A Frame-Invariant Phasor Formalism for Natal, Progressed, and Transit Configurations

With population baselines and a solar-return confound. Three voices — the birth chart, the progressed chart and today’s sky — measured as one interference field. No claims about human outcomes are made in this paper.

Sayer Ji  ·  Scalar Flower Astrology Research Program, Miami Beach, FL, USA

Correspondence: scalarflower.com. Manuscript type: Methods / baselines. Target venue: Correlation (Astrological Association), alternatively Journal of Scientific Exploration. With computational collaboration by the Hermes agent (Nous Research).

§Abstract

We describe a coordinate-free way to represent an astrological configuration as a sum of unit phasors, each measured from the natal lunar node. The method extends to three simultaneous “voices”: the natal chart, the secondary-progressed chart (one day after birth = one year of life), and the transiting sky. Their sum, the chord, has a magnitude and a normalised coherence κ ∈ [0,1]. We prove that every reported quantity is invariant under a constant rotation of the zodiac. It therefore makes no difference whether the tropical zodiac, the sidereal zodiac, or any ayanamsa is used.

We show that classical aspects are exactly the harmonics of this representation: |m_p|² = N⁻² Σ_jk cos(pΔ_jk). From a seeded random population (N = 3,000 birth–date pairs, real ephemeris), we give reference quantiles for κ at harmonics p = 1–6.

We also report a structural confound that any temporal study of this kind must control. The solar-return phase (the date’s distance from the person’s birthday) alone explains η² = 0.23 of the variance in the first-harmonic chord, falling to 0.07 at p = 2 and 0.009 at p = 3. The cause is that the transiting Sun, Mercury and Venus return near their natal positions each year. Any claim that chord coherence tracks events in a life must beat a seasonal control. We give code, seeds, and baselines, and we leave efficacy to separate pre-registered studies — whose verdicts to date are summarised in §5.

1Introduction

Quantitative astrology research has mostly tested single factors: sign placements, Gauquelin sectors, aspect counts with arbitrary orbs. We propose a representation with three properties:

• (a) No arbitrary thresholds. Aspects contribute continuously; there are no orbs.
• (b) Frame invariance. The choice of zodiac cannot change any result.
• (c) One measure for three time scales. Natal, progressed and transit configurations combine into a single interference measure.

The phasor sum is not new mathematics. It is the mean resultant vector of circular statistics (Mardia & Jupp 2000), and its harmonics recover Addey’s harmonic astrology (Addey 1976). What we contribute is the specific construction: anchoring on the natal node, the three-voice combination, the invariance proofs, the population baselines, and an explicit account of the seasonal confound.

2Formalism

Let λ_j(X) be the ecliptic longitude of body j (Sun through Pluto, N = 10) in voice X ∈ {N, P, T}, and Ω the natal true node. Each body’s angle is measured from the node:

The voice at harmonic p is the phasor sum Z_X(p) = Σ_j exp(i·p·θ_j(X)), and its normalised coherence is |m_p(X)| = |Z_X(p)|/N. At p = 1, H_X = |Z_X(1)| is the “hub”. The three voices are timed as follows:

• Natal (N): fixed at the birth moment, Julian day t₀.
• Progressed (P): secondary progression. The positions used for date t are those at t₀ + (t − t₀)/365.2422.
• Transit (T): the sky at date t itself.

Chord. J(p,t) = |Z_N + Z_P + Z_T|. Coherence is κ = J / (|Z_N| + |Z_P| + |Z_T|). The pairwise chords (N+P, N+T, P+T) are defined the same way. A transit-reinforcement index is R_T = (J − |Z_N + Z_P|)/|Z_T|.

Reduction check. Dropping the progressed voice reduces J to the natal–transit joint hub already used in the Scalar Flower transit layer. Our implementation reproduces that value to within 2×10⁻³, which is rounding at ephemeris precision.

Optional spatial rendering (illustrative only). Assigning a wavelength to each body gives a 2-D field F(x) = Σ_j exp(i(k_j·x + θ_j)) that can be drawn as an interference pattern. The pattern depends on the chosen wavelengths. We use it only as visualisation and draw no inference from it.

3Population baselines

Method. We drew N = 3,000 random (birth, evaluation-date) pairs, with seed 42: birth instants uniform over 1900–2000; evaluation dates uniform over ages 20–70. Positions come from the Swiss Ephemeris and the node is the true node. For each pair we computed κ at p = 1–6.

Typical coherence is high: the median is about 0.65–0.71. This is because the natal and progressed voices are always partly aligned — the progressed planets start where the natal ones are. Raw κ is therefore not informative; only percentiles are. Any “high chord” statement must be made against this table, or better, against the same person’s own long series.

4The solar-return confound

We binned the 3,000 pairs by solar-return phase: the transiting Sun’s angular distance from the natal Sun, in 30° bins (month 0 = the birthday month).

Consequence. A person-level finding such as “my chord was high when I was productive” is confounded whenever the productive period falls near a birthday. Any temporal test must at least:

• include solar-return phase (sin and cos of the month angle at harmonic p) as a covariate;
• or compare each subject’s windows only with the same calendar windows in other years;
• and report effects for p ≥ 3 separately, where the confound is weak.

5The empirical record so far

This paper establishes a representation, not an effect. Three pre-registered efficacy studies have since been run and are reported here so that the method is never read as a validated timing tool. None of them found a usable effect.

Two things follow, and we state both. First, large effects are disfavoured: Paper 2 rules out effects of roughly 0.17 or more, and Paper 3 effects of about 0.22 or more. The Chord is not supported as a timing instrument, and we do not offer it as one. Second, these are not clean nulls: each study fell short of its own pre-registered sample floor, so small effects are not excluded. Reporting them as NULL would overclaim in the sceptical direction, which is the same error in the other sign.

All three verdicts are recorded in the Verdict Ledger as they stand. The design that would actually settle an N-of-1 claim is prospective and first-person: a pre-registered daily log, prediction locked before the dates are seen. That instrument is published at Test your own Chord.

6Discussion and limits

• What this paper establishes: a coordinate-free, orb-free, three-voice representation; proofs of invariance; reference distributions; and a dominant seasonal confound with its size measured.
• What it does not establish: that the chord relates to anything in a person’s life. That is an empirical question, and §5 is its current answer.
• Choices that remain conventions: equal weights across bodies and voices, secondary rather than solar-arc progression, the node anchor, and the 10-body set. Alternative choices are easy to compute and should be reported as robustness checks rather than tuned.
• Generational components. The slow-planet parts of the progressed and transit voices are shared by everyone born within a few years of each other. A study must compare people against same-cohort controls.

7Reproducibility

• Engine: scalar-flower-api/engine/triple_field.py. Its self-test checks a canonical hub value, rotation invariance, and the reduction to the transit overlay.
• Baselines: chord/paper1_baselines.py, seed 42, giving paper1_baselines.json.
• Ephemeris: Swiss Ephemeris via pyswisseph. Transits evaluated at 12:00 UT.
• Live API: GET /api/chord and GET /api/chord/series return every quantity in this paper for any birth moment and date.

§References

1. Addey, J. (1976). Harmonics in Astrology. Romford: Fowler.
2. Mardia, K. V., & Jupp, P. E. (2000). Directional Statistics. Wiley.
3. Carlson, S. (1985). A double-blind test of astrology. Nature 318, 419–425.
4. Dean, G., & Kelly, I. W. (2003). Is astrology relevant to consciousness and psi? Journal of Consciousness Studies 10(6–7), 175–198.
5. Ertel, S. (2009). Appraisal of Shawn Carlson’s renowned astrology tests. Journal of Scientific Exploration 23(2), 125–137.