State-Preserving Growth: Invariants in Expansion

• State-preserving growth is a concept where systems expand while rigorously maintaining key invariants such as energy laws, balanced distributions, or functional contracts.
• In tumour models and sparse mixture-of-experts architectures, tailored discretization schemes and function-preserving techniques ensure the operational fidelity during system expansion.
• Economic and discrete applications leverage invariant distributions or combinatorial invariants to achieve sustainable growth while preventing uncontrolled system state enlargement.

State-preserving growth denotes a mode of expansion in which a system grows, scales, or evolves while retaining the structural properties that define an admissible state. In recent arXiv usage, the preserved object varies sharply by field: conserved mass and dissipative energy laws in diffuse-interface tumour models, learned inter-module interfaces in sparse mixture-of-experts LLMs, invariant firm-size tails on balanced growth paths, and compact authenticated summaries that prevent linear on-chain state blow-up in automated market makers (Soenjaya et al., 17 Sep 2025, Shravan, 5 Jun 2026, Malevergne et al., 2010, Michel et al., 2024).

1. Semantic range and unifying idea

The term has no single canonical definition across disciplines. In tumour-growth numerics, it refers to exact preservation of mass conservation and energy dissipation, together with boundedness or maximum-bound principles for state variables. In large-model systems, it means preserving learned interfaces during architectural expansion. In economics and spatial demography, it denotes growth paths that keep a distributional state invariant or pull it back toward balance. In transaction systems and graph processes, it refers to scaling while preventing uncontrolled state-size growth or while leaving prescribed combinatorial invariants unchanged (Soenjaya et al., 17 Sep 2025, Shravan, 5 Jun 2026, Malevergne et al., 2010, Kripfganz et al., 15 Jan 2026, Michel et al., 2024, Erdős et al., 2022).

Domain	Preserved state	Growth mechanism
Diffuse-interface tumour models	Mass conservation, energy dissipation, bounds or $L^\infty$ control	Aggregation, chemotaxis, tumour evolution
Sparse MoE LLMs	Learned interfaces, routing health, activation-memory flatness, bounded optimizer state	Dense$\rightarrow$MoE, shallow$\rightarrow$deep, few$\rightarrow$many experts
Firm-size and population growth	Invariant tail law or balanced cross-state distribution	Balanced growth, conditional convergence, spillovers
AMMs and graph growth	Controlled ledger state or preserved existing degrees	Traffic scaling, DPG vertex insertion

This suggests a unifying template: growth is permitted provided invariants, interface contracts, or feasibility constraints are preserved. What changes across fields is the ontology of the preserved state. In PDEs it is thermodynamic or invariant-region structure; in ML systems it is the functional contract between modules; in economics it is the shape of a distribution or a balanced allocation rule; in discrete systems it is a ledger or degree sequence.

2. Thermodynamic and bounded-state growth in tumour models

In diffuse-interface tumour-growth models, state-preserving growth is formulated most explicitly as exact compatibility between the continuous PDE structure and the fully discrete solver. A representative system couples a Cahn–Hilliard-type equation for tumour cell volume fraction $u$ with a nonlinear reaction-diffusion equation for nutrient concentration $n$, with chemical potentials

$$\mu = -\epsilon^2 \Delta u + \lambda u - \chi_0 n + f'(u), \qquad \sigma = \delta^{-1} n - \chi_0 u,$$

and free energy

$$\mathcal{E}[u,n] := \int_{\Omega} \left( \frac{\epsilon^2}{2}|\nabla u|^2 + \frac{\lambda}{2}u^2 + f(u) + \frac{1}{2\delta}n^2 - \chi_0\,u\,n \right)\,dx .$$

The continuous system satisfies

$$\frac{d}{dt}\,\mathcal{E}[u,n] = -\|\nabla \mu\|_{L^2(\Omega)}^2 - \|\nabla \sigma\|_{L^2(\Omega)}^2 - \| \sqrt{P(u)}(\sigma-\mu)\|_{L^2(\Omega)}^2 \le 0,$$

together with conservation of $\int_\Omega (u+n)\,dx$. The fully discrete SAV-based finite-element scheme is linear, unconditionally energy-stable, mass-preserving, and first-order accurate in time with optimal-order accuracy in space; under suitable regularity assumptions it satisfies

$\rightarrow$0

and also derives $\rightarrow$1 estimates that control nonlinear growth dynamics (Soenjaya et al., 17 Sep 2025).

A second formulation incorporates surrounding fluid through Darcy’s law,

$\rightarrow$2

and couples advection, degenerate mobilities, chemotaxis, and reaction terms. Its continuous energy law contains the additional dissipation $\rightarrow$3, while the discrete upwind DG plus convex-splitting scheme inherits mass conservation, pointwise bounds $\rightarrow$4, and a discrete energy inequality. The paper emphasizes that these properties are not incidental numerical side effects but the organizing principle of the discretization: local incompressibility, monotone upwind fluxes, and special mobility splittings are chosen precisely so that the discrete dynamics remains within the physically admissible state set (Acosta-Soba et al., 20 May 2026).

A related phase-field model with extracellular matrix degradation generalizes the preserved state further. There the evolving variables are tumour fraction $\rightarrow$5, necrotic fraction $\rightarrow$6, nutrient $\rightarrow$7, MDE $\rightarrow$8, and ECM density $\rightarrow$9. The numerical objective is to preserve the maximum bound principle for $\rightarrow$0 and $\rightarrow$1, preserve bounds for $\rightarrow$2, $\rightarrow$3, and $\rightarrow$4, and keep $\rightarrow$5 in $\rightarrow$6. The ETDRK construction uses stabilizing linear operators for the PDE components, trapezoidal updates for the ODE components, and a single cut-off $\rightarrow$7 after each ETD stage. Under the stated parameter conditions, it yields unconditional discrete MBP for nutrient and MDE, while $\rightarrow$8 and $\rightarrow$9 remain bounded under the stated time-step restrictions (Huang et al., 24 Mar 2025).

Taken together, these works define state-preserving growth in tumour modelling as growth under thermodynamic consistency, invariant-region control, and discretization-induced nonphysical-state exclusion. This suggests that, in this literature, the phrase is less about monotone increase in tumour size than about preserving admissible evolution during aggregation, chemotaxis, necrosis, and fluid-mediated transport.

3. Interface-preserving scaling in sparse mixture-of-experts models

In large-scale language-model systems, the term is used in a fundamentally different but equally technical sense. “State-preserving growth” is defined as “the preservation of learned interfaces” across model expansions: a larger model should begin from a state that already functions, rather than relearning what the smaller model knew. The preserved state includes learned functions and weights, routing distributions and router state, optimizer state where applicable, and layer-local or recurrent state. LightningLM 0.1V implements this principle in a four-stage lineage: a 2B dense seed, a 5B MoE, a 9B MoE, and a 120B MoE with 460 routed experts plus 1 shared expert under top-12 routing. Verified counts rise from 1.78B active and stored parameters at the dense seed to 5.93B active per token and 118.67B stored parameters at 120B; the released 120B checkpoint reports trailing-100 training loss $\rightarrow$0 at step $\rightarrow$1 on a single 8-GPU node (Shravan, 5 Jun 2026).

The preservation rules are concrete. For dense$\rightarrow$2MoE conversion, the trained dense FFN is copied exactly into an always-active shared expert, while routed experts are initialized by overlapping random partitions of the dense intermediate neurons. For shallow$\rightarrow$3deep expansion, the 8$\rightarrow$420 layer mapping “1–8, 1–4, 1–8” is selected so that the deeper stack still terminates on the original terminal layer $\rightarrow$5, preserving the hidden-state distribution expected by the LM head. For few$\rightarrow$6many experts, each of the 20 source experts is cloned into 23 variants, with “drop-upcycling” reinitializing half of intermediate positions while reusing the same mask across gate/up/down weights. Routing symmetry is then broken by router tiling plus small bias noise $\rightarrow$7, together with probabilistic early selection so that near-duplicate clones receive gradients before hard top-$\rightarrow$8 routing collapses onto a single winner (Shravan, 5 Jun 2026).

The paper also treats state preservation as a systems constraint. Reversibility keeps activation memory approximately flat with depth through the midpoint update

$\rightarrow$9

with backward reconstruction instead of stored activations. TQP bounds optimizer-state growth through

$u$0

so that Adam moments are carried only on adapters. Since experts alone exceed 100B parameters, naive Adam would require approximately $u$1 GB for expert moments, whereas the adapter-only optimizer state is approximately $u$2 GB, a reduction of about $u$3 (Shravan, 5 Jun 2026).

A notable feature of this usage is its emphasis on silent failure. Spectral or SVD upcycling destroys the dense function and yields “near-random cold loss” at step 1; non-strict checkpoint loading can leave target parameters random while cheap checks still pass; hard top-$u$4 routing over near-duplicate clones produces dead experts and clone-family collapse. The corrective actions—strict target-keyspace loading, function-preserving initialization, router retuning, and probabilistic early selection—are presented as state-preserving requirements rather than optional heuristics. Here growth is architectural and parametric, but what is preserved is the operational contract among modules.

4. Distribution-preserving and convergence-preserving growth in economics and demography

In stochastic growth theory, state-preserving growth is explicitly distributional. In a reduced-form model of firm dynamics, firms are born according to $u$5, entrants arrive with initial size $u$6, incumbents follow geometric Brownian motion,

$u$7

and exit occurs either at an efficiency threshold or by size-independent hazard $u$8. The upper tail of the firm-size distribution is asymptotically Pareto with exponent

$u$9

Zipf’s law, $n$0, holds if and only if

$n$1

That equality is the balance condition equating the effective growth of incumbents and the growth of investment into entrants, and it is also the condition for maximum sustainable growth of aggregate size. In this setting, state-preserving growth means that the economy scales while preserving the cross-sectional shape of the firm-size distribution (Malevergne et al., 2010).

A different mechanism appears in the human-capital redistribution model. Individual human capital evolves by a stochastic multiplicative process; when $n$2, individual human capital is destroyed in the long run. Taxation at rate $n$3 with administrative loss rate $n$4 pools risk through equal redistribution of the surviving public good. The aggregate law is

$n$5

and the paper interprets the resulting improvement in long-run growth as a portfolio effect. Progressive taxation yields the highest growth-maximizing $n$6, proportional taxation is second, and regressive taxation is last; by contrast, a government maximizing revenue under the model may prefer regressive taxation because it supports a higher growth-optimal tax rate, even though it delivers lower growth (Lorenz et al., 2012).

Spatial demography uses the term in yet another way. In a dynamic spatial panel for 49 U.S. states over 1965–2017, with data-inferred network $n$7, heterogeneous slopes, IV estimation, and interactive fixed effects, state-preserving growth refers to growth dynamics that pull the cross-state population distribution back toward balanced levels. The paper reports that $n$8 of states have $n$9 in $$\mu = -\epsilon^2 \Delta u + \lambda u - \chi_0 n + f'(u), \qquad \sigma = \delta^{-1} n - \chi_0 u,$$0, while a smaller cluster mildly diverges. Under MGIV with $$\mu = -\epsilon^2 \Delta u + \lambda u - \chi_0 n + f'(u), \qquad \sigma = \delta^{-1} n - \chi_0 u,$$1, the mean convergence coefficient is $$\mu = -\epsilon^2 \Delta u + \lambda u - \chi_0 n + f'(u), \qquad \sigma = \delta^{-1} n - \chi_0 u,$$2 with implied half-life approximately $$\mu = -\epsilon^2 \Delta u + \lambda u - \chi_0 n + f'(u), \qquad \sigma = \delta^{-1} n - \chi_0 u,$$3 years, and indirect spillover effects are roughly one-third of total impacts. The preferred network is sparse, about $$\mu = -\epsilon^2 \Delta u + \lambda u - \chi_0 n + f'(u), \qquad \sigma = \delta^{-1} n - \chi_0 u,$$4 dense, but diffusion reaches beyond contiguous neighbors through higher powers of $$\mu = -\epsilon^2 \Delta u + \lambda u - \chi_0 n + f'(u), \qquad \sigma = \delta^{-1} n - \chi_0 u,$$5 (Kripfganz et al., 15 Jan 2026).

Across these economic and demographic uses, the preserved state is not a conserved scalar but a balanced distributional configuration. This suggests that state-preserving growth, in these settings, is a doctrine of scale without distributional destabilization.

5. State-size control and combinatorial invariants in discrete systems

In automated market makers, state-preserving growth is tied to ledger scalability. ammBoost splits functionality between a minimal L1 TokenBank contract and a purpose-built sidechain. Swaps, mints, burns, and collects are executed on the sidechain and summarized at fixed epochs into concise per-user and per-position aggregates; the summary is then authenticated to L1 through a threshold BLS signature. If $$\mu = -\epsilon^2 \Delta u + \lambda u - \chi_0 n + f'(u), \qquad \sigma = \delta^{-1} n - \chi_0 u,$$6 is the number of sidechain operations in an epoch, $$\mu = -\epsilon^2 \Delta u + \lambda u - \chi_0 n + f'(u), \qquad \sigma = \delta^{-1} n - \chi_0 u,$$7 the number of active users, and $$\mu = -\epsilon^2 \Delta u + \lambda u - \chi_0 n + f'(u), \qquad \sigma = \delta^{-1} n - \chi_0 u,$$8 the number of positions touched, the reduction in L1 writes is from $$\mu = -\epsilon^2 \Delta u + \lambda u - \chi_0 n + f'(u), \qquad \sigma = \delta^{-1} n - \chi_0 u,$$9 to $$\mathcal{E}[u,n] := \int_{\Omega} \left( \frac{\epsilon^2}{2}|\nabla u|^2 + \frac{\lambda}{2}u^2 + f(u) + \frac{1}{2\delta}n^2 - \chi_0\,u\,n \right)\,dx .$$0, with compression ratio

$$\mathcal{E}[u,n] := \int_{\Omega} \left( \frac{\epsilon^2}{2}|\nabla u|^2 + \frac{\lambda}{2}u^2 + f(u) + \frac{1}{2\delta}n^2 - \chi_0\,u\,n \right)\,dx .$$1

After Sync is confirmed on L1, the sidechain prunes temporary meta-blocks and retains only permanent summary-blocks. In the reported evaluation, throughput rises to $$\mathcal{E}[u,n] := \int_{\Omega} \left( \frac{\epsilon^2}{2}|\nabla u|^2 + \frac{\lambda}{2}u^2 + f(u) + \frac{1}{2\delta}n^2 - \chi_0\,u\,n \right)\,dx .$$2 tx/s at $$\mathcal{E}[u,n] := \int_{\Omega} \left( \frac{\epsilon^2}{2}|\nabla u|^2 + \frac{\lambda}{2}u^2 + f(u) + \frac{1}{2\delta}n^2 - \chi_0\,u\,n \right)\,dx .$$3 daily volume, approximately $$\mathcal{E}[u,n] := \int_{\Omega} \left( \frac{\epsilon^2}{2}|\nabla u|^2 + \frac{\lambda}{2}u^2 + f(u) + \frac{1}{2\delta}n^2 - \chi_0\,u\,n \right)\,dx .$$4 Uniswap daily traffic; for $$\mathcal{E}[u,n] := \int_{\Omega} \left( \frac{\epsilon^2}{2}|\nabla u|^2 + \frac{\lambda}{2}u^2 + f(u) + \frac{1}{2\delta}n^2 - \chi_0\,u\,n \right)\,dx .$$5, gas cost reduction versus Uniswap on Sepolia is $$\mathcal{E}[u,n] := \int_{\Omega} \left( \frac{\epsilon^2}{2}|\nabla u|^2 + \frac{\lambda}{2}u^2 + f(u) + \frac{1}{2\delta}n^2 - \chi_0\,u\,n \right)\,dx .$$6, and mainchain state growth reduction is $$\mathcal{E}[u,n] := \int_{\Omega} \left( \frac{\epsilon^2}{2}|\nabla u|^2 + \frac{\lambda}{2}u^2 + f(u) + \frac{1}{2\delta}n^2 - \chi_0\,u\,n \right)\,dx .$$7 versus Sepolia and $$\mathcal{E}[u,n] := \int_{\Omega} \left( \frac{\epsilon^2}{2}|\nabla u|^2 + \frac{\lambda}{2}u^2 + f(u) + \frac{1}{2\delta}n^2 - \chi_0\,u\,n \right)\,dx .$$8 versus Uniswap on production Ethereum. The paper formalizes this as preserving AMM safety and liveness while preventing state growth proportional to transaction count (Michel et al., 2024).

Graph theory supplies a purely combinatorial analogue. In Degree-Preserving Growth, a new vertex is inserted into a simple graph by choosing a $$\mathcal{E}[u,n] := \int_{\Omega} \left( \frac{\epsilon^2}{2}|\nabla u|^2 + \frac{\lambda}{2}u^2 + f(u) + \frac{1}{2\delta}n^2 - \chi_0\,u\,n \right)\,dx .$$9-matching $$\frac{d}{dt}\,\mathcal{E}[u,n] = -\|\nabla \mu\|_{L^2(\Omega)}^2 - \|\nabla \sigma\|_{L^2(\Omega)}^2 - \| \sqrt{P(u)}(\sigma-\mu)\|_{L^2(\Omega)}^2 \le 0,$$0, deleting the $$\frac{d}{dt}\,\mathcal{E}[u,n] = -\|\nabla \mu\|_{L^2(\Omega)}^2 - \|\nabla \sigma\|_{L^2(\Omega)}^2 - \| \sqrt{P(u)}(\sigma-\mu)\|_{L^2(\Omega)}^2 \le 0,$$1 matched edges, and connecting the new vertex to all $$\frac{d}{dt}\,\mathcal{E}[u,n] = -\|\nabla \mu\|_{L^2(\Omega)}^2 - \|\nabla \sigma\|_{L^2(\Omega)}^2 - \| \sqrt{P(u)}(\sigma-\mu)\|_{L^2(\Omega)}^2 \le 0,$$2 endpoints. The new vertex has degree $$\frac{d}{dt}\,\mathcal{E}[u,n] = -\|\nabla \mu\|_{L^2(\Omega)}^2 - \|\nabla \sigma\|_{L^2(\Omega)}^2 - \| \sqrt{P(u)}(\sigma-\mu)\|_{L^2(\Omega)}^2 \le 0,$$3, while every existing vertex preserves its degree because it loses one incident edge and gains one edge to the new vertex. In degree-sequence language, if $$\frac{d}{dt}\,\mathcal{E}[u,n] = -\|\nabla \mu\|_{L^2(\Omega)}^2 - \|\nabla \sigma\|_{L^2(\Omega)}^2 - \| \sqrt{P(u)}(\sigma-\mu)\|_{L^2(\Omega)}^2 \le 0,$$4 is a non-increasing graphic sequence and $$\frac{d}{dt}\,\mathcal{E}[u,n] = -\|\nabla \mu\|_{L^2(\Omega)}^2 - \|\nabla \sigma\|_{L^2(\Omega)}^2 - \| \sqrt{P(u)}(\sigma-\mu)\|_{L^2(\Omega)}^2 \le 0,$$5 with $$\frac{d}{dt}\,\mathcal{E}[u,n] = -\|\nabla \mu\|_{L^2(\Omega)}^2 - \|\nabla \sigma\|_{L^2(\Omega)}^2 - \| \sqrt{P(u)}(\sigma-\mu)\|_{L^2(\Omega)}^2 \le 0,$$6, then $$\frac{d}{dt}\,\mathcal{E}[u,n] = -\|\nabla \mu\|_{L^2(\Omega)}^2 - \|\nabla \sigma\|_{L^2(\Omega)}^2 - \| \sqrt{P(u)}(\sigma-\mu)\|_{L^2(\Omega)}^2 \le 0,$$7 is graphic if and only if some realization of $$\frac{d}{dt}\,\mathcal{E}[u,n] = -\|\nabla \mu\|_{L^2(\Omega)}^2 - \|\nabla \sigma\|_{L^2(\Omega)}^2 - \| \sqrt{P(u)}(\sigma-\mu)\|_{L^2(\Omega)}^2 \le 0,$$8 has a matching of size $$\frac{d}{dt}\,\mathcal{E}[u,n] = -\|\nabla \mu\|_{L^2(\Omega)}^2 - \|\nabla \sigma\|_{L^2(\Omega)}^2 - \| \sqrt{P(u)}(\sigma-\mu)\|_{L^2(\Omega)}^2 \le 0,$$9. The paper further identifies the potential matching number $\int_\Omega (u+n)\,dx$0 and forcible matching number $\int_\Omega (u+n)\,dx$1 as sequence-level capacities for such insertions (Erdős et al., 2022).

These two uses sit at different levels of abstraction but share a common logic. ammBoost preserves operational correctness while changing the storage geometry of transaction execution; DPG preserves the entire degree profile of existing vertices while adding a new one. In both cases, growth is feasible only because the update rule is engineered to leave designated state variables invariant.

6. Mathematical extensions, misconceptions, and open directions

The phrase should not be read as implying static behavior. In conservative surface dynamics, area preservation can coexist with arbitrarily fast growth of periodic-point counts. For open subsets of real-analytic Hamiltonian diffeomorphisms of $\int_\Omega (u+n)\,dx$2, maps with

$\int_\Omega (u+n)\,dx$3

are dense for any given sequence $\int_\Omega (u+n)\,dx$4; in smooth area-preserving dynamics on closed surfaces, the analogous fast growth is $\int_\Omega (u+n)\,dx$5-typical in an open set (Asaoka, 2016). Preservation of measure or symplectic structure therefore does not suppress combinatorial or orbit-complexity growth.

A second misconception is that state preservation must always be exact positivity or exact boundedness. The tumour-model literature distinguishes several levels: exact conservation of $\int_\Omega (u+n)\,dx$6, discrete energy decay, pointwise bounds $\int_\Omega (u+n)\,dx$7, continuous MBP for transformed variables, and $\int_\Omega (u+n)\,dx$8 bounds derived by analysis rather than by hard constraints (Soenjaya et al., 17 Sep 2025, Acosta-Soba et al., 20 May 2026, Huang et al., 24 Mar 2025). The preserved state is therefore model-dependent: exact invariant, invariant region, modified-energy law, or boundedness estimate.

More abstractly, geometry offers a threshold formulation. For a complete Riemannian manifold $\int_\Omega (u+n)\,dx$9 and conformal change $\rightarrow$00, completeness is preserved if and only if $\rightarrow$01 is not $\rightarrow$02 on any $\rightarrow$03-divergent curve. In radial terms, decay like $\rightarrow$04 is critical: $\rightarrow$05 still preserves completeness, whereas $\rightarrow$06 or $\rightarrow$07 yields incompleteness and even finite $\rightarrow$08-diameter (Dirmeier, 2012). This provides a mathematically sharp example of state-preserving growth or decay controlled by an integral threshold rather than a conservation law.

Open directions in the current literature are correspondingly domain-specific. In tumour numerics, proposed extensions include higher-order time stepping such as BDF2-SAV and Crank–Nicolson-SAV, adaptive meshes, logarithmic potentials, and multi-species couplings such as Cahn–Hilliard–Brinkman, Navier–Stokes, or Keller–Segel variants (Soenjaya et al., 17 Sep 2025). In Darcy-coupled DG schemes, the stated limitations include lowest-order scalar spaces, mesh orthogonality assumptions, and the absence of an error analysis, with anisotropic media, 3D simulations, and richer couplings listed as extensions (Acosta-Soba et al., 20 May 2026). In large-model systems, open questions include formal capacity control at very high expert counts, mechanistic conditions under which periodic merge-and-reset succeeds or diverges at scale, and generalization of DroPE in recurrence-backbone hybrids (Shravan, 5 Jun 2026). In spatial growth econometrics, external validity beyond 1965–2017 U.S. data requires replicating data-driven network recovery and IV with interactive fixed effects (Kripfganz et al., 15 Jan 2026).

State-preserving growth is therefore best understood not as a single theory but as a recurrent research pattern: permit expansion only through update rules that preserve the state variables, invariants, or interface contracts deemed structurally essential. Across PDEs, ML systems, economics, spatial panels, blockchains, graphs, dynamics, and geometry, the phrase marks the same methodological ambition: scaling without loss of the laws that make the evolving state interpretable.