Attractor-Keyed Memory

Abstract

Physical selectors (lasers choosing a mode, Ising machines settling on a ground state, condensates occupying a spin state) produce high-dimensional signatures at the moment of decision: full field amplitudes, multimode interference patterns, or scattering responses. These signatures are richer than the winner’s index, yet they are routinely discarded. We show that when the signatures are repeatable across trials (stereotyped) and linearly independent across routes, a single linear decoder compiled from calibration data maps them to arbitrary payloads, merging selection and memory access into one event and eliminating the fetch that dominates latency and energy in sparse routing architectures. The construction requires one singular value decomposition of measured device responses, which certifies capability and bounds worst-case error for any downstream payload before the task is chosen. Runtime error separates into two independently diagnosable channels, decoding fidelity (controlled by dictionary conditioning) and routing reliability (controlled by the margin-to-noise ratio), each with a distinct physical origin and targeted remedy. We derive the full error decomposition, give Ising-machine selector constructions, and validate the predicted scalings on synthetic speckle-signature simulations across three measurement modalities. No hardware demonstration exists; we provide a falsifiable four-step experimental protocol specifying what a first experiment must measure. Whether real device signatures satisfy stereotypy is the central open question.

Architectures built around discrete selection, including mixture-of-experts models [7], neuromorphic processors [31, 14], and photonic classifiers [29, 32], perform two operations per input: a router selects which of $K$ experts to activate, then the associated $D$-dimensional payload is fetched from digital memory [35, 12, 10, 34]. The fetch is the bottleneck: the payload read dominates both latency and energy [12, 31]. Yet the physics of selection has already produced a rich observable at the moment of decision.

When a laser selects a mode, a condensate occupies a state [1], or an Ising machine relaxes toward a ground-state configuration [18, 11, 30], the winning attractor carries a high-dimensional physical signature far richer than the winner’s index. This signature is discarded. We show that a fixed linear decoder can map it to arbitrary data, so that selection and memory access become a single event. We call this primitive attractor-keyed memory (AKM); retrieval without a separate memory fetch we term fetchless lookup.

The framework rests on one physical assumption and one algebraic condition. The assumption is route-conditioned repeatability (stereotypy): conditioned on route $k$ winning, the measured signature is approximately the same across inputs. Given stereotypy, the algebraic condition is that the $K$ calibrated mean signatures must be linearly independent. When they are, a minimum-norm pseudoinverse decoder [25] $W=Y\Phi^{+}$ recovers any desired payload table $Y$ exactly, where $\Phi\in\mathbb{R}^{M_{\mathrm{sig}}\times K}$ collects the $K$ mean signatures of dimension $M_{\mathrm{sig}}$. Since $\Phi$ depends only on the hardware, changing the payload requires only recompiling $W$.

For hardware used as a selector, AKM adds a design objective beyond producing a reliable winner: engineer the post-selection state so its signatures form a well-conditioned dictionary. This added objective yields three practical consequences. First, a task-independent certification protocol: a single singular value decomposition (SVD) of measured device responses determines $\operatorname{rank}(\Phi)$ and $\sigma_{\min}(\Phi)$, certifying universal payload realizability and bounding worst-case error before deployment. Second, a two-channel error decomposition with distinct diagnostics and remedies: decoding fidelity controlled by $\sigma_{\min}(\Phi)$, routing reliability controlled by the ratio $\Delta/T_{\mathrm{eff}}$ of the selection margin to effective noise temperature. Third, concrete hardware design targets: three predeployment failure modes (rank loss, conditioning collapse, margin collapse), each measurable and separately remediable.

The linear algebra is standard [25]; the rank criterion for $\Phi$ is the finite-dictionary analogue of the full-rank condition in reservoir readout training [15, 5]. In reservoir computing [3, 5], errors fold into a single empirical error budget; in Hopfield retrieval [27], the attractor is the stored object. The novelty is the object the algebra acts on: $\Phi$ is compiled once from measured device responses and predicts capability for any payload. Timing-based address selection in spiking networks [2] and driven-dissipative mode competition [1, 30] provide candidate physical selectors.

Figure 1: Attractor-keyed memory versus reservoir computing. Both use a linear readout from a physical state, but differ structurally: reservoirs operate on a continuous driven trajectory, AKM on a discrete set of attractor signatures; reservoir states are input-sensitive (separability), AKM signatures are stereotyped (same pattern each time route $k$ wins); the reservoir decoder is trained on input–output pairs, the AKM decoder is compiled from calibration data via $W=Y\Phi^{+}$; reservoir analyses fold routing-like failures into an overall readout error budget, AKM decomposes into two separately diagnosable channels (decoding fidelity $\sigma_{\min}(\Phi)$, routing reliability $\Delta/T_{\mathrm{eff}}$); reservoir analysis yields a capacity bound, AKM yields a predeployment certification protocol. In AKM the payload $y_{k}$ need not resemble the attractor state; the attractor is the key, not the stored value.

Evidence hierarchy. Four layers, in decreasing rigor: (i) exact theorem (full-rank dictionary gives universal payload realizability); (ii) approximate theory (conditioning, drift, and stereotypy bounds); (iii) phenomenology (Gibbs routing fit); (iv) synthetic validation (Monte Carlo on speckle-signature models, not physical experiment). Synthetic validation confirms internal consistency; it does not test physical assumptions. Figure 1 contrasts the architecture with reservoir computing; Fig. 2(a) illustrates a photonic realization. Whether real competitive selectors satisfy stereotypy is the central open experimental question; the formal statement and its quantitative relaxation appear below.

Route–decode abstraction.

Fetchless lookup factors into routing and readout:

$$x\;\longmapsto\;\hat{k}(x)\;\longmapsto\;\tilde{\phi}_{\hat{k}(x)}\;\longmapsto\;y=W\tilde{\phi}_{\hat{k}(x)}.$$ (1)

Here $x\in\mathbb{R}^{N}$ is an $N$-dimensional input; $\hat{k}(x)$ is the discrete route selected by the physical competition; $\tilde{\phi}_{\hat{k}(x)}\in\mathbb{R}^{M_{\mathrm{sig}}}$ is the single-shot measured signature of the winning state; and $W\in\mathbb{R}^{D\times M_{\mathrm{sig}}}$ is a fixed linear decoder mapping signatures to $D$-dimensional payloads. The theory requires only that routing returns a winner $\hat{k}(x)$ with a measurable margin; the internal structure of the selector is an implementation detail. In these terms, a route is the index $\hat{k}(x)$ returned by the selector; the payload $y_{k}$ is the data assigned to route $k$; a fetch is the separate memory read that returns $y_{k}$ from a stored table once the route is known; and fetchless lookup eliminates that read by obtaining $y$ directly from the winner’s signature through $W$, as in Eq. (1).

Score generation and routing.

One realization maps an input $x$ to a bank of $M_{\mathrm{score}}\geq K$ candidate scores:

$$h(x)=B_{\mathrm{wide}}\,x,\qquad B_{\mathrm{wide}}\in\mathbb{R}^{M_{\mathrm{score}}\times N}.$$ (2)

In photonic platforms $B_{\mathrm{wide}}$ may be realized by a programmable linear optical transformation [19, 4, 22]. A fixed routing map $R_{\mathrm{route}}\in\mathbb{R}^{K\times M_{\mathrm{score}}}$ compresses the wide scores into $K$ competing routes:

$$g(x)=R_{\mathrm{route}}\,h(x)\in\mathbb{R}^{K},\qquad E_{k}(x)=-g_{k}(x),$$ (3)

where $g_{k}(x)$ is the score for route $k$ and $E_{k}(x)$ the corresponding selector energy. The selector returns $\hat{k}(x)\in\operatorname*{arg\,max}_{k}g_{k}(x)$ with winner-to-runner-up margin

$$\Delta(x)=g_{(1)}(x)-g_{(2)}(x),$$ (4)

where $g_{(1)}\geq g_{(2)}\geq\cdots$ are the ordered scores. Since $E_{k}=-g_{k}$, this score-space margin equals the energy-space gap $E_{(2)}-E_{(1)}$; we use $\Delta$ for both throughout. Equations (2)–(4) describe one realization; the theory requires only a winner $\hat{k}(x)$, a margin $\Delta(x)$, and the testable assumption that misrouting decreases monotonically with $\Delta(x)$.

Signature emission and decoding.

After routing converges to state $u^{(k)}$, a fixed measurement map $\mathcal{M}$ produces the signature

$$\tilde{\phi}_{k}=\mathcal{M}\,u^{(k)}+\delta\in\mathbb{R}^{M_{\mathrm{sig}}},$$ (5)

where $\delta$ is zero-mean measurement noise (decomposed in Supp. Mat., Sec. S3D into an input-dependent residual $\varepsilon_{k}(x)$ and a stochastic component $\delta_{k}$). Conditioned on selecting route $k$, the measured signature has mean $\bar{\phi}_{k}$ and covariance $\Sigma_{k}$.

Assumption (route-conditioned repeatability / stereotypy). Conditioned on route $k$ winning, shot-to-shot variation in $\tilde{\phi}_{k}$ is dominated by device noise, not by differences in the triggering input $x$. Strong competition funnels all initial conditions toward a single final state; weak competition lets the winning state retain input memory, making $\Phi$ a poor summary. Mode hopping or near-degeneracy can break stereotypy and must be diagnosed (Supp. Mat., Sec. S3).

Calibration forces each route in turn and collects mean signatures and payloads:

$$\Phi=[\bar{\phi}_{1}\cdots\bar{\phi}_{K}]\in\mathbb{R}^{M_{\mathrm{sig}}\times K},\qquad Y=[y_{1}\cdots y_{K}]\in\mathbb{R}^{D\times K}.$$ (6)

The dictionary $\Phi$ is empirical, compiled from measured device responses. Runtime readout uses a fixed linear decoder $y=W\tilde{\phi}_{\hat{k}(x)},$ with $W\in\mathbb{R}^{D\times M_{\mathrm{sig}}}.$

Block-parallel architecture.

$B$ independent blocks operate in parallel, each running its own physical $\operatorname*{arg\,max}$ over $K$ routes; the rank condition applies per block. The layer output is the sum of decoded signatures across all $B$ blocks (Supp. Mat. [28]). Figure 2(a) illustrates the pipeline.

Proposition 1 (Universal payload realizability).

The system $W\Phi=Y$ admits a solution for every $Y\in\mathbb{R}^{D\times K}$ if and only if $\operatorname{rank}(\Phi)=K$. When this holds, $M_{\mathrm{sig}}\geq K$ and the minimum-norm exact decoder is $W_{\star}=Y\Phi^{+}$ (proof and full decoder family in Supp. Mat. [28]). The condition is well known; what matters here is its physical interpretation: the $K$ routes must produce $K$ linearly independent measured patterns, and a single SVD of $\Phi$ checks this before any task is specified.

If $\operatorname{rank}(\Phi)=r<K$, exact decoding may still hold for a particular $Y$ whose rows lie in $\operatorname{Row}(\Phi)$. Rank deficiency rules out universal fetchless lookup, not every structured task.

Selector realizations.

The decoder theory is selector-agnostic, but two Ising constructions show the framework generates concrete hardware designs.

One-hot QUBO (quadratic unconstrained binary optimization) selector. Binary variables $z_{i}\in\{0,1\}$ define the energy

$$E_{\mathrm{WTA}}(z;x)=\lambda\Big(\sum_{i=1}^{K}z_{i}-1\Big)^{\!2}-\sum_{i=1}^{K}g_{i}(x)\,z_{i},\qquad\lambda>0,$$ (7)

where $\lambda$ is a penalty weight enforcing the one-hot constraint. If $\lambda$ exceeds the largest score magnitude, every global minimizer is one-hot: on the one-hot manifold, $E_{\mathrm{WTA}}(e_{i};x)=-g_{i}(x)$, recovering the selector energy of Eq. (3). The standard spin transformation $z_{i}=(1+s_{i})/2$ yields an equivalent Ising Hamiltonian with dense antiferromagnetic couplings and local fields proportional to $g_{i}(x)$. Each block in the parallel architecture realizes its own such QUBO. The full proof is given in the Supp. Mat. [28].

Binary comparator ($K\!=\!2$). An antiferromagnetically coupled spin pair with coupling $J>0$ and local fields $h_{a}(x),h_{b}(x)$ returns $\operatorname{sign}(h_{a}-h_{b})$ [14]; when $J$ exceeds the field magnitudes, the energy gap is $\Delta_{\mathrm{cmp}}(x)=2|h_{a}(x)-h_{b}(x)|$. Chaining $n_{c}$ such comparators produces an $n_{c}$-bit address into a $2^{n_{c}}$-entry lookup table. Sweeping the field difference and comparing the empirical misrouting rate against $\Delta_{\mathrm{cmp}}/T_{\mathrm{eff}}$ is the simplest experimental test of fetchless lookup.

Driven-dissipative oscillators offer a third realization (Supp. Mat. [28]).

Robustness: two separable failure channels.

At run time two error channels remain: signature perturbation after the correct route wins, and selection of the wrong route. The decomposition is a diagnostic tool, not a claim of statistical independence (Supp. Mat. [28], Remark 3).

Conditional decoding error. If route $k$ wins and the single-shot signature is $\tilde{\phi}_{k}=\bar{\phi}_{k}+\delta$ (with perturbation $\delta$ aggregating shot noise, detector noise, and drift), the minimum-norm decoder gives (using $W\bar{\phi}_{k}=y_{k}$)

$$\|W\tilde{\phi}_{k}-y_{k}\|_{2}=\|Y\Phi^{+}\delta\|_{2}\leq\frac{\|Y\|_{2}}{\sigma_{\min}(\Phi)}\,\|\delta\|_{2},$$ (8)

where $\|Y\|_{2}$ is the spectral norm of the payload table. For zero-mean fluctuations with covariance $\Sigma_{k}$,

$$\mathbb{E}\|W\tilde{\phi}_{k}-y_{k}\|_{2}^{2}=\operatorname{Tr}\!\big[Y\Phi^{+}\Sigma_{k}(\Phi^{+})^{\!\top}Y^{\!\top}\big]\leq\frac{\|Y\|_{2}^{2}}{\sigma_{\min}(\Phi)^{2}}\,\operatorname{Tr}\Sigma_{k}.$$ (9)

When stereotypy is imperfect and the residual input-dependent shift has norm at most $\varepsilon_{\mathrm{stereo}}$, the additional decoding error is bounded by $\|Y\|_{2}\,\varepsilon_{\mathrm{stereo}}/\sigma_{\min}(\Phi)$ (Supp. Mat. [28]), so stereotypy need only hold to within the device noise floor.

Dictionary drift. If the dictionary drifts from $\Phi_{0}$ at calibration to $\Phi_{0}+\delta\!\Phi(t)$ while the decoder remains $W_{0}=Y\Phi_{0}^{+}$,

$$\|W_{0}[\Phi_{0}+\delta\!\Phi(t)]-Y\|_{2}\leq\frac{\|Y\|_{2}}{\sigma_{\min}(\Phi_{0})}\,\|\delta\!\Phi(t)\|_{2}.$$ (10)

Recalibration is needed once the drift exceeds tolerance $\varepsilon_{\mathrm{tol}}\,\sigma_{\min}(\Phi_{0})$. Each recalibration costs $O(KR)$ forced-route measurements ($R$ trials per route), so the duty-cycle overhead is this cost divided by the drift interval $\tau_{\mathrm{drift}}$ over which $\|\delta\!\Phi(t)\|_{2}$ stays within tolerance. Since $\tau_{\mathrm{drift}}$ is device-set and presently uncharacterised, the overhead cannot be quantified without a drift measurement on a specific platform; the experimental protocol below measures it directly.

Routing error. If the intended route is $k^{\star}$ but the selector returns $\ell\neq k^{\star}$, no decoder can repair the mistake. We use the Gibbs form as a phenomenological fit; the theory needs only that larger margin means lower misrouting:

$$P(k)=\frac{e^{-E_{k}/T_{\mathrm{eff}}}}{\sum_{j}e^{-E_{j}/T_{\mathrm{eff}}}},$$ (11)

where $E_{k}(x)=-g_{k}(x)$ is the selector energy defined in Eq. (3) and $T_{\mathrm{eff}}$ is an effective temperature fitted from repeated trials [18, 9, 30], the misrouting probability satisfies

$$P_{\mathrm{mis}}\leq\frac{(K-1)\,e^{-\Delta/T_{\mathrm{eff}}}}{1+(K-1)\,e^{-\Delta/T_{\mathrm{eff}}}},$$ (12)

where $\Delta=\min_{k\neq k^{\star}}(E_{k}-E_{k^{\star}})$ is the minimum energy gap to the next competing route (equal to the score-space margin of Eq. (4), since $E_{k}=-g_{k}$). The bound follows from bounding each competitor’s Boltzmann weight by $e^{-\Delta/T_{\mathrm{eff}}}$; for $\Delta\gg T_{\mathrm{eff}}$, log-odds of correct routing scale linearly in $\Delta/T_{\mathrm{eff}}$. Near-degeneracy, where route statistics in Ising machines and SPIMs are known to depart from Boltzmann behaviour, is where the Gibbs form is least reliable; there the structural results (the two-channel decomposition and the certification) still hold, as they assume only monotonicity, while the specific bound (12) should be replaced by the empirically measured misrouting-versus-margin curve. With payload diameter $d_{Y}=\max_{k,\ell}\|y_{k}-y_{\ell}\|_{2}$, the triangle inequality gives a routing contribution at most $P_{\mathrm{mis}}\,(d_{Y}+\|Y\|_{2}\,\bar{\delta}/\sigma_{\min}(\Phi))$, where $\bar{\delta}=\max_{k}\sqrt{\operatorname{Tr}\Sigma_{k}}$ is the worst-case per-route noise level. A tighter second-moment decomposition is given in the Supp. Mat. (Remark 3); it requires the additional modeling assumption that route-conditioned emission noise has zero mean and covariance $\Sigma_{\ell}$ on each route $\ell$, independently of which route was intended (see Supp. Mat. for the precise statement). Figure 2(b–f) illustrates the decomposition on a controlled speckle-signature model (Monte Carlo; see Supp. Mat., Fig. S1). Because the simulation draws routes from the same Gibbs model used to derive Eq. (12), the agreement confirms the algebra, not the physical adequacy of the Gibbs assumption; that requires a goodness-of-fit test on real route frequencies, not yet available.

Calibration and training.

Fetchless lookup operates on two time scales. A slow calibration stage forces each route, measures signature statistics, forms $\Phi$, and compiles $W=Y\Phi^{+}$. Between recalibrations, $\Phi$ is fixed and online learning updates only $Y$ (one column per sample, rank-1 update in $W$); backpropagation through the $\operatorname*{arg\,max}$ uses a top-two surrogate gradient (Supp. Mat. [28]).

Figure 2: Photonic implementation and numerical validation of attractor-keyed memory. (a) Schematic photonic pipeline. A spatial light modulator (SLM) encodes the input $x$; a Fourier mask implements $B_{\mathrm{wide}}$. A routing map $R_{\mathrm{route}}$ compresses scores into $K$ competing routes. A spatial photonic Ising machine (SPIM) [26] selects the winner $\hat{k}_{b}$ with margin $\Delta$. The selected mode’s field propagates through a scattering medium; a detector array records the speckle signature $\tilde{\phi}_{\hat{k}}\in\mathbb{R}^{M_{\mathrm{sig}}}$. A pre-compiled decoder $W^{(b)}$ maps each signature to output; summing over $B$ blocks yields $y$. (b)–(f) Error decomposition on a controlled speckle-signature model (Monte Carlo, not experiment; $K\!=\!64$, $M_{\mathrm{sig}}\!=\!128$, $D\!=\!16$; $R$: calibration trials per route; $\sigma_{\mathrm{read}}$: readout noise s.d.). (b) Relative reconstruction error $\|W_{\hat{\Phi}}\,\Phi-Y\|_{F}/\|Y\|_{F}$ versus calibration trials $R$; four dictionary types (amplitude $|z|$, intensity $|z|^{2}$, heterodyne $[\Re\,z;\Im\,z]$, orthogonal); $\sigma_{\min}(\Phi)$ controls convergence rate. (c)–(f) All four panels use the amplitude-speckle dictionary. Solid lines: analytic RMS prediction $\sqrt{\mathbb{E}\|e\|_{2}^{2}}$ from the two-channel second-moment decomposition (Supp. Mat., Remark 3, Eq. (S17)); filled circles: Monte Carlo RMS ($10{,}000$ trials); faint circles: Monte Carlo mean $\mathbb{E}[\|e\|_{2}]$; dashed: per-trial worst-case bounds [Eq. (8) for decoding; $d_{Y}+\|Y\|_{2}\,\bar{\delta}/\sigma_{\min}(\Phi)$ for misrouting] weighted by $P_{\mathrm{mis}}$ from Eq. (12). (c) Routing-dominated regime (varying $T_{\mathrm{eff}}$ at fixed $\sigma_{\mathrm{read}}\!=\!0.03$). (d) Decoding-dominated regime (varying $\sigma_{\mathrm{read}}$ at fixed $T_{\mathrm{eff}}\!=\!0.05$). The two error channels are separable and each matches its predicted scaling. (e) Decoding error under stereotypy violation $\varepsilon_{\mathrm{stereo}}$ (correct routing enforced). (f) Full pipeline at a realistic operating point; the crossover at $T_{\mathrm{eff}}\!\approx\!0.71$ separates the decoding-limited from the routing-limited regime. (g), (h) Dictionary conditioning $\sigma_{\min}(\Phi)$ versus table size $K$ at fixed aspect ratio $M_{\mathrm{real}}/K$. Three measurement modalities, 30 random seeds per point; shaded bands: $\pm 1$ s.d. (g) $M_{\mathrm{real}}/K\!=\!2$. (h) $M_{\mathrm{real}}/K\!=\!4$. From $K\!=\!64$ onward, $\sigma_{\min}$ tracks the Bai–Yin prediction (dashed). At $K\!=\!1024$ the heterodyne value lies within $1\%$ of the asymptote at both aspect ratios. See Supp. Mat. [28] for full panel specifications.

Experimental protocol and falsifiability.

Four steps translate the theory into a falsifiable experiment: (1) Force each route in turn and record $R$ repeated signature measurements. This determines the sample-mean dictionary $\hat{\Phi}$ (the finite-sample estimate of $\Phi$), the covariances $\Sigma_{k}$, the empirical rank, $\sigma_{\min}(\hat{\Phi})$, and the stereotypy diagnostic $\varepsilon_{\mathrm{stereo}}/\!\sqrt{\operatorname{Tr}\Sigma_{k}}$ (Supp. Mat., Sec. S3). Full rank is confirmed only if a bootstrap confidence interval for $\sigma_{\min}(\hat{\Phi})$ excludes zero; the interval’s lower endpoint provides a conservative bound on dictionary conditioning. Under the hypothesis that centered signatures are sub-Gaussian with $\sigma_{\mathrm{sg},k}^{2}\leq\|\Sigma_{k}\|_{2}$, the number of trials needed to resolve $\sigma_{\min}(\Phi)$ scales as $R=O(K\,\|\Sigma\|_{2}\,(M_{\mathrm{sig}}+\log K)/\sigma_{\min}(\Phi)^{2})$ (Supp. Mat., Proposition 2). If $\operatorname{rank}(\Phi)<K$, no linear decoder can realize universal fetchless lookup. (2) Compile $W=Y\Phi^{+}$ for test payloads and verify $W\Phi\approx Y$ on forced-route means; compare single-shot error scaling with $\sigma_{\min}(\Phi)$ via Eqs. (8)–(9). (3) Release the selector, record winner frequencies versus the measured top-two margin, and fit $T_{\mathrm{eff}}$. Assess the Gibbs fit by a goodness-of-fit test (e.g., $\chi^{2}$ on binned route frequencies); test Eq. (12). A necessary consistency check: if forced-route and free-running signatures differ significantly, the calibration model requires correction. (4) Monitor drift in mean signatures over time. Recalibrate when $\|\delta\!\Phi(t)\|_{2}$ exceeds $\varepsilon_{\mathrm{tol}}\,\sigma_{\min}(\Phi_{0})$, the tolerance derived from Eq. (10).

No hardware demonstration exists to date; the protocol defines the criteria a first experiment must satisfy. The required ingredients exist separately in current platforms [3, 13, 19, 4, 22, 23, 18, 16, 30]; integrating them into a single device remains open. A forced-routing dictionary measurement is the natural first experiment, followed by a $K=2$ routing test of the predicted $\Delta_{\mathrm{cmp}}/T_{\mathrm{eff}}$ dependence.

Scope and outlook.

How far the scheme scales depends on the oversampling ratio $M_{\mathrm{sig}}/K$. The Bai–Yin law keeps $\sigma_{\min}(\Phi)$ bounded away from zero when $M_{\mathrm{sig}}/K$ exceeds unity by a finite factor, yielding low error amplification at $M_{\mathrm{sig}}/K=2$ [Eqs. (8)–(9)]. Real device signatures may exhibit spatial correlations that reduce the effective degrees of freedom below $M_{\mathrm{sig}}$, worsening conditioning; the SVD diagnostic detects this. Among the three readout modalities tested [Fig. 2(b) and (g, h)], heterodyne achieves the best conditioning, improving $\sigma_{\min}$ by $3.4\times$ over amplitude-only measurement at matched complex-mode count, and remains within $1\%$ of the Bai–Yin asymptote up to $K\!=\!1024$ (Supp. Mat., Fig. S2).

AKM replaces an $O(D)$ memory read with an $M_{\mathrm{sig}}$-channel measurement and linear decode, favourable when the decode is absorbed into the measurement optics or when data-movement cost dominates compute [12, 31]; at $M_{\mathrm{sig}}/K=2$ with digital decode, break-even requires native optical decode or DRAM-resident payloads (Supp. Mat., Sec. S13).

Any competitive physical selector that produces high-dimensional signatures and settles to stereotyped attractor states is a candidate platform: coherent Ising machines, polariton condensate networks, laser arrays, and spatial photonic Ising machines (SPIMs). Among these, SPIMs are closest to the requirements: focal-plane division now enables fully programmable Ising selection [33], and full-aperture wavefront correction removes the aberration bottleneck that previously limited effective $M_{\mathrm{sig}}$ [17]. What remains untested is the quantity AKM requires: within-attractor variance of the full speckle signature, conditioned on the same winning route.

Indirect evidence is encouraging: speckle physical unclonable functions (PUFs), photonic reservoirs, and polariton condensates achieve $\gtrsim\!94\%$ reproducibility of attractor-level observables [8, 24, 21, 3, 20, 6, 9], but the continuous high-dimensional state within a given attractor is almost never quantified [28]. A first experiment need only record continuous-valued output conditioned on the same attractor, yielding the four quantities $\Phi$, $\sigma_{\min}(\Phi)$, $\Sigma_{k}$, and $\varepsilon_{\mathrm{stereo}}$ that determine whether a platform supports fetchless lookup at a target scale.

Patent and Implementation Notice. Certain systems, methods, hardware configurations, acceleration techniques, implementation architectures, and commercial applications related to the work described in this manuscript are the subject of pending or patent applications in progress. This manuscript is intended to describe the scientific concepts and experimental framework at a research level and does not disclose all proprietary engineering, hardware, system-integration, optimization, or commercial implementation details.