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Representation Independence Experiment

Purpose

This experiment tests one of the highest-priority theory-hardening questions:

does the main inferential Shape Budget result survive a meaningful change in the boundary representation, or is too much of the experimental record tied to the centroid-normalized radial-signature pipeline?

The alternative representation used here is a centroid-normalized support-function profile.

The forward family stays the same.

Only the boundary encoding changes.

That makes this a clean test of representation dependence rather than a different geometry problem.

The experiment script is run_representation_independence_experiment.py.

Pre-Benchmark Logic Audit

Before the benchmark, the new script was checked explicitly.

Code sanity:

  • script compiled cleanly

Support-encoding identity audit:

  • audit cases: 30
  • canonical exact recovery fraction: 1.0
  • pose-free exact recovery fraction: 1.0
  • max canonical fit RMSE: 0.0
  • max pose-free fit RMSE: 0.0

That matters because the support profile is only useful as a comparison if it behaves cleanly under the same bank machinery. It does.

Method

The experiment compares two encodings on matched trials:

  • radial
  • support

For each encoding, it runs the same inverse structure in two settings:

  • canonical
  • pose_free

And in each setting it compares:

  • anisotropy-aware bank
  • Euclidean baseline bank

So the experiment is testing three things at once:

  1. can the alternative representation still recover the latent control object?
  2. does the anisotropy-aware advantage survive?
  3. does the selective pose penalty on alpha survive?

Boundary encodings

Radial:

  • centroid-centered radius as a function of angle
  • normalized by mean radius

Support:

  • centroid-centered support value as a function of angle
  • normalized by mean support

Benchmark scale

  • anisotropic bank size: 220
  • Euclidean baseline bank size: 120
  • trials per regime: 20
  • observation regimes: all 5 standard regimes

Main Result

The result is strong.

The core inferential BGP result survives the representation swap. Under the support encoding, canonical recovery remains useful, the anisotropy-aware inverse still decisively beats the Euclidean baseline, and pose-free observation still degrades alpha much more than geometry. The hard pose-free alpha recovery challenge therefore does not look like a radial-signature artifact, even though its severity is representation-sensitive in some sparse regimes.

This is exactly the kind of result that hardens the theory while narrowing the solver challenge.

The summary file is representation_independence_summary.json.

Canonical Pose: Core Recovery Survives

Under the support encoding, canonical recovery stays in the same general quality band as the radial encoding.

Examples:

  • full_clean

    • radial alpha: 0.0202
    • support alpha: 0.0187
    • radial fit improvement over Euclidean: 14.13x
    • support fit improvement over Euclidean: 14.48x

  • partial_arc_noisy

    • radial alpha: 0.0315
    • support alpha: 0.0343
    • radial fit improvement: 9.34x
    • support fit improvement: 10.14x

  • sparse_partial_high_noise

    • radial alpha: 0.0729
    • support alpha: 0.0625
    • radial fit improvement: 4.25x
    • support fit improvement: 5.50x

That is already enough to say:

the operational latent-variable result is not confined to the radial signature.

Pose-Free Observation: Selective Alpha Penalty Survives

This is the most important theory-hardening result.

Under the support encoding, the pose-free penalty still lands much more on alpha than on geometry.

Support-encoding pose selectivity:

  • best alpha-over-geometry selectivity: 4.88x
  • worst alpha-over-geometry selectivity: 11.96x

That means:

even after swapping encodings, hidden pose still hurts alpha far more than geometry.

Examples:

  • full_clean

    • support geometry penalty: 0.931
    • support alpha penalty: 10.959
    • selectivity: 11.77x

  • partial_arc_noisy

    • support geometry penalty: 1.004
    • support alpha penalty: 5.556
    • selectivity: 5.53x

  • sparse_partial_high_noise

    • support geometry penalty: 0.816
    • support alpha penalty: 3.978
    • selectivity: 4.88x

So the selective alpha recovery challenge is not just a quirk of the radial representation.

What Changed With The Representation Swap

The support encoding does not reproduce every number.

It changes the severity of the hard pose-free alpha penalty in some sparse regimes.

The most visible differences are:

  • sparse_full_noisy, pose-free

    • radial alpha: 0.1650
    • support alpha: 0.2461
    • support over radial alpha ratio: 1.49

  • sparse_partial_high_noise, pose-free

    • radial alpha: 0.1307
    • support alpha: 0.2486
    • support over radial alpha ratio: 1.90

So the solver challenge magnitude is representation-sensitive.

That matters.

But the key point is what did not disappear:

  • the anisotropy-aware baseline advantage
  • useful canonical recovery
  • the selective pose penalty on alpha

That is why this experiment strengthens BGP even though it does not “solve” the solver challenge.

Interpretation

This experiment changes the diagnosis in a very useful way.

What it establishes:

  • the core BGP inferential result is representation-robust across at least two genuinely different encodings
  • the pose-free alpha recovery challenge is not just a radial-signature artifact
  • the solver challenge magnitude still depends on representation, so practical inference design remains important

What it does not support:

  • a claim that the current solver challenge has nothing to do with representation

The strongest reading is:

BGP itself now looks substantially more representation-independent than before, while the hard pose-free anisotropic solver challenge looks partly representation-sensitive in magnitude but not in kind.

That is a strong theory-hardening result.

Why This Matters For Priority

This experiment answers the triage question directly.

The current solver challenge does not look like a general threat to BGP.

Why:

  • the core latent-variable and baseline-improvement result survived the representation swap
  • the selective alpha penalty also survived the swap
  • only the difficulty level of the hardest pose-free sparse cases changed materially

So the solver challenge still matters, but it now looks more like:

  • an inference-design problem
  • with representation-sensitive severity
  • rather than a theory-level failure of BGP

That means it is reasonable to move higher-value theory-hardening work ahead of solver challenge cleanup for a while.

Figures

Key figures:

The first figure shows that alpha recovery remains in the same broad quality band under both encodings in canonical pose, while both encodings suffer a larger alpha hit in pose-free mode.

The second figure is the main theory figure:

  • left: anisotropy-aware improvement over the Euclidean baseline survives the swap
  • right: pose-free alpha remains much more fragile than geometry under both encodings