Foundation Period Depends On Dimension No Period Circularity

The non-circular order

A period is a repeating cycle. In the Recognition Science framework, the central cycle has eight steps, and a natural question is whether the number eight was assumed to get the result. The declaration no_period_circularity answers that question directly. It proves two facts together: any dimension that supports a certain kind of linking must be three, and the period for dimension three is eight. The proof of the first fact never mentions the number eight, and the period is then defined as a function of the dimension.

The period is defined by the equation PeriodFromDimension D = 2^D. For one dimension the period is 2, for two it is 4, for three it is 8, and for four it is 16. The theorem period_eq_eight_iff_D_eq_three shows the equivalence is exact: the period equals eight if and only if the dimension is three. This is a two-way street, so either statement can be treated as the hypothesis and the other as the conclusion. The point of the module is to make the honest direction explicit: the dimension is forced first, and the period follows.

The dimension itself is forced by a topological argument, not by the cycle. The predicate SupportsNontrivialLinking asks whether a sphere admits a certain kind of circle linking, which is a question about the reduced cohomology group. The theorem period_at_D3 proves that this linking condition forces the dimension to be three, with no reference to the number eight or to the eight-step cycle. The declaration no_period_circularity packages this with the period calculation to exhibit the non-circularity in a single statement.

In Recognition Science, the framework models the physical world as a discrete record of recognition events with a forced cost. The eight-step cycle is a consequence of the three-dimensional structure, not a premise for it. This ordering matters because it answers a specific objection about the chain of theorems. The framework's library of machine-checked formal theorems contains this proof with zero sorry and zero new axiom, meaning the reasoning is fully checked by the kernel.

/-- **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⟩

/-- 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

/-- 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

/-- For `D = 3`, the period is 8. -/
theorem period_at_D3 : PeriodFromDimension 3 = 8 := rfl

What this page does not claim

The declaration does not prove that three-dimensional space is physically real, only that the topological linking condition forces the dimension in the framework. The declaration does not prove that the eight-step cycle is the only possible period, only that it is the period for dimension three. The declaration does not derive the value of any physical constant or coupling.

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)

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.

Derived articles

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