Retention Law (S2)

Universal, confirmed: the same stretched-exponential loss of contrast and correlation with probe scale across disparate domains.

Formal statement

Retention quantifies how much information or structure survives coarse-graining to probe scale λ (native units).

R(\lambda)\;=\;\exp\!\left[-\left(\frac{\lambda}{\lambda_q}\right)^{D_{\mathrm{eff}}}\right]

Semigroup → Weibull (gist)

If instrument blurs compose approximately by scale addition, $K_\lambda\circ K_\mu\!\approx\!K_{\lambda+\mu}$, and survival of features multiplies under composition, then $R(\lambda+\mu)\!\approx\!R(\lambda)R(\mu)$. Log-convex, scale-stable decay with a finite cliff uniquely selects the stretched-exponential (Weibull) family.

Linearization & kernel invariants

Plotting the standard transform exposes slope and intercept that are invariant for a given kernel class.

y(\lambda):=\ln\!\big(-\ln R(\lambda)\big) \;=\;D_{\mathrm{eff}}\ln\lambda\;-\;D_{\mathrm{eff}}\ln\lambda_q

\widehat D_{\mathrm{eff}}=\frac{d\,y}{d\ln\lambda},\qquad \widehat\lambda_q=\exp\!\Big(-\tfrac{\text{intercept}}{\widehat D_{\mathrm{eff}}}\Big)

After rescaling $ \lambda\to\lambda/\widehat\lambda_q $, disparate domains collapse to a common master curve.

Operational observables

Any admissible low-pass smooth_λ tied to instrument resolution is acceptable; bandwidth must grow monotonically with λ.

R_{\mathrm{corr}^2}(\lambda) \;=\; \mathrm{corr}^2\!\big(\,\text{data}\big|_{\lambda\to0},\;\text{data}\big|_{\lambda}\big)

R_{\mathrm{HF}}(\lambda) \;=\; 1-\frac{\mathrm{Var}\!\big(\text{data}-\mathrm{smooth}_\lambda(\text{data})\big)}{\mathrm{Var}(\text{data})}

Spectral calibration

\mathrm{smooth}_\lambda = K_\lambda * \text{data},\qquad H_\lambda(f)=\mathcal{F}\{K_\lambda\}

R_{\mathrm{HF}}(\lambda)= \frac{\int |H_\lambda(f)|^2 S(f)\,df}{\int S(f)\,df}, \quad \sigma_{\text{gauss}}=\frac{\mathrm{FWHM}}{2\sqrt{2\ln2}}, \quad \sigma_{\text{box,eq}}=\frac{w}{\sqrt{12}}

Scale-hazard & diagnostics

The “hazard” of information loss per unit scale is power-law under S2.

h(\lambda):=-\frac{d\ln R}{d\lambda} \;=\; \frac{D_{\mathrm{eff}}}{\lambda_q^{D_{\mathrm{eff}}}}\, \lambda^{D_{\mathrm{eff}}-1}

\beta(\lambda)=\frac{d}{d\ln\lambda}\,y(\lambda),\quad b(\lambda)=y(\lambda)-\beta(\lambda)\ln\lambda,\quad \lambda_q(\lambda)=\exp\!\big(-b(\lambda)/\beta(\lambda)\big)

S2 window: flat $ \beta(\lambda) $ and stable implied $ \lambda_q(\lambda) $; curvature flags regime changes or artifacts.

Estimation recipes

Log-spaced grid for λ in native units.
Compute $R_{\mathrm{HF}}$ and/or $R_{\mathrm{corr}^2}$.
Robust fit of $y=\ln(-\ln R)$ vs $\ln\lambda$.

Percentile-pair (no regression)

R(\lambda_1)=e^{-1},\; R(\lambda_2)=e^{-\alpha} \Rightarrow \widehat D_{\mathrm{eff}}=\frac{\ln(1/\alpha)}{\ln(\lambda_1/\lambda_2)},\quad \widehat\lambda_q=\lambda_1

Report $ \widehat D_{\mathrm{eff}}, \widehat\lambda_q $, CIs, λ-window, and smoother family.

Universality clause

After rescaling $ \lambda\to\lambda/\lambda_q $, curves from disparate domains align without re-fit. The law is insensitive to the low-pass family (box/gauss/PSF) once bandwidth is calibrated.

Falsification (technical)

Persistent curvature of $y(\lambda)$ after unit/smoother calibration.
Irreconcilable $D_{\mathrm{eff}}$ from $R_{\mathrm{HF}}$ vs $R_{\mathrm{corr}^2}$ within the same context.
Failure of collapse under $ \lambda/\lambda_q $ across platforms.

Universality demo (official live datasets)

Fit S(W) ≈ exp(−(W/λ)^β). Toggle x → x/λ to align cliffs at x=1 (λ always reported in native units).

What was tested

Sun & Space Weather

GOES X-rays, magnetometer, electrons ≥2 MeV, protons ≥10 MeV.

Earth Systems

USGS Potomac flow (IV), earthquakes (all_week hourly).

Society & Economy

Wikimedia enwiki “Earth”, World Bank GDP/CPI/Unemployment, CoinGecko BTC daily, iNaturalist observations.

Cross-domain collapse procedure

Express λ in native units (pixel, cadence, bin size, PSF FWHM).
Fit $y=\ln(-\ln R)$ vs $\ln\lambda$; extract $\widehat D_{\mathrm{eff}}, \widehat\lambda_q$.
Replot all $R$ curves vs $ \lambda/\widehat\lambda_q $ and verify alignment.

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Projection model (10→4)

4D observables are push-forwards of Meta-Manifold fields through a single, fixed, non-invertible projection kernel with finite resolution. The kernel compresses hidden 10D structure into our 4D data, averaging sub-resolution detail; only its invariants are measurable from within 4D.

\phi(x)\;=\;\int_{\mathcal{M}_{10}} K_{\lambda}(x;X)\,\Phi(X)\,d^{10}X, \qquad K_{\lambda}\!\circ\! K_{\mu}\;\approx\;K_{\lambda+\mu}\;.

Because blur scales compose approximately by addition and survival under composition multiplies, fine structure decays in a universal, stretched-exponential manner (S2). In practice the standard transform of retention, $y=\ln(-\ln R)$ vs $\ln \lambda$, exposes a slope and intercept that act as kernel invariants across observables and datasets.

What to report & how to calibrate

Invariants (operational): coherence scale $ \lambda_q $ (“cliff”), effective exponent $ D_{\mathrm{eff}} $, and the native-unit locality radius set by your instrument/analysis.
Units: always express $ \lambda $ in native instrument units (pixel, cadence, bin size, PSF). Typical bandwidth conversions: $\sigma_{\text{gauss}}=\frac{\mathrm{FWHM}}{2\sqrt{2\ln 2}}, \qquad \sigma_{\text{box,eq}}=\frac{w}{\sqrt{12}}.$
Pointer: full derivation and geometry live on the Kernel page.

Only invariants of the kernel class are empirical; the detailed microstructure of $K$ is not recoverable from 4D data.

Key consequences (summary)

Objective cliff: a measurable coherence scale λ_q exists across observables; it separates persistence from rapid erasure.
Universal decay law: the same stretched–exponential form governs both HF detail and structural correlation after unit calibration.
Bounded fractality: self-similar structure is a sub-λ_q phenomenon; large scales homogenize.
Irrecoverability: information strictly below λ_q cannot be restored by any algorithmic post-processing (physics, not software).
Cross-observable constraints: slopes D_eff from R_HF and R_corr² must agree within a calibrated context.
Design guidance: sampling/PSF should target λ \lesssim λ_q to preserve science yield; analysis should report the standard transform and λ-units.
Anomaly detection: failure of collapse under λ/λ_q flags pipeline/instrument issues or genuinely new physics.

Replication checklist

Declare native units and smoother family (with bandwidth calibration).
Show $y=\ln(-\ln R)$ vs $\ln\lambda$ with a flat-slope window and implied λ_q(λ) stability.
Provide collapse plot after rescaling by λ̂_q.

Fractal background