On the Necessity of a Liquid Substrate for Mesh Intelligence

# On the Necessity of a Liquid Substrate for Mesh Intelligence
Source: [https://arxiv.org/html/2606.28413](https://arxiv.org/html/2606.28413)
\(June 2026\)

###### Abstract

A mesh of sovereign agents has no center: no shared clock, no shared model, and no coordinator to gather data or retrain\. Its competence rests on each agent folding the projections its peers emit into a single internal state, online, from observations that arrive at irregular, unscheduled times, on a substrate whose weights it cannot retrain\. Any one of these constraints is tractable on its own; folding optimally under all three at once is not\. We ask what such a substrate must be, and prove two necessary conditions from one model of a self\-evolving latent observed at irregular, exogenous times\. Because the latent changes, its optimal estimator is time\-varying: an adaptive timescale is necessary, and every fixed\-gain filter is strictly suboptimal\. And because arrivals are clock\-free, the optimal estimate depends on the elapsed gap between them, which no gap\-blind network recovers at any width or depth\. This second condition is*capacity\-independent*: scale cannot substitute for the missing dependence\. The two conditions intersect in the continuous\-time*liquid*class\. An LSTM satisfies the first, a fixed continuous\-time filter the second, and a multi\-timescale liquid network both\. Synthetic experiments confirm each: the network attains the timescale, and the separation is computed exactly\. The characterization is necessary, not sufficient, and binds fixed\-weight substrates: a network free to retrain reaches the class by other means\. Proved per agent, the necessity binds every agent of a mesh, a structural condition on mesh intelligence\.

## 1Introduction

In a*mesh*, sovereign agents infer and learn with no center\. Each agent keeps a private, evolving state\. It*admits*only what it finds relevant in what its peers*emit*, and emits only typed projections of that state, never the state itself\. No agent and no coordinator holds the whole: the intelligence lies in what each agent admits from its peers and integrates into its own state\. We call this*mesh intelligence*, and it rests entirely on that per\-agent integration\.

Three constraints make that integration hard, and a mesh imposes all three at once\. The observations arrive at irregular times that no agent can schedule\. The quantity each describes keeps moving, so the stream is*non\-stationary*and any fixed summary goes stale\. And the agent must operate*online*, on a substrate whose weights it cannot retrain, because a live mesh has no training loop, no dataset, and no objective to optimize\. Each constraint alone is tractable; their conjunction is what makes optimal folding hard\. Because arrivals are clocked, the elapsed gap between them shapes each update111Information\-theoretically: under exogenous \(state\-independent\) sampling the elapsed gap is non\-informative about the latent’s*value*s⋆s^\{\\star\}\(I​\(Δ;s⋆\)=0I\(\\Delta;s^\{\\star\}\)=0\[[23](https://arxiv.org/html/2606.28413#bib.bib23)\]\), yet it constrains the prior covariance, hence the gain: the covariance channel Thm[1](https://arxiv.org/html/2606.28413#Thmtheorem1)shows is necessary information no gap\-blind estimator can forgo \(the gap sharpens the posterior given noisy observations while saying nothing about where the latent is\)\.I​\(Δ;s⋆\)\>0I\(\\Delta;s^\{\\star\}\)\>0would require informative, state\-modulated sampling, which we do not assume\.; because the target keeps moving, the substrate must both retain past structure and follow new change\.

Coupling has two levels, and only one is ours\. A*content*gate decides*which*of a peer’s projections to admit\[[2](https://arxiv.org/html/2606.28413#bib.bib2)\]; that is a separate problem\. The*temporal*substrate decides*how*the admitted observations accumulate into the agent’s state, and no prior work settles what it must be\. We study it in the regime a live mesh forces: weights fixed once and never retrained, with only the state evolving as data arrives, the reservoir or Liquid State Machine view\[[6](https://arxiv.org/html/2606.28413#bib.bib6),[7](https://arxiv.org/html/2606.28413#bib.bib7)\]\. This is the mesh’s operating condition, not a modeling convenience: an agent cannot know its future peers, the latent it tracks is never observed, and no coordinator gathers data or ships new weights\. The question is then sharp:*what must a fixed\-weight substrate be to integrate a non\-stationary, irregularly timed stream optimally?*

We answer with two necessities, both read from one model of a self\-evolving latent observed at irregular times \(§[4](https://arxiv.org/html/2606.28413#S4)\)\. First, because the latent moves, the optimal estimator is time\-varying, so the substrate needs an*adaptive*timescale; a fixed\-gain filter is strictly suboptimal \(Prop\.[1](https://arxiv.org/html/2606.28413#Thmproposition1); Lemma[1](https://arxiv.org/html/2606.28413#Thmlemma1)gives the explicit one\-pole case\)\. Second, because the same latent is seen at clock\-free times, the optimal estimate depends on the elapsed gap between arrivals, and*no*gap\-blind network matches it at any width or depth \(Thm[1](https://arxiv.org/html/2606.28413#Thmtheorem1)\)\. This second result is*capacity\-independent*: added size cannot close the gap\. Together the two properties define the*liquid*class\[[3](https://arxiv.org/html/2606.28413#bib.bib3),[4](https://arxiv.org/html/2606.28413#bib.bib4)\]: an LSTM has the first, a fixed continuous\-time filter the second, and a multi\-timescale liquid network has both \(Fig\.[1](https://arxiv.org/html/2606.28413#S1.F1)\)\. We confirm the adaptive\-timescale necessity against fixed\-timescale and naive\-adaptive baselines, and compute the gap\-awareness separation exactly \(§[6](https://arxiv.org/html/2606.28413#S6)\)\.

![Refer to caption](https://arxiv.org/html/2606.28413v1/x1.png)Figure 1:The two necessities in one cell\.Every substrate forms the same updateht=ht−1\+a​\(ot−ht−1\)h\_\{t\}=h\_\{t\-1\}\+a\\,\(o\_\{t\}\-h\_\{t\-1\}\); only the gainaadiffers\.\(a\)A fixed gain is one timescale: strictly suboptimal \(Prop\.[1](https://arxiv.org/html/2606.28413#Thmproposition1)\)\.\(b\)An LSTM\-type gatea=σ​\(W​\[ot,ht−1\]\)a=\\sigma\(W\[o\_\{t\},h\_\{t\-1\}\]\)\(a deliberate reduction; the full cell is richer but gap\-blind all the same\) adapts to content but is indexed by*step*, not elapsed time: it keeps state through*steps*, never sees the gapΔt\\Delta\_\{t\}, and so matches the optimum at*no*width or depth \(Thm\.[1](https://arxiv.org/html/2606.28413#Thmtheorem1)\)\.\(c\)A liquid cell \(CfC/LTC\) admitsΔt\\Delta\_\{t\}into the gain,a=σ​\(W​\[ot,ht−1\]​Δt\)a=\\sigma\(W\[o\_\{t\},h\_\{t\-1\}\]\\,\\Delta\_\{t\}\), adaptive*and*gap\-aware: the necessary class\. Capacity is the wrong axis: long step\-memory does not encode elapsed duration\.Both necessities are proved for a single agent, but a mesh places every agent under the same conditions, so the result binds each one\. The conditions also cannot be evaded within a mesh: no agent can force its peers onto a regular clock, the one case in which the gap stops mattering, and with no center to correct a gap\-blind agent the shortfall is present at every node\. A liquid substrate is therefore required at every agent\. This is a structural condition on mesh intelligence rather than a property of any single filter, obtained by reading the single\-agent theorems across the agents of a mesh\.

This necessity is one layer of a larger characterization\. A mesh couples its agents on two orthogonal axes\. The companion Mesh Inference\[[1](https://arxiv.org/html/2606.28413#bib.bib1)\]treats the*spatial*axis, with the latent held static: whether a center\-free population can recover the collective answer at all\. This paper treats the*temporal*axis, for a single agent: what its substrate must be to keep its own state current as the latent evolves\. With the content gate that decides which observations a receiver admits\[[2](https://arxiv.org/html/2606.28413#bib.bib2)\], the three name the layers a mesh agent runs on:*which*to admit,*whether*the collective can recover the answer, and*how*to track it in time\. We characterize the last, and take the other two as the companions’ domain\.

The result is*scoped, not universal*\. It states what optimal integration requires*among fixed\-weight substrates*\. A network free to retrain, such as GRU\-D or Time\-LSTM, can reach both properties by learning, and a fully trainable learner is not constrained at all\. We also do not claim that a liquid substrate*attains*the optimum, only that nothing outside the liquid class can; whether it does is the empirical question of §[6](https://arxiv.org/html/2606.28413#S6)\.

The ingredients are classical: time\-varying\-gain filtering and change detection\[[17](https://arxiv.org/html/2606.28413#bib.bib17),[18](https://arxiv.org/html/2606.28413#bib.bib18),[19](https://arxiv.org/html/2606.28413#bib.bib19),[21](https://arxiv.org/html/2606.28413#bib.bib21)\], continuous\-time and liquid networks\[[3](https://arxiv.org/html/2606.28413#bib.bib3),[4](https://arxiv.org/html/2606.28413#bib.bib4),[5](https://arxiv.org/html/2606.28413#bib.bib5)\], and reservoir computing\[[6](https://arxiv.org/html/2606.28413#bib.bib6),[7](https://arxiv.org/html/2606.28413#bib.bib7)\]\. What is new is the pairing: two necessities drawn from one model, the second a capacity\-independent separation, whose intersection*defines*the liquid class for fixed\-weight online integration, with the scope drawn honestly\.

## 2Related work

#### Irregularly\-sampled time series\.

The standard route to irregular sampling feeds the elapsed gap to a discrete recurrence: GRU\-D decays the hidden state by the elapsed gap\[[8](https://arxiv.org/html/2606.28413#bib.bib8)\], Time\-LSTM gates on it\[[9](https://arxiv.org/html/2606.28413#bib.bib9)\], and ODE\-RNN / latent\-ODE models evolve the state by a learned ODE between observations\[[10](https://arxiv.org/html/2606.28413#bib.bib10)\]\. These*attain*gap\-awareness, and by Thm[1](https://arxiv.org/html/2606.28413#Thmtheorem1)they must to be optimal, but they do so by*training*weights to the gap\. Our separation concerns what a*fixed\-weight*substrate can do without that training, where a continuous\-time cell is the native realization \(Cor\.[1](https://arxiv.org/html/2606.28413#Thmcorollary1)\)\.

#### Fixed\-weight vs\. trained substrates\.

The fixed\-weight premise also separates our setting from distributed learning, which jointly optimizes a shared model: federated averaging exchanges weights\[[14](https://arxiv.org/html/2606.28413#bib.bib14)\], decentralized SGD averages models with neighbors\[[15](https://arxiv.org/html/2606.28413#bib.bib15)\], and multi\-agent RL centralizes training\[[16](https://arxiv.org/html/2606.28413#bib.bib16)\]\. The substrate here is frozen during operation \(§[3](https://arxiv.org/html/2606.28413#S3)\): no weights are optimized, online or off, so there is nothing to share\.

#### Continuous\-time, liquid, and adaptive\-timescale networks\.

The substrate is a closed\-form continuous\-time network\[[3](https://arxiv.org/html/2606.28413#bib.bib3)\], the closed\-form approximation to the liquid time\-constant ODE\[[4](https://arxiv.org/html/2606.28413#bib.bib4)\]in the neural\-ODE family\[[5](https://arxiv.org/html/2606.28413#bib.bib5)\]\. Unlike a fixed\-rate recurrence, its time constants are input\-dependent and multi\-timescale, and that is what §[4](https://arxiv.org/html/2606.28413#S4)turns on: integrating a self\-evolving latent*requires*an adaptive timescale, the estimation\-theoretic reason a time\-varying\-gain filter dominates a fixed\-gain one\[[17](https://arxiv.org/html/2606.28413#bib.bib17),[18](https://arxiv.org/html/2606.28413#bib.bib18)\]and change must be detected to be followed\[[19](https://arxiv.org/html/2606.28413#bib.bib19)\]\. We do not claim the specific closed\-form network is the*unique*such substrate; any adaptive continuous\-time network satisfies the necessity\. What we add is to make the liquid \(input\-dependent, multi\-timescale\) property load\-bearing, where prior treatments leave it incidental\. From reservoir computing, the Liquid State Machine\[[6](https://arxiv.org/html/2606.28413#bib.bib6)\]and echo\-state network\[[7](https://arxiv.org/html/2606.28413#bib.bib7)\]lineage, we borrow only the fixed\-substrate idea \(a fixed continuous\-time recurrence carrying computation in its dynamics\), not its offline\-trained readout; we keepρ\\rhofixed too, so the only adaptation is of the online state, not the weights \(Remark[1](https://arxiv.org/html/2606.28413#Thmremark1)\)\.

#### Deep state\-space models\.

Structured state\-space models, HiPPO\[[11](https://arxiv.org/html/2606.28413#bib.bib11)\], S4\[[12](https://arxiv.org/html/2606.28413#bib.bib12)\], and the selective\-scan Mamba\[[13](https://arxiv.org/html/2606.28413#bib.bib13)\], are recurrences derived from a continuous\-time model and discretized with a stepΔ\\Delta, and that step is where our characterization bites\. The sameΔ\\Deltacan play two distinct roles: it yields an*adaptive timescale*\(L1\) when it is input\-dependent, but*elapsed\-gap dependence*\(L2\) only when it*is*the gap\. Mamba’s selection makesΔ\\Deltaa function of*content*, not of elapsed time \(Δ=softplus​\(Linear​\(x\)\)\\Delta=\\mathrm\{softplus\}\(\\mathrm\{Linear\}\(x\)\)\); its scalar reduction \(\[[13](https://arxiv.org/html/2606.28413#bib.bib13)\], Thm\. 1\) is the content\-gated recurrenceht=\(1−gt\)​ht−1\+gt​xth\_\{t\}=\(1\{\-\}g\_\{t\}\)h\_\{t\-1\}\+g\_\{t\}x\_\{t\}withgt=σ​\(Linear​\(xt\)\)g\_\{t\}=\\sigma\(\\mathrm\{Linear\}\(x\_\{t\}\)\), precisely the adaptive\-but\-gap\-blind cell of Fig\.[1](https://arxiv.org/html/2606.28413#S1.F1)\(b\)\. On an irregularly\-timed stream it is therefore gap\-blind, and Thm[1](https://arxiv.org/html/2606.28413#Thmtheorem1)binds it: even trained, no width or depth reaches the optimum untilΔ\\Deltais fed the elapsed gap\. A frozen SSM*joins*the liquid class precisely by carrying the gap in its discretization, the continuous\-time route of Cor\.[1](https://arxiv.org/html/2606.28413#Thmcorollary1), a different choice ofΔ\\Deltathan content selection\. Their weights are trained where our necessity is fixed\-weight; the result then tells an SSM designer the one structural property a frozen, deployed model must keep under irregular sampling:Δ\\Deltamust carry the gap, not only the content\. The same holds for a frozen Transformer: an LLM run on an irregularly\-timed stream is step\-indexed, hence gap\-blind, so the separation binds it at any scale; it needs its estimate to depend on the gap, supplied structurally by a liquid substrate or extrinsically by feeding the gap as input \(Cor\.[1](https://arxiv.org/html/2606.28413#Thmcorollary1)\)\.

## 3Setup: the model and the substrate

This section formalizes the setting and the substrate: the mesh \(Def\.[1](https://arxiv.org/html/2606.28413#Thmdefinition1)\), the single\-agent integration problem each of its agents faces by construction \(Def\.[2](https://arxiv.org/html/2606.28413#Thmdefinition2)\), and the fixed\-weight continuous\-time substrate that integrates the resulting asynchronous stream into one internal state\. The substrate is a fixed\-weight mechanism: what moves is the state, never the weights\. Section[4](https://arxiv.org/html/2606.28413#S4)then shows what that substrate*must*be\.

###### Definition 1\(Mesh; mesh intelligence\)\.

A*mesh*is a set of sovereign agents with no coordinator and no shared model\. Each holds a private, evolving cognitive state,*emits*typed projections of it on its own clock \(never the state itself\), and*admits*, through a content gate, the projections it finds relevant from others, integrating them into one internal state\.*Mesh intelligence*is the inference and learning the collective performs from these admissions alone, with no center\. Three properties then hold for*each*agent by construction: its admitted stream arrives at times the emitters set and it cannot schedule \(*exogenous*\); the cognitive states those projections describe keep evolving \(*non\-stationary*\); and its integration substrate is fixed during operation, since no coordinator gathers data, sets an objective, or retrains it\.

An agent thus estimates an evolving latent from a stream of observations arriving at irregular times: in a mesh, the admitted projections of its peers\. Our results concern the integration of whatever stream results; let𝒮​\(t\)\\mathcal\{S\}\(t\)denote it up to timett\. The three properties of Def\.[1](https://arxiv.org/html/2606.28413#Thmdefinition1)are exactly the hypotheses of the model below, so the necessities of §[4](https://arxiv.org/html/2606.28413#S4)hold for*each agent*of a mesh by construction\. The theorems are single\-agent; the mesh is the setting that forces their conditions rather than assuming them\. That is why a liquid substrate is necessary*for mesh intelligence*, and equally why the result stands on its own\.

###### Definition 2\(Latent and observation model\)\.

A single*family*of models underlies the necessities of §[4](https://arxiv.org/html/2606.28413#S4)\. The latents​\(t\)s\(t\)is*self\-evolving*: the uncertainty accumulated over an elapsed interval of lengthΔ\\Delta,G​\(Δ\):=Var⁡\(s​\(t\+Δ\)−s​\(t\)\)G\(\\Delta\):=\\operatorname\{Var\}\\\!\\big\(s\(t\{\+\}\\Delta\)\-s\(t\)\\big\), is*strictly increasing*inΔ\\DeltawithG​\(0\)=0G\(0\)=0\. Two realizations recur, and we name whichever makes a mechanism most transparent: a*change\-point*process \(piecewise\-stationary, change rateλ\>0\\lambda\>0: stable for stretches, with changes at unknown times; over a gapΔ\\Deltaa change has occurred with probability1−e−λ​Δ1\-e^\{\-\\lambda\\Delta\}, soGGrises from0toward the inter\-level variance\), and a*Brownian drift*withG​\(Δ\)=q​ΔG\(\\Delta\)=q\\Delta\. Observations arrive at irregular timest1<t2<⋯t\_\{1\}<t\_\{2\}<\\cdotsdrawn*exogenously*\(independently of the latent path and the noise\), with gapsΔk=tk−tk−1\\Delta\_\{k\}=t\_\{k\}\-t\_\{k\-1\}, asok=s​\(tk\)\+εko\_\{k\}=s\(t\_\{k\}\)\+\\varepsilon\_\{k\},εk\\varepsilon\_\{k\}zero\-mean, varianceσ2\\sigma^\{2\}\. The two legs lean on complementary facets: Prop\.[1](https://arxiv.org/html/2606.28413#Thmproposition1)uses the*change\-point*realization specifically \(regime alternation requiresλ\>0\\lambda\>0, making the optimal gain time\-varying\); Thm[1](https://arxiv.org/html/2606.28413#Thmtheorem1)uses onlyGGstrictly increasing under irregular sampling \(the elapsed gap*sets*the accumulated uncertainty, making the optimal gain gap\-dependent\)\.

#### The continuous\-time substrate\.

The agent carries an internal*state*h∈ℝnh\\in\\mathbb\{R\}^\{n\}, its evolving estimate, and updates it from the stream𝒮\\mathcal\{S\}through a closed\-form continuous\-time network\[[3](https://arxiv.org/html/2606.28413#bib.bib3)\]\. With the arrival times and gapsΔk\\Delta\_\{k\}of Def\.[2](https://arxiv.org/html/2606.28413#Thmdefinition2), andxk=enc​\(ok\)x\_\{k\}=\\mathrm\{enc\}\(o\_\{k\}\)the encoding of thekk\-th observation, the state advances by

h​\(tk\)=f1​\(uk\)⊙\(𝟏−σk\)\+f2​\(uk\)⊙σk,σk=sig​\(a​\(uk\)​Δk\+b​\(uk\)\),uk=ϕ​\(\[xk;h​\(tk−1\)\]\),h\(t\_\{k\}\)=\\;f\_\{1\}\(u\_\{k\}\)\\odot\\big\(\\mathbf\{1\}\-\\sigma\_\{k\}\\big\)\\;\+\\;f\_\{2\}\(u\_\{k\}\)\\odot\\sigma\_\{k\},\\qquad\\sigma\_\{k\}=\\textstyle\\mathrm\{sig\}\\\!\\big\(a\(u\_\{k\}\)\\,\\Delta\_\{k\}\+b\(u\_\{k\}\)\\big\),\\quad u\_\{k\}=\\phi\\big\(\[\\,x\_\{k\};\\,h\(t\_\{k\-1\}\)\\,\]\\big\),\(1\)whereϕ\\phiis a fixed backbone,f1,f2f\_\{1\},f\_\{2\}are fixed maps,a,ba,bare fixed elapsed\-time gates, andσk∈\(0,1\)n\\sigma\_\{k\}\\in\(0,1\)^\{n\}interpolates per coordinate between the “fast” responsef1f\_\{1\}and the “slow” responsef2f\_\{2\}as a function of the real elapsed timeΔk\\Delta\_\{k\}\. Equation \([1](https://arxiv.org/html/2606.28413#S3.E1)\) is the closed\-form CfC update\[[3](https://arxiv.org/html/2606.28413#bib.bib3)\]: the only time input is the elapsed gap, which is what makes the summary continuous\-time\. A fixed read\-outr=ρ​\(h\)r=\\rho\(h\)maps the state to the agent’s output\.

## 4Two necessities: an adaptive timescale and an elapsed\-gap update

Section[3](https://arxiv.org/html/2606.28413#S3)presented the substrate as a fixed\-weight continuous\-time recurrence; here we show what it*must*be, at the generality the claim needs\. Optimal integration of a self\-evolving latent, observed asynchronously, requires two properties at once: an*adaptive*\(input\-dependent\) timescale and a*continuous\-time*\(elapsed\-gap\) update\. Neither alone suffices: an LSTM forget gate is an adaptive timescale that is not continuous\-time, and a fixed continuous\-time filter is timescale\-rigid\. We prove each necessary here \(Prop\.[1](https://arxiv.org/html/2606.28413#Thmproposition1), Thm[1](https://arxiv.org/html/2606.28413#Thmtheorem1)\); the class realizing both, the*liquid*network, is defined once they are in hand \(§[5](https://arxiv.org/html/2606.28413#S5)\)\. These are necessity claims, not sufficiency or uniqueness, scoped to fixed\-weight substrates \(attainment is examined empirically in §[6](https://arxiv.org/html/2606.28413#S6)\); the full scope is in §[7](https://arxiv.org/html/2606.28413#S7)\.

#### The requirement\.

The agent integrates its observation stream into a statehhthat estimates an evolving latents​\(t\)s\(t\): its current best estimate\. Two demands sit onhhat once\.*Retention*: whilessis stable,hhmust reject observation noise and hold the accumulated value\.*Responsiveness*: whenssgenuinely changes, the latent shifting to a new value,hhmust move to it without long lag\. An estimator of a self\-evolving latent lives in both regimes, often alternating; a substrate that fails either does not follow it\. We exhibit the tension concretely on the exponential filter, the form the CfC’s gate takes, then state the two necessities, which do not depend on that form and both rest on the latent and observation model of Def\.[2](https://arxiv.org/html/2606.28413#Thmdefinition2)\.

###### Lemma 1\(Noise–lag floor of a fixed gain\)\.

Let the integrator be the fixed\-gain updatehk=\(1−a\)​hk−1\+a​okh\_\{k\}=\(1\-a\)h\_\{k\-1\}\+a\\,o\_\{k\},a∈\(0,1\)a\\in\(0,1\), with observationsok=sk\+εko\_\{k\}=s\_\{k\}\+\\varepsilon\_\{k\}whereεk\\varepsilon\_\{k\}are zero\-mean, varianceσ2\\sigma^\{2\}, uncorrelated\. Define its*noise*as the stationary error variance whenssis constant,V​\(a\)=limk→∞Var⁡\(hk−s\)V\(a\)=\\lim\_\{k\\to\\infty\}\\operatorname\{Var\}\(h\_\{k\}\-s\), and its*lag*as the integrated squared error of the noiseless unit\-step response \(the state starting at the old value whenssjumps\),L​\(a\)=∑k≥0\(hk−snew\)2L\(a\)=\\sum\_\{k\\geq 0\}\\big\(h\_\{k\}\-s\_\{\\mathrm\{new\}\}\\big\)^\{2\}for a unit jump\. Then

V​\(a\)=a2−a​σ2,L​\(a\)=1a​\(2−a\),V​\(a\)​L​\(a\)=σ2\(2−a\)2∈\(σ24,σ2\)\.V\(a\)=\\frac\{a\}\{2\-a\}\\,\\sigma^\{2\},\\qquad L\(a\)=\\frac\{1\}\{a\(2\-a\)\},\\qquad V\(a\)\\,L\(a\)=\\frac\{\\sigma^\{2\}\}\{\(2\-a\)^\{2\}\}\\in\\Big\(\\tfrac\{\\sigma^\{2\}\}\{4\},\\,\\sigma^\{2\}\\Big\)\.HenceV​\(a\)​L​\(a\)\>σ2/4V\(a\)\\,L\(a\)\>\\sigma^\{2\}/4for everya∈\(0,1\)a\\in\(0,1\), withinfaV​\(a\)​L​\(a\)=σ2/4\\inf\_\{a\}V\(a\)L\(a\)=\\sigma^\{2\}/4: the noise–lag product is bounded below by a positive constant no fixed gain beats, soV​\(a\)≤ϵV\(a\)\\leq\\epsilonforcesL​\(a\)≥σ2/\(4​ϵ\)L\(a\)\\geq\\sigma^\{2\}/\(4\\epsilon\)\.

###### Proof\.

*Noise\.*Withssconstant the errorek=hk−se\_\{k\}=h\_\{k\}\-sobeysek=\(1−a\)​ek−1\+a​εke\_\{k\}=\(1\-a\)e\_\{k\-1\}\+a\\varepsilon\_\{k\}, a stable AR\(1\) \(\|1−a\|<1\|1\-a\|<1\) driven bya​εka\\varepsilon\_\{k\}; taking variances,Var⁡\(e\)=\(1−a\)2​Var⁡\(e\)\+a2​σ2\\operatorname\{Var\}\(e\)=\(1\-a\)^\{2\}\\operatorname\{Var\}\(e\)\+a^\{2\}\\sigma^\{2\}, soVar⁡\(e\)=a2​σ21−\(1−a\)2=a2​σ2a​\(2−a\)=a2−a​σ2\\operatorname\{Var\}\(e\)=\\dfrac\{a^\{2\}\\sigma^\{2\}\}\{1\-\(1\-a\)^\{2\}\}=\\dfrac\{a^\{2\}\\sigma^\{2\}\}\{a\(2\-a\)\}=\\dfrac\{a\}\{2\-a\}\\sigma^\{2\}\.*Lag\.*For a unit step with the state at the old value, the noiseless response ishk=1−\(1−a\)kh\_\{k\}=1\-\(1\-a\)^\{k\}, so the error ishk−1=−\(1−a\)kh\_\{k\}\-1=\-\(1\-a\)^\{k\}andL​\(a\)=∑k≥0\(1−a\)2​k=11−\(1−a\)2=1a​\(2−a\)L\(a\)=\\sum\_\{k\\geq 0\}\(1\-a\)^\{2k\}=\\dfrac\{1\}\{1\-\(1\-a\)^\{2\}\}=\\dfrac\{1\}\{a\(2\-a\)\}\.*Product\.*V​\(a\)​L​\(a\)=a2−a​σ2⋅1a​\(2−a\)=σ2\(2−a\)2V\(a\)L\(a\)=\\dfrac\{a\}\{2\-a\}\\sigma^\{2\}\\cdot\\dfrac\{1\}\{a\(2\-a\)\}=\\dfrac\{\\sigma^\{2\}\}\{\(2\-a\)^\{2\}\}\. Fora∈\(0,1\)a\\in\(0,1\),\(2−a\)2∈\(1,4\)\(2\-a\)^\{2\}\\in\(1,4\), givingV​\(a\)​L​\(a\)∈\(σ2/4,σ2\)V\(a\)L\(a\)\\in\(\\sigma^\{2\}/4,\\sigma^\{2\}\)with infimumσ2/4\\sigma^\{2\}/4asa→0\+a\\to 0^\{\+\}\. The stated implication is immediate\. ∎

###### Proposition 1\(Adaptivity is necessary\)\.

In the model of Def\.[2](https://arxiv.org/html/2606.28413#Thmdefinition2)with change rateλ\>0\\lambda\>0\(the latent stable for stretches, with changes at unknown times\), the minimum\-mean\-square\-error \(MMSE\) estimate ofs​\(t\)s\(t\)given the observations has a*time\-varying*effective gain that no*fixed*\-gain \(LTI\) filter attains; every constant\-gain filter is therefore strictly suboptimal\.

###### Proof\.

The Bayesian filter for a change\-point process maintains a posterior over the run length since the last change\[[21](https://arxiv.org/html/2606.28413#bib.bib21)\]; the optimal weight on a new observation is the posterior\-mean Kalman gain, a function of that run\-length distribution\. Just after a probable change the run\-length posterior concentrates near zero \(high uncertainty\), forcing a high gain; during a long stable run it concentrates at large run length \(low uncertainty\), forcing a low gain\[[20](https://arxiv.org/html/2606.28413#bib.bib20)\]\. Forλ\>0\\lambda\>0both regimes occur with positive probability, so the optimal gain sequence is non\-constant\. The exact MMSE estimator is this BOCPD posterior mean, a*nonlinear*function of the observations \(it reweights by the run\-length posterior\), whereas every constant\-gain choice is a linear time\-invariant filter\. Forλ\>0\\lambda\>0the run\-length posterior is non\-degenerate, so the conditional mean is genuinely nonlinear and no linear, hence no constant\-gain, filter can equal it \(the conditional mean is the*unique*MMSE estimator, so any filter differing from it on a positive\-probability set has strictly greater error\); every constant choice is therefore strictly suboptimal\. \(Atλ=0\\lambda=0, the degenerate stationary limit, the run length grows without bound and the optimal gain settles to a constant, the steady\-state Kalman/exponential filter, so the hypothesisλ\>0\\lambda\>0is necessary; this is exactly the degenerate stationary limit of §[6](https://arxiv.org/html/2606.28413#S6)\.\) The exponential\-filter floor \(Lemma[1](https://arxiv.org/html/2606.28413#Thmlemma1)\) is one explicit face of the resulting tension\. An input\-dependent \(adaptive\) timescale is therefore necessary\. ∎

The second leg strengthens from a suboptimality claim about one filter into a*separation*between two architecture classes, one that no amount of capacity can cross\. Its mechanism is elementary, the law of total variance applied to the gap variable, and its value is the architecture separation it yields, not its difficulty\. We first name the class precisely\.

###### Definition 3\(Gap\-blind estimator\)\.

An estimator is*gap\-blind*if its estimates^k\\hat\{s\}\_\{k\}is a measurable function of the observation*values*Ok=\(o1,…,ok\)O\_\{k\}=\(o\_\{1\},\\dots,o\_\{k\}\)and the step indexkkalone, not of the arrival timest1:kt\_\{1:k\}or the elapsed gapsΔ1:k\\Delta\_\{1:k\}\. This is exactly the per\-observation unrolling of*any*recurrencehk=Fθ​\(hk−1,ok\)h\_\{k\}=F\_\{\\theta\}\(h\_\{k\-1\},o\_\{k\}\),s^k=gθ​\(hk\)\\hat\{s\}\_\{k\}=g\_\{\\theta\}\(h\_\{k\}\), with no time input: any width, depth, nonlinearity, or trained weightsθ\\theta, so long as the gap is not supplied\. A vanilla per\-step RNN, LSTM, or GRU is gap\-blind; a continuous\-time cell, or a discrete cell*fed*Δk\\Delta\_\{k\}\(Time\-LSTM, GRU\-D\), is not\. Such a recurrence keeps state*through steps*, not through elapsed time: it advances one update per arrival and cannot tell a long gap from a short one\.

###### Theorem 1\(Gap\-awareness separation under irregular sampling\)\.

Adopt the model of Def\.[2](https://arxiv.org/html/2606.28413#Thmdefinition2)withGGstrictly increasing \(e\.g\. the random walkG​\(Δ\)=q​ΔG\(\\Delta\)=q\\Delta, or the change\-point process of Prop\.[1](https://arxiv.org/html/2606.28413#Thmproposition1)\), sampled withVar⁡\(Δ\)\>0\\operatorname\{Var\}\(\\Delta\)\>0and noiseσ2\>0\\sigma^\{2\}\>0\. Writesk=s​\(tk\)s\_\{k\}=s\(t\_\{k\}\)\. Then*every*gap\-blind estimator \(Def\.[3](https://arxiv.org/html/2606.28413#Thmdefinition3)\) obeys

𝔼​\[\(s^k−sk\)2\]≥𝔼​\[Var⁡\(sk∣Ok\)\]=𝔼​\[Var⁡\(sk∣Ok,Δ1:k\)\]⏟gap\-aware MMSE\+ℐk,\\mathbb\{E\}\\big\[\(\\hat\{s\}\_\{k\}\-s\_\{k\}\)^\{2\}\\big\]\\;\\geq\\;\\mathbb\{E\}\\big\[\\operatorname\{Var\}\(s\_\{k\}\\mid O\_\{k\}\)\\big\]\\;=\\;\\underbrace\{\\mathbb\{E\}\\big\[\\operatorname\{Var\}\(s\_\{k\}\\mid O\_\{k\},\\Delta\_\{1:k\}\)\\big\]\}\_\{\\text\{gap\-aware MMSE\}\}\\;\+\\;\\mathcal\{I\}\_\{k\},where the nonnegative gap term is the variance reduction from observing the gaps,

ℐk:=𝔼​\[Var⁡\(sk∣Ok\)\]−𝔼​\[Var⁡\(sk∣Ok,Δ1:k\)\]≥0\.\\mathcal\{I\}\_\{k\}:=\\mathbb\{E\}\\big\[\\operatorname\{Var\}\(s\_\{k\}\\mid O\_\{k\}\)\\big\]\-\\mathbb\{E\}\\big\[\\operatorname\{Var\}\(s\_\{k\}\\mid O\_\{k\},\\Delta\_\{1:k\}\)\\big\]\\;\\geq\\;0\.The lower bound holds for every gap\-blind estimator irrespective of its capacity or training\. Moreoverℐk\>0\\mathcal\{I\}\_\{k\}\>0whenever the optimal estimate’s gain depends on the gap \(in particular for both realizations of Def\.[2](https://arxiv.org/html/2606.28413#Thmdefinition2)whenVar⁡\(Δ\)\>0\\operatorname\{Var\}\(\\Delta\)\>0andσ2\>0\\sigma^\{2\}\>0\), whileℐk=0\\mathcal\{I\}\_\{k\}=0when sampling is regular \(Var⁡\(Δ\)=0\\operatorname\{Var\}\(\\Delta\)=0\)\. Hence under irregular sampling no gap\-blind architecture, at any size, attains the MMSE: matching it*requires*the estimate to depend on the elapsed gap, and network capacity does not substitute for that dependence\.

###### Proof\.

The minimum mean\-square estimate from a given information set is the conditional mean, with error the conditional variance\. A gap\-blind estimator is, by Def\.[3](https://arxiv.org/html/2606.28413#Thmdefinition3), a function ofOkO\_\{k\}alone; among all such functions the error is minimized by𝔼​\[sk∣Ok\]\\mathbb\{E\}\[s\_\{k\}\\mid O\_\{k\}\], of error𝔼​\[Var⁡\(sk∣Ok\)\]\\mathbb\{E\}\[\\operatorname\{Var\}\(s\_\{k\}\\mid O\_\{k\}\)\]\. This optimum is attained only in the infinite\-capacity limit, so it lower\-bounds every concrete gap\-blind estimator \(the first inequality\), and the bound is capacity\-free precisely because it bounds the*best*function ofOkO\_\{k\}, already granting the network any gap information recoverable from the values themselves\. Conditioning on the further variableΔ1:k\\Delta\_\{1:k\}cannot raise mean\-square error \(tower property / Rao–Blackwell\), giving the decomposition withℐk≥0\\mathcal\{I\}\_\{k\}\\geq 0\. For strictness, the law of total variance \(conditioning onΔ1:k\\Delta\_\{1:k\}givenOkO\_\{k\}\) gives

ℐk=𝔼​\[VarΔ1:k∣Ok⁡\(𝔼​\[sk∣Ok,Δ1:k\]\)\],\\mathcal\{I\}\_\{k\}=\\mathbb\{E\}\\big\[\\operatorname\{Var\}\_\{\\Delta\_\{1:k\}\\mid O\_\{k\}\}\\\!\\big\(\\mathbb\{E\}\[s\_\{k\}\\mid O\_\{k\},\\Delta\_\{1:k\}\]\\big\)\\big\],soℐk\>0\\mathcal\{I\}\_\{k\}\>0iff the gap\-aware posterior*mean*𝔼​\[sk∣Ok,Δ1:k\]\\mathbb\{E\}\[s\_\{k\}\\mid O\_\{k\},\\Delta\_\{1:k\}\]depends on the gaps given the values, on a positive\-probability set: it is the mean’s gap\-dependence, not the variance’s, that the decomposition isolates\. This holds in both realizations of Def\.[2](https://arxiv.org/html/2606.28413#Thmdefinition2), because the optimal weight on the latest observation is gap\-dependent\. For the Brownian drift the one\-step gainK​\(Δk\)=\(P\+q​Δk\)/\(P\+q​Δk\+σ2\)K\(\\Delta\_\{k\}\)=\(P\+q\\Delta\_\{k\}\)/\(P\+q\\Delta\_\{k\}\+\\sigma^\{2\}\)is strictly increasing inΔk\\Delta\_\{k\}, so the posterior mean\(1−K\)​s^k−1\+K​ok\(1\-K\)\\,\\hat\{s\}\_\{k\-1\}\+K\\,o\_\{k\}moves withΔk\\Delta\_\{k\}wheneverok≠s^k−1o\_\{k\}\\neq\\hat\{s\}\_\{k\-1\}; and the values leave the gap underdetermined \(Var⁡\(Δk∣Ok\)\>0\\operatorname\{Var\}\(\\Delta\_\{k\}\\mid O\_\{k\}\)\>0, increment and noise confounded atσ2\>0\\sigma^\{2\}\>0\), so this mean\-dependence survives conditioning onOkO\_\{k\}\. For the change\-point process a longer gap raises the change probability1−e−λ​Δk1\-e^\{\-\\lambda\\Delta\_\{k\}\}, up\-weighting the latest observation against the accumulated past, so again the mean depends onΔk\\Delta\_\{k\}\. Henceℐk\>0\\mathcal\{I\}\_\{k\}\>0whenever the optimal gain depends on the gap\. WhenVar⁡\(Δ\)=0\\operatorname\{Var\}\(\\Delta\)=0the gap is a known constant and adds nothing, soℐk=0\\mathcal\{I\}\_\{k\}=0\. ∎

## 5The liquid class

The two necessities pick out one class of fixed\-weight substrates: those realizing both an adaptive timescale and an elapsed\-gap update\. We name it, give its realization, and locate the standard recurrences relative to it\.

###### Definition 4\(Liquid substrate\)\.

The substrate \([1](https://arxiv.org/html/2606.28413#S3.E1)\) is a*liquid substrate*: a fixed\-weight \(no online weight update\) continuous\-time state recurrence whose update between successive observations has two properties, both of which the two results above establish a self\-evolving, asynchronously\-observed substrate*must*have\. \(L1\) an*adaptive timescale*: the effective integration time\-constant is a function of the input/state, so the estimator it realizes is time\-varying \(Prop\.[1](https://arxiv.org/html/2606.28413#Thmproposition1)\)\. \(L2\)*elapsed\-gap dependence*: the update is a function of the gapΔk\\Delta\_\{k\}since the last observation, entering the dynamics as native structure rather than as a learned input \(Thm[1](https://arxiv.org/html/2606.28413#Thmtheorem1)\)\. The gateσk=sig​\(a​\(uk\)​Δk\+b​\(uk\)\)\\sigma\_\{k\}=\\mathrm\{sig\}\(a\(u\_\{k\}\)\\Delta\_\{k\}\+b\(u\_\{k\}\)\)of \([1](https://arxiv.org/html/2606.28413#S3.E1)\) keeps the timescale input\-dependent with frozen weights, and a multi\-timescale state supplies \(L1\) without a hand\-tuned rule\. The definition is the property pair, not a parameterization: the canonical realization is the MIT liquid\-network line, liquid time\-constant networks\[[4](https://arxiv.org/html/2606.28413#bib.bib4)\]and their closed\-form variant CfC\[[3](https://arxiv.org/html/2606.28413#bib.bib3)\], but*any*fixed\-weight continuous\-time cell meeting \(L1\)\+\(L2\) is liquid in this sense \(the exclusions are itemized below\)\. This paper’s liquid substrate is therefore a*liquid network of the MIT family at the level of the cell, deployed in the reservoir / Liquid\-State\-Machine stance*\(Remark[1](https://arxiv.org/html/2606.28413#Thmremark1)\): the same dynamics in a different role, a fixed substrate, not a trained model\.

###### Corollary 1\(The fixed\-weight face: continuous\-time is the native escape\)\.

Gap\-awareness can be supplied two ways:*structurally*, by a continuous\-time cell whose update carriesΔk\\Delta\_\{k\}in its dynamics; or*extrinsically*, by feedingΔk\\Delta\_\{k\}to a discrete cell trained to use it \(Time\-LSTM, GRU\-D\)\. Among*fixed\-weight*substrates, with no training to the gap, only the structural route applies, since the extrinsic one’s compliance is learned\. Thus on the reservoir / fixed\-substrate stance of Rem\.[1](https://arxiv.org/html/2606.28413#Thmremark1), the elapsed\-gap dependence Thm[1](https://arxiv.org/html/2606.28413#Thmtheorem1)requires is native to a continuous\-time \(liquid\) cell and unavailable to a frozen discrete one\. Thm[1](https://arxiv.org/html/2606.28413#Thmtheorem1)forbids gap\-blindness at any capacity; the liquid cell of Def\.[4](https://arxiv.org/html/2606.28413#Thmdefinition4)is how a frozen substrate escapes the forbidden class\.

###### Corollary 2\(The mesh face: every agent must be liquid\)\.

Each agent of a mesh \(Def\.[1](https://arxiv.org/html/2606.28413#Thmdefinition1)\) integrates an admitted stream that is exogenously timed, non\-stationary, and folded on a substrate fixed during operation, so each agent satisfies the hypotheses of Def\.[2](https://arxiv.org/html/2606.28413#Thmdefinition2)by construction\. Prop\.[1](https://arxiv.org/html/2606.28413#Thmproposition1)and Thm[1](https://arxiv.org/html/2606.28413#Thmtheorem1)therefore bind every agent: a mesh’s temporal substrate must be liquid at every node, with no node exempt and no center to carry the dependence on its behalf\. The statement is the per\-agent necessity quantified over the population, not a claim about the collective’s dynamics: convergence of center\-free inference, identification, and the learning loop are the companion’s domain\[[1](https://arxiv.org/html/2606.28413#bib.bib1)\]and are not addressed here\.

These two results are the two legs, both read off the single model of Def\.[2](https://arxiv.org/html/2606.28413#Thmdefinition2), each turning on a complementary facet of it, and together they close the gaps a single “adaptive” claim leaves open\. The latent’s change rate forces an*adaptive*timescale \(Prop\.[1](https://arxiv.org/html/2606.28413#Thmproposition1), at full generality: a fixed filter is suboptimal, not only the one\-pole form\); irregular observation of the same latent forces an*elapsed\-gap*update \(Thm[1](https://arxiv.org/html/2606.28413#Thmtheorem1): no gap\-blind recurrence, at any capacity, matches the MMSE\)\. Their intersection is the liquid class \(Def\.[4](https://arxiv.org/html/2606.28413#Thmdefinition4)\), and the exclusions are clean\. A generic reservoir \(echo\-state network\[[7](https://arxiv.org/html/2606.28413#bib.bib7)\]\) has timescales set by its fixed leak rate and weights,*input\-independent*, so it fails the adaptive leg\. A*vanilla, per\-step*LSTM has an input\-dependent forget gate, adaptive, but updates once per observation regardless of elapsed time, so it fails the elapsed\-gap leg\. A discrete net*fed*the gap \(GRU\-D, Time\-LSTM\) attains both, but only by being*trained*to: a coerced member of the gap\-aware class, outside the fixed\-substrate commitment\. Among*fixed\-weight*recurrences, only a liquid one carries an adaptive, elapsed\-gap timescale without retraining: its gate \([1](https://arxiv.org/html/2606.28413#S3.E1)\) stays input\-dependent*even with frozen weights*, exactly what committing to a fixed substrate buys rather than costs\. Detecting change is the prerequisite of the adaptive leg\[[19](https://arxiv.org/html/2606.28413#bib.bib19),[17](https://arxiv.org/html/2606.28413#bib.bib17),[18](https://arxiv.org/html/2606.28413#bib.bib18)\], which the multi\-timescale realization below supplies\.

#### The multi\-timescale realization\.

The naive way to make the gain input\-dependent, raise it when the incoming observation deviates fromhh, fails, because a single observation’s deviation cannot separate noise from change: in noise it raises the gain and amplifies it \(the negative control, §[6](https://arxiv.org/html/2606.28413#S6)\)\. The liquid substrate’s actual structure resolves this\. Its time constants are*bimodal*, fast and slow neuron populations\[[3](https://arxiv.org/html/2606.28413#bib.bib3)\], so the state carries a fast trackhfh\_\{f\}and a slow trackhsh\_\{s\}at once; their*divergence*\|hf−hs\|\|h\_\{f\}\-h\_\{s\}\|is the change signal, since noise does not persistently separate the two while a real jump does\. The effective gain rises with the divergence \(trust the fast track at a change, the slow track in noise\), so adaptivity is supplied by the multi\-timescale state itself, not by a hand\-tuned rule\.

## 6Verification

We verify each necessity in turn: the adaptive timescale of Prop\.[1](https://arxiv.org/html/2606.28413#Thmproposition1)against baselines that lack it \(Fig\.[2](https://arxiv.org/html/2606.28413#S6.F2)\), and the gap\-awareness separation of Thm[1](https://arxiv.org/html/2606.28413#Thmtheorem1)as an exact, training\-free computation of the bound itself \(Fig\.[3](https://arxiv.org/html/2606.28413#S6.F3)\)\.

### 6\.1The adaptive timescale: liquid versus fixed

Figure[2](https://arxiv.org/html/2606.28413#S6.F2)tests Prop\.[1](https://arxiv.org/html/2606.28413#Thmproposition1)directly\. A scalar agent integrates an asynchronous, bursty, noisy stream of an evolving latent \(piecewise\-constant with occasional jumps\); we compare fixed\-fast, fixed\-slow, the naive single\-deviation liquid rule, and the multi\-timescale liquid integrator \(seeded, reproducible;liquid\_repro\.py\)\. \(a\) Fixed\-fast follows jumps but is noisy, fixed\-slow is smooth but lags, the multi\-timescale liquid does both\. \(b\) On the noise/lag plane the two fixed timescales sit on the tradeoff frontier Prop\.[1](https://arxiv.org/html/2606.28413#Thmproposition1)describes; the naive liquid rule sits beside fixed\-fast \(it mistakes noise for change\); and of the integrators tested only the multi\-timescale liquid reaches the low\-noise, low\-lag corner; no fixed timescale does, as Prop\.[1](https://arxiv.org/html/2606.28413#Thmproposition1)requires\. \(c\) Its joint noise\-and\-lag advantage shows up as the best in\-band fraction \(99%99\\%vs\.9696–98%98\\%\): it stays within a fixed band of the truth most often, though the margin over fixed\-slow \(98%98\\%\) is small, so the in\-band claim rests on the joint \(a,b\) advantage, not panel \(c\) alone\. Across stream regimes the multi\-timescale liquid has the lowest error wherever the latent*evolves*\(calm, volatile, storm\); only in the degenerate stationary limit, pure noise around a constant with nothing to evolve and hence nothing to integrate, does a fixed slow average match or beat it\. That limit is the boundary of the claim, not a counterexample: adaptivity is necessary exactly where there is something to adapt to\.

![Refer to caption](https://arxiv.org/html/2606.28413v1/x2.png)Figure 2:An adaptive timescale is necessary \(synthetic, reproducible\)\. A scalar agent integrates an evolving latent from a noisy, bursty, asynchronous stream\. \(a\) fixed\-fast is noisy, fixed\-slow lags, the multi\-timescale liquid does both\. \(b\) the fixed timescales sit on the noise/lag tradeoff frontier and the naive single\-deviation liquid rule fails \(it amplifies noise\); only the multi\-timescale liquid \(fast/slow divergence as the change signal\) reaches the low\-noise, low\-lag corner\. \(c\) the liquid integrator stays within a fixed band of the truth the most, though the margin over fixed\-slow is small; the in\-band claim rests on the joint \(a,b\) advantage, not \(c\) alone\.
### 6\.2The gap\-awareness separation

Figure[3](https://arxiv.org/html/2606.28413#S6.F3)tests Thm[1](https://arxiv.org/html/2606.28413#Thmtheorem1)\. Because the bound is informational, not architectural, it needs no training: in the one\-step Gaussian case \(the random\-walk realization of Def\.[2](https://arxiv.org/html/2606.28413#Thmdefinition2)\), the gap\-aware MMSE𝔼​\[v​\(Δ\)\]\\mathbb\{E\}\[v\(\\Delta\)\]and the*best*gap\-blind error𝔼​\[Var⁡\(s∣O\)\]\\mathbb\{E\}\[\\operatorname\{Var\}\(s\\mid O\)\], the floor every gap\-blind estimator obeys at any width or depth, are computed exactly by integrating out the random gap\. \(a\) The gap\-blind error stays strictly above the gap\-aware floor, and the shortfall widens with the dispersion of the arrival times\. \(b\) The separationℐk\\mathcal\{I\}\_\{k\}is zero under regular sampling \(Var⁡\(Δ\)=0\\operatorname\{Var\}\(\\Delta\)\{=\}0\) and rises withVar⁡\(Δ\)\\operatorname\{Var\}\(\\Delta\), tangent at the origin to the analytic leading\-order law of Remark[3](https://arxiv.org/html/2606.28413#Thmremark3)\(the coefficient12​\|v′′​\(Δ¯\)\|\\tfrac\{1\}\{2\}\|v^\{\\prime\\prime\}\(\\bar\{\\Delta\}\)\|, not a fit\)\. \(c\) It is widest in the noisy regime and closes at high SNR, where the values themselves reveal the gap, exactly as Remark[3](https://arxiv.org/html/2606.28413#Thmremark3)predicts\. This is the gap no gap\-blind network closes with scale; §[7](https://arxiv.org/html/2606.28413#S7)turns it into the dispersion sweep a deployed mesh would measure\.

![Refer to caption](https://arxiv.org/html/2606.28413v1/x3.png)Figure 3:The gap\-awareness separation is real and grows with irregularity \(exact, training\-free\)\. One\-step Gaussian model \(Def\.[2](https://arxiv.org/html/2606.28413#Thmdefinition2), random walk\); the gap\-aware MMSE and the*best*gap\-blind error are computed by integrating out the random gap, so the bound holds at any capacity\. \(a\) the best gap\-blind error stays above the gap\-aware floor, the shortfall widening withVar⁡\(Δ\)\\operatorname\{Var\}\(\\Delta\)\. \(b\) the separationℐk\\mathcal\{I\}\_\{k\}vanishes atVar⁡\(Δ\)=0\\operatorname\{Var\}\(\\Delta\)\{=\}0and rises with it, matching the analytic leading\-order law \(Rem\.[3](https://arxiv.org/html/2606.28413#Thmremark3)\) near the origin\. \(c\)ℐk\\mathcal\{I\}\_\{k\}peaks in the noisy regime and closes at high SNR\.

## 7Limitations and future directions

#### Assumptions\.

The results rest on assumptions whose violation changes them\.*Exogenous sampling*: arrivals are state\-independent, so the elapsed gap is non\-informative about the latent’s value and acts only through the gain \(Thm[1](https://arxiv.org/html/2606.28413#Thmtheorem1)\); informative, state\-modulated sampling would let the gap carry value too, and the decomposition would not hold\.*A self\-evolving latent*: Prop\.[1](https://arxiv.org/html/2606.28413#Thmproposition1)is proven for a piecewise\-stationary \(change\-point\) latent and Thm[1](https://arxiv.org/html/2606.28413#Thmtheorem1)for any strictly\-increasingG​\(Δ\)G\(\\Delta\)\(a Brownian drift included\); other non\-stationarity classes, and the multivariate, partially\-observed case, are open\.*A single agent*: the theorems are per\-agent necessities, lifted to every node of a mesh by Cor\.[2](https://arxiv.org/html/2606.28413#Thmcorollary2); they do not address the collective’s multi\-agent dynamics, such as the convergence of center\-free inference\[[1](https://arxiv.org/html/2606.28413#bib.bib1)\]\.

#### What is not claimed\.

The result is necessity, not sufficiency: that a liquid substrate*attains*the optimum is examined only empirically, on a synthetic benchmark \(§[6\.1](https://arxiv.org/html/2606.28413#S6.SS1)\), and is not proven; the separation of Thm[1](https://arxiv.org/html/2606.28413#Thmtheorem1), by contrast, is computed exactly \(§[6\.2](https://arxiv.org/html/2606.28413#S6.SS2)\)\. Real\-data validation remains future work\. The separation is informative only where it bites: its shortfall grows with the dispersion of the arrival times \(Remark[3](https://arxiv.org/html/2606.28413#Thmremark3)\) and is negligible under near\-regular sampling\.

#### Future directions\.

Relaxing the fixed\-weight premise is the bridge from inference to learning\. What an adapting substrate should move*toward*is, in a mesh, the collective’s intention, the*ask*, which reaches each agent the way its observations do; whether such ask\-driven adaptation is stable rather than drifting is open\. Beyond the substrate, further questions remain open: the content gate that admits what is relevant\[[2](https://arxiv.org/html/2606.28413#bib.bib2)\], resilient admission under adversaries, and what*grounds*the collective rather than merely keeping it coherent\. Figure[3](https://arxiv.org/html/2606.28413#S6.F3)computes that separation exactly in the Gaussian model; carrying it to real data is the next step\. A deployed edge mesh, whose agents integrate genuinely irregular peer streams, supplies the test in situ: the gap\-blind\-versus\-liquid tracking gap should widen with the measured dispersion of inter\-arrival times, the real\-data validation the synthetic computation stands in for\.

## 8Conclusion

Two facts about integrating a self\-evolving latent online determine what the substrate must be: the latent changes, and its observations arrive at clock\-free times\. The first forces a time\-varying optimal estimator \(Prop\.[1](https://arxiv.org/html/2606.28413#Thmproposition1)\); the second forces an elapsed\-gap gain that no gap\-blind recurrence, at any capacity, can match \(Thm[1](https://arxiv.org/html/2606.28413#Thmtheorem1)\)\. Among fixed\-weight substrates, the intersection of the two is the liquid class: an LSTM has the first property only and a fixed continuous\-time filter the second only, while a multi\-timescale liquid network supplies both, outperforming the fixed\-timescale and naive\-adaptive baselines, with the gap\-awareness separation it exploits computed exactly \(§[6](https://arxiv.org/html/2606.28413#S6)\)\. The characterization is scoped, not universal: it binds optimal integration on a fixed\-weight substrate, not a learner free to retrain its weights, and weight\-adapting recurrences attain both properties by other means \(Remark[1](https://arxiv.org/html/2606.28413#Thmremark1)\)\. The setting that makes the necessity bite is*mesh intelligence*: there the quantities an agent tracks keep moving, its arrivals cannot be scheduled, and its weights are fixed, so the conditions of both results hold by construction, and this liquid substrate is the temporal layer of its coupling\. Whether that substrate can go further and adapt its own weights online, beyond the fixed\-weight regime characterized here, is the direction this work opens\.

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