dm³ 101 · Week 05 · Constant: T* = 2π

Period T* = 2π — What the Reeb Flow Forces

π is not chosen. It is the derived period of the contact structure’s flow.
dm³ 101 · Week 05 · Constant: T* = 2π
Period T* = 2π — What the Reeb Flow Forces
Course: dm³ 101  ·  Constant: T* = 2π  ·  Source: Book 3, “The Recurrence Ladder — From π to Hexabonacci”, Zenodo 10.5281/zenodo.19117400
A framing note before Week 5 starts
The course calls Weeks 5–8 the “π Operator” phase, and that label is kept because it is how the existing course design names this stretch. Read precisely, though: Volume I defines exactly five operators (C, K, F, U, and E). \(\pi\) is not a sixth operator in that chain — it is a proved period constant, \(T^* = 2\pi\), that appears inside the canonical triple \((T^*, \mu_{{\max}}, \tau) = (2\pi, -2, 2)\). This week treats it as exactly that: a constant with a real derivation, not an operator with its own composition rule.

Theorem π.1 (Period, proved). On the contact manifold \(M = S^1 \times \mathbb{{R}}\) with contact form \(\alpha = dz - p\,dq\), the Reeb vector field \(R\) satisfies \(\alpha(R)=1\), \(d\alpha(R,\cdot)=0\), and its flow is periodic with period exactly \(T^* = 2\pi\). The derivation is compactification, not convention: the action coordinate \(z\) is taken on the circle \(S^1 = \mathbb{{R}}/2\pi\mathbb{{Z}}\), so the Reeb flow \(\dot z = 1\) necessarily closes after \(2\pi\) — any other period would require a different compactification of \(z\), which nothing in the model asks for.

This is worth sitting with, because “\(2\pi\) shows up everywhere” is normally treated as background noise — a byproduct of using radians. Here it is being claimed as a structural fact of the dm³ contact geometry: every cyclic dm³ process inherits this same period, because every cyclic dm³ process is, by construction, a closed Reeb orbit on the same underlying \((M, \alpha)\). The Lean structure from Week 4’s correspondence table already carries this: T_star := 2 * Real.pi is a field assignment, not a computed limit.

Falsifiability π.1
If a genuine dm³ cyclic instantiation is found whose fundamental period is not an integer multiple of \(2\pi\) in its natural action coordinate, the compactification claim fails and \(T^*\) is not universal — it would be a property of specific systems, not of the contact structure itself.

Keep this constant fixed in mind for the rest of Phase II: Week 6 uses it as the anchor point of the recurrence ladder (\(\pi\) sits outside the ladder proper, as the periodic “clock” against which the ladder’s constants are read), and Week 8’s milestone asks you to verify the Reeb-flow derivation by hand.

This week’s content is grounded in Principia Orthogona, Book 3 (“The Recurrence Ladder”) and AXLE_v6.lean — theorem names and Lean identifiers above point at the actual source files.
-- dm³ 101 · Week 05 · T* = 2π, from AXLE_v6.lean

structure Dm3Canonical where
  T_star : ℝ
  mu_max : ℝ
  tau    : ℝ

def canonical : Dm3Canonical :=
  { T_star := 2 * Real.pi, mu_max := -2, tau := 2 }

-- T_star is an assignment, not a derived limit, in this top-level
-- structure — the derivation (Reeb flow period on M = S¹×ℝ) lives
-- in the Book 3 chapter (Theorem π.1). This file records the result.
-- Pending: Mathlib Floquet/periodic-orbit API to formalize the
-- Reeb-flow argument itself inside Lean (see AXLE_v6.lean note
-- near line 725: "Pending: Mathlib Floquet theory / periodic
-- orbit API").
← Week 4 ↑ dm³ 101 Course Week 6 →