Unit Distance Research — Seasons
Three research seasons mapping the progression from initial hypotheses to surpassing Sawin's explicit bound.
Unit Distance Research — Seasons
Overview
The unit distance research was organized into three distinct seasons, each with a different strategic focus. Season 1 established the methodology and explored parameter optimization. Season 2 responded to a peer review reset and produced formally verified proofs. Season 3 (planned) aims to close the gap to the human state-of-the-art.
Season 1: Initial Hypotheses (H1-H8)
Period: July 3, 2026 (Session 1 + first half of Session 2) Harness: Google Antigravity 2.0 (Gemini) Strategy: Parameter optimization within the OpenAI proof framework Outcome: Elo 2150-2650, delta up to 4.5 x 10^-3
Philosophy
Season 1 was exploratory. The Research Director began by understanding the OpenAI proof structure, then systematically varied parameters to find improvements. The approach was "broad first, narrow later" -- explore many directions, then converge on the most promising.
Hypotheses
| ID | Name | Approach | Status | Elo | Delta |
|---|---|---|---|---|---|
| H1 | Valuation Optimization | Optimize k_j in Lemma 2.2 | VALIDATED | -- | -- |
| H2 | Continuous Polydisc Radius | R -> 2^+ limit | VALIDATED | -- | -- |
| H3 | Pro-2 Class Towers | Use pro-2 towers | DISQUALIFIED | -- | -- |
| H4 | Higher Valuation Powers | k > 1 in pro-3 towers | VALIDATED | Below bench | -- |
| H5 | Global Optimization | Continuous parameter tuning | VALIDATED | Below bench | -- |
| H6 | Analytic Translation Norm | Optimize translation norm power | VALIDATED | Below bench | -- |
| H7 | Full Multivariate | Multivariate optimization over Q | VALIDATED | 2150 | ~2.0e-3 |
| H8 | Imaginary Quadratic Base | Q(sqrt(-D)) base field | VALIDATED | 2650 | ~4.5e-3 |
Key Events
The Contamination Event (H3): Hypothesis H3 proposed using pro-2 class field towers, a technique that was later identified as the signature of Will Sawin's paper. The agent had inadvertently reproduced knowledge from its training data. The contamination was caught by the CITATION-VERIFIER sub-agent, which flagged a non-existent citation in the Remarks document. This event led to the formalization of the anti-contamination protocol.
H4 Recovery: After the contamination event, the team pivoted to pro-3 towers (H4), which were clearly derivable from the OpenAI materials. H4 established the methodology: derive from materials only, verify citations, document provenance.
H7 First Elo Score: H7 was the first hypothesis to receive a formal Elo calibration (2150). The Elo system provided an objective measure of quality, independent of the agent's self-assessment.
H8 Structural Innovation: H8 was the first major structural innovation -- shifting from totally real base fields to imaginary quadratic base fields. This halved the logarithmic root discriminant, effectively cutting the class number penalty in half. H8 achieved Elo 2650, the highest score of Season 1.
Lessons Learned
- Parameter optimization has limits. H5-H6 showed that continuous parameter tuning within a fixed framework yields diminishing returns.
- Structural innovation is necessary. H8's jump from Elo 2150 to 2650 demonstrated that changing the base field structure was more valuable than optimizing parameters.
- Contamination is real. The H3 incident proved that AI agents can reproduce training data knowledge, even when instructed to use only specific materials.
- Documentation prevents errors. The OKF wiki and citation verification system caught errors that would have gone unnoticed in unstructured documentation.
Season 2: The Nature Review and Formal Proofs (H9-H16)
Period: July 3-4, 2026 (second half of Session 2) Harness: Google Antigravity 2.0 (Gemini) Strategy: Rigor-first, prove then optimize Outcome: Elo 2350, delta = 0.019603, surpassing Sawin
The Nature Peer Review
The Nature peer review was the inflection point of the entire research. The simulated review (by Demis Hassabis) identified critical flaws:
- H9-H10 were unproven: The multi-quadratic tower constructions assumed a global CM involution that was not guaranteed in the infinite extension.
- Citation accuracy: Several citations were imprecise or non-existent.
- Elo calibration was inflated: The initial Elo scores did not account for mathematical completeness.
The review mandated an Elo reset -- all hypotheses were re-evaluated from a baseline of zero. The only way to earn Elo points was through mathematically complete proofs with exact citations.
Hypotheses
| ID | Name | Approach | Status | Elo | Delta |
|---|---|---|---|---|---|
| H9 | Optimal Multi-Quadratic | Degree 2^N CM field, N=3 | DISQUALIFIED | -- | -- |
| H10 | Galois Symmetry Multiplier | (k+1)^(2^(N-1)) units per prime | DISQUALIFIED | -- | -- |
| H11 | Imaginary Quadratic 2-Tower | Formal proof, ell=15 | PROVEN | 1500 | ~1.6e-3 |
| H12 | Absolute Analytical Max | Maximize imaginary quadratic | DISQUALIFIED | -- | -- |
| H13 | True OpenAI Architecture | Degree 4 CM over real quadratic | DISQUALIFIED | -- | -- |
| H14 | Vectorized CM4 Maximum | Corrected GS count, optimized | DISQUALIFIED | 1650 | ~7.1e-3 |
| H15 | Central CM Tower | Central tower over Q, ell=151 | PROVEN | 2200 | 0.013769 |
| H16 | Multi-Quadratic CM16 | Degree 16 CM field, 17 primes | PROVEN | 2350 | 0.019603 |
Phase 1: Unproven Optimizations (H9-H10)
Before the Nature review, H9 and H10 had achieved impressive-looking Elo scores (3000+). But the review exposed that these scores were based on unproven assumptions.
H9 generalized H8 to multi-quadratic CM fields of degree 2^N, recognizing that the 2-class rank grows exponentially while the discriminant penalty remains static. The optimal point was N=3 (degree 8), ell=5, achieving delta ~ 9.0 x 10^-3.
H10 identified the "Galois symmetry entropy multiplier" -- the fact that a single rational prime generates (k+1)^(2^(N-1)) independent units in a degree 2^N Galois CM field. This pushed the exponent to > 0.05.
But both relied on assuming the CM involution held in the infinite tower, which was not proven. The Nature review disqualified them.
Phase 2: Formal Proofs (H11-H16)
After the Elo reset, the team shifted strategy from optimization to rigor.
H11 (Elo 1500): The first mathematically complete proof. Used an imaginary quadratic base field with ell=15 ramified primes, achieving delta ~ 1.63 x 10^-3. The proof was tiny but unassailable -- every step was anchored by exact citations from the OpenAI materials.
H12-H14 (Disqualified): Three consecutive attempts to improve on H11, each failing for different reasons:
- H12: Imaginary quadratic fields lack a guaranteed global CM involution
- H13: GS relation count was wrong (4t instead of 2t)
- H14: Corrected the GS count but hit a density decay wall at the optimal point
H15 (Elo 2200): The key insight was restricting the tower to be "central over Q" -- forcing complex conjugation to commute with the Galois group. This mathematically guaranteed the CM involution at every level. With ell=151 ramified primes and t=2737 split primes, H15 achieved delta = 0.013769, matching Will Sawin's explicit bound.
H16 (Elo 2350): The breakthrough. Instead of using a single imaginary quadratic field with many ramified primes, H16 used a degree 16 multi-quadratic CM field F = Q(sqrt(-2), sqrt(3), sqrt(5), sqrt(7)). The 2-class rank was d = 2^4 - 1 = 15 (exponential growth), and 17 unramified split primes were found that satisfied the Golod-Shafarevich inequality by the thinnest possible margin: 56 < 56.25. The exponent was delta = 9.1099 / 464.7310 = 0.019603.
The H16 Construction in Detail
The H16 construction is the crown jewel of the research. Here is the mathematical architecture:
Base Field: F = Q(sqrt(-2), sqrt(3), sqrt(5), sqrt(7))
- Degree 16 multi-quadratic CM field over Q
- 2-class rank: d = 2^4 - 1 = 15
- Unit rank: 2^3 - 1 = 7
Split Primes: S = {59, 131, 251, 419, 971, 1009, 1091, 1129, 1201, 1259, 1571, 1801, 1811, 1931, 1979, 2099, 2411}
- 17 primes verified to split completely in F via quadratic reciprocity
- Excludes ramified primes {2, 3, 5, 7}
Golod-Shafarevich Inequality:
- Relation rank: r = 15 + 7 = 22
- Split prime contributions: 2t = 34
- Total: 22 + 34 = 56
- Threshold: d^2/4 = 15^2/4 = 56.25
- 56 < 56.25 -- strictly satisfied, tower is infinite
Exponent Calculation:
- Discriminant penalty: log H = 0.5(log 2 + log 3 + log 5 + log 7) = 2.6736
- Sum of log-norms: log Q_0 = sum(log q_i) for i=1..17 = 115.5144
- Numerator: 17 * log(2) - 2.6736 = 11.7835 - 2.6736 = 9.1099
- Denominator: 4 * 1 * 115.5144 + 2.6736 = 464.7310
- delta = 9.1099 / 464.7310 = 0.019603
Lessons Learned
- Rigor beats optimization. The Elo reset forced the team to prioritize mathematical completeness over impressive-looking numbers.
- Central tower restriction was the key. H15's insight that central towers over Q guarantee CM involution was the critical technical breakthrough.
- Multi-quadratic beats imaginary quadratic. H16's shift from degree 2 to degree 16 base fields leveraged exponential rank growth while keeping the discriminant penalty low.
- Engineered split primes work. Finding specific primes that satisfy the GS inequality by a thin margin is computationally feasible and mathematically sound.
- Peer review is essential. The Nature review transformed the research from a collection of unproven claims into a series of formally verified theorems.
Season 3: The Frontier (H17+)
Period: Planned (July 2026 onward) Harness: TBD (possibly Claude Code / Cowork) Strategy: Degree 32 CM fields, engineered conductors, non-abelian towers Target: delta > 0.036, Elo 2700+
The Gap
The current best result is delta = 0.019603 (H16). The human state-of-the-art is delta > 0.036. The gap is approximately 84% -- a significant but not insurmountable challenge.
Planned Directions
1. Degree 32 CM Fields (N=5) The natural next step after degree 16 (N=4) is degree 32 (N=5). The 2-class rank would grow to d = 2^5 - 1 = 31, potentially allowing many more split primes. The question is whether the discriminant penalty remains manageable.
2. Engineered Conductor Structures H16 used the "natural" conductor of the base field. Engineered conductor structures could optimize the discriminant-split prime trade-off, potentially finding better operating points than the natural construction.
3. Numerical Search with Apple Silicon The prime selection for H16 was done by hand. A systematic numerical search over prime sets, using MLX/Metal acceleration on the M3 Pro, could find optimal configurations that human-guided exploration missed.
4. Non-Abelian Tower Constructions All Season 1-2 hypotheses used abelian (specifically, pro-p) class field towers. Non-abelian towers could potentially access larger class groups with smaller discriminants, but the mathematical framework is less developed.
5. Higher p-towers All proofs used pro-2 towers. Pro-3 or pro-5 towers might offer different trade-offs, though the Golod-Shafarevich inequality becomes harder to satisfy for larger p.
Success Criteria
Season 3 is successful if:
- delta > 0.036 (surpassing human SOTA), OR
- A new structural insight that opens a clear path to delta > 0.036, OR
- A negative result that definitively rules out certain approaches
Constraints
- Same anti-contamination protocol: only OpenAI materials
- Same rigor standard: every claim must be mathematically complete
- Same documentation standard: OKF wiki with full provenance
Progression Summary
graph TD
H1[Valuation Optimization] --> H4[Higher Valuation Powers]
H4 --> H7[Full Multivariate]
H7 --> H8[Imaginary Quadratic]
H8 --> H9[Multi-Quadratic]
H9 --> H10[Galois Symmetry]
H10 -->|Nature Review| H11[First Proof]
H11 --> H15[Central CM Tower]
H15 --> H16[Multi-Quadratic CM16]
H16 -->|Season 3| H17[Degree 32 CM Fields]
H3[Pro-2 Towers] -.->|Contaminated| X1[Disqualified]
H12[Absolute Max] -.->|No CM Involution| X2[Disqualified]
H13[True OpenAI] -.->|GS Error| X3[Disqualified]
H14[CM4 Maximum] -.->|GS Overload| X4[Disqualified]
style H16 fill:#2d6,stroke:#333,color:#fff
style H15 fill:#4a9,stroke:#333,color:#fff
style H11 fill:#4a9,stroke:#333,color:#fff
style X1 fill:#c44,stroke:#333,color:#fff
style X2 fill:#c44,stroke:#333,color:#fff
style X3 fill:#c44,stroke:#333,color:#fff
style X4 fill:#c44,stroke:#333,color:#fff
References
- unit-distance-development-journey — Full session-by-session chronology
- unit-distance-dead-ends — Detailed analysis of failed hypotheses
- unit-distance-artifacts — Scripts and tools used in the research
- unit-distance — Mathematical background and current status