fs) = false := by
  decide

/-- **d = 2 excluded**: no closed 2-tick walk is census-complete (cardinality). -/
theorem no_subperiod_two :
    ∀ (s : FaceCfg) (fs : Fin 2 → Fin 4),
      walkEnd s (List.ofFn fs) = s → censusComplete s (List.ofFn fs) = false := by
  decide

set_option maxRecDepth 4096 in
set_option maxHeartbeats 1600000 in

/-- **d = 4 excluded**: no closed 4-tick walk is census-complete. The mechanism is
parity: single-bit flips alternate popcount parity, so a closed 4-walk sees at most
2 distinct even-parity configurations, short of the 4 required orbits. The proof is
the full kernel enumeration of all 16 × 4⁴ = 4096 walks. -/
theorem no_subperiod_four :
    ∀ (s : FaceCfg) (fs : Fin 4 → Fin 4),
      walkEnd s (List.ofFn fs) = s → censusComplete s (List.ofFn fs) = false := by
  decide

/-- **The subperiod-exclusion capstone**: among the divisors of 8, the census-complete
closed walk lengths begin exactly at 8. Every proper divisor fails; 8 succeeds. This
is the discrete deficit-free-period statement: identifying the recognition cycle
modulo a proper divisor of 8 (the discrete conical deficit `2π/n`) destroys the
admissible sector census, so the census-preserving period is the full 8-tick turn. -/
theorem minimal_census_period_eight :
    (∀ (s : FaceCfg) (fs : Fin 1 → Fin 4),
        walkEnd s (List.ofFn fs) = s → censusComplete s (List.ofFn fs) = false) ∧
    (∀ (s : FaceCfg) (fs : Fin 2 → Fin 4),
        walkEnd s (List.ofFn fs) = s → censusComplete s (List.ofFn fs) = false) ∧
    (∀ (s : FaceCfg) (fs : Fin 4 → Fin 4),
        walkEnd s (List.ofFn fs) = s → censusComplete s (List.ofFn fs) = false) ∧
    (∃ (s : FaceCfg) (fs : Fin 8 → Fin 4),
        walkEnd s (List.ofFn fs) = s ∧ censusComplete s (List.ofFn fs) = true) :=
  ⟨no_subperiod_one, no_subperiod_two, no_subperiod_four, eight_tick_census_witness⟩

What this page does not claim

The physical Euclidean continuation must preserve the census. The Bekenstein-Hawking coefficient is derived from this declaration alone. The recognition walk model itself is derived rather than chosen.

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

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