# Structure-preserving learning: audit the actual operator, not its label

## Verified result and scope

The pre-verification design and executable were committed at `ede5b76c1`.
The completed audit and execution log are backed at `99dfc9900`; all assertions
pass. The frozen executable SHA-256 is
`cfdbdf00c4fe17329d743799a52dedff538c0fd42ecc3e6fd29291742d7e2be3`.
The result is [the immutable numerical receipt](apps_industrial_breakthrough/structure_preservation_audit_20260913_outputs/audit.json).

| Independent check | Verified result |
| --- | --- |
| Nonnegative-potential witness | Exact energy increase `9/64` at velocity `3/4`. |
| Homogeneous residual scale | Residual norm decreases from 16.0347358 to 0.0000160347358 as positive mass scale falls from 1 to 1e-6; normalized residual and represented ODE are unchanged. |
| Displayed uniqueness condition | Printed product .02 is below 1, but scalar roots are approximately -1.915008048, 0 and +1.915008048; the correctly scaled sufficient bound is 2 and asserts no uniqueness. |
| Invariant-coordinate force Gram | All sixteen exact weighted-moment entries agree with independent physical-velocity quadrature within 4.44e-16; the Gram is SPD and unequal diagonal entries exclude Toeplitz/circulant structure. |
| Passive, nonconvex potential | Exact second derivative `-61/240` while dissipated power is `29/160`. |
| Constrained coefficient recovery | One 121-observation synthetic nonnegative fit recovers four coefficients within 1.67e-16. |

There were zero real-data fits, neural models, physics rollouts or protected
test queries. The synthetic recovery is a representable noiseless fixture,
not a statistical identification, generalization or speed comparison.

This is an analytic design audit, not another performance gate, a neural
reproduction, or evidence that a published trained model failed. It checks
three premises that matter when replacing neural optimization with exact
coefficient-space calculus. The constructive spline example is elementary
calculus, not a claim of a newly discovered class of dissipative systems.

Sources read: LOpInf-SpML [arXiv v1](https://arxiv.org/abs/2404.05040v1), all
24 pages, plus the [published paper](https://kramer.ucsd.edu/img/pubs/Lagrangian-operator-inference-enhanced-with-Structure-preserving-ML-2024.pdf)
pages 6--9 to confirm the objective; published page 7 also rendered.
[Geo-NeW v2](https://arxiv.org/abs/2602.02788v2), all text and captions of
22 pages; page 6 rendered to confirm the displayed uniqueness condition.
Figure fields were not quantitatively digitized and no external training
code, model checkpoint or experimental beam data was executed or inspected.

## 1. Nonnegative Rayleigh potential is not enough

For M*qdd + grad(U) + grad(F)(v) = 0 with constant SPD M and v=qd,
the mechanical energy obeys E_dot=-v*grad(F)(v). Thus the needed sign is
v*grad(F)(v)>=0, not merely F(v)>=0. In one dimension,

    F(v)=v^2*(v-1)^2 >= 0, F(0)=0,
    v*F'(v) at v=3/4 is -9/64.

The corresponding energy increases at that state. This quartic is compatible
with an unrestricted polynomial correction to a positive quadratic prior.
The displayed nonnegative-potential penalty in LOpInf-SpML therefore is not
by itself a global dissipativity certificate. Its empirically useful fits
are a separate question; they have not been reproduced here. Convexity with
a minimum at zero is one sufficient extra condition, but is stronger than
radial nondecrease and is not required for every physically dissipative law.

## 2. Fix the scale of a learned homogeneous equation

If mass, potential and dissipation can all be scaled by epsilon>0, the
unforced Euler--Lagrange residual scales by epsilon while the resulting ODE
after division by mass is unchanged. Hence an unnormalized residual can tend
to zero without improving predictions. The flexible mass/correction model
displayed in LOpInf-SpML permits this family through its quadratic corrections.
Its soft nonnegativity penalty also admits the singular all-zero limit.

This criticism is restricted to that displayed flexible objective: the paper
explicitly fixes mass to the identity in its rod and membrane experiments,
which excludes this particular collapse. No claim is made that its beam
optimizer reached the singular limit, or that uninspected code lacks further
constraints. Our next coefficient learner must fix known mass/leading operator
scale, impose a justified normalization and positive mass floor, or use a
scale-invariant objective. A positive-definite inequality without a scale
constraint does not itself eliminate the positive-epsilon sequence.

## 3. Derive the perturbation bound from the assembled equation

For G(u)=epsilon*K*u+D^T*F(u)-b, with K SPD and F L-Lipschitz, the fixed-point
contraction bound involves ||K^-1*D^T||*L/epsilon<1. It cannot generally be
replaced by epsilon*||K^-1||*L<1, the dependence printed in Geo-NeW Eq. 12
alongside its Eq. 9 operator. The scalar choice K=D=1, epsilon=.1,
F(u)=-.2*tanh(u), b=0 satisfies that printed product (.02<1), yet G has
three roots. The correctly scaled sufficient bound is 2, so it makes no
uniqueness assertion. This checks the displayed equations together, not the
condition for a differently normalized operator in another reference.

The paper additionally states that its running-average implementation does
not guarantee its condition on every sample. We do not infer that its reported
solves failed. Any project certificate must use the actual assembled scaling,
norms and boundary constraints; a model family name is not the certificate.

## Constructive coefficient-space safeguard

Let s=v^2, g(s)=eta+sum_i c_i*B_i(s), eta>0, c_i>=0, where B_i are nonnegative
cardinal cubic generators. Define

    F(v) = (1/2)*integral_0^(v^2) g(s) ds,
    F'(v) = v*g(v^2),
    v*F'(v) >= eta*v^2.

The force is linear in c, so with known normalization and observed regressors,
quadratic fitting plus c>=0 remains convex. This allows some nonconvex F:
g=.1+B3(s) has F''<0 at v^2=1.5 while still dissipating energy everywhere.
For multiple fixed directions a_j, sum such potentials of a_j^T*v;
the power is a sum of nonnegative directional contributions. This does not
represent every anisotropic or history-dependent dissipation law.

The physical-velocity inner product is **not** the unweighted cardinal Gram.
For force features v*B_i(v^2),

    integral_(-sqrt(S))^(sqrt(S)) v^2*B_i(v^2)*B_j(v^2) dv
      = integral_0^S sqrt(s)*B_i(s)*B_j(s) ds.

Each product has polynomial pieces and its weighted moments are analytic.
The support remains local, but the sqrt(s) weight destroys translation
invariance: do not apply a circulant inverse. At deployment, a cardinal
index in squared velocity and fixed local polynomial evaluation suffice.
Numerical audits check the derivative, nonnegative power, physical-space
Gram and an independent nonnegative fit. This safeguards a **model's** energy
balance only; observation error, missing memory, control performance and
transfer to a new physical system still require separate evidence.