Rust Kerr Capacity Instrument and First Control
Status
The normalized Kerr characterization requested by the previous ledger entry now exists as an
executable Rust package at packages/kerr-capacity. pnpm only orchestrates the Cargo commands; the
simulator, estimator, command-line interface, tests, and report generation are implemented in Rust.
The first paired run passes the fixed-observation capacity-bound check but does not support Conjecture 5. At this parameter point, enabling Kerr dynamics increases the numerical rank of the observed mode set without increasing corrected total capacity or useful delayed-input capacity. The Kerr-disabled control exposes the main apparent nonlinear capacity as square-law detector capacity, not evidence of Kerr computation.
This is one normalized, noiseless parameter point and a severe control, not a LiTaO3 device claim.
Executable process model
The crate evolves an odd, centered window of complex cavity modes with
[ \dot a_\mu= \left[-\frac{\kappa}{2} +i\left(\delta-\theta-\frac{D_2\mu^2}{2}\right)\right]a_\mu +ig!\sum_{p+q-r=\mu}a_pa_qa_r^* +\sqrt{\kappa_{\mathrm{ex}}},s_\mu(t). ]
The declared drive has a continuous pump in mode zero and a bounded data signal in a separately declared mode. The scalar thermal detuning obeys
[ \dot\theta=\eta_T\sum_\mu |a_\mu|^2-\gamma_T\theta. ]
Integration uses fixed-step fourth-order Runge-Kutta. Optional complex Gaussian state noise has a seed independent of the input-stream seed. A run fails instead of silently continuing if the state becomes non-finite.
Two independently implemented Kerr operators represent the coupled-mode and mean-field/Fourier views:
direct-modalevaluates the frequency-selection sum explicitly;pseudospectralconstructs the angular field, evaluates (|\psi|^2\psi), and projects it back to the retained modes.
For (N=2h+1) modes, the pseudospectral path uses a grid of length (4N). Since the cubic spectrum lies in ([-3h,3h]), this is long enough to prevent circular aliases from returning to the requested window ([-h,h]). It is deliberately a transparent discrete Fourier implementation rather than an optimized FFT at this stage.
The accessible bus field is fixed as
[ s_{\mathrm{out},\mu}=s_{\mathrm{in},\mu} -\sqrt{\kappa_{\mathrm{ex}}},a_\mu. ]
Each symbol contributes exactly one observation snapshot. The configured observation is mode intensity, mode quadratures, or both. No virtual time nodes are introduced without increasing the declared real observation dimension.
Resource boundary
Every run records normalized integrals for:
- elapsed evolution time;
- incident continuous-pump energy;
- incident data-signal energy;
- coupling-weighted drive energy;
- intrinsic dissipation;
- external state out-coupling;
- expected energy injected by the declared noise process.
It also records mean and peak intracavity power and the final thermal state. Intrinsic loss and external out-coupling are kept separate because the latter belongs to the accessible optical interface rather than automatically to waste.
Raman response is not replaced by an arbitrary scalar correction. Every nonzero raman_fraction
is rejected until a material- and orientation-specific response kernel is implemented.
Capacity estimator
The input samples are seeded and independently uniform on ([-1,1]). Targets are products of probability-normalized Legendre polynomials over the declared input-history window. The finite target family is grouped by total polynomial degree, maximum lag, and number of interacting lags.
The estimator applies the following protocol:
- Align observations and targets only after the maximum requested lag.
- Split time sequentially into training and held-out partitions.
- Center and scale observations using training statistics only.
- Fit a linear ridge readout, including a training-only target mean.
- Score capacity on the held-out partition relative to that trained-mean baseline.
- Shuffle each target repeatedly with a separate deterministic seed, pool positive null scores, and take the configured global null quantile.
- Subtract that threshold from every raw target score and floor each corrected score at zero.
- Compare the unaltered corrected sum with the numerical rank of the training observation matrix.
The program does not clamp the total to the rank. A total exceeding rank by more than the declared finite-sample tolerance is reported as an empirical capacity-bound violation to investigate, not rewritten into a passing result.
Numerical verification
The Rust package has twelve passing tests covering:
- configuration and Raman-boundary validation;
- probability normalization of the Legendre basis;
- direct versus zero-padded cubic convolution;
- direct versus pseudospectral trajectories;
- the declared passive loss rate;
- deterministic input, noise, and permutation seeds;
- recovery of two known linear memories from a rank-two synthetic observation;
- the corresponding corrected-capacity/rank bound;
- complete-right-hand-side cross-checking;
- resource and report fields.
At the nonzero-Kerr smoke configuration, the maximum absolute direct/pseudospectral right-hand-side difference was
[ 5.389158\times 10^{-16}. ]
cargo fmt --check, cargo clippy --all-targets -- -D warnings, Cargo tests, and the pnpm package
checks pass.
Frozen first comparison
The two committed configurations are identical except for the Kerr coefficient:
| Parameter | Value |
|---|---|
| Seed | 20260810 |
| Retained modes | 7, indexed (-3) through (3) |
| Pump mode / data mode | 0 / 1 |
| Kerr coefficient, enabled / control | 0.35 / 0 |
| Intrinsic loss / external coupling | 0.65 / 0.35 |
| Detuning / second-order dispersion | 1.4 / -0.02 |
| Pump amplitude / input scale | 1.25 / 0.16 |
| Thermal coupling / decay | 0 / 0.08 |
| Step / steps per symbol | 0.02 / 4 |
| Warm-up / recorded symbols | 128 / 2400 |
| Observation | 7 bus-output intensities |
| Target degree / maximum lag | 3 / 4 |
| Training fraction / ridge | 0.7 / (10^{-6}) |
| Null trials / null quantile | 8 / 0.99 |
Lag alignment leaves 2,396 samples: 1,677 training and 719 held out. There are 55 population-orthonormal targets through degree three and lag four. State noise is zero; detector noise remains outside this model.
Result
| Measurement | Kerr 0.35 | Kerr-disabled | Kerr minus control |
|---|---|---|---|
| Declared observation dimension | 7 | 7 | 0 |
| Effective observation rank | 5 | 2 | +3 |
| Positive raw capacity sum | 1.031373 | 1.022198 | +0.009175 |
| Corrected total capacity | 0.979571 | 0.985532 | -0.005961 |
| Corrected degree-one capacity | 0 | 0 | 0 |
| Corrected degree-two capacity | 0.979152 | 0.985532 | -0.006381 |
| Corrected degree-three capacity | 0.000419 | 0 | +0.000419 |
| Current-input capacity | 0.977995 | 0.978269 | -0.000275 |
| Capacity involving history | 0.001576 | 0.007263 | -0.005687 |
| Effective-rank margin | 4.020429 | 1.014468 | — |
Both runs are within the effective-rank bound. Their independently estimated global permutation thresholds are 0.003934 for the Kerr run and 0.003235 for the control.
The normalized input-resource integrals are matched exactly: pump energy 316.000000, signal energy 1.721150, and coupling-weighted drive energy 111.202402 in each condition. The Kerr run has mean intracavity power 0.224683; the control has 0.247716. Thermal coupling is explicitly disabled in both, so the final thermal state is zero. These figures are useful for internal accounting only and have not been converted into joules or watts for a LiTaO3 device.
Critique of the result
The capacity bound survived this attempted falsification
The corrected totals are below their effective ranks without post hoc clipping. This is consistent with the observation-space bound. It is not an empirical proof of the theorem, and the evaluated target basis is incomplete beyond degree three and lag four.
Additional active modes did not create additional readable capacity
Kerr mixing raised the numerical observation rank from two to five, yet corrected total capacity fell slightly. Mode activation and rank availability therefore cannot be counted as task-relevant capacity without showing input-conditioned readable functions.
The detector explains the apparent nonlinearity
With Kerr and thermal feedback disabled, the control dynamics is affine in the field but its intensity observation is quadratic. It retains 0.985532 corrected degree-two capacity, almost entirely for the current input. The Kerr run’s similar 0.979152 degree-two result therefore cannot be credited to the material nonlinearity. This is the executable version of the ledger’s warning that decoder and detector nonlinearities are resources.
Raw capacity would have suggested the wrong ordering
The positive raw sum is larger with Kerr, while the bias-corrected total is smaller. This one case does not validate the chosen correction universally, but it demonstrates why an uncorrected sum over many targets is not an admissible success metric.
No useful memory gain appeared
Capacity involving any historical input is effectively absent in the Kerr run and is lower than in the Kerr-disabled control. This parameter point fails the gate requiring recurrence-enhanced task-relevant nonlinear delayed capacity while preserving input memory.
Limits of this entry
The comparison has only one seed and one parameter point. Thermal dynamics is implemented but is disabled here to isolate the Kerr coefficient. The comparison does not include Raman response, observation noise, detector bandwidth, physical-unit calibration, out-of-window leakage, orientation, a pump-only control, direct input-feature controls, a matched feed-forward Kerr cascade, or a parameter sweep. The numerical-rank threshold is also an analysis convention and must be tested for stability. Direct feedthrough of the current data field is part of the declared bus observation and is why current-input controls remain essential.
Decision
The implementation gate is passed: there is now a reproducible Rust instrument that represents the open driven process, cross-checks coupled-mode and LLE/Fourier formulations, charges fixed observations and normalized resources, and estimates held-out bias-corrected capacity.
The empirical gate for Conjecture 5 is not passed. The first controlled result is negative: Kerr does not improve corrected total or delayed nonlinear capacity at this operating point, and the dominant nonlinear score already exists with Kerr disabled.
The next experiment should add explicit linear-input, square-law-input, pump-only, and matched cascade controls, then sweep pump, detuning, Kerr strength, symbol duration, input scale, and quadrature versus intensity observation. Conjecture 5 remains deferred unless a connected robust region survives those controls.