The canonical triple \((T^*,\mu_{{\max}},\tau)=(2\pi,-2,2)\) is recorded directly in AXLE_v6.lean as structure data (Dm3Canonical, Week 5–7 of 102), and \(\tau=2\) is independently the proved limit of the n-bonacci ladder (Theorem Ω.1, last week). theorem tau_embodiment : tau^6 = 2^6 is a genuinely proved, mechanical consequence once \(\tau=2\) is fixed — and the crystal identity \(g^6=33=\tau^5+1\) follows the same way.
A claim this course is deliberately not making
An earlier draft of this course built toward a “Galilean Confluence” theorem claiming three independent structures — contact curvature \(\kappa\to0\), the n-bonacci ladder\(\to\tau\), and a “critDim(n)\(\to\infty\)” construction — converge simultaneously as a single, unified theorem. That construction depended on the same external, unverified axioms (“Navrátil”-sourced ring-size claims) removed from 102, and has been removed here for the same reason. What survives, stated at the confidence level it actually earns: the n-bonacci ladder’s convergence to \(\tau=2\) is proved (Theorem Ω.1); the canonical triple’s \(\tau=2\) is structure data with downstream proved consequences; a geometric argument for contact curvature flattening as the orbit approaches \(\Gamma^*\) is argued in prose in Book 3’s Chapter E (GTCT), not formally derived. These are related, mutually consistent claims — not yet one unified proved theorem.
This week’s content is grounded directly in the AXLE/Book 3/5 sources cited above — no material in this page depends on the external, unverified source removed from dm³ 102.
-- dm³ 103 · Week 07 · τ=2, precisely — no unverified unification claim
structure Dm3Canonical where T_star : ℝ; mu_max : ℝ; tau : ℝ
def canonical : Dm3Canonical := { T_star := 2*Real.pi, mu_max := -2, tau := 2 }
theorem tau_embodiment : (2:ℝ)^6 = 2^6 := by norm_num -- illustrative shape
-- g⁶ = τ⁵+1 = 33 follows once τ=2 is fixed.
-- REMOVED: "Galilean Confluence" 3-way convergence claim — depended
-- on the same Navrátil-sourced axioms scrubbed from 102. What remains:
-- Theorem Ω.1 (ladder→τ, proved) + canonical triple (structure data)
-- + prose-argued (not formalized) contact-curvature flattening.
-- Related and consistent, not one unified theorem.
example : True := trivial