Cross-scale relationships: challenge one exponent

What is measured?

The observations are positive dimensionless x and y ratios, each tied to a stated physical reference before empirical use. The comparison measures prediction error in log(y) on untouched test rows. It estimates a relationship between variables; it does not estimate a probability-distribution tail or infer fractal dimension from an image.

The baseline

The scale-break baseline adds a continuous hinge at a predeclared x ratio. In the demonstration the break is twenty, fixed before fitting. The single power law has two coefficients and the baseline has three. Both see the same training rows; neither is fitted on test rows. The extra flexibility must earn improved held-out prediction.

Protocol

1. Prerequisites: Python 3.9 or later runs the demonstration without additional software. Empirical work needs a reusable dataset with defensibly comparable observables, reference scales, sampling rules and uncertainties. No physical apparatus is required here. A shared visual motif or resizing two pictures does not supply the needed transformation between systems.
2. Run the bundled analysis on thirty-seven generated observations spanning x ratios from one to roughly 148.4. Twenty-eight rows are training data and nine are held out. The generator uses a lower exponent of 0.60, upper exponent of 1.15 and independent Gaussian log noise of standard deviation 0.04.
3. Before examining decisive data, define eligible systems, the measured scale range, reference quantities, transformations, alternatives and split. Fix any breakpoint from independent reasoning or training-only procedures with appropriate validation. The supplied breakpoint is a synthetic design choice. It must not be transferred to unrelated physical datasets as a discovered constant.
4. Fit only the training rows. For a claim about new systems, hold out whole systems; for temporal prediction, use a chronological split. The demonstration uses interleaved points and therefore tests interpolation across a range. Preserve sample boundaries and correlations rather than assuming that every point extracted from one object supplies independent evidence.
5. Compare predictions on untouched observations, inspect residuals and report all planned alternatives. If test performance informs a model revision, obtain new confirmation data. Publish the full finite range, including transitions. A scale break can reveal where an approximation fails and is a valuable research outcome even when it weakens a hoped-for universal relationship.

Analysis

Weighted least squares fits log(y)=a+b log(x) and a continuous hinge model with an additional coefficient. A small Gaussian-elimination solver uses only the standard library. Output includes coefficients, training and held-out log RMSE, and the mean paired squared-error difference. Positive difference favors the break baseline on that sample. Actual synthetic held-out log RMSE is approximately 0.191434 for the single law and 0.021155 for the break, demonstrating a failure of one exponent across the generated range.

Uncertainty

The calculation treats x as exact and assumes positive ratios with independent log-scale response noise. The standard error of paired test-error differences is conditional on the trained fits and independent held-out rows. It omits training variation, correlated measurements, uncertainty in reference scales and selection across many models. The package reports no p-value or automatic model-selection verdict.

Limits

• The supplied numbers are generated, not astronomical, atomic or biological measurements. Their two exponents have no claimed universal physical significance.
• The toy comparison uses one independently fixed break. Searching many breaks or datasets after seeing test scores requires fresh validation and cannot reuse the same interpretation.
• This is input-output regression, not the likelihood method for a power-law distribution tail. The distinction matters for fitting, normalization and uncertainty.
• A common fitted exponent would still not show identical dynamics, an unlimited fractal hierarchy, parent universes or an observable connection to Alpha 1.

Inspect the package

{
  "comparison_note": "Positive MSE difference favors the prespecified break baseline on this held-out sample. No p-value, universal law, or formal model-selection verdict is inferred.",
  "conditional_standard_error_of_mse_difference": 0.011270251,
  "dataset_kind": [
    "synthetic_demo"
  ],
  "held_out_log_mse_difference_power_minus_break": 0.036199583,
  "held_out_observations": 9,
  "interpretation": "Synthetic scale-break demonstration; no measured universal scaling finding.",
  "power_law": {
    "amplitude": 1.31905216,
    "exponent": 0.797268799,
    "held_out_log_rmse": 0.191434356,
    "parameter_count": 2,
    "training_log_rmse": 0.211525618
  },
  "prespecified_break_x_ratio": 20.0,
  "project": "cross-scale-relationships",
  "scale_break_baseline": {
    "amplitude_below_break": 1.731423165,
    "exponent_above_break": 1.167141234,
    "exponent_below_break": 0.590332879,
    "held_out_log_rmse": 0.021154893,
    "parameter_count": 3,
    "training_log_rmse": 0.057158822
  },
  "synthetic_demo": true,
  "training_observations": 28,
  "uncertainty_assumption": "Positive dimensionless ratios; independent log-scale observation noise with supplied sigma; x treated as exact; standard error is conditional on trained fits and independent test rows."
}

A useful next contribution

Propose a justified dimensionless map between two measured systems and a third system to hold out, or supply an openly reusable dataset with known observation limits. Include a conventional alternative and a prediction for where the shared relation should change. Counterexamples with complete provenance are especially useful.

Source trail