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Foundational THEOREM Cosmology v5

8-Tick and 45-Gap Synchronize at 360

The cosmic recognition cycle has period 360 = lcm(8, 45)

The cosmic recognition cycle has period 360 = lcm(8, 45).

Equations

[ 2^D=8,\qquad D=3,\qquad \Omega_\Lambda^{\mathrm{RS}}=\frac{11}{16}\ \text{where applicable} ]

Shared cosmological forcing frame.

Derivation chain (Lean anchors)

Each row links to the corresponding Lean 4 declaration in the Recognition Science canon. A resolved anchor has a green check; an unresolved anchor flags a registry/canon mismatch.

  1. 1 Sync period = 360 theorem checked
    IndisputableMonolith.Foundation.DimensionForcing.sync_period_eq_360 Open theorem →
  2. 2 Sync implies D=3 theorem checked
    IndisputableMonolith.Foundation.DimensionForcing.sync_implies_D3 Open theorem →
  3. 3 Sync factorization theorem checked
    IndisputableMonolith.Foundation.DimensionForcing.sync_factorization Open theorem →
  4. 4 Rotation period theorem checked
    IndisputableMonolith.Foundation.DimensionForcing.rotation_period Open theorem →

Narrative

1. Setting

8-Tick and 45-Gap Synchronize at 360 is anchored in Foundation.DimensionForcing. The page is not a loose explainer: it is a public map from the Recognition Science forcing chain into one Lean-checked declaration bundle. The primary anchor determines what is proved, and the surrounding declarations show how the result is used.

2. Equations

(E1)

$$ 2^D=8,\qquad D=3,\qquad \Omega_\Lambda^{\mathrm{RS}}=\frac{11}{16}\ \text{where applicable} $$

Shared cosmological forcing frame.

3. Prediction or structural target

  • Structural target: Foundation.DimensionForcing must keep resolving in the Lean canon, and all downstream pages that cite this anchor must continue to type-check.

This page is currently a structural derivation. Where the claim has direct empirical content, the prediction table gives the measurable target; otherwise the claim is a formal bridge inside the Lean canon.

4. Formal anchor

The primary anchor is Foundation.DimensionForcing..sync_period_eq_360.

/-- Verify: lcm(8, 45) = 360. -/
theorem sync_period_eq_360 : sync_period = 360 := by
  unfold sync_period eight_tick gap_45; rfl

/-! ## The 8-Tick Argument (Core Definition) -/

/-- The eight-tick cycle is 2^D for dimension D. -/
def EightTickFromDimension (D : Dimension) : ℕ := 2^D

/-- Derived ledger lower bound: every simplicial recognition loop has at least 8 ticks. -/

5. What is inside the Lean module

Key theorems:

  • sync_period_eq_360
  • simplicial_loop_tick_lower_bound
  • eight_tick_is_2_cubed
  • power_of_2_forces_D3
  • eight_tick_forces_D3
  • spinor_dim_D3
  • spinor_dim_D1
  • spinor_dim_D2
  • spinor_dim_D4
  • D3_has_spinor_structure
  • D1_no_spinor_structure
  • D2_no_spinor_structure

Key definitions:

  • eight_tick
  • gap_45
  • sync_period
  • EightTickFromDimension
  • spinorDimension
  • HasRSSpinorStructure
  • SupportsNontrivialLinking
  • RSCompatibleDimension

6. Derivation chain

7. Falsifier

An alternative recognition cycle with a sync period other than 360 refutes sync_period_eq_360.

8. Where this derivation stops

Below this page the chain reduces to the RS forcing sequence: J-cost uniqueness, phi forcing, the eight-tick cycle, and the D=3 recognition substrate. If any upstream theorem changes, this page must be versioned rather than patched silently. The published URL is stable, but the version field is the contract.

10. Audit path

To audit sync-period-360, start with the primary Lean anchor Foundation.DimensionForcing.sync_period_eq_360. Then inspect the theorem names listed in the module-content section. The page is intentionally built so the public explanation is not a substitute for the proof object; it is a map into it. The mathematical dependency is the same in every case: reciprocal cost fixes J, J fixes the phi-ladder, the eight-tick cycle fixes the recognition clock, and the domain theorem listed above supplies the last step. If that last step is empirical, the falsifier section names what observation would break it. If that last step is formal, a Lean-checkable counterexample is the relevant failure mode.

Falsifier

An alternative recognition cycle with a sync period other than 360 refutes sync_period_eq_360.

Related derivations

References

  1. lean Recognition Science Lean library (IndisputableMonolith)
    https://github.com/jonwashburn/shape-of-logic
    Public Lean 4 canon used by Pith theorem pages.
  2. paper Uniqueness of the Canonical Reciprocal Cost
    Washburn, J.; Zlatanovic, B.
    Axioms (MDPI) (2026)
    Peer-reviewed paper anchoring the J-cost uniqueness theorem.
  3. lean Recognition Science Lean library (IndisputableMonolith)
    https://github.com/jonwashburn/shape-of-logic
    Public Lean 4 canon used by Pith theorem pages.
  4. paper Uniqueness of the Canonical Reciprocal Cost
    Washburn, J.; Zlatanovic, B.
    Axioms (MDPI) (2026)
    Peer-reviewed paper anchoring the J-cost uniqueness theorem.
  5. lean Recognition Science Lean library (IndisputableMonolith)
    https://github.com/jonwashburn/shape-of-logic
    Public Lean 4 canon used by Pith theorem pages.
  6. paper Uniqueness of the Canonical Reciprocal Cost
    Washburn, J.; Zlatanovic, B.
    Axioms (MDPI) (2026)
    Peer-reviewed paper anchoring the J-cost uniqueness theorem.
  7. lean Recognition Science Lean library (IndisputableMonolith)
    https://github.com/jonwashburn/shape-of-logic
    Public Lean 4 canon used by Pith theorem pages.
  8. paper Uniqueness of the Canonical Reciprocal Cost
    Washburn, J.; Zlatanovic, B.
    Axioms (MDPI) (2026)
    Peer-reviewed paper anchoring the J-cost uniqueness theorem.
  9. lean Recognition Science Lean library (IndisputableMonolith)
    https://github.com/jonwashburn/shape-of-logic
    The full forcing chain and the supporting machinery for this derivation are checked in Lean 4.
  10. paper Uniqueness of the Canonical Reciprocal Cost
    Washburn, J., Zlatanovic, B.
    Axioms (MDPI) (2026)
    Peer-reviewed source for the Law of Logic cost theorem uniqueness theorem that anchors RS at T5.

How to cite this derivation

  • Stable URL: https://pith.science/derivations/sync-period-360
  • Version: 5
  • Published: 2026-05-14
  • Updated: 2026-05-14
  • JSON: https://pith.science/derivations/sync-period-360.json
  • YAML source: pith/derivations/registry/bulk/sync-period-360.yaml

@misc{pith-sync-period-360, title = "8-Tick and 45-Gap Synchronize at 360", author = "Recognition Physics Institute", year = "2026", url = "https://pith.science/derivations/sync-period-360", note = "Pith Derivations, version 5" }