This experiment asks how stable the control-knob consequences are near the two extreme regimes:
e -> 0, where the normalized family approaches a circlee -> 1, where the family approaches degeneration
The core question is not just “what do the observables equal there?” It is:
how much does a small change in
eactually matter near each edge, and how much uncertainty inewould be induced by small measurement error in an observable?
How well-conditioned are the main normalized Shape Budget observables near the circular edge and the degenerate edge?
The experiment tracks three distinct normalized observables:
- width residue
b/a = sqrt(1 - e^2) - exact normalized perimeter
P / (2 pi a) = (2 / pi) E(e)withEthe complete elliptic integral of the second kind with moduluse - normalized major-tip response
a kappa_major = 1 / (1 - e^2)
The normalized minor-tip response is not plotted separately because under the current normalization it exactly matches width residue:
[ a \kappa_{minor} = \frac{b}{a} ]
The experiment uses two complementary notions of stability:
- forward sensitivity
[ \kappa(e) = \left| \frac{e}{q(e)} \frac{dq}{de} \right| ]
This is a relative condition number. It measures how strongly a relative change in e is amplified into a relative change in the observable.
- inverse recoverability
For a fixed 1 percent relative measurement error in an observable, the experiment estimates the implied local uncertainty in e:
[ \delta e \approx 0.01 \left| \frac{q(e)}{dq/de} \right| ]
This makes the result directly operational.
The experiment script is run_edge_regime_stability_experiment.py.
It uses exact formulas and exact derivatives for all three observables, then evaluates them on a dense grid concentrated near both edges.
The main output is not a single score. It is a conditioning map.
The result has a very clean shape:
The circular edge is the poorly conditioned end for shape-only inference, while the degenerate edge splits observables into “very sensitive” and “moderately sensitive” families.
That is a stronger and more useful statement than a generic “things get unstable near the edge.”
Numerically and analytically, the circular end is locally insensitive.
The leading expansions are:
[ \frac{b}{a} = 1 - \frac{e^2}{2} + O(e^4) ]
[ \frac{P}{2 \pi a} = 1 - \frac{e^2}{4} + O(e^4) ]
[ a \kappa_{major} = 1 + e^2 + O(e^4) ]
So every one of these observables changes only at second order in e near zero.
That means small nonzero eccentricities are hard to resolve from shape alone in the near-circular regime.
With 1 percent relative measurement noise, the implied local uncertainty in e is:
-
at
e = 0.01- width residue:
9.9990e-01 - normalized perimeter:
1.9999 - major-tip response:
4.9995e-01
- width residue:
-
at
e = 0.1- width residue:
9.9000e-02 - normalized perimeter:
1.9875e-01 - major-tip response:
4.9500e-02
- width residue:
So the circular end is genuinely low-information if the measurement comes only from these geometric summaries.
At the degenerate edge, the result splits.
The width residue and major-tip response become sharply sensitive to e, while the normalized perimeter remains much more moderate.
With 1 percent relative measurement noise:
-
at
e = 0.99- width residue:
2.0101e-04 - normalized perimeter:
4.3734e-03 - major-tip response:
1.0051e-04
- width residue:
-
at
e = 0.999- width residue:
2.0010e-05 - normalized perimeter:
2.8726e-03 - major-tip response:
1.0005e-05
- width residue:
So perimeter remains the most edge-stable observable, while width and especially major-tip response become much sharper probes of very high eccentricity.
The relative-condition crossover kappa(e) = 1 occurs at:
- major-tip response:
e = 0.5773502692 - width residue:
e = 0.7071067812 - normalized perimeter:
e = 0.9089085575
This means perimeter stays relatively mild much farther into the high-e regime, while major-tip response becomes strongly sensitive earliest.
This experiment gives the control-knob result a conditioning map.
It says:
- the circular regime is not “easy” from the point of view of shape-only inference; it is actually the flattest regime
- the degenerate regime is not uniformly unstable; different observables split into different sensitivity classes
- perimeter is the smoothest high-
eobservable - major-tip response is the sharpest high-
eobservable - width residue sits between them and shares its exact conditioning with normalized minor-tip curvature
That establishes:
The control knob is globally meaningful, but the observables it governs are not equally informative everywhere. Near
e = 0, they are all weakly informative. Neare = 1, they separate into coarse and sharp probes.
Before this experiment, the project knew that e organized geometry and was recoverable in the known-source setting.
After this experiment, the project also knows where different geometric summaries are well-conditioned and poorly conditioned as measurements of e.
That is a practical upgrade, not just a conceptual one.
This experiment does show:
- exact conditioning behavior for the symmetric Euclidean normalized observables tested here
- strong low-information behavior near the circular end
- differentiated high-
ebehavior near the degenerate end
This experiment does not address:
- conditioning under asymmetry or anisotropy
- conditioning when source positions are unknown
- how these same conclusions transfer to non-elliptic family members
The clearest figures are:
- edge_conditioning_overview.png for the full-range result
- edge_zoom_panels.png for the asymptotic edge behavior
Data:
Code:
The hardening side of the roadmap is now in good shape.
The next natural move is to broaden the family and test whether the same budget logic has a clean twin on the fixed-difference side.
That means the next experiment should be the hyperbola flip.