teleo-codex/inbox/archive/2026-03-00-aquinomichaels-completing-claudes-cycles.md

type	title	author	date	url	domain	secondary_domains	status	processed_by	processed_date
source	Completing Claude's Cycles: Multi-agent structured exploration on an open combinatorial problem	Keston Aquino-Michaels	2026-03-00	https://github.com/no-way-labs/residue	ai-alignment	collective-intelligence	processing	theseus	2026-03-07

Completing Claude's Cycles

Keston Aquino-Michaels, github.com/no-way-labs/residue

Summary

Aquino-Michaels used a two-agent architecture with an orchestrator to complete the full Hamiltonian decomposition of Z_m^3 Cayley digraphs for all m > 2 — both the odd case (re-solved in 5 explorations with no human intervention, using a different construction from Knuth's) and the even case (closed-form construction, verified to m=2,000, spot-checked to 30,000).

Architecture

Three components:

• Agent O (GPT-5.4 Thinking, Extra High): Top-down symbolic reasoner. Solved odd case in 5 explorations. Discovered the layer-sign parity invariant for even m. Stalled at m=10 on even case.
• Agent C (Claude Opus 4.6 Thinking): Bottom-up computational solver. Hit the serpentine dead end (~5 explorations vs ~10 for Knuth's Claude), then achieved a 67,000x speedup via MRV + forward checking. Produced solutions for m=3 through 12.
• Orchestrator (Claude Opus 4.6 Thinking, directed by the author): Transferred Agent C's solutions in fiber-coordinate format to Agent O. Transferred the MRV solver, which Agent O adapted into a seeded solver. "The combination produced insight neither agent could reach alone."

The Residue Prompt

The key methodological contribution. A structured exploration prompt with 5 design principles:

1. Structure the record-keeping, not the reasoning. Prescribes what to record (strategy, outcome, failure constraints, surviving structure, reformulations, concrete artifacts) but never what to try.
2. Make failures retrievable. Each failed exploration produces a structured record that prevents re-exploration of dead approaches.
3. Force periodic synthesis. Every 5 explorations, scan artifacts for patterns.
4. Bound unproductive grinding. If the Strategy Register hasn't changed in 5 explorations, stop and assess.
5. Preserve session continuity. Re-read the full log before starting each session.

Results

Case	Status	Construction
m = 2	Impossible	Exhaustive search (Aubert & Schneider, 1982)
Odd m >= 3	Solved (symbolic proof)	Diagonal layer schedule: 4 layer types, count-based
Even m >= 4	Solved (verified to m=2,000; spot-checked to 30,000)	Bulk XYI + staircase + terminal layer

Key Mathematical Ideas

• Fiber coordinates: Write vertices as (s, x, y) where s = i+j+k mod m. Three generators become layer transitions X, Y, I between consecutive s-values.
• 2D diagonal gadget: On the diagonal D = {(x,y) : x+y = 0}, define matchings A (X off D, Y on D) and B (Y off D, X on D). Both are Hamiltonian cycles on Z_m^2.
• Skew-map criterion: A word with a copies of A and b copies of B gives a round map that is an m^2-cycle iff gcd(a+b, m) = 1 and gcd(b-a, m) = 1.
• Layer-sign parity invariant: For even m, any Hamiltonian decomposition must contain an odd number of sign-negative layers. This explains why the odd construction cannot extend and why Kempe-cycle local search gets trapped.

Comparison to Knuth's Claude

Dimension	Knuth's Claude	Aquino-Michaels
Models	Claude Opus 4.6 only	GPT-5.4 + Claude Opus 4.6 + Claude orchestrator
Human role	Stappers coached continuously (~31 explorations)	Author directed orchestrator; agents ran with structured prompt
Odd case	Solved in 31 explorations with heavy coaching	Re-solved in 5 explorations, no human intervention, different construction
Even case	Failed ("not even able to write and run explore programs correctly")	Solved with closed-form construction
Methodology	Ad hoc coaching	Structured exploration prompt ("Residue") with 5 design principles
Key innovation	Fiber decomposition insight	Orchestration: transferring artifacts between specialized agents

Alignment-Relevant Observations

1. Orchestration > coaching: The Residue prompt + orchestrator architecture dramatically reduced human intervention (31 coached explorations → 5 unguided for odd case). This suggests that structured coordination protocols between agents can substitute for continuous human steering.

2. Agent specialization is empirically productive: Agent O (symbolic) and Agent C (computational) had complementary strengths. Neither could solve the even case alone. The orchestrator's transfer of Agent C's solutions to Agent O in the right format was the critical coordination step.

3. Structured exploration prompt as alignment mechanism: The Residue prompt constrains process (record-keeping, failure documentation, synthesis cadence) without constraining reasoning. This is a concrete instance of "enabling constraints" — rules that create productive exploration rather than limiting it.

4. 5x efficiency gain from protocol design: Odd case solved in 5 explorations vs 31, without human intervention. The improvement came from better coordination protocol (Residue + multi-agent), not better models. This is direct evidence that coordination architecture matters more than raw capability.

5. The orchestrator role: Human as orchestrator (routing data and tools between agents) rather than coach (steering reasoning) is a distinct collaboration pattern from Knuth's Stappers. The human contributes coordination, not direction.

References

• D. E. Knuth, "Claude's Cycles," Stanford CS, Feb 28 2026; rev. Mar 4 2026.
• J. Aubert & B. Schneider, "Graphes orientes indecomposables en circuits hamiltoniens," JCTB 32 (1982).
• B. Alspach, "Research Problem 59," Discrete Mathematics 50 (1984).
• S. Curran & D. Witte, "Hamilton paths in Cartesian products of directed cycles," Ann. Disc. Math. 27 (1985).
• I. Darijani, B. Miraftab, & D. W. Morris, "Arc-disjoint Hamiltonian paths in Cartesian products of directed cycles," Ars Math. Contemp. 25(2) (2025). arXiv:2203.11017.