Simulation Study · Companion to the Coexistence Criterion Note · CC0

Meeting Every Condition and Still Not Computing

A coupled-mode test of the coexistence criterion for driven disordered optical cavities: the five published conditions are simultaneously satisfiable while the substrate computes worse than its own passive limit. Two missing conditions are identified and formalized.

STATUS — SIMULATION RESULT (CMT LEVEL) LICENSE — CC0 / PUBLIC DOMAIN CODE — INCLUDED, ALL SIX MODEL GENERATIONS

The companion criterion note reduces the question "can a driven, disordered, open optical cavity compute?" to five simultaneous conditions on one complex spectrum. This study tests that criterion in the note's own governing model: forty quasinormal modes, disorder coupling, saturable gain inside the recurrence, impulsive injection, intensity readout. The result is a constructive counterexample. All five conditions were met at once — nonzero intracavity second-order Volterra kernel with support on the photon lifetime, effective rank up to 27 of 40, matched fading memory, Thouless parameter of order unity, modest Petermann factor — and task performance monotonically degraded as the nonlinearity strengthened, across three distinct gain mechanisms and two coupling structures. Above threshold, the note's predicted mode-competition collapse was observed directly (participation 3 of 40 modes); spatially distributed gain restored multimode operation and computation still failed, because the autonomous lasing dynamics overwhelmed the input. Two conditions absent from the published criterion follow: consistency (input-dominance over autonomous dynamics) and kernel diversity (the second-order kernel must be high-rank as an operator, not merely nonzero). The criterion is necessary-only, and now demonstrably so.

§ 1

Scope

A full Stage-1 test of the criterion requires 3D vectorial FDTD of the cavity with explicit scatterers. That is not what this is. This study works at the level of the criterion note's own Eq. (4) — the coupled-mode expansion in quasinormal modes — which is precisely where the coexistence question is posed: the conditions are statements about modal amplitudes, decay rates, and the nonlinear coupling between them. A counterexample at this level is a counterexample to the criterion as written. What this level cannot do is settle the spatial-field questions (eigenfunction statistics from real scatterer geometry, evanescent readout correlations); those remain for FDTD (§ 9).

Everything here is deterministic and noiseless. Noise can only make the negative result stronger.

§ 2

Model

\(N=40\) modes with detunings \(\delta_m\) drawn uniformly on \([-10,10]\) (mean spacing \(\Delta\omega\approx0.5\)) and decay rates \(\gamma_m\) on \([0.3,1.0]\), giving a Thouless parameter \(\bar\gamma/\Delta\omega\approx1.3\) — the \(\mathcal O(1)\) crossover the criterion asks for. Disorder coupling \(\kappa\) is a random Hermitian matrix of element scale \(\sigma_\kappa\). The governing equation is the note's Eq. (4) with a saturable-gain nonlinearity in the recurrence:

(1) \[ \dot a_m=\bigl(i\delta_m-\gamma_m+G_m\bigr)\,a_m+i\sum_n\kappa_{mn}a_n+b_m\,u(t), \]

with three gain mechanisms tested:

(2) \[ G_m=\frac{p\,w_m}{1+S_{\rm tot}},\qquad G_m=\frac{p\,w_m}{1+S_m^{\rm loc}},\qquad \tau_g\dot G_m=p\,w_m-G_m\bigl(1+S\bigr), \]

i.e. instantaneous saturation on the total intensity \(S_{\rm tot}=\sum_n c_n|a_n|^2\) (rank-1, mode-symmetric); instantaneous saturation on a local intensity \(S_m^{\rm loc}=(C|a|^2)_m\) with a sparse row-normalized overlap matrix \(C\) (high-rank, mode-diverse — spatially distributed gain); and class-B dynamics in which the gain is a slow population variable with \(\tau_g=2.0\approx5\) symbol periods. Drive: an impulsive kick of amplitude \(\propto s_k\) at the start of each symbol period \(T_{\rm sym}=0.4\) — broadband (it excites all modes through \(b\)) and clock-synchronized. Per-symbol retention \(e^{-\gamma_m T_{\rm sym}}\in[0.67,0.89]\): genuine multi-symbol fading memory. Readout: \(N_{\rm det}=20\) random projections, intensity-detected, sampled twice per symbol.

§ 3

Metrics and controls

Four quantities per operating point. Dimensionality: \(r_{\rm eff}\), the participation ratio of the readout matrix's singular spectrum (note Eq. 6). Intracavity nonlinearity: the impulse-pair Volterra residual with a linear field readout, which excludes the detector by construction —

(3) \[ \epsilon_{h_2}(\tau)=\frac{\max_t\bigl|y_{12}(t)-y_1(t)-y_2(t)+y_{00}(t)\bigr|}{\max_t\bigl|y_1(t)-y_{00}(t)\bigr|}, \]

where \(y_{1},y_{2},y_{12},y_{00}\) are responses to pulse one, pulse two, both, and neither. For any linear system \(\epsilon_{h_2}\equiv0\); measured passive values were \(\sim10^{-16}\) (machine precision), confirming the diagnostic. Excess noise: the maximum Petermann factor \(K\) of the linearized operator. Computation: NARMA-10 NMSE and parity-3 accuracy with ridge readout, against the control the criterion note demands — a linear readout on a tapped delay line of the raw input (20 taps): NMSE \(0.18\), parity \(\approx\) chance.

§ 4

Getting to a working reservoir: two encoding lessons

Two failed model generations preceded the working one, and both failures are physics, not code. A spectrally narrow drive (constant within the symbol, i.e. DC in the rotating frame) addressed only the few modes within \(\sim\gamma\) of resonance: \(r_{\rm eff}\approx2\) regardless of disorder. A frequency-comb drive fixed the addressing (\(r_{\rm eff}\to20\!-\!28\)) but its beat period was incommensurate with the symbol clock, and the unsynchronized carrier phase scrambled the features: both tasks fell to chance. The lesson generalizes: the injection must be simultaneously broadband across the modal spectrum and synchronized to the symbol clock. An impulsive kick per symbol satisfies both, and produced a genuine reservoir: passive NARMA-10 NMSE \(0.25\) (\(\sigma_\kappa=0.1\)), within reach of the linear control, with \(r_{\rm eff}\approx11\!-\!12\).

§ 5

Result I — the conditions hold; the computation degrades

With the working encoding, the pump sweeps from passive to \(2.2\,p_{\rm th}\). Every diagnostic behaves exactly as the criterion intends: \(\epsilon_{h_2}\) rises from machine zero to \(0.44\!-\!0.70\) — large, intracavity-sourced, supported over the photon lifetime — and \(r_{\rm eff}\) increases with pump (gain offsets loss and lengthens lifetimes). Yet:

Fig. 1 — NARMA-10 NMSE (solid, left axis; lower is better) and \(\epsilon_{h_2}\) (dashed, right axis) versus pump, \(\sigma_\kappa=0.1\) ensemble. The nonlinearity switches on exactly as designed, and performance monotonically worsens. Dotted line: linear delay-line control.

mechanism	p/p_th=0	0.9	2.2	ε_h₂ @2.2	r_eff @2.2
instantaneous, global S	0.360	0.347	0.388	0.69	26.8
instantaneous, local S (high-rank)	0.360	0.379	0.464	0.67	26.7
class-B, global (τ_g≈5 symbols)	0.360	0.376	0.403	0.75	26.2
class-B, local	0.360	0.399	0.484	0.60	26.7
linear delay-line control	0.180			—	—

NARMA-10 NMSE at σ_κ=0.30; lower is better. Parity-3 stayed at chance for every mechanism and pump.

Three nonlinearity mechanisms, two coupling structures, one verdict: the intracavity nonlinearity never helps and usually hurts, while every published condition is satisfied. The high-rank (local) variant — the natural structural fix — performs worst.

§ 6

Result II — above threshold: collapse observed, and a second failure behind it

Operating above threshold around the free-running lasing state, with the input as a weak (\(\sim\)15%) perturbation — the regime of the published microcavity-laser reservoir results — produced the cleanest pair of observations:

Fig. 2 — Steady-state mode participation of the lasing field versus pump. Global gain (rust): the predicted mode-competition collapse, ~3 of 40 modes. Spatially distributed gain (blue): spatial-hole-burning analog sustains 9–16 modes. Computation is at chance in both (NMSE ≈ 1.02–1.09, parity ≈ 0.51): the autonomous field overwhelms the input either way.

The collapse the criterion note predicts in § 5 appears on cue and quantitatively: shared gain saturation crushes the lasing field onto a participation ratio of \(3.0\) out of 40 modes. Distributing the saturation spatially — the coupled-mode analog of spatial hole burning — restores multimode lasing (participation \(9\!\to\!15.5\), perturbation \(r_{\rm eff}\) up to \(23.6\)). And computation fails completely in both cases, at every pump: NMSE \(\approx1\) (the mean predictor), parity at chance. The reason is not dimensionality and not the kernel. The free-running field has its own autonomous dynamics — the lasing modes beat at their pulled frequencies regardless of the input — and the readout is dominated by activity uncorrelated with the symbols. The echo-state property fails while every condition in the published list holds.

§ 7

Why a nonzero kernel can be sterile

The below-threshold failure has a structural explanation. Expand the global saturable gain to first order in intensity:

(4) \[ G_m=\frac{p\,w_m}{1+S}\;\approx\;p\,w_m\bigl(1-S+\dots\bigr) \;\;\Rightarrow\;\; N_m[\{a\}]\;\approx\;-\,p\,w_m\,a_m\sum_n c_n|a_n|^2 . \]

This is nonlinear, intracavity, and generates a bona fide second-order kernel: with \(a_m(t)=\int G_m(t-t')s(t')dt'\) linear in the drive history, the product \(a_m\cdot S\) contains terms \(\propto s(t')\,s(t'')\,s(t''')\) supported over \(\tau_{\rm photon}\). The criterion's Eq. (10) is satisfied. But viewed as an operator on the mode space, the coupling is common-mode: one scalar, \(S(t)\), multiplies every amplitude through nearly identical coefficients \(p\,w_m\). The nonlinear feature it offers the readout is essentially a single function — total stored energy — times the linear state. Writing the second-order kernel with its output structure, \(h_2^{(m)}(\tau_1,\tau_2)\approx -p\,w_m\,[G\!\ast\!\cdot\,](\tau_1)\,\Phi(\tau_2)\), the mode index enters only through the scalar \(w_m\): the kernel is rank-one across the readout. One nonlinear degree of freedom cannot supply the diverse cross-products that NARMA-10 and parity require — but it does corrupt all \(N\) linear memory channels with a common multiplicative fluctuation. Hence the monotone degradation: the nonlinearity spends the substrate's linear memory and buys one feature.

The local-saturation variant was built to break exactly this symmetry, and its failure is therefore the sharpest data point in the study. The overlap matrix \(C\) diversifies the saturation inputs, but each region still saturates on a weighted sum of intensities — smooth, averaging, and strongly correlated across regions through the shared mode field. Diversity of the kernel requires more than distributing the medium; it plausibly requires nonlinear elements that respond to individual field configurations — point-like Kerr scatterers, antinode-placed saturable absorbers — rather than to integrated intensity. That is an open, testable design question (§ 9), not a conclusion.

§ 8

The corrected criterion

Two conditions must be added. Consistency (the echo-state property): trajectories from different initial conditions, driven by the same input, must converge —

(5) \[ \bigl\|a(t)-a'(t)\bigr\|\;\xrightarrow{\;t\;}\;0 \quad\text{under identical }u(t),\ \text{different }a(0),a'(0), \]

equivalently, the input-driven response must dominate the autonomous dynamics. Below threshold this holds trivially; above threshold it fails while all five published conditions hold. Kernel diversity: the second-order kernel, viewed as a map to the readout vector, must itself be high-rank —

(6) \[ r_{\rm eff}\!\left[h_2^{(m)}(\tau_1,\tau_2)\right]\gg1 \quad\text{across the readout index }m, \]

not merely \(h_2\neq0\). Gain saturation gives \(h_2\neq0\) with rank \(\approx1\) and is computationally sterile.

Revised simultaneous conditions — seven, of which this study tested and broke five-alone

1 · Nonlinearity in the recurrence\(h_2\neq0\), intracavity-sourced, \(\operatorname{supp}h_2\sim\tau_{\rm photon}\).
2 · Dimensionality\(r_{\rm eff}=\mathcal O(N_{\rm modes})\).
3 · Memory matched to drivespread of \(\{\gamma_m\}\) comparable to the symbol rate.
4 · Modal individuality\(\delta=\bar\gamma/\Delta\omega=\mathcal O(1)\).
5 · Readout above the excess-noise floorper-mode SNR degraded by \(\sqrt K\) (Petermann).
6 · Consistency / echo-stateinput-driven response dominates autonomous dynamics; Eq. (5). Fails above threshold even when collapse is cured.
7 · Kernel diversity\(h_2\) high-rank across the readout; Eq. (6). Gain saturation fails this while passing condition 1.

Status of the existence question. This study does not show the favorable region is empty — a wave-chaotic microcavity laser has computed in experiment, at a finely tuned "edge of stability" that is exactly the knife-edge of condition 6. It shows the region is not where conditions 1–5 alone point, and that the published criterion admits constructive counterexamples. Necessary, demonstrably not sufficient.

§ 9

Limitations and next steps

Limitations, plainly: this is the coupled-mode caricature, one cavity ensemble, one encoding family, noiseless, with a coarse pump grid that cannot resolve a narrow consistency window. The spatial physics — real scatterer geometry, eigenfunction statistics, evanescent patch correlations — is outside the model by construction.

Test condition 7 constructively. Replace distributed gain with point-like nonlinear elements in the mode basis — a single Kerr-type scatterer (\(N_m\propto\psi_m(\mathbf r_0)\sum_{n,n'}\psi_n\psi_{n'}a_na_{n'}^*a_{n'}\) at one location \(\mathbf r_0\)) or a saturable absorber at an antinode. Prediction: kernel rank, and parity-3, rise together. This is the single highest-information experiment available at CMT level.
Map the consistency window. A fine pump sweep across \(p_{\rm th}\) with the consistency correlation of Eq. (5) as the order parameter, locating the edge-of-stability band and measuring its width against disorder — i.e., quantify how much tuning condition 6 demands.
Add the noise floor. Quantum-limited noise weighted by the per-mode Petermann factor (condition 5), to test whether the consistency window survives realistic excess noise — the criterion note's own conjecture that high-\(r_{\rm eff}\) is intrinsically high-noise.
Then, and only then, Stage-1 FDTD. The 3D simulation inherits a sharper question than the original: not "is the field high-rank?" but "does the scatterer geometry admit a high-rank \(h_2\) with the input dominant?" — conditions 6 and 7 reformulated spatially.
An \(h_2\)-extraction protocol. The impulse-pair residual of Eq. (3) worked cleanly in simulation; specifying its experimental counterpart (pulse-pair pump–probe with homodyne readout) would make the criterion practically measurable, answering the standing review critique.

§ 10

Method note

This study was conducted by an AI system (Claude Fable 5, Anthropic) in a single interactive session at the request of the project, including model construction, the diagnosis and correction of the two encoding failures of § 4, the design of the falsifying comparisons of §§ 5–6, and this write-up. All six model generations — including the failed ones — are released with the paper, unedited, in the spirit of documenting rather than silently fixing. Everything is placed in the public domain under CC0; build on it without permission or attribution.