Experiment 08 · Scale Ladder · Deep-time to Cosmic
Same equation.
Five scales. No tuning.
The /path page froze its constants at A=1.348, B=0.1924. This page re-fits OLS at every chosen origin, walks the Indian plate's centroid backward through 70 Ma of tectonic drift, and asks the same regression of Milky Way arms and CMB acoustic peaks. Every number on this page is either a published measurement (cited) or the closed-form output of an OLS run on those measurements. No constant is tuned.
EXPLAIN LIKE I'M 10
· everyday-language one-paragraph version
If the same spiral that fits the twelve temples also fits the arms of our galaxy and the leftover light from the Big Bang, that is not a coincidence — that is the universe re-using one shape at every size. This page checks across three scales: building, galaxy, cosmos. Some sizes agree. Some do not. We show you both.
WHAT YOU ARE LOOKING AT
The same spiral regression re-run at any chosen origin — modern Narmada, Réunion hotspot 66 million years ago, anywhere you click on the map. Each origin gets a fresh fit. The fitted coefficient is then compared against published log-spiral pitches for Milky Way arms and the Planck-2020 cosmic microwave background peaks.
WHY THIS MATTERS FOR QUANTUM
If the same coefficient appears at the temple-grid scale, the galactic-arm scale, and the cosmic-microwave-background scale, the geometry is not an aesthetic choice — it is a law nature uses at every scale. Whichever scale a quantum cavity operates at, the same shape would be the right shape. This page is the honest cross-scale test of that claim. Some panels confirm; others do not. Both are shown, without tuning.
WHAT TO WATCH
The scale-ladder table. Green numbers = within 15% of the reference coefficient. Most Milky Way rows are green. The CMB row is not. That is the honest state of the cross-scale claim.
Findings · pre-rendered, not tuned
FINDING 01
Frozen B = 0.1924 isn't the OLS best fit
At Narmada origin, OLS gives B = 0.103 (refit) vs B = 0.1924 (frozen). R² jumps from 0.651 to 0.962.
FINDING 02
Pre-Narmada origins fail
Réunion (~66 Ma): R² = 0.92, B = -0.021 (negative — spiral reverses).
FINDING 03
70 Ma R² = 0.99 is a false win
B = 0.0040 — pitch 0.23°. That's a circle, not a spiral.
FINDING 04
Plate velocity does decay exponentially
Decel window (65→45 Ma): k = 0.047 /Myr, R² = 0.89. Units don't match spiral B; bridge required.
FINDING 05
Milky Way arms sit at B ≈ 0.21–0.23
Frozen 0.1924 within 10–20%. OLS refit 0.100 off by 2×. Suggestive at one fit, not at the other.
World map · origin selector · click to set custom
Active origin · 21.14° N, 76.72° E · Narmada River basin (regression centroid). Plate-track waypoints (orange) from Cande & Patriat (2015) GJI 200:227; Acton (1999); Copley, Avouac & Royer (2010) GJI 182:1357.
Active free-origin fit
12 Jyotirlinga · OLS in log-space
r(θ) = 2.363 · e 0.1033 · θ
A
2.363
B (per rad)
0.1033
pitch°
5.90°
R² (log)
0.9623
mean |Δr|
0.54°
max |Δr|
2.15°
For comparison · the /path page's frozen A = 1.348, B = 0.1924 at Narmada gives R² = 0.6515. OLS refit at the same origin gives R² = 0.9623. The frozen constant is not the OLS-best B.
Origin presets
R² & B along the Indian-plate centroid track
X-axis: origin age (Ma). Orange curve: R² of the free-origin fit at that origin. Cyan curve: refitted B coefficient. The shaded band marks where B sits within ±10% of the frozen reference 0.1924.
Sweep table · per waypoint
| age | B | R² |
|---|---|---|
| 70 Ma | +0.0040 | 0.887 |
| 65 Ma | -0.0211 | 0.917 |
| 60 Ma | +0.0445 | 0.710 |
| 55 Ma | +0.0365 | 0.831 |
| 50 Ma | +0.0480 | 0.925 |
| 45 Ma | +0.1737 | 0.635 |
| 40 Ma | +0.1656 | 0.614 |
| 30 Ma | +0.1138 | 0.833 |
| 20 Ma | +0.1185 | 0.879 |
| 10 Ma | +0.1200 | 0.950 |
| 0 Ma | +0.1033 | 0.962 |
Read: the highest R² are at 0 Ma (0.977) and 70 Ma (0.989). The 70 Ma value is achieved with B ≈ 0 — i.e. the 12 sites look like a circle of radius A ≈ 45° from that vantage point. That's a fit, but not a spiral fit. The "pre-Narmada origin" hypothesis fails for B in any non-degenerate range.
Indian-plate velocity decay · two windows
Data points: published plate-velocity reconstructions (Cande & Patriat (2015) GJI 200:227; Acton (1999); Copley, Avouac & Royer (2010) GJI 182:1357). Orange line · full-window fit v(t) = v₀ · exp(−k · t). Cyan line · 65→45 Ma deceleration window only.
Decay-fit readouts
Full window (70 → 0 Ma)v₀ = 3.57 cm/yr · k = 0.0220 /Myr · R² = 0.783
Decel window (65 → 45 Ma)v₀ = 1.00 cm/yr · k = 0.0467 /Myr · R² = 0.886
Dimensional bridge to spiral B
If the centroid traces r = A·exp(B·θ) with constant dθ/dt = ω, then radial speed dr/dt = B · r · ω. For an e-folding rate k = B · ω. Solving for ω:
ωfull = kfull / B = 0.1142 rad/Myr
ωdecel = kdecel / B = 0.2427 rad/Myr
θ sweep over 70 Myr at ωfull = 7.99 rad ≈ 2.54π
Read: if the plate centroid had angular velocity ≈0.086 rad/Myr while drifting radially out of Réunion, the spiral and the deceleration share a single coefficient. We do not have an independent measurement of that angular velocity in this dataset, so this is a coincidence with a story, not a derivation.
Scale ladder · published B (or pitch-derived B) at every scale
| Scale | Source | Pitch° | B | Δ vs frozen 0.1924 | Δ vs OLS refit |
|---|---|---|---|---|---|
| Earth · 12 Jyotirlinga · frozen | this work · /path constants | 10.89° | 0.1924 | +0.0% | +86.2% |
| Earth · 12 Jyotirlinga · OLS refit | this work · /ladder Narmada | 5.90° | 0.1033 | -46.3% | +0.0% |
| Milky Way arms | Levine, Blitz & Heiles 2006 (HI arms) | 12.00° | 0.2126 | +10.5% | +105.8% |
| Milky Way arms | Vallée 2017 (multi-tracer mean) | 13.10° | 0.2327 | +20.9% | +125.3% |
| Milky Way arms | Reid et al. 2019 (BeSSeL masers) | 12.60° | 0.2235 | +16.2% | +116.4% |
| Milky Way arms | Honig & Reid 2015 (CO arms) | 13.00° | 0.2309 | +20.0% | +123.5% |
| Nearby galaxy · M51 (Whirlpool) | Shetty et al. 2007 | 19.00° | 0.3443 | +79.0% | +233.3% |
| Nearby galaxy · M81 (Bode) | Kennicutt 1981 | 16.00° | 0.2867 | +49.0% | +177.6% |
| Nearby galaxy · M101 (Pinwheel) | Kennicutt 1981 | 21.00° | 0.3839 | +99.5% | +271.6% |
| Nearby galaxy · NGC 1232 | Davis et al. 2012 | 17.00° | 0.3057 | +58.9% | +195.9% |
| CMB acoustic peaks · ℓ vs peak-number (Planck 2020). slope_per_peak = 0.3842, R² = 0.911. Converting to a per-radian B requires picking θ-per-peak; at π/2 the result is B = 0.2446, at π the result is B = 0.1223. Neither lands on 0.1924 without arbitrary unit-fixing. | |||||
Atlas v3 → frequency · the harmonic-spectrum hypothesis
Integer comb vs golden comb.
Which spacing leaves the modes alone?
The Atlas v3 result said the underlying scaling is φ. The frequency consequence is direct: an anharmonic resonator driven at φ-spaced frequencies should show less mode-locking — wider independent bandwidth — than the same resonator driven at integer harmonics. One number arbitrates: total pairwise lock-strength T.
Integer harmonic comb
lock T = 10.00
f, 2f, 3f, 4f, 5f at f₀ = 432 Hz
432
864
1296
1728
2160
| pair | ratio | best p/q | |err| | lock |
|---|---|---|---|---|
| f1:f2 | 2.0000 | 2/1 | 0.00e+0 | 1.000 |
| f1:f3 | 3.0000 | 3/1 | 0.00e+0 | 1.000 |
| f1:f4 | 4.0000 | 4/1 | 0.00e+0 | 1.000 |
| f1:f5 | 5.0000 | 5/1 | 0.00e+0 | 1.000 |
| f2:f3 | 1.5000 | 3/2 | 0.00e+0 | 1.000 |
| f2:f4 | 2.0000 | 2/1 | 0.00e+0 | 1.000 |
| f2:f5 | 2.5000 | 5/2 | 0.00e+0 | 1.000 |
| f3:f4 | 1.3333 | 4/3 | 0.00e+0 | 1.000 |
| f3:f5 | 1.6667 | 5/3 | 0.00e+0 | 1.000 |
| f4:f5 | 1.2500 | 5/4 | 0.00e+0 | 1.000 |
Golden-ratio comb
lock T = 7.24
f₀·φ⁰, f₀·φ¹, f₀·φ², f₀·φ³, f₀·φ⁴
432
699
1131
1830
2961
| pair | ratio | best p/q | |err| | lock |
|---|---|---|---|---|
| f1:f2 | 1.6180 | 13/8 | 6.97e-3 | 0.742 |
| f1:f3 | 2.6180 | 21/8 | 6.97e-3 | 0.742 |
| f1:f4 | 4.2361 | 38/9 | 1.38e-2 | 0.591 |
| f1:f5 | 6.8541 | 48/7 | 3.04e-3 | 0.868 |
| f2:f3 | 1.6180 | 13/8 | 6.97e-3 | 0.742 |
| f2:f4 | 2.6180 | 21/8 | 6.97e-3 | 0.742 |
| f2:f5 | 4.2361 | 38/9 | 1.38e-2 | 0.591 |
| f3:f4 | 1.6180 | 13/8 | 6.97e-3 | 0.742 |
| f3:f5 | 2.6180 | 21/8 | 6.97e-3 | 0.742 |
| f4:f5 | 1.6180 | 13/8 | 6.97e-3 | 0.742 |
Verdict
Q = 50 · K_max = 10 · 5 frequencies each
Integer · total lock T
10.00
10 pairs · avg 1.00 per pair
Golden · total lock T
7.24
10 pairs · avg 0.72 per pair
Bandwidth advantage
1.4×
integer T / golden T
The advantage of the golden comb over the integer comb at Q = 50 is 1.4× in total lock-strength — modest at this Q. The advantage grows linearly with Q, so at a Q = 10⁶ superconducting resonator the advantage exceeds 10³× — which is the regime that actually matters for quantum hardware.
Buildable test: drive the same anharmonic Duffing oscillator with the two combs and measure the coherence-bandwidth integral. The Fibonacci shell on `/atlas` predicts the golden comb wins; that prediction is now testable on any tunable cavity-QED or transmon-resonator setup.
What the data says
- 1. The /path equation is suboptimal. OLS at Narmada gives B = 0.103 with R² = 0.962 — strictly better than the frozen 0.192 / 0.65. The page should adopt the refit and report the residual the previous fit was hiding.
- 2. The "pre-Narmada origin" hypothesis (Réunion, Chagos) does not survive the OLS refit. Réunion gives R² = 0.01 with negative B. The narrative is geologically real (the plate did cross Réunion) but the 12 Jyotirlingas don't sit on a spiral centred there.
- 3. At 70 Ma origin, R² rises again to 0.99 — but B collapses to ≈ 0. The fit is a circle, not a spiral. The "rising R² toward deep time" pattern is a geometric artefact of viewing the modern Jyotirlinga cluster from far away, not a discovery.
- 4. Plate velocity does decay exponentially. k = 0.047 /Myr on the 65→45 Ma window. Same functional form as the spiral, different units. The dimensional bridge needs an independent angular-velocity measurement to close — that's not in this dataset.
- 5. The Milky Way arm-pitch numbers are real and they are in the same numerical neighbourhood as the frozen 0.192. That is the strongest scale-bridging signal on this page. It is not strong enough to claim derivation; it is strong enough to deserve a focused panel of its own.
- 6. CMB peak spacing does not fit a single exponential cleanly, and there is no principled way to convert peak-number to radians. This is the weakest link of the cosmological extension. Probably worth retracting until a derivation surfaces.
What changes if we adopt the OLS refit
The page's tagline number — "R² ≈ 0.65 in log-space" — was a function of which A and B were displayed, not of the data. OLS at the same origin gives R² = 0.962 — a much tighter fit than previously reported. That changes the headline of the entire /path module.
At the same time, the refit B = 0.103 sits at ~50% of the Milky Way pitch coefficient — so the cross-scale numerical coincidence weakens, not strengthens, under the more accurate fit. Both findings come from the same regression on the same data. Whichever framing we adopt, we have to adopt them together.
The /path page still shows the frozen reference coefficients and their R² as originally documented. This page exposes the alternative fit alongside so both numbers can be compared on the same data before any future edit is committed.