Geometry → resonance: predict the mode shift
Start with a relationship that can be calculated and challenged: an ideal one-dimensional cavity has frequencies fₙ=n v/(2L). This package supplies a reproducible simulation, an uncertainty-aware fit, and a blank measurement schema. It provides a local benchmark for the geometry program. Recovering this known relation would not establish that geometry came before matter or energy.
What is measured?
The observable is resonant frequency in hertz as a function of cavity length in metres and integer mode number. The main fitted parameter is propagation speed in metres per second. Every reading includes separate standard uncertainties for length and frequency. Boundary conditions and mode identity must be documented before interpreting a real dataset.
The baseline
The baseline is ordinary scalar-wave behavior with fixed boundary conditions and independently characterized propagation speed. The demonstration uses generating speed 343.2 m/s; that is a simulation input, not a measured constant supplied for an arbitrary apparatus. A separate free-intercept fit checks an offset-like discrepancy without automatically treating it as a new physical mechanism.
Protocol
- Prerequisites: Python 3.9 or later runs the package without installations, network access or equipment. For an empirical extension, obtain a calibrated resonance dataset or arrange a reviewed study with known boundary conditions, geometry, mode assignments and environmental records. The package does not assume that every physical cavity obeys the ideal model.
- Run analysis.py on synthetic-demo.csv and compare the output with expected-results.json. The generator uses twelve lengths, independent length noise of 0.00012 m and frequency noise of 0.35 Hz. These are explicitly selected demonstration assumptions. Regenerating with the fixed seed should reproduce the CSV and calculated metrics.
- Complete the preregistration template before inspecting decisive measurements. Specify the length range, mode-selection rule, independent speed reference, calibration uncertainties, meaningful residual, exclusions and stopping rule. Reserve new lengths or modes for prediction after calibration. Archive what was already examined so exploration and confirmation remain distinguishable.
- Copy blank-measurements.csv and enter independently documented observations. Randomize acquisition order, repeat separate runs, and preserve temperature, instrument settings and raw spectra outside the minimal schema. Include plausible boundary corrections, damping, dispersion and geometry errors in the physical model before attributing a disagreement to an unfamiliar effect.
- Fit the declared model, inspect residuals against length and run order, then predict the withheld observations. Report uncertainty and failed predictions. Share the dataset version, code output and calibration rationale so another researcher can test a different conventional explanation without reconstructing undocumented preprocessing or mode-selection decisions.
Analysis
The script fits frequency against x=n/(2L) through the origin. For one fixed mode, this is a frequency-versus-inverse-length regression. Twenty effective-variance iterations incorporate small independent length and frequency errors. It reports fitted speed, its conditional standard uncertainty, frequency RMSE, reduced chi-squared, maximum standardized residual and a free-intercept diagnostic. Expected results are captured from an actual run, rather than typed predictions. The synthetic fit returns about 343.131286 m/s and RMSE 0.425232 Hz.
Uncertainty
The fit assumes small Gaussian errors, known supplied standard uncertainties and no correlated calibration drift. Effective variance is an approximation, not a full errors-in-variables likelihood. The reported speed uncertainty, approximately 0.092128 m/s in the demonstration, excludes temperature, boundary-condition and material-model uncertainty. Those missing contributions must be included or bounded in a real study before a residual is physically interpreted.
What would count as success?
The computational milestone is regeneration of the data and expected metrics. A separate empirical milestone would be correctly predicting withheld mode shifts within a complete, predeclared uncertainty budget. An additional geometric mechanism would require its own advance prediction and an advantage over adequate conventional alternatives.
What would count against it?
The proposed extra mechanism is weakened if its predicted residual is absent at sufficient sensitivity, follows calibration or temperature drift, or disappears after appropriate boundary corrections. An imprecise measurement is inconclusive. A poor ideal-model fit alone cannot identify new physics.
Limits
- The model is one-dimensional, linear and idealized. Real resonators can require end corrections, losses, dispersion, material stress and higher-dimensional mode calculations.
- The supplied CSV is generated. It contains no laboratory readings and cannot establish any empirical support for the Hill framework.
- The free-intercept result is a diagnostic on the same data, not a held-out validation or an automatic model-selection verdict.
- A familiar geometry-dependent frequency shift is not evidence for universal fractality, geometry-first ontology, Alpha 1, additional energy or propulsion.
Inspect the package
Captured synthetic output
{
"conditional_standard_uncertainty_speed_m_s": 0.092128355,
"dataset_kind": [
"synthetic_demo"
],
"fitted_speed_m_s": 343.131285867,
"fixed_reference_rmse_hz": 0.432791858,
"free_intercept_conditional_standard_uncertainty_hz": 0.24127024,
"free_intercept_diagnostic_hz": -0.254423551,
"free_intercept_slope_m_s": 343.303794827,
"frequency_rmse_hz": 0.425231748,
"interpretation": "Synthetic benchmark only; no empirical finding.",
"max_abs_standardized_residual": 1.777945961,
"model": "f_n = n v / (2 L); fitted intercept fixed at zero",
"project": "geometry-resonance",
"reduced_chi_squared": 1.24534355,
"reference_speed_m_s": 343.2,
"sample_count": 12,
"synthetic_demo": true,
"uncertainty_assumption": "Small independent Gaussian errors; effective variance; supplied sigmas known; no correlated calibration, boundary correction, or temperature uncertainty."
}A useful next contribution
Contribute an openly reusable resonance dataset with independently known dimensions, boundary conditions and uncertainty records, or a correction to the model. The most useful next analysis predicts new modes or lengths before reading their measured frequencies and explains where the ideal approximation should fail.