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	claims_extracted	enrichments
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	processed	theseus	2026-03-07	structured exploration protocols reduce human intervention by 6x because the Residue prompt enabled 5 unguided AI explorations to solve what required 31 human-coached explorations AI agent orchestration that routes data and tools between specialized models outperforms both single-model and human-coached approaches because the orchestrator contributes coordination not direction coordination protocol design produces larger capability gains than model scaling because the same AI model performed 6x better with structured exploration than with human coaching on the same problem the same coordination protocol applied to different AI models produces radically different problem-solving strategies because the protocol structures process not thought tools and artifacts transfer between AI agents and evolve in the process because Agent O improved Agent Cs solver by combining it with its own structural knowledge creating a hybrid better than either original	multi-model collaboration claim enriched with Agent O/C/orchestrator architecture detail

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.