README.md

Latent Ambiguity Experiment

Purpose

This experiment asks a narrower question than the previous inverse artifacts.

The pose-free weighted anisotropic inverse and the alpha-refinement pilot already showed that:

• geometry and weights remain operational under unknown rotation plus unknown anisotropy
• alpha is the weakly identified part of the latent state
• better local fit does not uniformly translate into better alpha

The next question is not whether the inverse still works at all.

It is:

what kind of ambiguity is hiding inside the inverse once rotation is unknown, and does that ambiguity broaden mostly in alpha, mostly in geometry, or across the whole latent object?

This experiment targets that question directly.

Research Question

If we compare matched canonical-pose and pose-free observations of the same true latent state, does hiding rotation broaden the near-optimal latent family, and is that broadening concentrated in alpha rather than in geometry or weights?

Pre-Benchmark Logic Audit

Before running the benchmark, the script was checked in three ways.

First, the script compiled cleanly.

Second, the matched-observation construction was tested directly. The canonical and pose-free observations were built from the same true latent state and the same relative mask-plus-noise pattern on the underlying boundary, with only the pose nuisance changed.

Third, the scoring logic was checked with a forced-shift identity test. When the pose-free scorer is forced to use the true observation shift rather than minimizing over all cyclic shifts, its per-candidate scores agree with the canonical scorer down to floating-point noise:

• forced-shift max score difference: 8.33e-17

That matters because it means the benchmark is comparing matched inference conditions rather than silently mixing in a different observation pattern.

The experiment script is run_latent_ambiguity_experiment.py.

Method

The setup uses the same anisotropic weighted three-source forward family as the recent inverse artifacts.

For each trial:

1. sample one true latent state
2. generate one clean radial-signature boundary
3. create a matched pair of observations from that same boundary:
   • one canonical observation
   • one pose-free observation with unknown cyclic rotation
4. use the same relative observation regime on both views
5. score the same anisotropy-aware reference bank against both observations
6. measure not only the best candidate, but the top-10 near-optimal candidate envelope

The ambiguity metrics are:

• best-candidate alpha error
• top-10 alpha span
• top-10 geometry dispersion
• top-10 weight dispersion
• score-gap width of the top-10 envelope
• a near-tie-and-alpha-diverse flag for cases where the top-10 family stays both close in score and broad in alpha

This is still a bank-based ambiguity profile, not a continuous posterior.

Parameter Sweep

Reference bank:

• anisotropy-aware bank size: 300

Test set:

• 40 matched trials per observation regime

Envelope:

• top-10 candidates per trial

Near-tie rule:

• a trial is marked near-tie-and-alpha-diverse when the top-10 score spread is at most max(noise_sigma^2, 5e-5) and the top-10 alpha span is at least 0.20

Observation regimes:

• full_clean
• full_noisy
• partial_arc_noisy
• sparse_full_noisy
• sparse_partial_high_noise

Main Result

The result is sharp.

Hiding rotation does not broaden the near-optimal latent family uniformly. It broadens the alpha envelope dramatically while leaving geometry dispersion nearly unchanged and weight dispersion only modestly larger.

The summary file is latent_ambiguity_summary.json.

Global summary:

• bank size: 300
• trials per regime: 40
• envelope size: 10
• smallest mean pose-over-canonical alpha span ratio: 3.7419
• largest mean pose-over-canonical alpha span ratio: 5.5697
• smallest mean pose-over-canonical best-alpha error ratio: 11.3755
• largest mean pose-over-canonical best-alpha error ratio: 30.9833
• largest increase in near-tie-and-alpha-diverse fraction: 0.725

These ratios are much larger than the corresponding geometry and weight broadening factors:

• geometry-dispersion ratio range: 0.9286 to 1.0054
• weight-dispersion ratio range: 1.0891 to 1.1631

That is the core result.

By Regime

• full_clean

  • mean top-10 alpha span: 0.1231 -> 0.5922
  • mean best-candidate alpha error: 0.0158 -> 0.2500
  • mean top-10 geometry dispersion: 0.0711 -> 0.0681
  • mean top-10 weight dispersion: 0.1472 -> 0.1650
  • near-tie-and-alpha-diverse fraction: 0.000 -> 0.000

• full_noisy

  • mean top-10 alpha span: 0.1327 -> 0.5275
  • mean best-candidate alpha error: 0.0199 -> 0.2647
  • mean top-10 geometry dispersion: 0.0738 -> 0.0676
  • mean top-10 weight dispersion: 0.1556 -> 0.1723
  • near-tie-and-alpha-diverse fraction: 0.000 -> 0.000

• partial_arc_noisy

  • mean top-10 alpha span: 0.1572 -> 0.5220
  • mean best-candidate alpha error: 0.0316 -> 0.1294
  • mean top-10 geometry dispersion: 0.0724 -> 0.0670
  • mean top-10 weight dispersion: 0.1580 -> 0.1715
  • near-tie-and-alpha-diverse fraction: 0.000 -> 0.500

• sparse_full_noisy

  • mean top-10 alpha span: 0.1323 -> 0.5527
  • mean best-candidate alpha error: 0.0249 -> 0.2734
  • mean top-10 geometry dispersion: 0.0746 -> 0.0685
  • mean top-10 weight dispersion: 0.1527 -> 0.1668
  • near-tie-and-alpha-diverse fraction: 0.000 -> 0.625

• sparse_partial_high_noise

  • mean top-10 alpha span: 0.1407 -> 0.5410
  • mean best-candidate alpha error: 0.0384 -> 0.2871
  • mean top-10 geometry dispersion: 0.0753 -> 0.0737
  • mean top-10 weight dispersion: 0.1544 -> 0.1641
  • near-tie-and-alpha-diverse fraction: 0.125 -> 0.850

Two features stand out:

• the alpha envelope expands by about fourfold to sixfold in every regime
• the hardest partial and sparse regimes frequently produce genuinely near-tied candidate families that differ materially in alpha

Interpretation

This result sharpens the earlier pose-free anisotropic findings.

The key issue is not just that alpha gets noisier.

It is that hidden rotation creates a materially broader family of near-optimal latent explanations, and that broadening concentrates much more in alpha than in geometry.

That is why local alpha refinement helped fit but did not rescue alpha uniformly:

• the inverse was often navigating a real ambiguity envelope
• not just a slightly too-coarse alpha grid

In plain language:

• the shape is still telling us a lot about normalized geometry
• it is telling us somewhat less about weights
• but once rotation is hidden, it can support several materially different anisotropy values that all score nearly as well

That is a more precise diagnosis of the solver challenge than the earlier inverse artifacts alone.

What This Establishes

This experiment does show:

• matched evidence that the pose nuisance broadens the latent candidate family
• that the broadening is strongly concentrated in alpha
• that geometry dispersion stays comparatively stable under the same nuisance
• that the hardest regimes often contain near-tied but alpha-diverse candidate families

This experiment does not address:

• whether that alpha ambiguity is fundamentally irreducible
• whether a better representation or alignment scheme can collapse the alpha envelope again
• how much of the remaining ambiguity comes from pose-anisotropy aliasing versus finite-bank discretization

Figures

• latent_ambiguity_overview.png
• latent_ambiguity_spectra.png

The clearest figure is latent_ambiguity_overview.png, because it shows the asymmetry directly:

• alpha broadens a lot
• geometry barely broadens
• near-tie diverse families emerge mainly under pose-free partial and sparse observation

Artifacts

Data:

• latent_ambiguity_trials.csv
• latent_ambiguity_summary.csv
• latent_ambiguity_summary.json

Code:

• run_latent_ambiguity_experiment.py

Recommended Next Step

The highest-leverage next step is not a larger alpha sweep.

This experiment already shows that the pose-free anisotropic solver challenge is an ambiguity-structure problem.

The next experiment should try to reduce that ambiguity structure directly, for example by:

• a pose-alignment representation that separates cyclic orientation from anisotropic deformation more cleanly
• an uncertainty or multimodality map over (geometry, weights, alpha, pose)
• a representation that exposes anisotropy with rotationally stable features rather than relying only on the current radial-signature encoding