Claude Code: Antikythera-Maths Research Build

Framing

The Antikythera mechanism (Greek, ca. 150–60 BCE, recovered 1901, reconstructed through Freeth/UCL 2021 and subsequent work) is not a chess-like problem we need to discover structure in. It is a physical instantiation of coprime-indexed phase-space addressing, designed deliberately 2100 years ago to solve the exact class of Diophantine approximation problems that docs/addressing-maths/ now characterizes formally. Every gear is a cyclic group ℤ/nℤ; every mesh is a rational map between cyclic groups; every shared gear-train is an empirical solution to the multi-dataset packing problem (A-H1 in the addressing-maths research plan); every celestial pointer is an HDC-style hypervector whose components are the phase angles on the various dials. The Greeks built a resonant HDC object before Plate wrote HRR, before Kanerva wrote SDM, before Chung wrote Spectral Graph Theory.

The HDC state is rendering-agnostic — orrery and Antikythera are sibling projections. The angular dynamic state captured by the encoder is the complete input to any rendering of the mechanism's output. The Antikythera's dial display projects each body's angle onto a concentric circular scale at a fixed dial radius chosen at instrument-design time. A classical orrery projects the same angle onto a scaled orbital radius chosen for visual fit. Both renderings consult a static radial-parameter table that is rendering-specific, not dynamic; both expose a free scale parameter that does not enter the phase-space computation. Perspective is the scale invariance. Consequently, what the project is reconstructing is not the Antikythera qua dial-calculator but the parent HDC state that the Antikythera's dial rendering and the Archimedean-tradition orrery rendering are both projections of. Cicero (De re publica, Tusculan Disputations) describes Archimedes' Syracuse planetarium as an orrery-like device built from related gearing principles — whether that historical tradition and the Antikythera share a lineage is DISPUTED in the archaeology literature; the mathematical equivalence of the dynamic computations underlying both device classes is not.

This project documents the shared parent structure.

The methodological difference from chess, Othello, logo

Chess, Othello, and logo were discovery projects: structure was present in the game/language and we used spectral tools to extract it. Antikythera is a reconstructive/descriptive project: the structure was designed in by named historical agents (plausibly in the Archimedean tradition), and our job is to document it in the vocabulary the addressing-maths thread has now assembled. The encoding is not something we invent. It is something we recognize.

This means the honest-science tags shift:

• KNOWN — published in the archaeology/historical literature (Price 1974, Freeth et al. 2006, Freeth et al. 2021, Freeth UCL 2025 lecture, etc.)
• NOVEL — the HDC/phase-space framing itself, where we name a property of the mechanism in addressing-maths vocabulary that the archaeology literature has not used
• CONFIRMED — computationally verified that the mechanism-as-HDC-object reproduces the astronomical cycles it was designed to
• FAILED — our HDC encoding cannot reproduce something the physical mechanism does
• DISPUTED — where archaeological reconstructions disagree (and there are several — 2021 Freeth vs. 1974 Price vs. 2006 Freeth vs. Wright, and the 2025 Guillermo manufacturing-tolerance simulation)

Ground truth documents

1. docs/chess-maths/chess_spectral_research_notebook.md — the template this notebook mirrors.
2. docs/addressing-maths/ADDRESSING_MATHS_RESEARCH_PLAN.md — the theoretical substrate. Every claim in the Antikythera notebook should be readable as "the Greeks instantiated sub-question X of the addressing-maths plan in configuration Y." This is the most important cross-reference in the entire project.
3. docs/othello-maths/othello_spectral_research_notebook.md — the second-instance template, especially for the Phase 1 hypothesis-battery format.
4. Freeth et al. (2021), "A Model of the Cosmos in the ancient Greek Antikythera Mechanism," Scientific Reports 11:5821 — current authoritative reconstruction. This is the primary source for tooth counts and gear topology. Open-access at nature.com.
5. Freeth, Jones, Steele, Bitsakis (2008), "Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism," Nature 454:614 — back-dial decipherment.
6. de Solla Price (1974), "Gears from the Greeks: The Antikythera Mechanism — A Calendar Computer from ca. 80 B.C.," Transactions of the American Philosophical Society 64(7) — foundational but superseded on specific tooth counts.
7. Wright (2005–2012) papers — alternative reconstruction. Cite where it differs from Freeth.
8. Guillermo & Szigety (2025, arXiv) — manufacturing-tolerance simulation suggesting the mechanism may not have run smoothly in practice.

Folder layout

Create docs/antikythera-maths/ with structure parallel to docs/othello-maths/:

docs/antikythera-maths/
├── antikythera_spectral_research_notebook.md
├── ANTIKYTHERA_SPECTRAL_INSTRUCTIONS.md
├── research/
│   ├── __init__.py
│   ├── gear_database.py              # Tooth counts, gear topology, sources
│   ├── astronomical_cycles.py        # Metonic, Saros, Callippic, etc.
│   ├── cyclic_group_algebra.py       # ℤ/nℤ operators for each gear
│   ├── rational_approximation.py     # Continued fraction / best-rational tools
│   ├── packing_analysis.py           # Empirical A-H1 analysis of shared-gear-train solutions
│   ├── pin_and_slot.py               # The T-breaking epicyclic analog of the pawn
│   ├── encode_ant.py                 # The resonant HDC encoder
│   ├── dial_decoder.py               # Unbinding: recover which cycle from a combined state
│   ├── astronomical_ground_truth.py  # Compare encoder output to NASA ephemeris
│   └── consolidated_tests.py         # All-pass sanity battery
├── results/
│   └── .gitkeep
└── figures/
    └── .gitkeep

The eventual production package would live at antikythera-spectral/python/antikythera_spectral/ but is not created in this pass — matches the Othello precedent of research-first, package-later.

Phase 0 — Data extraction and infrastructure

Known gear tooth counts (from Freeth 2021, unless noted)

This is the empirical substrate. Hard-code it in research/gear_database.py with provenance notes. Some counts are uncertain; mark them.

Main drive train and calendar (known): - b1 (main sun gear): 224 teeth (4-spoked, the largest surviving gear; some reconstructions report 223) - a1 (crown gear, hand-crank input): count uncertain, likely ~48 - e-series (Metonic calendar train): 53, 96, 53 (Freeth 2021 argues for this arrangement) - Metonic output: 235-tooth display (5 turns of 47-tooth inner gear)

Lunar system (known): - 127-tooth gear (= 254/2; encodes sidereal month count in Metonic cycle) - 50-tooth × 4 (the pin-and-slot epicycle for variable Moon motion) - 223-tooth gear (Saros eclipse cycle pointer) - The pin-and-slot mechanism is at Fragment B — this is the key T-breaking element

Planetary system (Freeth 2021 reconstruction, treat as NOVEL for the mechanism itself): - 63-tooth gear in Fragment D (critical for Venus: 462-year period relation) - Proposed Venus train uses period relation (289, 462); 462 = 2·3·7·11 - Proposed Saturn train uses period relation (427, 442); 442 = 2·13·17 - Freeth 2021 argues that factors 7 and 17 are shared across multiple planetary trains — this is exactly the multi-dataset packing structure of A-H1

Back-dial cycles (known from inscriptions + gear topology): - Metonic: 19 tropical years = 235 synodic months - Callippic: 4 Metonic = 76 years - Saros: 223 synodic months ≈ 18 years 11 days (eclipse cycle) - Exeligmos: 3 Saros = 54 years (eclipse-triple cycle, aligns eclipses to same time of day) - Olympic: 4 years (Panhellenic games) - Lunar anomalistic period encoded in the pin-and-slot epicycle: ≈8.85 years (apsidal precession)

research/astronomical_cycles.py — the "channel" layout

The mechanism's dial set is the analog of chess's 10-channel 640-dim encoding. Each dial is one channel:

Dial	Cycle	Integer encoding	Prime factors
Front zodiac	360° / year	365 days ≈ 1 tropical year	—
Front calendar	Egyptian civil year	365	5 · 73
Metonic spiral	235 synodic months / 19 years	235 = 5 · 47	5, 47
Callippic	4 Metonic	940 = 4 · 235	2², 5, 47
Olympic	4 years	4	2²
Saros spiral	223 synodic months	223 (prime)	223
Exeligmos	3 Saros	669 = 3 · 223	3, 223
Lunar anomaly	8.85 years	fractional; epicyclic	—
Lunar nodes	18.6 years (nodal regression)	—	—
Sun pointer	1 year	365 or year-fraction	—
Moon pointer	variable (elliptical)	—	—
Mercury pointer	synodic 116 days	—	—
Venus pointer	synodic 584 days; period relation (289, 462)	462 = 2·3·7·11	2, 3, 7, 11
Mars pointer	synodic 780 days	—	—
Jupiter pointer	synodic 399 days	—	—
Saturn pointer	synodic 378 days; period relation (427, 442)	442 = 2·13·17	2, 13, 17

Observe the shared-prime-factor structure: 7 and 17 appear in multiple planetary trains per Freeth 2021, exactly the shared-generator pattern the addressing-maths plan names in A-H1.

Sanity checks Phase 0 must pass

Put these in research/consolidated_tests.py:

1. Tooth-count consistency. For every gear pair, the ratio encodes a known cycle to within Greek-attainable precision (≤1/1000). E.g., Metonic: 19 years ↔ 235 lunar months is reproduced by the e-series train to machine precision.
2. Prime-factor catalog. All gear tooth counts factorize into known primes; list the prime spectrum the mechanism actually uses. This is the "alphabet" of the HDC encoding.
3. Gear-topology graph. Build the gear-composition DAG. Leaves are dial pointers; root is the crank. Each internal node carries its tooth count and the cycle it encodes.
4. Astronomical cycle residuals. For each cycle (Metonic, Saros, Callippic, …), compute the cycle length produced by the gear train and compare against the modern astronomical value. Report the error per cycle. Expected: <0.1% for the Greek-solved cycles, larger for Mars (the Greeks didn't have equants).

Phase 1 — Hypothesis battery

Same format as Othello Phase 1: each hypothesis is a concrete computation with prediction, threshold, prior art, status. Emit results/phase1_hypotheses.csv and results/phase1_detail.json.

A. Coprime addressing is the mechanism's native language

A-H1: Every gear ratio is a best rational approximation under a tooth-count budget. For each astronomical cycle, compute the best rational approximation p/q with max(p, q) ≤ some budget (say 500, reflecting the physical feasibility of cutting >500 teeth in bronze). Compare to the ratio the mechanism actually uses. Prediction: the mechanism's ratios are within the top-3 convergents of the continued-fraction expansion for every cycle. Prior art: Freeth 2021 supplementary material does this informally; we do it systematically. Threshold: ≥90% of cycles within top-3 convergents.

A-H2: Shared prime factors across planetary trains are Pareto-optimal. Freeth 2021 claims 7 and 17 are shared. Enumerate all prime factors ≤ 50 and compute, for each set of candidate shared factors, the total tooth budget required to achieve the stated per-planet precision. Prediction: the Freeth 2021 choice of shared primes is Pareto-optimal on the (precision, total-tooth-count) frontier. Threshold: no candidate factoring achieves lower total tooth count at equal or better precision. Status: NOVEL — this reframes the Freeth reconstruction as an optimization result.

A-H3: The prime spectrum of the mechanism is non-random. Compute the histogram of primes appearing in the gear tooth counts. Compare to a null model (random tooth counts drawn from [10, 300]). Prediction: the observed spectrum is heavily biased toward small primes (2, 3, 5, 7) plus a handful of large primes needed for specific irrationals (47, 53, 127, 223). Prior art: Freeth 2008 notes the 223-prime is "strange"; we contextualize it as the signature of a genuinely irrational cycle requiring a prime denominator.

B. The mechanism as a group-algebra element

B-H1: Every cycle is an element of ℂ[ℤ/D_Antℤ] for some D_Ant. Compute D_Ant = lcm of all gear tooth counts (and their compositions). Each dial pointer corresponds to a specific residue class. Prediction: D_Ant is factorizable and of moderate size (expected: under 10¹⁰ but significantly structured). Threshold: compute D_Ant and its prime factorization explicitly.

B-H2: Crank-turn = single generator of the phase-space. One turn of the hand-crank advances the full system by one day. In the phase-space ℤ/D_Antℤ, this is a single generator σ_day. Every astronomical prediction is a power of σ_day projected onto a specific dial. Prediction: σ_day is a unit in ℤ/D_Antℤ (coprime to D_Ant). Status: likely CONFIRMED — follows from design.

B-H3: The HDC binding operation = gear composition. If piece A has hypervector h_A encoding its phase on ℤ/n_Aℤ, and piece B has h_B on ℤ/n_Bℤ, then meshing gear A with gear B at ratio n_A/n_B corresponds to the HDC bind h_A ⊗ R_{n_A/n_B}, where R is a roll/rotation operator. Prediction: the chess §9f coprime roll binding is exactly the Antikythera gear-meshing operation transposed to an abstract HDC space. Novel: this is the structural claim the project most wants to establish.

C. Bounds, aliasing, and the Greeks' error-correction strategy (they had none)

C-H1: The mechanism has zero intrinsic error correction. Apply the addressing-maths §3D theorem: coprime addressing is a bijection; bijections add zero correction capacity. Prediction: any mis-counting of teeth or single-gear-slip propagates with no correction. Implication: the Greeks knew this — they compensated by using exact integer tooth counts with astronomical precision at design time rather than error-correction at runtime. The Guillermo & Szigety 2025 simulation result — that the mechanism may not have run smoothly — is a direct empirical demonstration of this theorem at the manufacturing level.

C-H2: Aliasing horizon as the spiral-dial return-to-start. The Metonic and Saros dials are physical spirals rather than circles because the cycles are longer than the visible spiral-arc length. The dial wraps after a full spiral traversal. This is the physical instantiation of the §11.3.3 torus-clip: when the pointer reaches the end of the spiral, it re-enters at the start, signaling a cycle boundary. Prediction: this is formally identical to the "phase-op shifts that carry an origin off the lattice land in the off-image complement" mechanism of chess §11. Novel: no one has framed the spiral dial as an aliasing-horizon detector before. Status: likely CONFIRMED by construction — the spiral is the physical mechanism for boundary detection.

D. T-breaking and the pawn-analog

D-H1: The pin-and-slot mechanism is the Antikythera's antisymmetric fiber. In chess, the pawn's directed Laplacian breaks Z₂ time-symmetry (§9m, Hatano-Nelson t_L = 0). In Antikythera, the pin-and-slot epicycle breaks time-symmetry of the lunar motion — it approximates Kepler's second law, so the Moon moves faster at perigee and slower at apogee. The pin-and-slot is mechanical T-breaking: a component that does not commute with the time-reversal operator. Prediction: the ratio ||pin_and_slot_asymmetric|| / ||pin_and_slot_symmetric|| approaches 1.0 in the relevant decomposition, matching the chess pawn ratio ||A_anti|| / ||A_sym|| = 1.0. Status: NOVEL — the pawn/pin-and-slot correspondence has not been drawn in the literature.

D-H2: All other gear trains are T-symmetric. Running the mechanism backward (turning the crank the other way) gives valid astronomical predictions for past dates, except where the pin-and-slot is involved. Prediction: the Sun dial, the Metonic, the Saros, and the outer planets (Jupiter, Saturn) all run cleanly in reverse. The Moon dial and (to a lesser extent) the inner planets (Mercury, Venus) carry T-breaking in their epicyclic approximations.

E. Ground-truth validation

E-H1: The encoder reproduces ancient eclipse predictions. Pick a known Hellenistic-era eclipse (e.g., the 120 BCE lunar eclipse). Run the encoded mechanism forward from a known reference date. Compare the Saros dial output against NASA historical ephemeris. Prediction: within ~1 day for the Greek-solved cycles, within ~1 month for Mars. Threshold: Saros match to ±1 day across any 20 eclipses in 200 BCE – 100 CE.

E-H2: Planetary-position errors match the documented Greek-astronomy limits. The Mars pointer is up to 38° wrong at retrograde nodes (Wikipedia; Freeth 2021) because Greek astronomy lacked equants. Reproduce this error exactly: our encoder should be wrong in the same way the mechanism was wrong. Prediction: the encoder's error at Mars-retrograde is within a few degrees of the mechanism's. Status: this is a fidelity test, not a correctness test — we want to match the historical device, not modern astronomy.

F. Open exploration

F-E1: Does the mechanism's prime spectrum match any modern VSA/HDC encoding? Compare to Residue-HDC (Kymn et al. 2025), which explicitly uses CRT moduli. Prediction: the Antikythera is a 1^st-century-BCE Residue-HDC instance with moduli chosen from astronomy rather than number theory.

F-E2: Is there a natural D_Ant at which every cycle the mechanism supports becomes a single integer? If yes, this is the mechanism's "640-dim" — the natural ambient modulus for an encode_Ant function.

F-E3: What are the "failed" cycles? Cycles the mechanism attempted to approximate but got wrong. Mars is the canonical example. Open: are these failures attributable to specific approximation-precision tradeoffs the Greeks made, or to genuine limits of their astronomical theory?

Phase 2 — encode_Ant: the resonant HDC encoder

Build research/encode_ant.py to produce a hypervector for any date (or ℝ-valued "time" variable) that encodes the full mechanism state.

Architecture

The encoding has one channel per dial, matching the mechanism's physical structure. Dimension is determined by the largest gear tooth count (or its lcm composition, depending on H2/H3 outcomes). Candidate dimensions:

• D = 940 (Callippic cycle × 4 for safety) — encodes annual + multi-year cycles cleanly but may not fit planets.
• D = lcm(all cycles) — exact and natural, but possibly huge.
• D = chess-style ambient (e.g. 2⁷ × 3 × 5 × 7 = 13440) — engineered for packing. Probably best for HDC operations.

Commit to one primary and one ablation, like Othello's D=768 + rank-8 ablation.

Rendering-agnostic design constraint. encode_Ant(t) returns angular dynamic state only — residue-class phases for each cycle, bundled into the D-dim HDC vector. It does NOT return pre-baked (x, y) pointer positions or spatial coordinates. Radial parameters (dial radii for Antikythera-style rendering, orbital radii for orrery-style rendering) are kept in a separate static lookup module, e.g. research/rendering.py, with two modes: render_dial(state, dial_layout) and render_spatial(state, orbital_radii, orrery_scale). The encoder output is the parent; the renderers are projections. This matters for Phase 4 validation: the same encode_Ant(t) output must validate against Freeth 2021 dial-pointer positions AND against NASA JPL Horizons angular ephemeris, because both ground truths are projections of the same underlying astronomy. If the encoder is implemented correctly, writing render_dial vs render_spatial is ~20 lines of code each and neither involves re-running the mechanism.

Fiber structure

The Antikythera's "fiber" is the shared gear-train structure (H-A2). Unlike chess's piece Laplacians (static) or Othello's ray Laplacians (dynamic with flanking), the Antikythera fiber is static but shared across species. Each celestial body has its own dial (species), but the internal gearing draws from a shared pool of generators (7, 17, and a few others). This is the cleanest "fiber bundle" in the project so far — completely static, empirically verified by Freeth 2021.

Unbinding

The Phase 2 deliverable must include research/dial_decoder.py: given a full-mechanism hypervector, recover which dial is pointing where. This is the chess-style unbinding test. Given the hypervector is constructed from shared generators, unbinding across multiple planetary dials should work cleanly (if the Greeks got the packing right — which A-H2 tests).

Phase 3 — Phase-operator preflight

Minimal. The Antikythera's phase operator is trivial: it's the single σ_day = crank-turn operator. All predictions are powers of σ_day projected onto specific dials. There are no distinct "pieces" with different movement rules; there is one universal operator (time advance) and many projections.

This is a feature, not a bug. The Antikythera is the simplest possible phase-operator system in the project's triad-plus: - Chess: 6 piece types × 8 D₄ orbits = rich operator set - Othello: 1 piece type × 8 ray directions × dynamic gating = rich but different - Logo: command set + grammar = structured but different again - Antikythera: 1 operator, multiple projections = minimal

Write OTHELLO_PHASE_OP_PREFLIGHT.md equivalent as ANTIKYTHERA_PHASE_OP_PREFLIGHT.md. Short. Argue that the phase-operator build for this domain is ~20 lines of code: advance_day(state) = state ⊗ σ_day.

Phase 4 — Historical validation (WTHOR equivalent)

The Antikythera's WTHOR is the NASA JPL Horizons ephemeris, computed backward to the Hellenistic era. Horizons is free and covers 15 BCE – present; several historical eclipse catalogs extend it further back.

Tests: 1. Compare encoder-produced Sun positions to Horizons' Sun positions over 100 BCE – 100 CE. 2. Compare encoder-produced Moon positions to Horizons' Moon positions over the same range. 3. Compare encoder-produced eclipse times to known Hellenistic eclipse records (Ptolemy's Almagest lists several). 4. Reproduce the Mars-retrograde error pattern.

Ground-truth availability is much better for this domain than for chess or Othello. The research-budget question is just whether to run modern astronomy against Phase 2 encoder, or whether to stop at validating against Freeth 2021's own astronomical validation (which already exists).

Recommendation: do it. NASA Horizons has a Python API (astroquery.jplhorizons); the marginal cost is low.

Research notebook scaffold

Create docs/antikythera-maths/antikythera_spectral_research_notebook.md with this skeleton. Fill in §1, §2, §3 from Phase 0–4 as they complete.

# The Antikythera Mechanism as a Resonant HDC Object

**Authors:** Steven (mlehaptics Project) & Claude (Anthropic)
**Date:** April 2026
**Status:** Active research — reconstructive/descriptive project; the Greeks did the math, we're reading it off.

> Living document. Sibling to:
> - [../chess-maths/chess_spectral_research_notebook.md](../chess-maths/chess_spectral_research_notebook.md) — the template this notebook mirrors.
> - [../othello-maths/othello_spectral_research_notebook.md](../othello-maths/othello_spectral_research_notebook.md) — the second-instance template, Phase 1 battery format.
> - [../logo-maths/logo_research_notebook.md](../logo-maths/logo_research_notebook.md) — the non-board generalization.
> - [../addressing-maths/ADDRESSING_MATHS_RESEARCH_PLAN.md](../addressing-maths/ADDRESSING_MATHS_RESEARCH_PLAN.md) — the formal substrate. Every result in this notebook should be readable as "the Greeks instantiated X of the addressing-maths framework in configuration Y."

Every claim is tagged **KNOWN / NOVEL / CONFIRMED / FAILED / DISPUTED** (see Framing for DISPUTED tag definition specific to this domain).

---

## 0. Framing

The Antikythera mechanism is coprime-indexed phase-space addressing executed in bronze, designed 2100 years ago. This notebook documents it as such. The math is already in the artifact; our job is to name it in the vocabulary the addressing-maths thread has now assembled. This is the first mlehaptics project where the encoding is *descriptive*, not *prescriptive* — we are not inventing an encoder, we are recognizing one.

### 0.1 The HDC state is rendering-agnostic

The encoded state is angular dynamic information: each celestial body's phase in its respective cyclic group. That state is the complete input to any rendering of the mechanism's output. The Antikythera's dial display and a classical orrery's spatial model are both projections of the same state, differing only in which static radial-parameter table is consulted at rendering time (concentric dial radii for the Antikythera; scaled orbital radii for the orrery) and in which free scale parameter is exposed (dial-ring layout vs. overall model scale). Neither parameter participates in the dynamic computation; both are rendering-time choices. **Perspective is the scale invariance.** A single `encode_Ant(t)` output can drive either rendering — which is why, historically, the same Hellenistic gearing tradition that plausibly produced the Antikythera (dial calculator) also, per Cicero, produced the Archimedean planetarium (orrery-like model). The HDC state is the shared parent; the two device families are sibling renderings. The Archimedean attribution is DISPUTED in the archaeology literature; the mathematical equivalence of the underlying computations is not.

## 1. Infrastructure (Phase 0)

### 1.1 The artifact
Brief physical description. Size, date, recovery, fragment count. Reference to Freeth 2021 as the canonical reconstruction.

### 1.2 The gear database
Hard-coded tooth counts with provenance. Freeth 2021 as primary, Wright where it differs, de Solla Price for historical context.

### 1.3 The astronomical cycle layout
Metonic, Saros, Callippic, Exeligmos, Olympic, lunar anomaly, lunar nodes, 7 planets. Each as a row in a table with integer encoding and prime factorization.

### 1.4 Sanity checks
Tooth-count consistency; prime spectrum; gear-topology graph; per-cycle astronomical residuals.

## 2. Phase 1 battery

§2.A, §2.B, §2.C, §2.D, §2.E, §2.F matching Phase 1 above. Each subsection: prediction, experiment, measured numbers, status tag, interpretation, lesson.

## 3. `encode_Ant` — the resonant encoder (Phase 2)

Architecture decision log: which D, which fiber structure, which unbinding test suite passes.

## 4. Phase-operator preflight (Phase 3)

Short. `advance_day` is the entire operator set for this domain.

## 5. Validation against NASA Horizons (Phase 4)

Eclipse comparison, planetary-position comparison, Mars-retrograde error reproduction.

## 6. The Archimedes question

Cicero, in *De re publica* and *Tusculan Disputations*, describes a planetarium built by Archimedes (captured by Marcellus at Syracuse, 212 BCE). The device he describes matches the Antikythera functionally. **Open:** is the Antikythera a descendant of an Archimedean design tradition? If yes, the mechanism we have is ~150 years of iterative design away from its progenitor, and the coprime-factoring choices may have been refined over generations by an unbroken tradition of craftsmen. The HDC encoding we extract is thus a **distilled** design, not a single inventor's flash of insight. This framing is DISPUTED in the literature; Freeth (2021) leans toward Archimedean origin, others argue for Rhodian astronomical schools.

## 7. Vocabulary collisions specific to Antikythera

- "Mechanism" (the device) vs "mechanism" (the causal process).
- "Gear" (physical wheel) vs "gear" (HDC generator — we use "generator" or "channel" where possible).
- "Cycle" (astronomical period) vs "cycle" (graph-theoretic closed walk) — we commit to the astronomical usage in this notebook.
- "Fiber" — adopted from chess §7 with the refinement that here the fiber is static *and* shared across species, unlike Othello's dynamic fiber.
- "Phase" — adopted from chess/addressing-maths. In this notebook "phase" means angular position on a dial (equivalently: residue class in ℤ/n_dialℤ).
- **"Rendering"** — a projection from the `encode_Ant(t)` dynamic state to a user-visible spatial or dial display, parameterized by a static radial-parameter table and a free scale parameter. Distinct from the "rendering" in computer graphics (ray tracing, rasterization, shading). This project uses the term in the specific sense "projecting an HDC angular state through a parameter table into a spatial or graphical form."
- **"Orrery"** — any device or simulation that renders planetary positions in 2D or 3D spatial arrangement with bodies at their scaled orbital radii. Contrast with "Antikythera-style" which renders angular positions on concentric circular dials. Both are renderings of the same HDC state in this project's framing. The word "orrery" historically derives from the 4th Earl of Orrery's 1704 Tompion/Graham clockwork model; this project uses it genericly for any spatial-position renderer, acknowledging the anachronism for ancient devices (the Archimedean planetarium is described as orrery-like by Cicero without the word being available in antiquity).

## 8. Appendix: Environment and reproducibility

Python 3.12, NumPy, SciPy, `astroquery.jplhorizons` for Phase 4. All seeds explicit. All gear tooth counts hard-coded with provenance notes.

---

Reporting discipline (same as Othello)

At end of session: 1. results/phase1_hypotheses.csv — structured per-hypothesis status table. 2. results/phase1_detail.json — per-hypothesis numeric detail. 3. results/session_summary.md — ~1 page narrative. 4. Notebook sections §1–§3 filled in. 5. ANTIKYTHERA_PHASE_OP_PREFLIGHT.md — short handoff.

Common pitfalls anticipated

1. Tooth-count disputes. Freeth 2021 vs Wright vs Price disagree on specific counts. Document all three where they differ; run the hypothesis battery primarily against Freeth 2021 as the most recent authoritative reconstruction; note sensitivity to the disputed counts.

2. Don't overstate the HDC framing. The Greeks did not think of this as HDC. They thought of it as gear ratios chosen to approximate astronomical cycles. The HDC recognition is our framing. Keep the language clean: "the gear ratios, when expressed in modern vocabulary, constitute an HDC encoding" — not "the Greeks built an HDC encoder."

3. The 2025 manufacturing-tolerance result is real. Guillermo & Szigety's arXiv finding that the mechanism may not have run smoothly matters methodologically. It's the domain's equivalent of the chess §9a character-table bug: the design was clean, the implementation was tolerance-limited. Don't treat design cleanness as implementation success.

4. Mars is intentionally wrong. The Greeks didn't have equants. The mechanism is doing the best it could with epicycles-on-deferents, which is why Mars is up to 38° off at retrograde nodes. This is not a "bug" — it's the ceiling of Greek astronomical theory, faithfully encoded. Our encoder should reproduce the error, not correct it.

5. The Archimedes attribution is DISPUTED. Tag it DISPUTED and move on. Don't overclaim, don't dismiss.

6. NASA Horizons before 15 BCE. The API doesn't cover pre-15 BCE cleanly. Use historical eclipse catalogs (e.g., NASA's own pre-computed eclipse catalog, which extends to 2000 BCE) for validation of the Hellenistic-era test cases. skyfield with JPL DE441 is the standard Python tool.

7. Integer vs modular arithmetic. Every gear ratio is a rational number p/q. The cyclic-group structure comes from taking the ratio mod 1 (i.e., keeping only the fractional part). Be explicit: the HDC encoding lives in ℝ/ℤ (or ℤ/Dℤ after discretization), not ℤ itself.

How this fits the Rosetta Stone

Project	Structure type	Encoding type	Phase operator complexity	Rendering
chess-maths	discovered	prescriptive	rich (6 piece types × 8 orbits)	implicit (board diagram)
othello-maths	discovered	prescriptive	rich but different (1 piece, 8 rays, dynamic)	implicit (board diagram)
logo-maths	discovered	prescriptive	command set + grammar	implicit (turtle canvas)
addressing-maths	formalized foundation	—	— (the substrate itself)	—
antikythera-maths	recognized/documented	descriptive	trivial (1 operator, many projections)	explicit, factored (dial / orrery / ephemeris)

Antikythera is the first project where: - The structure was deliberately designed into the artifact. - The encoding is recognized, not invented. - The phase operator is minimal. - Ground truth is external (actual astronomy) rather than internal (game rules). - Rendering is explicitly factored out of encoding. The dial display (Antikythera), spatial model (orrery), and angular ephemeris (NASA Horizons) are all projections of the same encode_Ant(t) output through different static radial-parameter tables. Perspective is the scale invariance; the encoding is rendering-agnostic.

The "rendering" column deserves a sentence on its own: the three game projects all treat rendering as implicit — chess diagrams, Othello boards, and LOGO canvases are how results are displayed, not separate projection modes. Antikythera is the first project in the triad-plus where a single encoding has multiple culturally-independent rendering traditions (Hellenistic dial calculator, Renaissance-to-modern orrery, astronomical almanac) all drawing from the same underlying dynamic computation. Factoring rendering out cleanly from encoding is both a design discipline and a methodological statement about what the HDC object IS — angular state, not displayed state.

These simplifications make Antikythera the best pedagogical entry point for the framework. A reader coming to mlehaptics fresh may find Antikythera easier to enter than chess because every claim has an external grounding — one can check "is this really how the 127-tooth gear works?" against the physical mechanism, rather than against our construction of piece Laplacians.

Consequence for the framework narrative. When the addressing-maths paper is eventually written, the Antikythera example is probably the best motivating vignette — it gives a concrete, visual, historically documented instance of every concept the paper introduces. Consider writing the mlehaptics framework paper with Antikythera as §1 introduction, addressing-maths as §2 formal substrate, and chess/Othello/logo as §3 empirical case studies.

What this prompt is not

• Not a claim that the Greeks "invented HDC." They invented a specific geared astronomical calculator. The HDC recognition is our framing, valid as a description but not as an anachronistic attribution.
• Not a commitment to reconstructing the mechanism in hardware. The deliverable is computational/descriptive.
• Not a commitment to one reconstruction as ground truth. Where Freeth 2021, Wright, and Price disagree, all three are documented and the sensitivity is reported.
• Not a replacement for the excellent archaeology literature on the mechanism. We cite and build on it; we do not supplant it.
• Not a commitment to the Archimedes attribution. DISPUTED tag, stays DISPUTED.

Definition of done

• docs/antikythera-maths/ exists with the file layout above.
• Gear database, astronomical cycle database, and cyclic-group algebra utilities pass sanity checks.
• Phase 1 battery run (A-H1 through E-H2 + F-E1–F-E3) with numbers.
• Phase 2 encode_Ant produces hypervectors that reproduce the mechanism's astronomical predictions to within Greek-attainable precision.
• Phase 3 preflight is short and argues for the trivial phase operator.
• Phase 4 validation against NASA Horizons (or skyfield) completes on a test set of 20+ eclipses in 200 BCE – 100 CE.
• Research notebook §1–§3 prose + numbers matching the CSV.
• Session summary ~1 page.
• ANTIKYTHERA_PHASE_OP_PREFLIGHT.md drafted.

Extra credit! All protoplanet named celestial bodies in our solar system: - Reproduce the Mars retrograde error to within a few degrees of the mechanism's documented 38° error. - Cross-check the Freeth 2021 shared-prime-factor claim (7, 17) is Pareto-optimal per A-H2. - Identify whether any ancient astronomical table (Babylonian MUL.APIN, Ptolemy's Almagest tables) shows evidence of the same prime-factorization choices the mechanism makes. - How many more gears? Find out first if it's practically feasible to build a physical Antikythera with the tooth counts we document. If not, how many more gears would be needed to achieve the same astronomical precision?