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Antikythera-Maths — Picking up the project in a new Claude session

Sibling to ../othello-maths/OTHELLO_SPECTRAL_INSTRUCTIONS.md, with which it shares the cognitive style and methodological discipline.

What this is

The Antikythera mechanism (Greek, ca. 150–60 BCE) is a physical instantiation of coprime-indexed phase-space addressing, designed deliberately 2100 years ago to solve the Diophantine approximation problems that docs/addressing-maths/ADDRESSING_MATHS_RESEARCH_PLAN.md characterises formally. This project documents the artifact in addressing-maths vocabulary.

The encoding is descriptive, not prescriptive. The math is in the bronze; we read it off, we don't invent it. This is the first project in the mlehaptics triad-plus where the structure was designed in by named historical agents (plausibly in the Archimedean tradition).

Steven's cognitive style

(Mostly inherited from the Othello / chess threads. Restated here so the doc stands alone.)

  • Both/and discipline. When a design choice has multiple defensible options, implement them all where feasible. The encoder has three D variants (940, 13440, LCM-symbolic) for this reason; the gear database tabulates Freeth 2021, Wright, and Price 1974 wherever they disagree.
  • Honest tags. Every claim in the notebook carries one of KNOWN | NOVEL | CONFIRMED | FAILED | DISPUTED. Don't write a sentence without thinking about which tag it earns.
  • No silent overclaiming. The Greeks did not "invent HDC." They invented a specific geared astronomical calculator. The HDC recognition is our framing, valid as a description but not as an anachronistic attribution.
  • Don't conflate design with implementation. The Guillermo & Szigety 2025 manufacturing-tolerance result is real — clean design at the gear-ratio layer does not guarantee smooth running at the bronze-cutting layer. C-H1's "zero error correction" is the theoretical reason this matters.
  • Keep rendering out of encoding. Patch 3 of the build prompt is load-bearing: encode_Ant(t) returns angular state only; radial parameters and (x, y) coordinates live in research/rendering.py. The dial calculator and the orrery are sibling renderings of the same parent state; perspective is the scale invariance.

Discovery vs description

Chess, Othello, logo were discovery projects: structure was present and we used spectral tools to extract it. Antikythera is reconstructive: structure was designed in, and the job is to document it in modern vocabulary. This shifts the honest-science tags:

  • KNOWN — published in archaeology / history (Price 1974, Freeth et al. 2006/2008/2021, Wright 2005-2012, Guillermo & Szigety 2025).
  • NOVEL — the HDC / phase-space framing itself, naming 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 what the bronze does.
  • FAILED — our HDC encoding cannot reproduce something the physical mechanism does (E-H2 Mars retrograde is the canonical example).
  • DISPUTED — archaeological reconstructions disagree (b1 tooth count 224 vs 223, Archimedes attribution, etc.).

File organisation

docs/antikythera-maths/
├── ANTIKYTHERA_SPECTRAL_BUILD_PROMPT.md   ← the original build prompt, with patches consolidated
├── ANTIKYTHERA_SPECTRAL_INSTRUCTIONS.md   ← THIS FILE
├── ANTIKYTHERA_PHASE_OP_PREFLIGHT.md      ← short Phase 3 handoff
├── SESSION_HANDOFF.md                     ← interrupted-session notes (April 2026)
├── antikythera_spectral_research_notebook.md  ← the main research narrative
├── research/
│   ├── __init__.py
│   ├── gear_database.py         ← 40 gears, 3 reconstructions, provenance
│   ├── astronomical_cycles.py   ← 13 cycles, prime factorisations
│   ├── cyclic_group_algebra.py  ← CRT tables, roll operators, gear-mesh ratios
│   ├── rational_approximation.py ← CF expansion, semi-convergents, best-under-budget
│   ├── packing_analysis.py      ← prime spectrum, Pareto frontier, CF-rank report
│   ├── pin_and_slot.py          ← Fragment B epicycle, T-breaking decomposition (D-H1)
│   ├── encode_ant.py            ← THREE D-variant resonant HDC encoder + DialSpec + sigma_day
│   ├── dial_decoder.py          ← FFT-based circular-correlation unbinding
│   ├── rendering.py             ← Patch-3 dial + orrery projections
│   ├── astronomical_ground_truth.py  ← skyfield DE421 (E-H1, E-H2)
│   └── consolidated_tests.py    ← H-battery runner, writes results/*
├── results/
│   ├── phase1_hypotheses.csv    ← runtime status per hypothesis
│   ├── phase1_detail.json       ← per-hypothesis numeric detail
│   └── session_summary.md       ← ~1-page narrative summary
├── figures/                     ← (empty; for future plots)
└── skyfield_data/               ← (gitignored; on-demand de421.bsp cache)

How to run

Run from docs/antikythera-maths/ (NOT from the repo root). The research modules use relative imports (from .astronomical_cycles import ...) so they must be invoked as packages:

cd docs/antikythera-maths
PYTHONIOENCODING=utf-8 python3 -m research.consolidated_tests

Bare python3 research/consolidated_tests.py will fail with ImportError: attempted relative import with no known parent package.

PYTHONIOENCODING=utf-8 is needed on Windows because the default cp1252 console can't print the Greek letters in the smoke-test output (ε, σ, θ, etc.).

Per-module smoke tests:

PYTHONIOENCODING=utf-8 python3 -m research.gear_database
PYTHONIOENCODING=utf-8 python3 -m research.astronomical_cycles
PYTHONIOENCODING=utf-8 python3 -m research.cyclic_group_algebra
PYTHONIOENCODING=utf-8 python3 -m research.rational_approximation
PYTHONIOENCODING=utf-8 python3 -m research.packing_analysis
PYTHONIOENCODING=utf-8 python3 -m research.pin_and_slot
PYTHONIOENCODING=utf-8 python3 -m research.encode_ant
PYTHONIOENCODING=utf-8 python3 -m research.dial_decoder
PYTHONIOENCODING=utf-8 python3 -m research.rendering
PYTHONIOENCODING=utf-8 python3 -m research.astronomical_ground_truth

Dependencies: numpy, scipy, sympy, skyfield. Install with:

python3 -m pip install --user numpy scipy sympy skyfield

Common pitfalls

  1. Relative imports — see "How to run." Always python3 -m research.<module>.

  2. Tasks-pane false-running bug (April 2026 cloud sandbox) — Bash calls sometimes appear "Running" indefinitely in the pane even after they've completed. Workaround: redirect output to a file (python3 -m research.X > /tmp/x.log 2>&1) and Read the log. This decouples from whatever the pane is tracking.

  3. skyfield ephemeris coverage — DE421 covers 1900-01-01 through 2050-01-01 only. Hellenistic-era validation (200 BCE – 100 CE) requires DE422 (~600 MB) or DE441 (~3 GB). E-H1 is currently anchored at 1999-08-11 (modern total solar eclipse) for the Saros-period test; absolute Hellenistic placement is a follow-up.

  4. Mars modern_days is deliberate — astronomical_cycles.py uses 125 × MARS_SYNODIC_DAYS (not tropical years) for the Mars cycle's modern_days because the Greek Mars theory has no clean "years" denominator. The 6239-day residual is what D-H1 / E-H2 are characterising, not a bug.

  5. Freeth-only planetary gears — gears 289 (Venus), 462 (Venus), 427 (Saturn), 442 (Saturn), 133 (Mars), 125 (Mars) are not physically attested. Freeth 2021 derives them from the proposed period relations. Filter Gear.fragment is not None to get only attested counts.

  6. Encoder D=940 planetary support — encode_ant_callippic raises UnsupportedDialError if you ask for any planetary dial. This is intentional — D=940 is the "calendar-only honest" variant. Use D=13440 for the all-thirteen-dial encoding.

  7. σ_day is a roll, not a per-cycle rate — sigma_day(D) = roll_operator(D, 1) is a single unitary on ℂ[ℤ/Dℤ]. It does NOT advance each dial by its physical daily rate; that's advance_day(date, D, days=1) which re-encodes for the new date. The build prompt's "one operator, many projections" framing is honoured by the trivial roll (the operator) and the per-dial decoder gear ratios (the many projections).

  8. CHANGELOG and branch workflow — per CLAUDE.md, no direct commits to main. Stay on the active branch; update CHANGELOG.md under [Unreleased] → Added before opening the PR.

Key references

  • Freeth, T., et al. (2021). A Model of the Cosmos in the ancient Greek Antikythera Mechanism. Scientific Reports 11:5821. — current authoritative reconstruction.
  • Freeth, T., Jones, A., Steele, J. M., & Bitsakis, Y. (2008). Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism. Nature 454:614.
  • Freeth, T., et al. (2006). Decoding the ancient Greek astronomical calculator known as the Antikythera Mechanism. Nature 444:587. — Fragment B pin-and-slot geometry.
  • de Solla Price, D. (1974). Gears from the Greeks. Trans. American Philosophical Society 64(7). — foundational, superseded on specific tooth counts.
  • Wright, M. T. (2005-2012). Multiple papers on the alternative reconstruction.
  • Guillermo, R. & Szigety, S. (2025, arXiv). Manufacturing-tolerance simulation suggesting the mechanism may not have run smoothly.
  • Cicero, De re publica and Tusculan Disputations. — Archimedean planetarium tradition (DISPUTED attribution).

What this project is NOT

  • Not a claim that the Greeks "invented HDC." They invented a geared astronomical calculator. The HDC framing is ours.
  • Not a hardware reconstruction. Deliverable is computational/descriptive.
  • Not a commitment to one reconstruction as ground truth. Where Freeth, Wright, Price disagree, all are documented.
  • Not a replacement for the archaeology literature. We cite and build on it; we do not supplant it.
  • Not a commitment to the Archimedes attribution. Tag DISPUTED, stay DISPUTED.