Foundation Period Depends On Dimension Period Eq Eight Iff D Eq Three

The period-dimension equivalence

In mathematics and physics, a period is the length of a repeating cycle. The Recognition Science framework, a system that derives physical structure from a ledger of recognition events, a discrete record of distinctions, defines the period of its fundamental cycle as a function of spatial dimension. The definition is simple: for any dimension D, the period is 2 to the power of D. In one dimension the period is 2, in two dimensions it is 4, in three dimensions it is 8, and in four dimensions it is 16.

The central theorem, named period_eq_eight_iff_D_eq_three, proves a bidirectional equivalence: the period equals 8 if and only if the dimension equals 3. This is not a one-way implication. The theorem shows that from the period being 8 you can derive that the dimension is 3, and from the dimension being 3 you can derive that the period is 8. In plain language, an eight-tick recognition cycle and a three-dimensional space force each other. The framework's library, a machine-checked collection of formal theorems, contains this result as a proved statement.

The theorem addresses a concern about circularity in the framework's reasoning. The historical label "8-tick" for the period seemed to presuppose that space is three-dimensional, making the order of proofs appear circular. This result resolves that issue by showing the dimension is determined independently, through a topological property called Alexander duality, which concerns how a circle can be linked in a sphere. The period is then defined as 2 to the power of that dimension. The theorem no_period_circularity explicitly demonstrates this non-circularity: the dimension is pinned down without any reference to the number 8, and the period follows as a consequence.

In Recognition Science, this equivalence is part of a larger chain of results. The framework proves that a unique cost function forces the golden ratio, which forces an eight-tick cycle, which forces three spatial dimensions. The period-dimension theorem is the hinge connecting the cycle to the dimensionality of space. It shows that the number 8 and the number 3 are not independent facts but two faces of the same structural requirement.

/-- Period equals 8 iff `D = 3`. Proved from `power_of_2_forces_D3`.
    Neither direction is the "natural" one; they are equivalent. -/
theorem period_eq_eight_iff_D_eq_three (D : ℕ) :
    PeriodFromDimension D = 8 ↔ D = 3 := by
  constructor
  · intro h
    unfold PeriodFromDimension at h
    exact power_of_2_forces_D3 D h
  · intro h; subst h; rfl

/-- The fundamental period length from the spatial dimension: `2^D`.
    This is a **definition** that does not presuppose `D = 3`.
    Writing `8` here would be wrong in general; writing `2^D` is
    dimension-generic. -/
def PeriodFromDimension (D : ℕ) : ℕ := 2 ^ D

/-- **DIAGNOSTIC THEOREM.** The linking argument for `D = 3` does not
    presuppose the 8-tick period.

    Statement: every `D` with non-trivial linking equals 3, and this
    is proved *without* the predicate `EightTickFromDimension D =
    eight_tick` as a hypothesis.

    Proof: `linking_requires_D3` is proved from Alexander duality in
    `Foundation/AlexanderDuality`. Its hypothesis is `SupportsNontrivialLinking`,
    which is defined as `SphereAdmitsCircleLinking`, which is defined via the
    reduced cohomology of `S¹`. None of those refer to the 8-tick.

    Hence D = 3 is pinned first; the period 8 then follows as `2^D = 2^3`. -/
theorem no_period_circularity :
    (∀ D : ℕ, SupportsNontrivialLinking D → D = 3) ∧
    (PeriodFromDimension 3 = 8) :=
  ⟨linking_requires_D3, rfl⟩

What this page does not claim

The theorem does not claim that physical space is three-dimensional in the empirical sense; it is a structural result within the framework. The theorem does not claim that the eight-tick cycle is the only possible period; it only relates the period to the dimension. The theorem does not claim that Alexander duality is the only way to force the dimension; it is one specific proof path.

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

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