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Antikythera-Maths — Phase-Operator Preflight (Phase 3)

Sibling to ../othello-maths/OTHELLO_PHASE_OP_PREFLIGHT.md. This one is short for a reason — see below.

TL;DR

The Antikythera's phase operator is trivial. There is one operator, σ_day, applied uniformly. There are no piece-types, no orbit-types, no flip-gates, no ray-directions, no dynamic gating. The mechanism is the simplest possible phase-operator system in the mlehaptics triad-plus.

σ_day(D) = roll_operator(D, 1)              # single generator of ℤ/Dℤ
advance_day(date, D, days=1) =              # one day later
    encode_ant(date + days, D)              # re-encode (multi-dial state)

That's the entire phase operator: ~5 lines of code. The interesting work happens at decode time, where each dial extracts its own residue from the underlying day-counter via its gear ratio.

Triad-plus comparison

Project Pieces / channels Operator complexity
chess-maths 6 piece types × 8 D₄ orbits Rich operator set; per-piece per-orbit move kernels (§11.4–§11.6 build prompts).
othello-maths 1 piece type × 8 ray directions × dynamic gating Rich but different; flank-gating couples cells.
logo-maths Command set + grammar Structured but different; turtle state with motion-and-turn.
antikythera-maths 1 operator, 13 projections Trivial: one σ_day, dial-specific gear ratios at decode.

This is a feature, not a bug. The Antikythera is the best pedagogical entry point for the framework 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 a constructed Laplacian.

The dial → projection table

Each dial is one cycle in the astronomical_cycles.CYCLES list. Its decode-time projection is its cycle_modulus and cycle_period_days:

Dial cycle_modulus cycle_period_days Projection formula
Metonic 235 6939.69 residue = floor(D · (date − ε) / 6939.69) mod D
Callippic 940 27758.78 residue = floor(D · (date − ε) / 27758.78) mod D
Olympic 4 1460.97 residue = floor(D · (date − ε) / 1460.97) mod D
Saros 223 6585.32 residue = floor(D · (date − ε) / 6585.32) mod D
Exeligmos 669 19755.96 residue = floor(D · (date − ε) / 19755.96) mod D
SiderealMonth 254 6939.70 residue = floor(D · (date − ε) / 6939.70) mod D
DraconicMonth 242 6585.36 residue = floor(D · (date − ε) / 6585.36) mod D
LunarAnomaly 251 6916.19 residue = floor(D · (date − ε) / 6916.19) mod D
Mercury 145 16802.59 residue = floor(D · (date − ε) / 16802.59) mod D
Venus 289 168752.74 residue = floor(D · (date − ε) / 168752.74) mod D
Mars 133 103732.02 residue = floor(D · (date − ε) / 103732.02) mod D
Jupiter 76 30314.88 residue = floor(D · (date − ε) / 30314.88) mod D
Saturn 427 161425.06 residue = floor(D · (date − ε) / 161425.06) mod D

ε = REFERENCE_JD = 1684595.0 (≈ 205 BCE epoch).

B-H2 follows trivially: σ_day is a unit in ℤ/Dℤ for every D (gcd(1, D) = 1). The "physical" crank-turn is the day-advance via re-encoding; the "algebraic" σ_day is the canonical generator on the day-counter ambient. The two coincide at design time; the build prompt's CONFIRMED tag is by construction.

What a sequel could cover

This minimal phase operator means there is no follow-up "Phase Operator Build Prompt" for Antikythera the way there is for chess (§11.4, §11.5, §11.6) or Othello (PHASE_1C_PLAN). If a sequel were ever needed it would cover one of:

  1. An equant-bearing Mars encoder. Add a per-dial epicycle-and-equant model so E-H2's Mars retrograde error matches the Greek mechanism's documented 38° peak. ~50 LOC; meaningful research finding.
  2. DE422 / DE441 ground truth. Validate E-H1 against actual Hellenistic eclipses (200 BCE – 100 CE) instead of the modern Saros-anchor proxy. ~100 LOC plus ephemeris-cache management.
  3. Manufacturing-tolerance overlay. Reproduce the Guillermo & Szigety 2025 simulation: introduce per-gear tooth-count noise (e.g., ±0.5 tooth per gear) and measure how dial accuracy degrades.

None of these requires a new phase-operator framework. The phase operator IS already trivial. Time spent here is best spent at the decode-projection or ground-truth layer, not at the operator layer.