Periods of the reversible second-order cellular automaton RESOCA 115

Window transfer: a finite orbit search settles the period of a bi-infinite configuration

Proposition 4 (family 10²ᵏ⁻¹1) and Proposition 5 (family 1²ᵏ⁺¹) at k up to 12

Family 1(01)ᵏ at k = 0..9, including five values past the source's printed list

Negative controls: minimality, window adequacy, and the exceptional cases of the source's closed forms

Conjecture 3's closed form factors as (2*4ⁱ⁺¹ - 5) * (4ⁱ⁺¹ - 1)/3

Even-k splitting theorem: [u_bar; 1(01)²ⁱ⁺²] is the one-cell merge of E(i) and L(i+1) across the pinned cell -1, for every i

Two auxiliary families: E(m) = [0_bar; 1(01)ᵐ] with period 4ᵐ⁺²-1 (m = 0..9) and L(m) = [u_bar; 1(01)ᵐ], memory 1 at the word's LEFT end, with period 2²ᵐ⁺³-5 (m = 0..10)

Family 1(01)ᵏ at every even k from 2 to 20, by the split; the landed certificate frontier was k = 9

RESOCA 115 is reversible (step is injective), and at odd k the zero-memory configuration is on the same orbit as the source's

Time-column certificate machinery: notᵣealisedₒf kernel-clean; the encoding an equivalence; background anchoring with no containment lemmas

Question 1* of arXiv:2606.13159 answered NO: no finite configuration of minimal period 9, 18, 21 or 24

The section at deviation width w has size exactly 2ʷ with return times 1,2,3,6 and multiplicities 1, 1, 2ʷ-2-b, b where 3b = 2ʷ + 2(-1)ʷ – verified at w = 1..16, both parities; and the left-end half of the statement is PROVED at every w

Conjectures 2 and 3 of arXiv:2606.13159 are theorems at EVERY k, conditional on the section statement at the relevant widths (Conjecture 2 additionally on the odd-k shift)

Every column of a RESOCA 115 space-time diagram is one of exactly four sequences – the constant 1 and the three cyclic shifts of (001)ⁱnf – and it changes type only where the row reads 110

The star: a kick exchanges the constant-1 column with the shift carrying a 1 at the kick's own time, never one shift with another; hence over any period every cell is kicked an EVEN number of times

The kick-count law: over one period the cell at distance d >= 1 from the left end of the deviation interval is kicked exactly 2ʷ⁺¹⁻ᵈ times and the left end exactly (2ʷ⁺¹+4(-1)ʷ)/3 times – the counts halve exactly, cell by cell

No affine map over F₂ agrees with step³ on the orbit of E 1: four states at times 0,3,9,12 sum to zero and their images do not – the LFSR/primitive-polynomial route to Conjectures 2 and 3 is closed by proof, not by measurement

A row with no factor 011 inside its deviation interval has at most one factor 110 there, hence at most one kick; and the rows of E m and L m do avoid 011 there, certified at m <= 5

Question 2 answered NO: 3 is a period and 9 is not, under the source's definition AND with finiteness dropped

No configuration of Z at all has minimal period 9; every 9-periodic state is 3-periodic — CORRECTS the source's 9* entry

Two-sided trimming: surviving columns of period d force every p-periodic configuration to be d-periodic — no finiteness, no anchor

All-1 is the unique fixed point; no finite config has period 1, nothing has period 2; Lemma 1's finiteness is load-bearing

Periods closed under lcm but not products: 3, 6 periods, 18 not; nor under x2,3,4,7,8

Full period spectrum below 13 over ALL configurations: exactly 1, 3, 6, 12

Propositions 4 and 5 for every k (families 10²ᵏ⁻¹1 and 1²ᵏ⁺¹), by insertion-commutation induction, no native axiom