Holography Eight Tick Subperiod Exclusion

The minimum census period

Holography eight tick subperiod exclusion is a result about a small discrete model: a cube face with one recognition bit at each of its four corners. A recognition event, a single bit flip at one corner, is called a tick. The model studies closed walks, sequences of ticks that return to the starting corner configuration. The question is how long a closed walk must be to visit all four admissible loop types that the model's ledger, a discrete record of events, allows.

The four loop types are the empty loop, the adjacent-edge loops, the diagonal loops, and the full loop. A walk is census-complete when it visits at least one representative of each type. The framework's machine-checked library of formal theorems proves that no closed walk of length 1, 2, or 4 can be census-complete. For lengths 1 and 2 the reason is cardinality: such a walk visits at most two configurations, but four loop types must be met. For length 4 the reason is parity: each bit flip changes the parity of the number of set bits, so a closed 4-walk visits at most two even-parity configurations, again short of the four required types.

The same library proves that a closed walk of length 8 exists and is census-complete. The witness starts from the configuration 0000 and flips vertices in the order 0, 1, 2, 3, 0, 2, 1, 3, visiting the configurations 0, 1, 3, 7, 15, 14, 10, 8, and back to 0. This walk meets all four loop types, at configurations 0, 3, 15, and 10. The combined theorem, minimal_census_period_eight, states that among the divisors of 8, the census-complete closed walk lengths begin exactly at 8.

In Recognition Science, this result is the discrete half of a deficit-free-period argument. The full 8-tick cycle is identified with one full 2π turn of a continued clock. The exclusion theorem shows that identifying ticks modulo a proper divisor of 8, the discrete analog of a conical deficit 2π/n, destroys the admissible census. Therefore, once census preservation is granted, the Euclidean period must be the full 2π/κ, not a proper sub-multiple.

The theorem does not grant that census preservation. Whether the physical Euclidean continuation must preserve the census is an open question, owned by the derivation captain. This result removes the freedom of choosing a sub-period once preservation is assumed; it does not establish the assumption itself.

/-- **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⟩

/-- **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⟩

/-- **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

This result does not prove that the physical Euclidean continuation must preserve the census. This result does not derive the Bekenstein-Hawking coefficient or any other physical constant. This result does not apply to walks that are not closed or that start from a configuration outside the model's domain.

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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