artatopics · the exact phasor, solved analytically
The model this campaign was built around, solved in closed form — no optimiser, no seed — and tested on the finest record we have: every arXiv submission since 1991, day by day, 127 categories.
1 · The model and its exact solution
y(t) = | b + Σᵢ aᵢ · e i(θᵢ(t) − pᵢ) |²
The only inputs are θᵢ(t): the sidereal (Lahiri) longitudes of Sun, Moon, Mercury, Venus, Mars, Jupiter, Saturn and the true node from kerykeion (Swiss Ephemeris). The unknowns b, aᵢ, pᵢ are solved analytically. Expanding the square exactly:
y = b² + Σaᵢ² + Σᵢ 2b·aᵢ·cos(θᵢ−pᵢ) + Σᵢ<ₖ 2aᵢaₖ·cos((θᵢ−pᵢ)−(θₖ−pₖ))
Every term is linear in the fixed basis {1, cos θᵢ, sin θᵢ, cos(θᵢ−θₖ), sin(θᵢ−θₖ)}, so one least-squares solve gives its coefficients c. Then, exactly: pᵢ = atan2(βᵢ, αᵢ); Mᵢ = √(αᵢ²+βᵢ²) = 2b·aᵢ; and from c₀ = b² + ΣMᵢ²/4b² the quadratic 4b⁴ − 4c₀b² + ΣMᵢ² = 0 gives b² = (c₀ + √(c₀² − ΣMᵢ²))/2 — the '+' root, the only one that reproduces the fitted curve. Self-test on data generated by the exact model: b recovered to six decimals, aᵢ and pᵢ to 10⁻¹³; with 5% noise, R² 0.9976.
The form over-determines itself: the aspect coefficients must equal 2aᵢaₖcos(pᵢ−pₖ) with the aᵢ, pᵢ the transit terms already fixed. How far the freely fitted aspects sit from that is the aspect residual — the part of the data the phasor cannot be. If disc = c₀² − ΣMᵢ² < 0, no real (b, aᵢ) exists at all: the fit is infeasible.
2 · What it scored on the daily record
Each of 127 arXiv categories fitted independently on the first 80% of its reliable days (the series as a ratio to its own trailing-365-day level, so b carries the level and the arrows the timing), scored on the last 20% as a peak detector: does the forecast rank the days before a submission peak above the rest? One AUC per category, averaged.
| exact phasor, kerykeion sidereal | level AUC | rise AUC | shift-null | feasible | held-out R² (median) |
|---|---|---|---|---|---|
| all 8 bodies, unregularised | 0.4999 | 0.5002 | nan | 22% | -1.327 |
| Sun–Mars only (cycle-complete) | 0.5001 | 0.4958 | 0.4973 | 83% | -0.411 |
| all 8, L2, λ per category on an inner wall | 0.4932 | 0.4972 | 0.5068 | 95% | -0.052 |
Chance for these autocorrelated labels is the circular-shift null, ~0.50–0.52. Under L2 the inner walls chose λ=0.0001: 4 · λ=0.1: 3 · λ=10.0: 2 · λ=100.0: 118 — in 118 of 127 categories the strongest shrinkage on offer, i.e. the model prefers b alone. Aspect residual is 1.000 throughout: the aspect terms the data wants bear no relation to the ones the transit terms determine.
On the yearly citation-share task — where the campaign's record model scores −3.58 against carry-forward's −2.56 (per-field R² vs holdout mean) — the exact form with the same horizon anchor, selection on inner walls, scores −8.25 on the live board. The record model fits √y ≈ b + Σaᵢcos(θᵢ−pᵢ), which drops the (Σaᵢ sin)² term of |z|²; keeping it makes the fit stiffer, not better. That approximation was carrying the record model.
3 · Solve it yourself
Pick a category. The button loads Python (Pyodide + numpy, ~10 MB), runs the same closed-form solution on the same kerykeion sky and the same daily series, and prints b, every aᵢ, every pᵢ, the feasibility, the aspect residual and the held-out peak AUC. Nothing here is precomputed.