Conservation is usually stated as a prohibition: the system may not leave the energy shell. But if admissible states are viewed geometrically as a thin band around a fixed Hamiltonian level, one can ask not only whether a state lies outside that band, but how much must change to return it there, and how that correction is distributed. In that sense, conservation becomes not just a restriction on reachable states, but a law of repair.
It also extends the borrowing picture from an instantaneous redistribution law to a causal transport theory of repair: beyond the total decay of excess, the framework acquires a borrowing speed, a finite repair lag, and mode-dependent relaxation that may be monotone or damped-oscillatory. Vacuum fluctuations and detectability then admit a concrete interpretation: Lamb-shift-scale effects appear as temporally accumulated repair, while Casimir forces arise when confinement forces the borrowing response into an observable regime.
Let $H$ be time-independent and fix a reference energy $E_0$. For a tolerance $\tau \geq 0$, define the admissible band
This is the thickened energy shell. States in $\mathcal{C}_\tau$ are admissible; those outside it carry an energy mismatch. We treat the upper side of the band, so assume
Here $\beta$ is the excess beyond the admissible band. The lower-side case $H(w) - E_0 < -\tau$ is analogous, with signs reversed. Proofs are collected in Appendix A.
The starting point is a single, undivided state $w \in \mathbb{R}^n$ whose energy overshoots the admissible band by $\beta > 0$. The natural question is: what is the smallest perturbation that returns the state to admissibility?
To answer it, expand $H$ to first order around $w$. A perturbation $u$ changes the energy by
For the perturbed state to reach the upper boundary of the admissible band, the linear term must cancel the excess:
This is a linear equality constraint on $u$. Among all $u$ satisfying it, the one with smallest Euclidean norm is the most economical repair. Geometrically, $u^\star$ is the projection of the origin onto the affine hyperplane defined by (1), and it points directly opposite the energy gradient.
Assume $H \in C^1$ and $\nabla H(w) \neq 0$. Among all perturbations $u \in \mathbb{R}^n$ satisfying $\nabla H(w)^\top u = -\beta$, the unique minimum-norm correction is
Think of $\nabla H(w)$ as pointing uphill on the energy landscape. The state $w$ is already too high, so the cheapest route back is to step straight downhill — opposite $\nabla H(w)$. The required step size is $\beta / \|\nabla H(w)\|$: a steep Hamiltonian (large gradient) requires only a small geometric displacement, while a flat Hamiltonian requires a larger one. This is exactly the intuition that steep hills are easy to descend quickly but hard to climb — here, it is the system's own landscape that governs the repair cost.
In most physical systems of interest, the full state $w$ is not monolithic. It decomposes naturally into coupled subsystems — degrees of freedom belonging to distinct components, regions, or modes. The key question then shifts: must a subsystem that violates admissibility fix itself, or can other parts of the system share the burden?
Suppose the state is partitioned as $w = (w^{(1)}, \ldots, w^{(m)})$. Write the block gradient with respect to subsystem $i$ as $g_i := \nabla_{w^{(i)}} H(w)$, and decompose a perturbation as $u = (u_1, \ldots, u_m)$. Then
This decomposition suggests defining the repair contribution of subsystem $i$ by
Intuitively, $\alpha_i$ is the amount by which subsystem $i$ reduces the energy excess. A positive $\alpha_i$ means subsystem $i$ is lending energy toward the repair; a negative $\alpha_i$ would mean it is making things worse.
To first order, restoration to admissibility occurs whenever
This proposition is the conceptual heart of the paper. It says that no single subsystem need shoulder the entire repair. If subsystem 1 can contribute $\alpha_1 = 0.3\beta$, subsystem 2 contributes $\alpha_2 = 0.5\beta$, and subsystem 3 contributes $\alpha_3 = 0.3\beta$, the combined contribution of $1.1\beta > \beta$ is sufficient — even though no individual part could manage it alone. This collective repair is what we call distributed energy borrowing. The excess is not resolved by the most responsible component; it is shared across the coupled structure.
In practice, each subsystem has a limited capacity to change. Suppose subsystem $i$ may deform by at most $\|u_i\| \leq r_i$. By the Cauchy–Schwarz inequality, the maximum contribution that subsystem $i$ can make toward the repair is
This motivates the reserve capacity of subsystem $i$:
The reserve $R_i$ has a transparent physical meaning: it is large when the subsystem is both energetically sensitive (large $\|g_i\|$, meaning the Hamiltonian responds strongly to changes in $w^{(i)}$) and kinematically free (large $r_i$, meaning the subsystem is permitted to move substantially). A subsystem that is either insensitive to the Hamiltonian or tightly constrained contributes little regardless of how willing it is to participate.
If $\sum_{i=1}^m R_i \geq \beta$, then a first-order repair is feasible. Conversely, if $\sum_{i=1}^m R_i < \beta$, then no perturbation satisfying $\|u_i\| \leq r_i$ can repair the excess to first order.
The reserve criterion is a sharp threshold: the aggregate reserve either covers the deficit or it does not. When it does not, the system is genuinely stuck at first order; no redistribution of the repair burden can help.
Not all subsystems need to contribute. Sort the reserves in decreasing order: $R_{(1)} \geq R_{(2)} \geq \cdots \geq R_{(m)}$ and define
This is the smallest number of subsystems whose combined reserve covers the deficit. The ratio $k^\star / m$ measures what fraction of the total structure must participate in the repair. When $k^\star/m$ is small, the system can recover using only a localized subset of its components; when it approaches 1, essentially the whole system must be mobilised.
The analysis above asks only whether repair is possible. A more refined question is: among all feasible repairs, which is the cheapest? Suppose changing subsystem $i$ incurs a quadratic cost $\tfrac{1}{2}\lambda_i \|u_i\|^2$ with $\lambda_i > 0$. The most economical repair solves
This is a standard constrained quadratic program with a single linear constraint, so it has a closed-form solution via Lagrange multipliers.
The minimum-cost repair is
The borrowed fraction $\alpha_i^\star/\beta$ is proportional to $\|g_i\|^2/\lambda_i$. This ratio rewards subsystems that are energetically responsive (large $\|g_i\|$) relative to how expensive they are to move (large $\lambda_i$). A cheap-to-deform subsystem in a sensitive region of the Hamiltonian absorbs the lion's share of the repair. This mirrors intuitions from mechanics: a flexible component in a high-stress zone deforms preferentially, shielding the stiffer parts of the structure.
When the number of subsystems grows without bound and their sizes shrink, the discrete picture passes naturally to a continuum. Instead of a finite list of blocks $w^{(1)}, \ldots, w^{(m)}$, the state is now a field $w : \Omega \to \mathbb{R}^d$ defined on a spatial domain $\Omega \subseteq \mathbb{R}^n$, and the Hamiltonian is a functional $H[w]$ that assigns a scalar energy to the entire field configuration.
A field perturbation $u(x)$ changes the energy by
where the first variation is
The functional gradient $g(x) := \delta H / \delta w(x)$ plays the role that $\nabla H(w)$ played in the finite-dimensional setting. Physically, $g(x)$ measures how sensitive the total energy is to a small local perturbation of the field at the point $x$. The repair condition $\nabla H(w)^\top u = -\beta$ now becomes an integral condition:
Define the local borrowing density
This is the pointwise rate at which the field contributes to the repair at each location $x$.
To first order, a field perturbation $u$ restores admissibility whenever
The continuous borrowing law is the natural field-theoretic counterpart of the discrete law in Proposition 2.1. Where the discrete case summed contributions $\alpha_i$ over a finite collection of subsystems, here we integrate the density $\alpha(x)$ over the domain. The intuition is unchanged: no single point of the domain need carry the full repair. The contribution of each infinitesimal patch $dx$ is $\alpha(x)\, dx$, and these add up globally. A highly localised excursion outside the admissible band can, in principle, be healed by a diffuse, low-amplitude response spread across the entire domain.
Equip the space of perturbations with the $L^2$ norm,
Assume $g \not\equiv 0$. Among all perturbations $u$ satisfying $\int_\Omega g(x) \cdot u(x)\, dx = -\beta$, the minimum-$L^2$-norm repair is
The structure mirrors Proposition 1.1 precisely. In the finite-dimensional case the optimal perturbation was proportional to $-\nabla H(w)$; here it is proportional to $-g(x)$. The repair is spread across the domain in exact proportion to the local energy sensitivity: regions where $\|g(x)\|$ is large deform more, and regions where $\|g(x)\|$ is small deform less.
Suppose each point $x$ has a pointwise deformation budget $\|u(x)\| \leq r(x)$. Define the reserve density
If $\int_\Omega R(x)\, dx \geq \beta$, then a first-order repair is feasible. If $\int_\Omega R(x)\, dx < \beta$, no perturbation satisfying $\|u(x)\| \leq r(x)$ can repair the excess to first order.
The discrete threshold $k^\star/m$ now becomes a support fraction. Define $S^\star \subseteq \Omega$ to be a measurable region of minimal measure such that $\int_{S^\star} R(x)\, dx \geq \beta$. The ratio $\mu(S^\star)/\mu(\Omega)$ is the minimal fraction of the domain that must participate in the repair. A small support fraction indicates that the repair can be accomplished by a spatially concentrated response; a fraction close to 1 indicates that the entire domain must contribute.
The variational results in Sections 1–3 are essentially static: they characterise the geometry of the repair without saying how it unfolds over time. The natural next step is to promote borrowing from a one-shot correction to a continuous process that drives the system toward admissibility.
Define the excess functional
So $\beta(w) = 0$ on the admissible band and $\beta(w) > 0$ outside it. Choose a rate $\kappa > 0$ and a response function $f : [0, \infty) \to [0, \infty)$ with $f(0) = 0$ and $f(\beta) > 0$ for $\beta > 0$. Consider the gradient-like flow
Intuitively, when the state is outside the admissible band, this flow pushes it downhill along the energy landscape at a rate proportional to $f(\beta)$. When $\beta = 0$, the flow switches off: the system stops moving once admissibility is restored. The normalisation by $\|\nabla H(w)\|^2$ ensures that the actual energy decay rate is controlled by $\kappa$ and $f$, independently of how steep or flat the Hamiltonian is at $w$.
Assume $H \in C^1$ and $\nabla H(w) \neq 0$ whenever $\beta(w) > 0$. Then along any solution of the flow above,
Hence $\beta(t)$ decreases monotonically to $0$.
The choice of $f$ governs the temporal profile of the repair, independently of its spatial distribution.
Linear decay ($f(\beta) = 1$): The excess decreases at a constant rate,
reaching zero in finite time $T = \beta(0)/\kappa$. This is a hard landing: the system coasts into the admissible band and stops.
Exponential decay ($f(\beta) = \beta$): The excess obeys $\dot\beta = -\kappa\beta$, giving $\beta(t) = \beta(0)\,e^{-\kappa t}$. Admissibility is approached asymptotically. The repair is faster when the excess is large and slows as the system nears the boundary — a soft landing.
Other choices of $f$ interpolate between or extend these two extremes, offering a flexible toolkit for modelling different physical relaxation mechanisms.
In the discrete partition $w = (w^{(1)}, \ldots, w^{(m)})$, the flow decomposes as
Define the instantaneous borrowing current of subsystem $i$ by
The total repair rate satisfies $\sum_{i=1}^m \rho_i(t) = \kappa f(\beta(t))$, in agreement with Proposition 4.1. The fraction $\|g_i\|^2 / \sum_j \|g_j\|^2$ is the share of the total squared gradient norm attributable to subsystem $i$. Subsystems with high energy sensitivity carry a proportionally larger share of the instantaneous repair burden — the natural gradient-flow distribution that minimises the instantaneous kinetic cost of the repair.
The same construction extends to field theories. Define $g(x, t) := \delta H / \delta w(x, t)$ and set
Assume $H[w]$ is differentiable and $g \not\equiv 0$ whenever $\beta[w] > 0$. Then
The local borrowing current density is
In the continuum, borrowing manifests as a repair current field: at each instant, energy is being redistributed across the domain, with each point contributing according to its local sensitivity.
That construction determines the total repair rate once the coupled structure is participating. It does not yet determine how quickly that participation spreads across space, how long distant regions take to respond, or whether the return to admissibility is monotone or oscillatory. Those questions require a causal transport law for the borrowing current itself.
The borrowing flows of Section 4 describe how excess decays once a repair direction has been chosen, but they are still effectively instantaneous in one important sense: the moment the state lies outside the admissible band, every part of the coupled structure is allowed to participate at once. That is enough to determine the total repair rate, but not enough to say how the repair propagates across space, how long distant regions take to respond, or whether the return to admissibility is monotone or oscillatory.
To ask those questions, the formalism must be given one further constitutive ingredient. The scalar excess $\beta[w(t)]$ measures the total mismatch, but it does not say where that mismatch is concentrated. Introduce therefore a nonnegative excess density
together with a vector borrowing current $J(x,t)$ describing the transport of repair through the domain. The natural continuity law is
where $\Gamma > 0$ is a local repair rate. The meaning is simple: excess at a point may disappear either because it is redistributed elsewhere through the current $J$, or because it is locally repaired at rate $\Gamma$.
The continuity equation above is the local counterpart of the global borrowing flow. If the domain is closed and no repair current escapes through the boundary, that is,
then integrating over $\Omega$ immediately yields a closed equation for the total excess.
Assume $b$ and $J$ satisfy
on a bounded domain $\Omega$, together with the no-flux boundary condition $J\cdot n = 0$ on $\partial\Omega$. Then
This recovers, at the integrated level, the exponential repair schedule of Section 4. But it does more than that: it separates two distinct pieces of the dynamics. The scalar $\Gamma$ governs how fast excess is removed once it is locally available to be repaired, while the current $J$ governs how fast that excess can be redistributed across the structure in the first place.
At this stage the theory still lacks a law for the current itself. The simplest diffusive closure,
would imply instantaneous propagation, since any local disturbance would be felt everywhere immediately. If one wishes the borrowing process to be genuinely causal, the current must relax on a nonzero time scale.
A natural causal closure is the Maxwell-Cattaneo-type law
where $D > 0$ is a borrowing diffusivity and $\tau_J > 0$ is the current relaxation time. When $\tau_J = 0$, this reduces to the instantaneous diffusive law. When $\tau_J > 0$, the current cannot adjust infinitely fast; the borrowing response develops over time.
Combining the continuity law with this constitutive equation gives a second-order evolution equation for the excess density.
Assume $b$ and $J$ satisfy
Then $b$ satisfies
The significance of this equation is that the repair is no longer purely relaxational. It possesses an intrinsic propagation scale. In particular, the characteristic borrowing speed is
Thus a disturbance spread over a length scale $L$ cannot be redistributed faster than on the order of
This lag is new. In the instantaneous borrowing flow of Section 4, the excess begins to decay globally as soon as it is present. In the causal theory, by contrast, a sharply localised excess may remain locally visible for a finite time before the rest of the system has joined the accounting. The formalism therefore predicts not merely a repair rate, but a repair front.
The original borrowing flow determines how much excess must be repaired and how that burden is distributed once the full coupled structure participates. The causal extension determines how quickly that participation becomes available. The new observable is not only the total decay of $\beta(t)$, but the lag between a local excess and the arrival of the distributed repair.
The telegraph-type equation above becomes especially transparent when expanded in spatial modes. Let $\{\phi_n\}$ be Laplacian eigenfunctions on $\Omega$, satisfying
with boundary conditions compatible with the no-flux constraint, and write
Then each mode evolves independently according to
Under the hypotheses of Proposition 5.2, the modal amplitudes $b_n(t)$ satisfy
So the repair spectrum is geometry-dependent. Long-wavelength modes, corresponding to small $\lambda_n$, relax differently from short-wavelength modes, corresponding to large $\lambda_n$. In particular, the discriminant
controls whether the return to admissibility is monotone or oscillatory. When $\Delta_n < 0$, the $n$-th mode exhibits damped ringing before settling.
That point is important. The first-order borrowing flows of Section 4 force monotone decay of the total excess. The causal extension permits more structure: a localised surplus may launch a travelling repair wave, and sufficiently confined or high-frequency modes may overshoot before the repair settles. This is not a relaxation of conservation, but a refinement of how conservation becomes dynamically legible.
A confined geometry is therefore not merely a static constraint on the support of the repair, as in the Casimir-style picture. It also changes the spectrum $\{\lambda_n\}$, thereby changing which borrowing modes are available, how quickly they respond, and whether the repair is smooth or oscillatory. The narrower the geometry, the larger the relevant modal eigenvalues, and the more sharply time-resolved the redistribution can become.
The causal extension turns the borrowing picture from a purely geometric law of repair into a transport theory of repair. It predicts quantities that the static and instantaneous formulations do not specify:
These are not merely reinterpretive. They imply that two systems with the same total excess $\beta(0)$ need not repair that excess in the same observable way. A large diffuse system may show a delayed, front-like redistribution; a tightly confined system may show immediate modal participation and, in some regimes, damped oscillatory repair. In this sense the borrowing framework begins to predict not just whether an excess can be hidden below detectability, but when, where, and in what spectral order that hidden accounting must become visible.
With the causal extension in place, the framework suggests a more precise way to speak about quantum vacuum fluctuations. A fluctuation need not be treated as a literal violation of conservation. It may instead be viewed as a transient excess relative to the admissible band, measured by
together with a repair process governed by the borrowing dynamics
On this view, conservation remains exact, but its restoration may be temporally unresolved, spatially diffuse, or delayed by the finite time required for borrowing currents to form. What appears as a fleeting local surplus is simply a regime in which the repair has not yet become operationally visible.
A natural temporal example is the Lamb shift. For hydrogen, the measured splitting between the $2S_{1/2}$ and $2P_{1/2}$ levels is approximately
If the repair law is taken in the exponential form $f(\beta)=\beta$, then
Identifying the characteristic repair rate with the natural quantum scale $\kappa \sim \beta_0/\hbar$ gives
Thus the Lamb shift may be read as the observable imprint of a vacuum-scale excess whose repair is not instantaneously legible, but emerges only after sufficient temporal accumulation.
The Casimir effect is the geometric analogue. In the continuous borrowing law, the admissibility condition depends on the integrated contribution over the available domain. When the vacuum is confined between conducting plates separated by distance $a$, the support over which the repair can be distributed is sharply restricted. The characteristic vacuum energy scale then varies as $\hbar c/a$, and the resulting pressure varies as $a^{-4}$:
At a separation of $a = 10\,\mathrm{nm}$, this yields
about one atmosphere. In this setting the excess cannot remain diffusely hidden across a large support; the confinement drives the borrowing density upward until the repair appears as a macroscopic force.
These two cases give the formalism concrete physical legs. The Lamb shift is a case of temporally accumulated repair, while the Casimir effect is a case of geometrically forced repair. In both, the vacuum is not escaping conservation; it is inhabiting a regime in which conservation is exact but the excess $\beta$ becomes visible only once the distributed repair crosses a temporal or spatial threshold of detectability.
The picture assembled above is now fairly sharp, and it is worth pausing to draw out the threads.
The geometry of conservation. Time-translation symmetry constrains admissible evolution to an energy shell, or more practically, to a tolerance band $\mathcal{C}_\tau$ around it. A state outside that band carries an excess $\beta$. To first order, restoring admissibility is a variational problem: the system must move opposite the energy gradient.
Distributed repair. If the system is structured into subsystems or distributed over a domain, the repair need not be localised. Other parts of the system may absorb the burden. In the discrete setting this appears as borrowing from finite reserves $R_i$; in the continuous setting, as borrowing from a reserve density $R(x)$. The reserve $R_i = r_i \|g_i\|$ quantifies each part's lending capacity: how far it is allowed to move, weighted by how much that movement matters energetically.
Repair dynamics. Once dynamics is introduced, borrowing becomes a flow. At the integrated level the excess $\beta(t)$ still decays toward zero, but the local route back need not be instantaneous. The causal extension separates repair rate from repair availability: borrowing currents relax on a time scale $\tau_J$, disturbances propagate with characteristic speed $c = \sqrt{D/\tau_J}$, and confined modes may settle monotonically or with damped ringing. The temporal profile of the total repair remains governed by the choice of response function $f$, ranging from finite-time extinction to exponential relaxation.
What this is not. Nothing here relaxes true conservation of energy for the full closed system. The point is subtler. A local or component-wise excess need not be resolved locally. Conservation, viewed geometrically, permits distributed restoration. What looks like a local near-violation may simply be deferred accounting across a larger coupled structure.
That is the sense in which one may speak of distributed energy borrowing: not as a loophole in conservation, but as a first-order theory of how systems outside an admissible energy band can return to it by coordinated structural response.
Proofs of all propositions. Each relies primarily on the Cauchy–Schwarz inequality and standard Lagrange multiplier arguments.
Statement. Under the hypotheses of Proposition 1.1, the unique minimum-norm perturbation satisfying $\nabla H(w)^\top u = -\beta$ is
Statement. Under the hypotheses of Proposition 2.1, first-order restoration occurs whenever $\sum_{i=1}^m \alpha_i \geq \beta$, with $\alpha_i := -g_i^\top u_i$.
Statement. Under the hypotheses of Proposition 2.2, if $R_i := r_i\|g_i\|$ and $\sum_i R_i \geq \beta$, then a first-order repair is feasible; if $\sum_i R_i < \beta$, no repair satisfying $\|u_i\| \leq r_i$ exists to first order.
Statement. Under the hypotheses of Proposition 2.3, the minimum-cost repair is
Statement. Under the hypotheses of Proposition 3.1, a field perturbation restores admissibility whenever $\int_\Omega (-g(x) \cdot u(x))\, dx \geq \beta$.
Statement. Under the hypotheses of Proposition 3.2, the minimum-$L^2$-norm repair is $u^\star(x) = -(\beta / \|g\|^2_{L^2})\,g(x)$, with $\|u^\star\|_{L^2} = \beta/\|g\|_{L^2}$.
Statement. Under the hypotheses of Proposition 3.3, if $\int_\Omega R(x)\, dx \geq \beta$ then a first-order repair is feasible; if $\int_\Omega R(x)\, dx < \beta$, none exists under the pointwise bound $\|u(x)\| \leq r(x)$.
Statement. Under the hypotheses of Proposition 4.1, $\dot\beta(t) = -\kappa f(\beta(t))$ whenever $\beta(t) > 0$.
Statement. Under the hypotheses of Proposition 4.2, $\tfrac{d}{dt}\beta[w(t)] = -\kappa f(\beta[w(t)])$ whenever $\beta[w(t)] > 0$.
Statement. Under the hypotheses of Proposition 5.1, the total excess obeys $\dot{\beta}(t) = -\Gamma \beta(t)$ and hence $\beta(t) = \beta(0)e^{-\Gamma t}$.
Statement. Under the hypotheses of Proposition 5.2, the excess density satisfies $\tau_J \partial_{tt} b + (1+\Gamma\tau_J)\partial_t b = D \Delta b - \Gamma b$.
Statement. Under the hypotheses of Proposition 5.3, the modal amplitudes satisfy $\tau_J \ddot b_n + (1+\Gamma\tau_J)\dot b_n + (D\lambda_n+\Gamma)b_n = 0$.