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Structure-preservation audit

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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.

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, all 24 pages, plus the published paper pages 6--9 to confirm the objective; published page 7 also rendered. Geo-NeW v2, 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 $M\ddot q+\nabla U+\nabla F(v)=0$ with constant SPD M and $v=\dot q$ , the mechanical energy obeys $\dot E=-v\nabla F(v)$ . Thus the needed sign is $v\nabla F(v)\ge 0$ , not merely $F(v)\ge 0$ . In one dimension,

F(v)=v^{2}\,(v-1)^{2} \ge 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 $\lVert K^{-1}D^{T}\rVert L/\epsilon<1$ . It cannot generally be replaced by $\epsilon\lVert K^{-1}\rVert L<1$ , the dependence printed in Geo-NeW Eq. 12 alongside its Eq. 9 operator. The scalar choice K=D=1, $\epsilon=0.1,\quad F(u)=-0.2\tanh(u),\quad b=0$ 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 $s=v^2,\quad g(s)=\eta+\sum_i c_i B_i(s),\quad \eta>0,\ c_i\ge 0$ , where B_i are nonnegative cardinal cubic generators. Define

F(v) = (1/2)\,\int_{0}^{v^{2}} g(s) ds, F'(v) = v\,g(v^{2}), v\,F'(v) \ge \eta\,v^{2}.

The force is linear in c, so with known normalization and observed regressors, quadratic fitting plus $c\ge 0$ remains convex. This allows some nonconvex F: $g=0.1+B_3(s)$ has $F''<0\text{ at }v^2=1.5$ 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),

\int_{(-\sqrt{S})}^{\sqrt{S}} v^{2}\,B_i(v^{2})\,B_j(v^{2}) dv = \int_{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.

Original: STRUCTURE_PRESERVATION_AUDIT_2026-09-13.md · Raw source file

View raw MD source

# 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.