Semigroup.md

Device: the affine Heisenberg monoid

Agent 2 — algebra / semigroup. Not the free word on {odd, even}.

Exact definition

Let H be the set of triples (a, L, c) ∈ ℕ³, standing for the affine map

x  ↦  (3^a · x + c) / 2^L

on the module of numerators (the denominator is kept as an exponent). The product is composition, apply-right first:

(a, L, c) · (a', L', c')  =  (a+a',  L+L',  3^a · c' + 2^{L'} · c).

This is a monoid with unit (0,0,0). It is the semidirect product of the Parikh monoid ℕ² of exponents with the translation module ℕ in the c-slot.

Three generators:

name	triple	map
e	(0,1,0)	x ↦ x/2
o_d	(1,0,d)	x ↦ 3x+d
τ_t	(0,0,t)	x ↦ x+t (numerator translation)

The Collatz word submonoid is the image of {e, o_1}^*. That image is free (wC_inj) and every word is realised by an integer (word_realizable). The ambient monoid H is not that image: τ_1 has length 0 and translation 1, and the only length-0 Collatz word is the identity c = 0. So τ is an extra generator, not a renamed parity word.

The translation module acts on H by addTrans((a,L,c), t) = (a, L, c+t). Left multiplication by u scales a translation by 3^{a(u)}; right multiplication by v scales it by 2^{L(v)}.

The commutator defect of a split (u, v) at additive constant d is

K(u, v, d)  :=  3^{a(u)} · 2^{L(v)} · d.

Evaluated at a start n, this is the numerator defect of commuting o_d past e between the frozen prefix u and the frozen suffix v. It does not depend on n.

One nontrivial theorem

Insertion of the commutator (DeviceSemigroup.middle_commutator). For all u, v ∈ H, all d, n ∈ ℕ,

num(u · (o_d · e) · v, n)
  =  num(u · (e · o_d) · v, n)  +  3^{a(u)} · 2^{L(v)} · d

where num((a,L,c), n) = 3^a · n + c. The two frozen words have identical exponents; they differ by a constant translation. The start n cancels.

Special case u = v = 1: o_d · e and e · o_d share exponents (1,1) and differ in the c-slot by exactly d (seed_commutator). On numerators this is 3n+2d versus 3n+d.

Supporting laws, all Nat identities:

• τ_t · e = e · τ_{2t} (tau_conj_even)
• o_d · τ_t = τ_{3t} · o_d (tau_conj_odd), independent of d
• num(u, n+t) = num(u, n) + 3^{a(u)} · t (num_add) — frozen words act linearly on translations.

None of these compares 2^L to 3^a. The 2 and the 3 appear as conjugation weights of the translation module, not as a cycle-product inequality.

3x+d test

Replace o_1 by o_d.

• tau_conj_even is unchanged: the even map is still x ↦ x/2. Even conjugation is d-blind.
• tau_conj_odd is unchanged: odd conjugation scales by 3, independent of d.
• K(u, v, d) = 3^{a(u)} · 2^{L(v)} · d is homogeneous of degree one in d. Diagnostic 3 of CLOSURE.md does not fire: this is not a d-free quantity.
• Genuine cycles of 3x+5, 3x+7, … satisfy the same identities with their own d. The theorem does not exclude them, and is not claimed to.

The object sees d and does not pretend that d = 1 is special.

expStep test

expStep agrees with Collatz on odds and replaces the even branch by x ↦ 3x/2. In H that even generator is eExp = (1,1,0), not e = (0,1,0).

• eExp ≠ e (immediate).
• The even conjugation law fails: τ_1 · eExp ≠ eExp · τ_2. Direct computation: left side has c = 2, right side has c = 6 (tau_conj_even_fails_expStep). In the group this is the statement that eExp conjugates translations by 3/2, not by 1/2, so the monoid relation τ e = e τ² has no integer analogue for eExp.
• The seed commutator against o_d still exists (both maps remain affine), so mere noncommutativity does not distinguish expStep. The discriminator is the weight of even conjugation.

middle_commutator consumes the even generator e, i.e. the even branch's contraction x ↦ x/2. It is not a theorem of the affine law plus a saturated odd count, so it does not hold of expStep upon substituting eExp for e.

What the parity word misses

The parity word of n is one ray in {e, o_1}^*: at each point exactly one letter is legal. Three things live off that ray.

1. The extra generator τ. No parity word equals τ_1. Commuting an odd letter past an even letter is illegal on the dynamical diagonal (domains are disjoint: a number is not both even and odd). The commutator exists only after extending e and o_d to frozen affine maps on all of ℕ. word_realizable realises every finite word as some start's ray; it does not realise τ.

2. The module action. A frozen word acts linearly on translations: num(u, n+t) − num(u, n) = 3^{a(u)} · t. The dynamical step does not: T(7) = 11 and T(8) = 4, and 4 ≠ 11 + 3 (dynamical_step_not_module). The parity-selected map is not a map of modules.

3. n-independence of K. Along the dynamical diagonal the letters of u and v depend on n, so a word-level statistic computed from the orbit of n is n-dependent by construction. K is the defect of two frozen words, and the theorem is that the defect does not see n.

What expStep misses

expStep saturates a = L at every window and conjugates the translation module by 3/2 on both letters. The two generators become indistinguishable as linear actions on the module: the even/odd asymmetry of conjugation (×1/2 versus ×3) is exactly the even branch, and expStep has deleted it. tau_conj_even is the named form of that asymmetry, and expStep fails it.

A theorem proved from the affine law, the parity word, heaviness and positivity alone cannot mention tau_conj_even, because that law is false of eExp.

Decisive lemma

tau_conj_even: τ_t · e = e · τ_{2t} for every t.

This is the statement that the even Collatz generator is pure halving on the translation module. It is:

• true for every 3x+d (the even map never depends on d);
• false for expStep (eExp has a 3 in the linear part);
• invisible to the parity word (which never applies e and a translation through each other);
• not a cycle-product comparison;
• the even-branch contraction that every surviving divergence-half mechanism is required to consume.

The insertion theorem middle_commutator is this lemma transported across an arbitrary split, together with the odd conjugation ×3 and the seed defect d. The d-content and the even-branch content live in different slots of the same monoid (K versus tau_conj_even), which is why one object hits both filters without collapsing onto 2^L versus 3^a.

What this is not

• Not ExtensionClass. That file computes Ext¹ ≅ ℤ/G and identifies the class with C mod G — the cycle criterion. The identities here never mention G.
• Not SegmentMonoid.rank_compose. Rank still telescopes; this file does not multiply local bounds around a cycle.
• Not RunAlgebra.runA_comm. That commutator is runA(v)·G(u) − runA(u)·G(v), which is the gap again. K is 3^{a(u)} 2^{L(v)} d, a conjugation weight times d.
• Not a Collatz claim. Every identity holds for every d, and the even conjugation is a statement about the even generator, not about orbits reaching 1.