Lifted Equation in Mechanics & Optimization

• Lifted Equation is an advanced technique that reformulates physical and optimization problems by embedding them in extended spaces where potentials and constraints become linear or geodesic equations.
• It recasts classical mechanics, quantum systems, and PDE-constrained inverse problems into higher-dimensional formulations, improving solvability and numerical efficiency.
• The method leverages symmetry transformations, such as Noether to Killing mappings, and structural decompositions to bridge geometric, field theoretic, and algebraic lifting approaches.

In the cited literature, a lifted equation is an equation posed on an extended space—configuration space, field space, cotangent bundle, moment-matrix space, or a complete local principal ideal ring—such that the original system is recovered after projection, restriction, averaging, or passage to a limit. The common operation is to replace a potential, a bilinear constraint, a symmetry-redundant formulation, or a residue-field solution by a higher-dimensional or structurally enlarged formulation in which the governing equations become geodesic, Hamiltonian, linear in lifted variables, or inductively solvable. This usage appears in geometric mechanics and field theory, continuum dynamics, PDE-constrained inverse problems, convex optimization, and matrix-valued Henselian lifting (Finn et al., 2018, Esen et al., 2010, Fang et al., 2020, Panja et al., 4 Feb 2026).

1. Eisenhart lift in mechanics and homogeneous field theory

In the Eisenhart-lift formalism, the dynamics of a system evolving under a conservative force is recast as the dynamics of a free system embedded in a curved manifold with one additional generalized coordinate. For classical mechanics, one starts from

$$L(q,\dot q)=\frac12\,m\,\delta_{ij}\,\dot q^i\dot q^j - V(q),$$

with Euler–Lagrange equations

$m\,\ddot q^i + V_{,i}(q)=0.$

The configuration space is enlarged by one coordinate $v(t)$, and the lifted Lagrangian is taken to be purely kinetic,

$$L_{\rm lift}=\frac12\,g_{AB}(q)\,\dot x^A\dot x^B,\qquad x^A=(q^i,v),$$

with metric

$$g_{AB}(q)= \begin{pmatrix} m\,\delta_{ij} & 0 \ 0 & \dfrac{M^2}{V(q)} \end{pmatrix}.$$

The geodesic equations imply

$$\frac{d}{dt}\Bigl(\frac{M^2}{V}\,\dot v\Bigr)=0 \qquad\Longrightarrow\qquad \dot v=A\,\frac{V(q)}{M},$$

and substitution into the $q^i$ equations gives

$m\,\ddot q^i=-\frac{A^2}{2}\,V_{,i}(q).$

Choosing the affine-parameter normalization $A^2=2$ reproduces exactly the original Newton equation. In this construction, the potential is encoded in the geometry of the lifted manifold rather than appearing explicitly as a force term (Finn et al., 2018).

Finn, Karamitsos, and Pilaftsis extend the same construction to homogeneous field theories. For $N$ homogeneous scalar fields $m\,\ddot q^i + V_{,i}(q)=0.$0 with kinetic metric $m\,\ddot q^i + V_{,i}(q)=0.$1 and potential $m\,\ddot q^i + V_{,i}(q)=0.$2,

$m\,\ddot q^i + V_{,i}(q)=0.$3

one introduces a new field $m\,\ddot q^i + V_{,i}(q)=0.$4 and extended coordinates

$m\,\ddot q^i + V_{,i}(q)=0.$5

The lifted Lagrangian is again purely kinetic,

$m\,\ddot q^i + V_{,i}(q)=0.$6

with block-diagonal field-space metric

$m\,\ddot q^i + V_{,i}(q)=0.$7

The new variable is not an auxiliary field; it is fully dynamical and is therefore termed fictitious. Its equation of motion yields

$m\,\ddot q^i + V_{,i}(q)=0.$8

and with $m\,\ddot q^i + V_{,i}(q)=0.$9 the lifted equations reduce exactly to the original potential-driven equations of motion. The construction therefore removes the potential from the Lagrangian at the price of enlarging the field space (Finn et al., 2018).

A central structural result is that Noether symmetries of the original theory with potential become Killing symmetries of the lifted field-space metric. For an infinitesimal transformation $v(t)$0,

$v(t)$1

so invariance requires the Killing equation

$v(t)$2

If one restricts to $v(t)$3, this becomes

$v(t)$4

which are precisely the Noether conditions for the original theory with potential.

2. Higher-dimensional field theory and mixed vielbein formulation

The extension from homogeneous systems to four-dimensional field theory is nontrivial. For a standard multi-field action

$v(t)$5

a naive lift by adjoining a scalar field $v(t)$6 with term

$v(t)$7

fails. The corresponding equation,

$v(t)$8

does not imply $v(t)$9 because the $$L_{\rm lift}=\frac12\,g_{AB}(q)\,\dot x^A\dot x^B,\qquad x^A=(q^i,v),$$0-sum spoils constancy. A common misconception is therefore that a purely scalar lift extends directly from homogeneous mechanics to general four-dimensional field theory; in the formulation of Finn, Karamitsos, and Pilaftsis, it does not (Finn et al., 2018).

The remedy is to introduce a vector field $$L_{\rm lift}=\frac12\,g_{AB}(q)\,\dot x^A\dot x^B,\qquad x^A=(q^i,v),$$1 and use the kinetic combination

$$L_{\rm lift}=\frac12\,g_{AB}(q)\,\dot x^A\dot x^B,\qquad x^A=(q^i,v),$$2

Variation with respect to $$L_{\rm lift}=\frac12\,g_{AB}(q)\,\dot x^A\dot x^B,\qquad x^A=(q^i,v),$$3 gives

$$L_{\rm lift}=\frac12\,g_{AB}(q)\,\dot x^A\dot x^B,\qquad x^A=(q^i,v),$$4

which does reproduce the effect of $$L_{\rm lift}=\frac12\,g_{AB}(q)\,\dot x^A\dot x^B,\qquad x^A=(q^i,v),$$5 in the $$L_{\rm lift}=\frac12\,g_{AB}(q)\,\dot x^A\dot x^B,\qquad x^A=(q^i,v),$$6 equations when $$L_{\rm lift}=\frac12\,g_{AB}(q)\,\dot x^A\dot x^B,\qquad x^A=(q^i,v),$$7.

To make the geometry explicit, the vector is expressed through a mixed vierbein,

$$L_{\rm lift}=\frac12\,g_{AB}(q)\,\dot x^A\dot x^B,\qquad x^A=(q^i,v),$$8

The extended coordinates are then

$$L_{\rm lift}=\frac12\,g_{AB}(q)\,\dot x^A\dot x^B,\qquad x^A=(q^i,v),$$9

and the kinetic tensor becomes

$$g_{AB}(q)= \begin{pmatrix} m\,\delta_{ij} & 0 \ 0 & \dfrac{M^2}{V(q)} \end{pmatrix}.$$0

The lifted action is

$$g_{AB}(q)= \begin{pmatrix} m\,\delta_{ij} & 0 \ 0 & \dfrac{M^2}{V(q)} \end{pmatrix}.$$1

Its field equations take the generalized geodesic form

$$g_{AB}(q)= \begin{pmatrix} m\,\delta_{ij} & 0 \ 0 & \dfrac{M^2}{V(q)} \end{pmatrix}.$$2

with

$$g_{AB}(q)= \begin{pmatrix} m\,\delta_{ij} & 0 \ 0 & \dfrac{M^2}{V(q)} \end{pmatrix}.$$3

and reduce in the homogeneous limit to the ordinary one-dimensional geodesic equations.

The symmetry statement also generalizes. A transformation $$g_{AB}(q)= \begin{pmatrix} m\,\delta_{ij} & 0 \ 0 & \dfrac{M^2}{V(q)} \end{pmatrix}.$$4 leaves the lifted action invariant iff

$$g_{AB}(q)= \begin{pmatrix} m\,\delta_{ij} & 0 \ 0 & \dfrac{M^2}{V(q)} \end{pmatrix}.$$5

that is, ten Killing equations for the ten tensors $$g_{AB}(q)= \begin{pmatrix} m\,\delta_{ij} & 0 \ 0 & \dfrac{M^2}{V(q)} \end{pmatrix}.$$6. Restricting $$g_{AB}(q)= \begin{pmatrix} m\,\delta_{ij} & 0 \ 0 & \dfrac{M^2}{V(q)} \end{pmatrix}.$$7 reproduces the Noether conditions for the original four-dimensional action with potential (Finn et al., 2018).

3. Quantum lifted equations and lifted field-space quantization

The Eisenhart lift also admits a quantum formulation. For a particle of mass $$g_{AB}(q)= \begin{pmatrix} m\,\delta_{ij} & 0 \ 0 & \dfrac{M^2}{V(q)} \end{pmatrix}.$$8 in $$g_{AB}(q)= \begin{pmatrix} m\,\delta_{ij} & 0 \ 0 & \dfrac{M^2}{V(q)} \end{pmatrix}.$$9 dimensions with Hamiltonian

$$\frac{d}{dt}\Bigl(\frac{M^2}{V}\,\dot v\Bigr)=0 \qquad\Longrightarrow\qquad \dot v=A\,\frac{V(q)}{M},$$0

one introduces an extra coordinate $$\frac{d}{dt}\Bigl(\frac{M^2}{V}\,\dot v\Bigr)=0 \qquad\Longrightarrow\qquad \dot v=A\,\frac{V(q)}{M},$$1 and obtains the lifted Hamiltonian operator

$$\frac{d}{dt}\Bigl(\frac{M^2}{V}\,\dot v\Bigr)=0 \qquad\Longrightarrow\qquad \dot v=A\,\frac{V(q)}{M},$$2

The time-dependent Schrödinger equation on the lifted manifold is

$$\frac{d}{dt}\Bigl(\frac{M^2}{V}\,\dot v\Bigr)=0 \qquad\Longrightarrow\qquad \dot v=A\,\frac{V(q)}{M},$$3

or, in position space,

$$\frac{d}{dt}\Bigl(\frac{M^2}{V}\,\dot v\Bigr)=0 \qquad\Longrightarrow\qquad \dot v=A\,\frac{V(q)}{M},$$4

Because the lifted system is $$\frac{d}{dt}\Bigl(\frac{M^2}{V}\,\dot v\Bigr)=0 \qquad\Longrightarrow\qquad \dot v=A\,\frac{V(q)}{M},$$5-translation invariant, one may separate variables as

$$\frac{d}{dt}\Bigl(\frac{M^2}{V}\,\dot v\Bigr)=0 \qquad\Longrightarrow\qquad \dot v=A\,\frac{V(q)}{M},$$6

The reduced equation becomes

$$\frac{d}{dt}\Bigl(\frac{M^2}{V}\,\dot v\Bigr)=0 \qquad\Longrightarrow\qquad \dot v=A\,\frac{V(q)}{M},$$7

and choosing

$$\frac{d}{dt}\Bigl(\frac{M^2}{V}\,\dot v\Bigr)=0 \qquad\Longrightarrow\qquad \dot v=A\,\frac{V(q)}{M},$$8

recovers exactly the original Schrödinger equation with potential $$\frac{d}{dt}\Bigl(\frac{M^2}{V}\,\dot v\Bigr)=0 \qquad\Longrightarrow\qquad \dot v=A\,\frac{V(q)}{M},$$9. The extra quantum number $q^i$0, the eigenvalue of $q^i$1, is the lifted momentum. If $q^i$2 is compactified on a circle of length $q^i$3, then $q^i$4 (Finn et al., 2020).

This construction shows that the lifted manifold reproduces not only the classical effects of the potential but also its quantum-mechanical effects. The lifted Schrödinger equation is therefore not merely a geometric analogy; after projection onto a fixed lifted-momentum sector, it becomes dynamically equivalent to the original quantum system.

The extension to quantum field theory follows the same pattern. For a real scalar field,

$q^i$5

one introduces a fictitious scalar or vector field $q^i$6 and defines

$q^i$7

In field space $q^i$8, the metric is

$q^i$9

Classically, the $m\,\ddot q^i=-\frac{A^2}{2}\,V_{,i}(q).$0 equation implies $m\,\ddot q^i=-\frac{A^2}{2}\,V_{,i}(q).$1, and substitution recovers the original Klein–Gordon equation.

After canonical quantization, a distinctive new structure appears: $m\,\ddot q^i=-\frac{A^2}{2}\,V_{,i}(q).$2 commutes with $m\,\ddot q^i=-\frac{A^2}{2}\,V_{,i}(q).$3 and with the field operators at all $m\,\ddot q^i=-\frac{A^2}{2}\,V_{,i}(q).$4. Its eigenvalue $m\,\ddot q^i=-\frac{A^2}{2}\,V_{,i}(q).$5 is therefore a conserved quantum charge, uniform in space and time. Each value of this charge defines an independent Fock space; states with different eigenvalues of $m\,\ddot q^i=-\frac{A^2}{2}\,V_{,i}(q).$6 are orthogonal, and no operator in the physical $m\,\ddot q^i=-\frac{A^2}{2}\,V_{,i}(q).$7 sector can connect them. The full Hilbert space is thus an ensemble of disjoint Fock spaces labeled by the lifted charge. The relevance of these extended Fock spaces to the cosmological constant and gauge hierarchy problems is considered in the quantum-lift framework (Finn et al., 2020).

4. Cotangent lifts, vertical representatives, and continuum dynamics

A second major meaning of lifted equation arises in the theory of complete cotangent lifts developed by Esen and Gümral. Let $m\,\ddot q^i=-\frac{A^2}{2}\,V_{,i}(q).$8 be an $m\,\ddot q^i=-\frac{A^2}{2}\,V_{,i}(q).$9-dimensional smooth manifold with local coordinates $A^2=2$0, and let

$A^2=2$1

be a vector field on $A^2=2$2. Its flow induces a one-parameter family of diffeomorphisms on $A^2=2$3, and the infinitesimal generator is the complete cotangent lift

$A^2=2$4

In canonical coordinates $A^2=2$5, this is equivalently the Hamiltonian vector field of

$A^2=2$6

with respect to the canonical symplectic form (Esen et al., 2010).

The lifted field admits a canonical holonomic–vertical decomposition:

$A^2=2$7

$A^2=2$8

with

$A^2=2$9

The integral curves of the lifted system are the lifted equations

$N$0

or concisely,

$N$1

The significance of this construction is that special choices of the base vector field $N$2 recover standard continuum equations. For ideal incompressible fluid, taking $N$3 with fixed volume form $N$4 and restricting to divergence-free vector fields yields the Lie–Poisson equation

$N$5

which is equivalent to Euler’s equations in velocity or vorticity form. For collisionless plasma, taking $N$6 with symplectic form $N$7 and $N$8 Hamiltonian for a single-particle energy $N$9 gives

$m\,\ddot q^i + V_{,i}(q)=0.$00

which is exactly the Vlasov equation. For contact flows on a contact manifold $m\,\ddot q^i + V_{,i}(q)=0.$01, the resulting kinetic equation is

$m\,\ddot q^i + V_{,i}(q)=0.$02

The paper presents this as a single unifying mechanism: starting from a vector field on a configuration manifold, one forms its complete cotangent lift, decomposes it into holonomic and vertical parts, and reads off the lifted equations on $m\,\ddot q^i + V_{,i}(q)=0.$03 (Esen et al., 2010).

5. Lifted equations in PDE-constrained inverse problems

In seismic full waveform inversion, Fang and Demanet use “Lift and Relax” to reformulate a PDE-constrained inverse problem as a lifted equation on a higher-dimensional moment matrix. The standard frequency-domain Helmholtz constraint is

$m\,\ddot q^i + V_{,i}(q)=0.$04

with data misfit

$m\,\ddot q^i + V_{,i}(q)=0.$05

The classical formulation,

$m\,\ddot q^i + V_{,i}(q)=0.$06

becomes highly nonconvex after eliminating $m\,\ddot q^i + V_{,i}(q)=0.$07.

The lifting step introduces the moment matrix

$m\,\ddot q^i + V_{,i}(q)=0.$08

with block structure

$m\,\ddot q^i + V_{,i}(q)=0.$09

Exact rank-one factorization yields

$m\,\ddot q^i + V_{,i}(q)=0.$10

Polynomial and bilinear expressions in $m\,\ddot q^i + V_{,i}(q)=0.$11 become linear in the entries of $m\,\ddot q^i + V_{,i}(q)=0.$12:

$m\,\ddot q^i + V_{,i}(q)=0.$13

The exact lifted representation satisfies

$m\,\ddot q^i + V_{,i}(q)=0.$14

Dropping the nonconvex rank-one constraint while keeping $m\,\ddot q^i + V_{,i}(q)=0.$15 and $m\,\ddot q^i + V_{,i}(q)=0.$16 gives an SDP relaxation (Fang et al., 2020).

The relaxation step adopts wavefield reconstruction inversion. Instead of enforcing the wave equation exactly, one minimizes

$m\,\ddot q^i + V_{,i}(q)=0.$17

and in lifted form,

$m\,\ddot q^i + V_{,i}(q)=0.$18

Trace penalties are then added because $m\,\ddot q^i + V_{,i}(q)=0.$19 acts as a convex proxy for rank$m\,\ddot q^i + V_{,i}(q)=0.$20. The penalized SDP is

$m\,\ddot q^i + V_{,i}(q)=0.$21

As $m\,\ddot q^i + V_{,i}(q)=0.$22 one recovers the exact-constraint formulation; as $m\,\ddot q^i + V_{,i}(q)=0.$23 one enforces $m\,\ddot q^i + V_{,i}(q)=0.$24.

The method is not implemented as a full dense SDP. Direct interior-point solvers scale like $m\,\ddot q^i + V_{,i}(q)=0.$25 per iteration with $m\,\ddot q^i + V_{,i}(q)=0.$26 and are hopelessly large once $m\,\ddot q^i + V_{,i}(q)=0.$27 and $m\,\ddot q^i + V_{,i}(q)=0.$28 exceed a few hundred unknowns. The practical algorithm uses a low-rank factorization

$m\,\ddot q^i + V_{,i}(q)=0.$29

with $m\,\ddot q^i + V_{,i}(q)=0.$30 in LRWI. The factorization is parameterized as

$m\,\ddot q^i + V_{,i}(q)=0.$31

so that

$m\,\ddot q^i + V_{,i}(q)=0.$32

For fixed model factors and $m\,\ddot q^i + V_{,i}(q)=0.$33, the wavefield factors admit an explicit normal-equations solution of size $m\,\ddot q^i + V_{,i}(q)=0.$34. The algorithm alternates between solving for $m\,\ddot q^i + V_{,i}(q)=0.$35 exactly by $m\,\ddot q^i + V_{,i}(q)=0.$36 linear algebra, updating $m\,\ddot q^i + V_{,i}(q)=0.$37 by limited-memory BFGS, and updating the angle $m\,\ddot q^i + V_{,i}(q)=0.$38 by simple gradient descent, with per-iteration cost $m\,\ddot q^i + V_{,i}(q)=0.$39 rather than $m\,\ddot q^i + V_{,i}(q)=0.$40.

Numerically, LRWI increases the acceptable starting frequency from $m\,\ddot q^i + V_{,i}(q)=0.$41 Hz and $m\,\ddot q^i + V_{,i}(q)=0.$42 Hz to $m\,\ddot q^i + V_{,i}(q)=0.$43 Hz and $m\,\ddot q^i + V_{,i}(q)=0.$44 for the Marmousi model and the Overthrust model, respectively, in the cases of a linear gradient starting model. The detailed examples further state that standard FWI/WRI fail at $m\,\ddot q^i + V_{,i}(q)=0.$45 Hz on Marmousi and at $m\,\ddot q^i + V_{,i}(q)=0.$46 Hz on Overthrust, whereas LRWI succeeds from $m\,\ddot q^i + V_{,i}(q)=0.$47 Hz and $m\,\ddot q^i + V_{,i}(q)=0.$48 Hz, respectively (Fang et al., 2020).

6. Lifted convex quadratic programming by symmetry compression

In convex optimization, lifting can mean compression rather than dimensional enlargement. Mladenov, Globerson, and Kersting introduce lifted convex quadratic programming by exploiting fractional symmetries of the data. The ground problem is

$m\,\ddot q^i + V_{,i}(q)=0.$49

with $m\,\ddot q^i + V_{,i}(q)=0.$50 symmetric positive semidefinite. A partition $m\,\ddot q^i + V_{,i}(q)=0.$51 of the variable indices defines a partition matrix

$m\,\ddot q^i + V_{,i}(q)=0.$52

which is doubly stochastic, symmetric, and idempotent. A similar partition $m\,\ddot q^i + V_{,i}(q)=0.$53 is chosen for the constraints.

The pair $m\,\ddot q^i + V_{,i}(q)=0.$54 is a fractional automorphism of $m\,\ddot q^i + V_{,i}(q)=0.$55 if

$m\,\ddot q^i + V_{,i}(q)=0.$56

$m\,\ddot q^i + V_{,i}(q)=0.$57

These invariance conditions imply that averaging preserves feasibility,

$m\,\ddot q^i + V_{,i}(q)=0.$58

and does not increase the objective,

$m\,\ddot q^i + V_{,i}(q)=0.$59

Hence there always exists an optimal solution in the fixed subspace

$m\,\ddot q^i + V_{,i}(q)=0.$60

whose dimension is the number $m\,\ddot q^i + V_{,i}(q)=0.$61 of variable classes (Mladenov et al., 2016).

If $m\,\ddot q^i + V_{,i}(q)=0.$62 is the indicator matrix of $m\,\ddot q^i + V_{,i}(q)=0.$63, then

$m\,\ddot q^i + V_{,i}(q)=0.$64

and every $m\,\ddot q^i + V_{,i}(q)=0.$65 can be written as $m\,\ddot q^i + V_{,i}(q)=0.$66. Substituting into the original QP yields the lifted problem

$m\,\ddot q^i + V_{,i}(q)=0.$67

Under the fractional-automorphism conditions, this lifted QP is equivalent to the original one: every optimal $m\,\ddot q^i + V_{,i}(q)=0.$68 lifts to an optimal $m\,\ddot q^i + V_{,i}(q)=0.$69, and every ground optimum can be averaged to the reduced subspace.

The significance of the construction is computational. If the number of classes $m\,\ddot q^i + V_{,i}(q)=0.$70 is much smaller than $m\,\ddot q^i + V_{,i}(q)=0.$71, then optimization on the lifted QP is more compact and likely to be more efficient. The paper states that a typical interior-point or active-set solver scales roughly like $m\,\ddot q^i + V_{,i}(q)=0.$72 instead of $m\,\ddot q^i + V_{,i}(q)=0.$73, that the coarsest equitable partitions can be found by a color-refinement-style routine in time $m\,\ddot q^i + V_{,i}(q)=0.$74, and that one often obtains savings of $m\,\ddot q^i + V_{,i}(q)=0.$75–$m\,\ddot q^i + V_{,i}(q)=0.$76 on real-world symmetric QPs in machine learning (Mladenov et al., 2016).

7. Matrix-valued Henselian lifting of polynomial equations

A further notion of lifted equation appears in algebra: lifting solutions of polynomial equations on matrices from a residue field to a complete local principal ideal ring. Let $m\,\ddot q^i + V_{,i}(q)=0.$77 be a complete local principal ideal ring with maximal ideal $m\,\ddot q^i + V_{,i}(q)=0.$78 and residue field $m\,\ddot q^i + V_{,i}(q)=0.$79 of characteristic not $m\,\ddot q^i + V_{,i}(q)=0.$80, let

$m\,\ddot q^i + V_{,i}(q)=0.$81

and let $m\,\ddot q^i + V_{,i}(q)=0.$82 with reduction $m\,\ddot q^i + V_{,i}(q)=0.$83. The lifting problem is: given a commuting $m\,\ddot q^i + V_{,i}(q)=0.$84-tuple

$m\,\ddot q^i + V_{,i}(q)=0.$85

satisfying

$m\,\ddot q^i + V_{,i}(q)=0.$86

under what conditions can one find a commuting lift

$m\,\ddot q^i + V_{,i}(q)=0.$87

such that

$m\,\ddot q^i + V_{,i}(q)=0.$88

For $m\,\ddot q^i + V_{,i}(q)=0.$89 cyclic, Panja, Roy, and Singh prove a matrix-valued Hensel lemma. If the partial derivatives satisfy

$m\,\ddot q^i + V_{,i}(q)=0.$90

and

$m\,\ddot q^i + V_{,i}(q)=0.$91

then there exists a unique commuting tuple

$m\,\ddot q^i + V_{,i}(q)=0.$92

reducing to $m\,\ddot q^i + V_{,i}(q)=0.$93 modulo $m\,\ddot q^i + V_{,i}(q)=0.$94 and satisfying

$m\,\ddot q^i + V_{,i}(q)=0.$95

The lift is obtained by a convergent $m\,\ddot q^i + V_{,i}(q)=0.$96-adic limiting process; at each step one solves a linear system in the centralizer algebra of $m\,\ddot q^i + V_{,i}(q)=0.$97, and the coefficient matrix is invertible by the Jacobian hypothesis (Panja et al., 4 Feb 2026).

The proof proceeds by successive approximations in the quotients

$m\,\ddot q^i + V_{,i}(q)=0.$98

Writing

$m\,\ddot q^i + V_{,i}(q)=0.$99

and expanding to first order gives

$v(t)$00

The known error term

$v(t)$01

is corrected by solving

$v(t)$02

inside the centralizer. Cyclicity of $v(t)$03 ensures that the centralizer is a principal local ring at each stage, and completeness of $v(t)$04 passes the compatible system to the inverse limit.

The paper also gives a concrete $v(t)$05 example over $v(t)$06 with

$v(t)$07

$v(t)$08

and commuting residue-field matrices

$v(t)$09

The partial derivatives are invertible in $v(t)$10, so the theorem yields a unique lift $v(t)$11 with $v(t)$12 (Panja et al., 4 Feb 2026).

The algebraic setting differs sharply from geometric and optimization lifts, but the structural theme is the same: a difficult equation is solved by moving to an enlarged or more regular setting in which linearized correction, rather than direct nonlinear solution, controls existence and uniqueness.