README.md

Controlled Anisotropy Experiment

Purpose

This experiment asks whether the Shape Budget control-knob result survives a simple directional warp of distance.

Instead of standard Euclidean distance, it uses the axis-aligned quadratic metric

[ d_{\alpha}\big((x, y), (u, v)\big) = \sqrt{(x-u)^2 + \alpha^2 (y-v)^2} ]

with the two sources still placed on the x-axis.

This is a controlled anisotropy test, not the fully general warped-medium case.

Research Question

When distance is directionally stretched by one anisotropy parameter alpha, does the Shape Budget family stay low-dimensional, and can the original one-knob collapse be recovered by whitening?

Why This Setup Matters

The key feature of this metric is that it is exactly whitenable:

[ (x, y) \mapsto (x, \alpha y) ]

Under that transform, the anisotropic constant-sum locus becomes the standard Euclidean ellipse again.

So this experiment is testing a precise claim:

• in raw Euclidean coordinates, the family should depend on both e and alpha
• after whitening, the family should collapse back to the original one-knob e result

That is exactly the kind of controlled universality result we want before moving to harder media.

Experiment Design

The experiment script is run_anisotropy_experiment.py.

It performs four linked tests:

1. anisotropic process reconstruction
2. raw one-knob failure versus whitening recovery
3. response curves over e for multiple alpha
4. two-parameter raw parameter map in (e, alpha)

Parameter Sweep

• e values: 17 values from 0.10 to 0.90
• alpha values: 0.50, 0.75, 1.00, 1.50, 2.00
• scales a: 0.75, 1.0, 1.5, 2.5, 4.0

Main Result

The result is clean and exactly interpretable:

Controlled anisotropy does not destroy the Shape Budget program. It promotes raw geometry from a one-parameter family to a two-parameter family (e, alpha), and whitening restores the original one-knob collapse.

The summary file is anisotropy_summary.json.

Key numerical results:

• maximum anisotropic-equation residual: 6.2172e-15
• maximum RMS residual: 1.2391e-15
• maximum raw scale-collapse error at fixed (e, alpha): 7.9473e-08
• mean raw scale-collapse error: 1.9925e-11
• minimum raw family distance at fixed e across varying alpha: 5.6938e-02
• maximum raw family distance: 1.4925
• maximum whitened collapse error across varying alpha and scale at fixed e: 3.9736e-08
• mean whitened collapse error: 1.5095e-11

Those numbers say exactly what the theory predicts:

• raw one-knob sufficiency fails when anisotropy varies
• raw geometry remains low-dimensional and stable under (e, alpha)
• whitening recovers the original one-knob family to numerical precision

Interpretation

This is a strong controlled extension.

It means:

• e is still the right control variable once distance is expressed in the whitened coordinates
• the extra anisotropy parameter is a geometric warp descriptor, not uncontrolled variation
• the Shape Budget principle survives a simple directional distortion in a structured way

The raw geometry is no longer organized by e alone.

For example:

• raw vertical residue becomes sqrt(1 - e^2) / alpha
• raw major-tip response becomes alpha^2 / (1 - e^2)

So in raw Euclidean coordinates, anisotropy multiplies or divides the geometric consequences of e.

After whitening, those extra alpha factors disappear and the original e-only collapse returns.

That is a meaningful universality result.

What This Changes

Before this experiment, the strongest extension result was:

• asymmetry upgrades the family from one parameter to two
• the hyperbola twin gives a second one-knob family

After this experiment, there is now a controlled warped-distance result:

• one extra directional descriptor is enough in raw space
• whitening restores the original one-knob Euclidean result exactly

That is much better than either of the two simpler outcomes:

• “anisotropy destroys the idea”
• “anisotropy changes nothing”

Scope Of The Result

This experiment does show:

• exact process reconstruction under one quadratic anisotropic metric
• clean raw organization by (e, alpha)
• exact recovery of the e-only family after whitening

This experiment does not address:

• that arbitrary anisotropic media behave the same way
• that non-quadratic directional costs reduce to one extra descriptor
• that the same recovery result survives when the sources are not aligned with the principal anisotropy axis

So this is a controlled universality test, not the final word on warped spaces.

Figures

• anisotropy_process_reconstruction.png
• anisotropy_whitening_recovery.png
• anisotropy_response_curves.png
• anisotropy_parameter_map.png

The clearest figures are:

• anisotropy_whitening_recovery.png for the raw-failure / whitening-recovery result
• anisotropy_response_curves.png for the exact way alpha enters the raw observables

Artifacts

Data:

• anisotropy_metrics.csv
• anisotropy_scale_collapse.csv
• anisotropy_raw_family_distances.csv
• anisotropy_whitened_collapse.csv

Code:

• run_anisotropy_experiment.py

Recommended Next Step

The next natural step is multi-source generalization.

At this point the project has:

• a symmetric one-knob closed family
• a symmetric one-knob open twin
• a structured asymmetry result
• a structured anisotropy result

The next clean question is whether the two-source control-knob idea lifts into a low-dimensional allocation simplex once three or more centers share the budget.