README.md

Manifold-Dimension Experiment

Purpose

This experiment asks whether the symmetric Shape Budget family is only one-dimensional in a few chosen observables, or whether the full normalized boundary family itself forms a one-dimensional object in shape space.

That distinction matters.

If the family is genuinely one-dimensional in a natural boundary representation, then e = c/a is not only organizing widths, perimeters, and curvatures. It is organizing the entire normalized shape family.

Research Question

Do normalized boundaries generated by the symmetric constant-sum two-source process lie on a one-dimensional manifold in boundary space?

Representation Choice

To keep the test honest, this experiment does not use the original process parameter as the shape vector.

Instead, each normalized boundary is converted into a radial signature on a fixed polar grid:

[ \rho_e(\theta) ]

sampled at 360 evenly spaced angles.

This matters because it tests one-dimensionality in a natural shape representation rather than in the same coordinate system that generated the curve.

Experiment Design

The experiment script is run_manifold_dimension_experiment.py.

It performs three linked tests:

1. PCA spectrum Fit PCA to the family of normalized radial signatures and measure how much variance is captured by the first few components.

2. Nonlinear 1D embedding Fit a one-dimensional Isomap embedding and test whether its coordinate is monotonic in e.

3. Scale-overlay check Project multiple absolute scales into the same embedding space and measure whether same-e shapes collapse relative to the spacing between neighboring e values.

Parameter Sweep

Unique manifold family

• e values: 193 values from 0.02 to 0.98
• fixed scale: a = 1
• radial-signature resolution: 360 angles
• process samples per boundary: 800

Scale-overlay family

• e values: 37 values from 0.05 to 0.95
• scales a: 0.75, 1.0, 1.5, 2.5, 4.0

Main Result

The cleanest reading is:

In radial-signature space, the symmetric normalized Shape Budget family forms a one-dimensional curved manifold.

That wording is important.

The family is not best described here as a one-dimensional linear subspace, because one principal component does not reconstruct the entire family perfectly. But it is very strongly one-dimensional in the manifold sense.

The summary file is manifold_summary.json.

Spectrum results

• first principal component explained variance ratio: 9.8908e-01
• first two principal components cumulative ratio: 9.9976e-01
• first three principal components cumulative ratio: 9.9999e-01

Reconstruction results

Relative reconstruction error from PCA:

• 1 component mean: 1.6545e-02 max: 1.9712e-01

• 2 components mean: 2.4512e-03 max: 3.4534e-02

• 3 components mean: 4.0339e-04 max: 5.6501e-03

• 5 components mean: 1.2794e-05 max: 1.2229e-04

This is exactly what a thin curved one-dimensional family should look like:

• one dominant direction
• small but nonzero curvature away from a straight line
• near-perfect reconstruction once a second component is allowed

Nonlinear ordering result

The one-dimensional Isomap coordinate had absolute Spearman correlation

• |rho| = 1.0

with e over the dense sweep.

That means the nonlinear 1D embedding recovered the family ordering perfectly.

Scale-overlay result

In the PCA embedding plane:

• maximum within-e across-scale spread: 1.5221e-12
• mean within-e across-scale spread: 2.5540e-13
• smallest gap between neighboring e means: 1.8170e-02

So same-e shapes at different scales are effectively indistinguishable in embedding space, while neighboring e values remain cleanly separated.

Interpretation

This is a strong hardening result for the control-knob idea.

It says that in a natural boundary encoding:

• the normalized family is almost entirely organized by one dominant direction
• the residual complexity is small and curved rather than chaotic
• a nonlinear one-dimensional coordinate recovers the family ordering exactly
• scale collapse remains intact even in the embedding space

That establishes:

In the symmetric Euclidean constant-sum setting, e = c/a does not only organize chosen scalar observables. It organizes the full normalized boundary family as a one-dimensional curved manifold.

What This Changes

Before this experiment, the established statement was:

e is sufficient for normalized observables and recoverable from noisy data.

After this experiment, the stronger statement is:

e is the intrinsic ordering coordinate of the normalized symmetric family, at least in this natural radial-signature representation.

That is a meaningful upgrade.

Scope Of The Result

This experiment does show:

• strong one-dimensional organization in a natural boundary representation
• exact monotonic recovery of the family ordering under a nonlinear 1D embedding
• scale collapse in the embedding itself

This experiment does not address:

• that every possible shape encoding will yield the same quantitative spectrum
• that the same one-dimensional manifold result survives asymmetry, anisotropy, or multi-source settings
• that the family is a perfectly linear one-dimensional subspace

The correct reading is “one-dimensional curved manifold,” not “perfect straight line.”

Figures

• manifold_embedding.png
• manifold_spectrum.png
• manifold_isomap_and_reconstructions.png

The clearest figures are:

• manifold_embedding.png for the thin-curve geometry and scale-collapse separation
• manifold_spectrum.png for the one-dominant-direction result

Artifacts

Data:

• manifold_spectrum.csv
• manifold_reconstruction_errors.csv
• manifold_embeddings.csv

Code:

• run_manifold_dimension_experiment.py

Recommended Next Step

The next hardening step should be edge-regime stability.

At this point the program has:

• a process reconstruction result
• a symmetric one-knob sufficiency result
• a structured asymmetry result
• an identifiability and baseline-usefulness result
• a boundary-space manifold result

The next clean question is how stable or unstable the control-knob consequences become as e -> 0 and especially as e -> 1.