dm³ 102 · Week 07 · Criticality

The Criticality Bridge — Why c = 3, in Depth

The Whitney A₁ fold, worked in full, and why it forces the Collatz map
dm³ 102 · Week 07 · Criticality
The Criticality Bridge — Why c = 3, in Depth
Course: dm³ 102  ·  Criticality  ·  Source: Book 3, Criticality Bridge (Theorem C.1)

101 introduced \(c^*=3\) as the unique value producing a Whitney \(A_1\) fold in \(V(q)=q^3-cq\) at \(q=1\). This week works the full derivation. Setting \(V'(q)=3q^2-c=0\) gives critical points \(q^*=\pm\sqrt{{c/3}}\); for the fold to land exactly on the integer \(q=1\), we need \(q^*=1\), i.e. \(\sqrt{{c/3}}=1 \Rightarrow c=3\). At exactly \(c=3\): \(V(1)=1-3=-2\), \(V'(1)=3-3=0\), \(V''(1)=6q|_{{q=1}}=6\neq0\) — a genuine double root, not a higher-order degeneracy. Factoring confirms it directly: \(V(q)+2 = q^3-3q+2 = (q-1)^2(q+2)\), verified by direct multiplication.

Theorem C.1 (Criticality at c*=3) — why this is the unique value
No other value of \(c\) produces an integer-coherent fold this way. For \(c<3\): \(q^*=\sqrt{{c/3}}<1\), the fold sits short of the integer lattice (this is exactly \(\varphi\)’s situation from 101 Week 9: \(c=\varphi\approx1.618\) gives \(q^*\approx0.734\)). For \(c>3\): \(q^*>1\), the fold overshoots (this is \(\Delta\)’s situation from Week 9: \(c=\Delta\approx1.928\) still gives \(q^*<1\) since \(\Delta<3\), but larger \(c\) values would overshoot). \(c=3\) is the exact, unique crossing point.

The Collatz map \(T:n\mapsto3n+1\) (odd branch) has coefficient exactly \(c=3\) — making it, per this framework, the unique integer map sitting precisely at the fold threshold. This is the real, substantive connection this course cares about: not a numeric coincidence with an external physics constant, but the same \(c=3\) appearing in both the abstract fold-potential argument and the specific integer dynamical system this course has been building toward since 101.

This week’s content is grounded directly in the AXLE Lean sources cited above — no material in this page depends on the external, unverified source removed from earlier drafts of this course.
-- dm³ 102 · Week 07 · Theorem C.1, worked in full

-- V(q) = q³ - cq,  V'(q) = 3q² - c = 0  ⟹  q* = ±√(c/3)
-- Fold at integer q*=1  ⟺  √(c/3)=1  ⟺  c=3

-- At c=3 exactly:
example : (1:ℝ)^3 - 3*1 = -2 := by norm_num          -- V(1) = -2
example : 3*(1:ℝ)^2 - 3 = 0 := by norm_num           -- V'(1) = 0
example : (6:ℝ)*1 ≠ 0 := by norm_num                 -- V''(1) = 6 ≠ 0 (genuine double root)

-- Factorization: V(q)+2 = (q-1)²(q+2)
example (q : ℝ) : q^3 - 3*q + 2 = (q-1)^2*(q+2) := by ring

-- Collatz map T: n ↦ 3n+1 has coefficient c=3 exactly — the unique
-- integer map at the fold threshold, per this derivation.
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