71. Geometry can be proposed from the operators themselves
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An operator identity can reveal a useful coordinate geometry without specifying that geometry in advance. Suppose two fitted operators evaluated at an anchor \(u_*\) give an invertible ratio
This ratio is an algebraic object. It need not be a physically available action or a positive map by itself. In mass-first coordinates write
If \(I-A\) is invertible, its fixed center is \(c=(I-A)^{-1}b\). The centered spatial action is \(A\). A candidate invariant metric obeys
With \(A\) fixed, this is a linear equation in the independent entries of symmetric \(H\). No coordinate template appears in that equation.
Proposition 63: Conditional operator-only ellipsoid discovery
Suppose the symmetric solution space of \(A^{\mathsf T}HA=H\) is one-dimensional and contains a positive definite matrix \(H_0\). Let a rank-one fitted branch at a declared observation have a nonzero-mass image, normalized to \(v=(1,x_v)\). If \(x_v\neq c\), define
This gives a unique positive definite metric in that solution space for which \(v\) is on the normalized boundary. If this proposed geometry passes every hypothesis of Proposition 62 for the complete fitted family and its initial state, it is an internally valid probability-domain certificate. The proposal requires no supplied template. Failure at any step withholds this certificate and does not establish model invalidity.
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Proof
Invertibility of \(I-A\) gives the unique center. One-dimensionality makes every symmetric invariant form a scalar multiple of \(H_0\). The nonzero displacement has strictly positive squared \(H_0\) length, so the normalization is well-defined, positive and independent of the initially chosen positive scale. The final acceptance test is exactly Proposition 62. Neither invariance under the ratio nor the choice of a boundary image alone is asserted to certify the full family. \(\square\)
The rank-one image proposes a scale. It does not prove that every possible probability domain must place that image on its boundary. Some valid models have many admissible domains. The procedure can also fail because there is no suitable ratio, because invariant forms are not unique, because no rational rank-one anchor is found, or because the chosen ellipsoid fails another branch. These are explicit search outcomes.
71.1 The continuous example no longer needs its template
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Use the same exact population moment tables as Section 66. The existing moment learner first reconstructs the two polynomial operators in test coordinates. The new discovery routine receives those operators, their mass row and initial vector. It does not read continuous_template.json or import the population generator.
At \(u_*=0\), the operator ratio gives the required invariant-form equation. A rational rank-one observation supplies a boundary image. The recovered center and metric are
The two metric eigenvalues are \(40/9\) and \(160/9\). For both actions the exact certificate is
Every scalar on the right has an exact rational sum-of-squares representation. Integrated mass is unchanged. The fitted initial state is the ellipsoid center, and the two densities \(13/16\) and \(413/1040\) from Section 66 are unchanged. This is a new certificate construction for the same model, not a change to its predictions.
The exported JSON contains the fitted operators, center, metric, initial state and all witnesses. Independent replay needs only that JSON. The discovery calculation is also checked on three additional rational coefficient families and affine coordinate changes. Those finite examples exercise the stated sector and do not establish unrestricted discovery.
The procedure currently uses three predictive coordinates, a two-dimensional ellipsoid, a finite rational anchor list, a one-dimensional invariant-form space and rational rank-one roots. It proposes only exactly conformal polynomial branches whose scalar nonnegativity can be resolved by its factor-based sum-of-squares routine. General noisy or jointly optimized curved-domain learning remains outside this implementation.