slug: nersissian-2026-rule84-mod-three-center-column bibkey: nersissian2026diagonalbases doi: 10.48550/arXiv.2609.25078 url: https://arxiv.org/abs/2609.25078v1 triage: theorem motivation_gids:

• D5/S3/StatisticalMechanics/CellularAutomata/Rule84ModThreeCenterColumn.result

The Rule 84 center column modulo three

Problem

Tigran Nersissian, Diagonal Bases and Diagonal Periods of Elementary Cellular Automata, arXiv:2609.25078v1, Remark 6, page 12, states:

(The outstanding Rule 84 case) The polynomial of Rule 84 is (a + b + ab)(1 + c). Modulo three, its center column begins 1, 1, 2, 2, 1, 2, 2, … and follows the repeating block (1, 2, 2) after the first entry for the tested times 0 ≤ t < 2048. All center values are also units modulo 9 and 27 on that range, as Corollary 2 predicts. An all-time proof of the observed pattern is not supplied here. If Rule 84 is universal modulo three, it is universal at every power of three; three could not be the sole exceptional modulus.

Section 19, page 42, states:

The classification reduces the remaining single-seed universality question to the eight rules in E at odd primes. The observed Rule 84 pattern modulo three requires an invariant or an all-time recurrence proof. A finite nonzero prefix is insufficient.

This dossier anchors exactly that all-time pattern. Definition 1, page 4, fixes the canonical positive-coefficient polynomial lift and the synchronous integer-lattice orbit. For the modulus 3 and the single seed at zero, let A 0 x = if x = 0 then 1 else 0 and A (t+1) x = (a+b+a*b)*(1+c), where a = A t (x-1), b = A t x and c = A t (x+1) in ZMod 3. The statement is A 0 0 = 1 ∧ ∀ s : ℕ, A (3*s+1) 0 = 1 ∧ A (3*s+2) 0 = 2 ∧ A (3*s+3) 0 = 2. It includes every natural time, with no time cutoff or extra hypothesis.

Motivation

D5/S3/StatisticalMechanics/CellularAutomata/Rule84ModThreeCenterColumn.result proves the displayed pattern. Nonzero center values give the unit condition needed by the source’s all-window diagonal-basis universality criterion. Universality here means unique coordinates for every finite target vector in the diagonal family, as in the source; it is separate from computational universality of the full automaton dynamics.

Gap

Remark 6 supplies only a finite prefix and explicitly omits an all-time proof. The preregistration in issue #12485 contains the quantified claim, the finite invariant languages and the literature check before the Lean probe. The checked v1 text, the companion arXiv:2609.25077, and the MathDB queries recorded in #12485 supply no proof of this statement. Repository and pinned Mathlib searches find no prior matching settlement: not-found-in-searched-scope. This bounded check makes no worldwide priority claim.

Route

The half-plane x < 0 remains zero by induction, since (0+0+0*0)*(1+c) = 0. For positive time t, every length-seven window starting at x ≥ -5 belongs to a finite language indexed by t mod 3. The phase languages have sizes 35, 37 and 38 and share 31 words. The window at x = -5 is 0000021, 0000011 or 0000020 at phases 0, 1 or 2, respectively.

Every compatible length-nine word, whose three overlapping length-seven windows belong to one phase language, maps under the local rule to a length-seven word in the successor language. The boundary update uses the proved zero left input and membership of the actual neighboring window to constrain its right input. Kernel-checked finite closure certificates and the time-one base establish the invariant by unbounded induction. The sixth letter of the boundary window is the center value.

Falsifier

An incorrect certificate would give a compatible length-nine word whose image is outside the next language, or a boundary word with an admissible right extension that fails to map to the next boundary word. A time with center value different from the displayed three-phase pattern would contradict the settling theorem.

Evidence

The canonical source is D5/S3/StatisticalMechanics/CellularAutomata/Rule84ModThreeCenterColumn.lean. Its public surface is A, claim and result; the two private theorems are zero_left and invariant_all. The bulk and boundary certificates use decide +kernel inside the live induction. The result has only the standard axioms propext, Classical.choice and Quot.sound. The mathematical truth is carried by the Lean declarations and their kernel-checked proof terms. Prime-power transfer and the source’s diagonal-basis criterion are not additional Lean declarations here.

Triage

Tier 1 external named open problem, preregistered in #12485. admission_basis: open-problem-resolution; proof_shape: content for result, zero_left and invariant_all. The escape witness is the all-time window-language invariant. Utility is none: the mathematical conclusion is unbounded in time; the finite certificates support that induction and are not separately delivered positive finite instances.

What the settlement shows

• Proved in this module: the three-phase length-seven language and distinguished boundary word are forward invariant from time 1. The private invariant_all establishes this and is used by the public GID D5/S3/StatisticalMechanics/CellularAutomata/Rule84ModThreeCenterColumn.result. The zero half-plane supplies the boundary input; overlapping actual windows supply compatible bulk inputs. No finite time observation is used as a premise of the all-time conclusion.
• Proved in this module: the center column is 1 at time zero and repeats (1, 2, 2) from time 1 onward. The result covers the fixed canonical lift, modulus 3 and single seed; it does not assert a pattern for another modulus or seed, or a minimal period for the entire orbit.
• Proved consequence using source results, not separately formalized here: for each integer a ≥ 1, every center value of the canonical single-seed orbit modulo 3^a is a unit, and its diagonal basis is universal for every finite window length. Polynomial reduction identifies its reduction modulo 3 with the orbit proved here, whose center value is always 1 or 2. Divisibility by 3 is therefore excluded at every time. The identity p84(0,0,c)=0 gives triangularity; the source’s Corollary 1 then gives universality, and Corollary 2 transfers the prime case to every positive power. This removes the all-time Rule 84/modulo-three gap in Remark 6 and Section 19. It does not assert that the higher-power center sequence has period three.
• Open: the remaining odd-prime cases of the source’s eight-rule family E, including Rule 84 at odd primes other than 3. The modulus-3 certificate does not give a certificate at a different prime.
• Open: extensions to other initial seeds, minimal window-language size, and exact center periods at higher powers of 3. The argument provides no classification or sharpness statement for these questions.

ASSUMED-UNVERIFIED

The bounded literature search does not establish exhaustive worldwide novelty, priority or absence of an independent proof. The source’s Corollaries 1–2 are literature results and are not kernel-checked in this module. Other seeds and moduli remain outside its formal statement.