Reconciled 2026-10-23. The argument these tests support: cw_synchronized_wobble.md. Every number here is produced by a script in this directory (listed at the end). §8 records the claims made earlier in this work that the tests withdrew.
Raw IERS polar motion. These are the IERS Rapid Service daily x and y (USNO finals2000A.all, measured rows only,
1973.0–2026.8), averaged into 0.05-yr bins. fetch_eop.py downloads them to data/, which is not in the repository.
Polar motion is analysed as the complex series p = x − iy, in which the CW turns prograde (positive frequency). Two
other sources could not be used: the IERS datacenter rejected scripted downloads, and the Paris Observatory server
(C01, C04) timed out. So nothing before 1973 has been checked against raw data.
../cw.dat. This is a processed x-pole series, 1935–2018.6 at 0.05 yr, of unknown provenance. Over 1973–2018 it
equals −0.334 x + 0.05 y (r = −0.967 with x alone), but with signs of further processing:
cw.dat are products of its processing (cw_sidebands.py, cw_eop_closure.py, cw_peaks.py)What cw.dat shows. Peaks at 0.157 and 1.844 /yr, about 0.06–0.08 of the CW in amplitude, with phase closure
D₁ = φ(f+1) + φ(f−1) − 2φ(f) = 16° ± 19°. That is the signature of amplitude modulation (0°), not phase modulation
(180°), and the same in both halves of the record. Read as Earth physics, it would mean a ±14% annual swing in the
wobble’s amplitude.
Raw polar motion does not have them. Amplitudes are relative to the CW line:
| window | raw p, upper sideband (+1.843) | raw x, upper / lower | cw.dat, upper / lower |
|---|---|---|---|
| 1973–1990 | 0.010 | 0.011 / 0.001 | 0.064 / 0.064 |
| 1990–2026 | 0.0003 | 0.004 / 0.038 | 0.032 / 0.048 |
| 1973–2018.6 | 0.005 | 0.009 / 0.016 | 0.033 / 0.055 |
The raw phase closures are noise (spreads of 70–160°). The upper sideband sits in a clean part of the raw spectrum,
and there it is 6–100 times weaker in amplitude than in cw.dat. Line by line over 1973–2018, cw.dat ÷ (−0.334 × raw x):
| CW | annual | semiannual | 2 − CW | 1 − CW | 1 + CW | 2 + CW | 1.5 | 3.0 |
|---|---|---|---|---|---|---|---|---|
| 1.00 | 1.10 | 0.10 | 0.98 | 3.44 | 4.11 | 3.73 | 3.18 | 5.41 |
No fixed (time-invariant) filter jumps from a gain of 1 to 3.4–4.1 at selected lines. Some multiplicative step is involved.
The mechanism: multiplicative cross-terms from a calendar-locked gain. Multiplying a series that contains the CW by anything with an annual period puts equal lines at f_CW ± 1 with closure 0°, the observed signature. A semiannual factor adds 2 ± CW; annual × semiannual adds 3.0. Two candidate products were tested by regression on raw x and y. The table gives line power left over, relative to the CW:
model regressed onto cw.dat |
0.157 | 1.843 | 3.0 |
|---|---|---|---|
cw.dat itself |
2.2 × 10⁻³ | 5.6 × 10⁻⁴ | 7.5 × 10⁻⁴ |
| a·x + b·y + trend | 1.8 × 10⁻³ | 4.5 × 10⁻⁴ | 7.9 × 10⁻⁴ |
| + x², y², xy (squared amplitude) | 1.7 × 10⁻³ | 3.1 × 10⁻⁴ | 7.7 × 10⁻⁴ |
| + seasonal gain on x, y (4 terms) | 6.5 × 10⁻⁵ | 5.7 × 10⁻⁴ | 7.4 × 10⁻⁴ |
| + annual and semiannual gain (8 terms) | 5.7 × 10⁻⁵ | 5.6 × 10⁻⁴ | 2.2 × 10⁻⁴ |
Consequence. The --amp-step option of ../cw.py (annual rescaling of the wobble, s = 0.25) reproduces
cw.dat, not the Earth. Raw data limit any annual amplitude modulation of the CW to about 1% since 1990. The 2021
satellite result (../ChandlerWobbleForcing.pdf) appears to rest on a similar processed series. Before 1973 is
untested.
cw_peaks.py)Every local maximum of the cw.dat periodogram above 3 × 10⁻⁴ of the CW power, matched to n₁·CW + n₂ and to the lunar
alias lines, and checked against raw p:
| peak (/yr) | power / CW | raw p, prograde / retrograde | share left after a·x + b·y (1973–2018) | identification | verdict |
|---|---|---|---|---|---|
| 0.843 | 1 | 1 / 10⁻⁵ | 0.2% | CW: draconic B² alias 0.8440 | Earth, phase-locked |
| 1.000 | 0.32 | 0.39 / 9 × 10⁻⁴ | 1.5% | annual wobble | Earth |
| 0.737–0.748 | 5.7 × 10⁻³ | 1.2 × 10⁻² / 2 × 10⁻⁵ | 0.5% | Mm: 14 − anomalistic = 0.7448 (tropical B² 0.7365 is 0.008 away) | Earth, stationary line (§4) |
| 0.817 | 0.064 (beside the CW) | not resolved | — | — | slow modulation of the CW (§4) |
| 0.940, 0.680 | 2 × 10⁻³ – 8 × 10⁻³ | similar, prograde | 3–12% | — | Earth: broadened CW band |
| 2.0 | removed in cw.dat |
2.5 mas, steady | — | semiannual (comb) | Earth |
| 0.157, 1.844 | 4.5 × 10⁻³, 3.7 × 10⁻³ | ≤ 2 × 10⁻⁴ | 80% | 1 ∓ CW | processing (§2) |
| 1.524, 2.984 | ~10⁻³ | ≤ 4 × 10⁻⁵ | 50–100% | 3CW − 1; 3 | processing |
| 1.7–5 (about 30 peaks) | 3 × 10⁻⁴ – 1.3 × 10⁻³ | 10⁻⁶ – 10⁻⁵ | — | none | cw.dat noise floor, ~100× raw |
cw_peaks.py)A forced line keeps its absolute phase from window to window; a noise peak, or a sidelobe of a changing CW, does not. This is raw p in four 13.5-yr windows, with amplitudes in mas:
| line | 1973–86 | 1986–2000 | 2000–13 | 2013–27 | phase spread | p (random phases) | best-fit frequency |
|---|---|---|---|---|---|---|---|
| CW 0.8440 | 156 | 177 | 109 | 35 | ±18° | 0.002 | 0.8435 |
| Mm 0.7448 | 18 | 24 | 26 | 18 | ±34° | 0.017 | 0.7478 |
| tropical B² 0.7365 | 15 | 19 | 20 | 16 | ±54° | 0.065 | 0.7480 (the same peak) |
| Mm2 0.5105 | 3 | 3 | 7 | 1 | ±141° | 0.73 | 0.494 |
| 14 − tropical 0.6317 | 4 | 4 | 8 | 6 | ±135° | 0.68 | 0.621 |
| 2 − CW 1.1560 | 2 | 1 | 14 | 7 | ±104° | 0.41 | 1.163 |
| 1.1848 | 3 | 4 | 6 | 4 | ±86° | 0.25 | 1.165 |
| semiannual 2.0 | 3 | 4 | 3 | 1 | ±22° | 0.005 | 2.000 |
The 0.817 /yr feature was once a candidate for the free mode, but its phase is steady. This is cw.dat in
windows long enough to separate it from the CW:
| windows | 0.817 amplitude / CW | 0.817 phase | 0.871 phase | closure with the CW (AM = 0°) |
|---|---|---|---|---|
| 1935–77 / 1977–2018 | 0.31 / 0.35 | 48° / 61° (±6°) | −149° / −154° | 24° / 48° |
| 1935–63 / 1963–91 / 1991–2018 | 0.45 / 0.13 / 0.42 | 42° / 26° / 74° (±27°) | −146° / −105° / 157° | 23° / 42° / 18° |
A noise-driven free mode would wander. This feature holds its phase and closes with the CW like amplitude modulation, so it reads as a slow (~37-yr) modulation of the forced CW. With only two or three independent windows, that reading is suggestive.
cw_joint_lock.py)Both lines are named in advance from the alias rule, so there is no look-elsewhere penalty. The null model is a free wobble (432.3 d) excited by white noise, with a small white floor, analysed in the same four windows (40,000 runs per Q):
| null model | CW lock | Mm lock | both locks | both + Mm amplitude steady |
|---|---|---|---|---|
| Q = 50 | 0.080 | 0.018 | 0.0014 | ~5 × 10⁻⁵ (2 hits) |
| Q = 85 | 0.157 | 0.018 | 0.0024 | ~1.5 × 10⁻⁴ (6 hits) |
| Q = 150 | 0.282 | 0.017 | 0.0050 | ~1 × 10⁻⁴ (4 hits) |
The amplitude criterion uses the observed value as its threshold, and the last column rests on few hits. With three 18-yr windows the CW lock alone has p ≈ 0.05.
cw_bandpass.py)The device has to produce each line at its observed strength:
cw_flagship.py, which gates the astronomical torque through the pulses) reproduces
the analytic line ratios: Mm/CW 0.103 against 0.106. The semiannual comes out about 2× high.cw.dat: its source series, the splice near 2005, the semiannual removal and the seasonal gain.| earlier claim | status | where |
|---|---|---|
| The 1 ∓ CW satellites are amplitude-modulation products of the Earth’s wobble; they measure a ±14% N/S balance, largest in early December | withdrawn: they come from cw.dat’s processing |
§2 |
--amp-step 0.25 makes the model match the observed sidebands |
it matches cw.dat, not the Earth; default 0 |
§2 |
| The 0.737 peak is the tropical B² alias and is not a forced line | corrected: it is a stationary Mm line at 0.7448 | §4 |
| Mm is 15× too strong for the distance modulation | resolved: an artifact of fixing f₀ = 432 d, Q = 85 and sharp pulses | §6, lead §3 |
| The natural period is ≈ 450 d, with 0.817 as the free mode and a 37-yr free/forced beat | withdrawn: with pulse shape free, f₀ ≈ 425–440 d; 0.817 is phase-steady | §4, lead §3 |
| Low Q (single digits) is allowed once the wobble is forced | bounded: Q ≳ 30, set by the 2 − CW line | lead §3 |
| CW counter-torque from a finite-difference inverse | corrected: computed exactly in the frequency domain | cw_counter_torque.py |
| The peak near 1.16 /yr is the draconic comb partner 2 − CW (1.1560) | re-identified: it is F − M on the annual harmonic, 1 + (F − M) = 1.1668 ≈ 7/6, phase-steady (±31°); 2 − CW wanders | cw_suppression.md §6.2 |
| The Kittel (nanomagnet) mode as the working analog | kept only as a mathematical identity; rotating machinery is the analog | lead §1, cw_magnetized_spheroid.md |
| script | what | outputs |
|---|---|---|
fetch_eop.py |
download raw IERS polar motion | data/finals2000A.all |
cw_sidebands.py |
cw.dat sideband phase closure (the original AM reading) |
cw_spheroid_sidebands.png, cw_sidebands.json |
cw_eop_closure.py |
raw IERS against cw.dat: closures, sense test, line gains, seasonal residual |
cw_eop_closure.png, .json |
cw_peaks.py |
every peak; processing models; phase stability; blog lines; the 0.817 feature | cw_peaks.png, .json |
cw_joint_lock.py |
joint chance of the CW and Mm locks; Mm strength under fixed resonance | cw_joint_lock.json |
cw_bandpass.py |
free bandpass fitted to the line set | cw_bandpass.png, .json |