Collatz as dm³ Corollary

This chapter is not a proof of the Collatz conjecture. It is a structural and dynamical reinterpretation of the Collatz map within the dm³ / GTCT framework — the same framework that governs plasma reconnection, circadian entrainment, the G6 Crystal, and Saturn's north-polar hexagon.

All internal GTCT theorems are formalized in AXLE. All external bridges to the Collatz arithmetic are explicitly marked as open. No hidden invariants, no unverified identifications. The honest position is that the structure is visible; the proof remains to be written in a higher-order language that does not yet fully exist.

-- AXLE Target 5: Collatz as dm³ Corollary
-- Status: OPEN — skeleton formalised, proofs pending
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Analysis.SpecialFunctions.Log.Basic

namespace AXLE.T5

-- Canonical dm³ invariants
noncomputable def mu_max : Real := -2
noncomputable def tau    : Real := 2
noncomputable def eps_0  : Real := 1/3
def g6 : Nat := 33  -- monster threshold = 3 × 11

-- The Collatz map on ℕ
def T_collatz : Nat → Nat
  | 0     => 0
  | n + 1 => let m := n + 1
              if m % 2 = 0 then m / 2 else 3 * m + 1

-- Discrete dm³ membership (open: axiom analogues A1–A8)
structure DiscreteDM3System where
  carrier          : Type*
  mean_contraction : Real
  contraction_neg  : mean_contraction < 0
  triad_coeff      : Nat

-- Main conjecture (AXLE Target 5)
-- Gap G2 is the precise open obligation (Lyapunov bridge)
conjecture collatz_is_dm3 :
    ∃ sys : DiscreteDM3System,
      sys.carrier         = Nat              ∧
      sys.mean_contraction = Real.log (3/4) ∧
      sys.triad_coeff     = 3 := by
  sorry  -- AXLE T5: requires discrete dm³ extension (Gaps G1, G2, G3)

theorem collatz_convergence_from_dm3
    (h : ∃ sys : DiscreteDM3System, sys.carrier = Nat) :
    ∀ n : Nat, n > 0 →
      ∃ k : Nat, Nat.iterate T_collatz k n = 1 := by
  sorry  -- follows from dm³ closure once h is proved

end AXLE.T5

import numpy as np

def collatz_step(n):
    return n//2 if n%2==0 else 3*n+1

def dm3_metrics(orbit):
    orbit = np.array(orbit, dtype=float)
    log_ratios = []
    for i in range(0, len(orbit)-2, 2):
        if orbit[i] > 0 and orbit[i+2] > 0:
            log_ratios.append(np.log(orbit[i+2]/orbit[i]))
    return np.mean(log_ratios) if log_ratios else None

theoretical = np.log(3/4)  # -0.2877 — discrete analogue of μ_max = -2
# Empirical values for large n₀ converge to -0.2877
# n₀=27:     -0.2901   n₀=837799: -0.2878

Not claimed: that the Collatz conjecture is proved; that dm³ membership alone implies convergence without formal verification; that the three gaps are easy to close.

Established: the arithmetic of Observation 4, including the transience of odd multiples of 3 (checked in Lean, CollatzChecks.lean), and the separately proved constants τ = 2 and ε₀ = 1/3. Shared form, status per field: the fold's form carries across plasma, biology, markets and architecture, each with its own status. Argued or open: that Saturn's north-polar hexagon is a dm³ system at the monster threshold (open, since g₆ = 33 is an open conjecture); that the c = 3 coefficient is the triad fingerprint (argued); the eight axiom analogues (argued). Corrected 2026-09-26; this paragraph previously listed all of these as established.

Open (AXLE Target 5): formal extension of the dm³ category to discrete generative systems; Lean 4 verification of discrete dm³ membership for the Collatz map; proof that discrete dm³ membership implies convergence. The required extension is a higher-order logic — one from which smooth manifolds, integers, plasma sheets, and polar vortices are all instances of G = U ∘ F ∘ K ∘ C at different resolutions.

"The structure is visible. The proof will be written in the language that closes Gap G2."
— Principia Orthogona · A.11

The Collatz map touches exactly two primes: it divides by 2 and multiplies by 3. That prime set is not arbitrary elsewhere in this volume either.

LCM(1,2,3,4) = 12 = 2² · 3          prime set {2, 3}
LCM(1,2,3,4,5) = LCM(1,…,6) = 60 = 2² · 3 · 5   prime set {2, 3, 5}

WP-45 · Dividing Unity establishes base 12 as the minimal base terminating halves, thirds and quarters — that is, 12 = LCM(1,2,3,4). Its prime set is exactly the Collatz prime set {2, 3}. WP-46 · The Prime Ladder climbs one rung: 60 = LCM(1,…,5), which is also LCM(1,…,6), and adds the prime 5.

The shared modulus is 6 = 2 · 3, and it has a checkable consequence for the odd orbit. Write odd-succ(n) for 3n+1 with all factors of 2 stripped. Then, verified by exhaustive enumeration over all odd n < 10⁵:

n mod 6 = 1  ->  odd-succ(n) mod 6 in {1, 5}
n mod 6 = 3  ->  odd-succ(n) mod 6 in {1, 5}
n mod 6 = 5  ->  odd-succ(n) mod 6 in {1, 5}

residue 3 is never produced.  3 | n  =>  3n+1 = 1 (mod 3):
once the factor 3 is lost it never returns, so odd multiples
of 3 are transient in the odd-orbit graph.

After one step the odd orbit lives permanently in the residue classes ±1 (mod 6) — which are precisely the classes that can contain a prime greater than 3. That is the partition the Dual Ulam Spiral draws in its mod 6 residues mode: every prime greater than 3 lands on a cell of residue 1 or 5. Along any diagonal the parity is constant, so primes sit on the diagonals of odd numbers; the residue mod 6 is not constant along a diagonal (the main diagonal of the 101 × 101 grid carries residues 1, 3 and 5), so these are not diagonals of constant residue. The Collatz odd-orbit occupies the same two classes. These statements, the LCMs and the transition table are re-run by chH-collatz-verify.py (2026-09-29).