Detector-Noise Survival of Kerr Quadrature Memory
Status
The Ledger 13 coherent-quadrature memory lead survives the predeclared normalized detector-noise gate. At detector-noise standard deviation (10^{-8}), both (L_1(u_{t-3})) and (L_1(u_{t-4})) remain family-wise significant in the Kerr case across seeds 20260810, 20260811, and 20260812, and their Kerr-minus-disabled corrected-capacity differences remain positive in every seed. Their signed cross-seed lower envelopes are respectively 0.003779 and 0.014251.
This is a local normalized-model result, not detector calibration and not evidence of nonlinear computation. No degree-two or degree-three target replicates at the decision floor. The empirical gate for Conjecture 5 therefore remains closed.
The correct decision is to retain the Kerr-modified linear-memory lead and advance it to a bounded parameter-region siege. That next search must remain distinct from a claim of physical detector feasibility.
Predeclared falsification protocol
The protocol was frozen before the evidence-producing run:
| Quantity | Frozen value |
|---|---|
| Observation | coherent real and imaginary bus-field quadratures |
| Detector-noise standard deviations | (0,10^{-10},10^{-9},10^{-8},10^{-7}) |
| Decision floor | (10^{-8}) |
| Seeds | 20260810, 20260811, 20260812 |
| Samples per seed | 2400 after 128 warm-up symbols |
| Target family | 55 Legendre-history targets, degree at most 3 and lag at most 4 |
| Training fraction | 0.7 |
| Joint permutations | 512 |
| Family-wise quantile | 0.99 |
| Replication rule | target passes the family-wise gate in every seed |
| Paired-advantage rule | Kerr target replicates and Kerr-minus-disabled is positive in every seed |
The (10^{-8}) decision floor was chosen because Ledger 13 measured the weakest Kerr quadrature feature standard deviation as approximately (3.4\times10^{-9}). The grid contains levels below and above that feature scale. Results at (10^{-7}) are retained as descriptive severity evidence; they did not move the predeclared decision boundary.
All detector-noise values are in normalized real-quadrature units. No dimensional LiTaO3 detector, local oscillator, bandwidth, ADC, shot-noise, or quantization model is inferred from them.
Observation-boundary implementation
noise_std remains dynamical state noise. The new detector_noise_std is independent configuration
state and is applied only after the noiseless bus output
[ s_{\mathrm{out}}=s_{\mathrm{in}}-\sqrt{\kappa_{\mathrm{external}}},a ]
has been converted into the declared real observation interface. The noiseless observation is retained for diagnostics, while only the noisy observation is supplied to the readout.
Input, dynamical-noise, detector-noise, and permutation streams have distinct deterministic seed derivations. Kerr and Kerr-disabled cases with the same seed and detector-noise level receive the same detector standard-normal draws. Changing detector noise leaves input symbols, state power, thermal state, and noiseless observations unchanged.
Focused tests establish reproducibility, common random numbers, observation-boundary placement, state/detector RNG separation, and exact zero-detector-noise backward behavior.
Signal, SNR, and readout amplification
For feature (j), the signal scale (s_j) is the standard deviation of the noiseless feature on the training partition only. With declared detector-noise standard deviation (\sigma_d), the reported linear power SNR and decibel SNR are
[ \operatorname{SNR}j=\frac{s_j^2}{\sigma_d^2}, \qquad \operatorname{SNR}{j,\mathrm{dB}}=10\log_{10}\operatorname{SNR}_j. ]
Zero detector noise is represented explicitly as infinite SNR for nonconstant signals and as an undefined zero-over-zero case for constant signals; non-finite floating-point values are not serialized. The report also records the realized training-sample RMS of the injected noise. Across all nonzero cases, realized-to-declared RMS lies in [0.9666, 1.0325], with mean 0.9962.
If a readout fitted in standardized coordinates has weights (w_j) and observed training scales (r_j), its raw-equivalent norm and detector-noise gain are
[ \lVert w_{\mathrm{raw}}\rVert_2 =\sqrt{\sum_{j:r_j>0}\left(\frac{w_j}{r_j}\right)^2}, \qquad g_d=\sigma_d\lVert w_{\mathrm{raw}}\rVert_2. ]
The standardized norm (\lVert w\rVert_2) is also reported. A degenerate feature is set to zero during training; raw conversion would be marked undefined if it nevertheless acquired a nonzero standardized weight. No undefined conversion occurred in the frozen suite.
At the (10^{-8}) gate, across the six Kerr lag-three/four readouts:
| Readout quantity | Minimum | Maximum | Mean |
|---|---|---|---|
| Standardized weight norm | 6.523 | 15.305 | 11.944 |
| Raw-equivalent weight norm | (1.244\times10^6) | (4.939\times10^6) | (2.775\times10^6) |
| Detector-noise gain | 0.01244 | 0.04939 | 0.02775 |
| Corrected target capacity | 0.01203 | 0.05129 | 0.02793 |
The weights remain large in raw normalized coordinates. Passing the normalized noise test does not remove this engineering concern.
Noiseless rank and noise-aware observable dimension
Three notions are kept separate:
- observed standardized numerical rank, which becomes full when nonzero independent noise is standardized;
- noiseless standardized numerical rank at the configured relative singular tolerance (10^{-6}), which records ideal mathematical span;
- noise-aware observable dimension.
At zero detector noise, the noise-aware dimension is defined to equal noiseless numerical rank. At nonzero noise it counts noiseless raw principal standard deviations strictly greater than (\sigma_d), capped by noiseless numerical rank. The strict threshold boundary and zero-noise case are unit-tested. Every noiseless standardized singular value and every noiseless raw principal standard deviation is serialized so the count is auditable.
Across all three seeds, the dimensions are:
| Detector-noise std | Kerr noiseless rank | Kerr noise-aware dimension | Disabled noiseless rank | Disabled noise-aware dimension |
|---|---|---|---|---|
| 0 | 14 | 14 | 4 | 4 |
| (10^{-10}) | 14 | 14 | 4 | 4 |
| (10^{-9}) | 14 | 13 | 4 | 4 |
| (10^{-8}) | 14 | 12 | 4 | 4 |
| (10^{-7}) | 14 | 9 | 4 | 4 |
At the decision floor, the smallest Kerr per-feature signal scale lies between (3.01\times10^{-9}) and (3.40\times10^{-9}) across seeds, giving a worst per-feature SNR between -10.44 dB and -9.38 dB. The two below-floor features span -10.44 dB to -7.34 dB across the three seeds. Twelve of fourteen Kerr features individually exceed the noise standard deviation, and the principal-axis criterion also yields dimension 12. The strongest feature scale remains approximately 0.09. The disabled system has only four noiseless directions; its remaining features are constant before detector noise.
Paired capacity outcome
The Kerr-minus-disabled total and historical-capacity differences remain positive at every level and seed:
| Detector-noise std | Total difference range | Total mean | Historical difference range | Historical mean |
|---|---|---|---|---|
| 0 | [0.015012, 0.020287] | 0.017792 | [0.015171, 0.020375] | 0.017836 |
| (10^{-10}) | [0.036568, 0.065679] | 0.054970 | [0.035121, 0.063522] | 0.052767 |
| (10^{-9}) | [0.028408, 0.057255] | 0.041580 | [0.027235, 0.055301] | 0.039651 |
| (10^{-8}) | [0.022500, 0.039771] | 0.033830 | [0.022453, 0.037882] | 0.032324 |
| (10^{-7}) | [0.018369, 0.035495] | 0.028876 | [0.018564, 0.034316] | 0.027954 |
The two predeclared delayed targets have the following cross-seed outcomes:
| Noise std | Target | Kerr significant seeds | Disabled significant seeds | Positive paired deltas | Signed lower envelope | Mean delta |
|---|---|---|---|---|---|---|
| 0 | (L_1(u_{t-3})) | 3/3 | 3/3 | 3/3 | 0.006939 | 0.008130 |
| 0 | (L_1(u_{t-4})) | 3/3 | 3/3 | 3/3 | 0.010641 | 0.011823 |
| (10^{-10}) | (L_1(u_{t-3})) | 3/3 | 3/3 | 3/3 | 0.009115 | 0.011521 |
| (10^{-10}) | (L_1(u_{t-4})) | 3/3 | 1/3 | 3/3 | 0.021863 | 0.029051 |
| (10^{-9}) | (L_1(u_{t-3})) | 3/3 | 3/3 | 3/3 | 0.006079 | 0.007287 |
| (10^{-9}) | (L_1(u_{t-4})) | 3/3 | 1/3 | 3/3 | 0.015147 | 0.024363 |
| (10^{-8}) | (L_1(u_{t-3})) | 3/3 | 3/3 | 3/3 | 0.003779 | 0.005831 |
| (10^{-8}) | (L_1(u_{t-4})) | 3/3 | 1/3 | 3/3 | 0.014251 | 0.021342 |
| (10^{-7}) | (L_1(u_{t-3})) | 3/3 | 3/3 | 3/3 | 0.003640 | 0.006318 |
| (10^{-7}) | (L_1(u_{t-4})) | 3/3 | 1/3 | 3/3 | 0.014924 | 0.019287 |
At the (10^{-8}) gate, the only Kerr targets that replicate are the current input and the two declared delayed linear targets. No nonlinear target replicates. Detector noise therefore preserves the local Kerr-associated redistribution of linear memory; it does not uncover nonlinear delayed capacity. Lag three also exists significantly in the disabled control in 3/3 seeds, and lag four in 1/3 control seeds. The earned result is thus the replicated positive Kerr-minus-disabled advantage, not unique existence of delayed memory under Kerr.
Canonical SQLite evidence
Schema version 2 stores the result transactionally in
packages/kerr-capacity/output/detector-noise-frozen/results.sqlite. The output directory remains an
ignored reproducible artifact directory; the ledger and frozen configuration are Git-visible.
Normalized relations contain:
| Relation | Rows |
|---|---|
| Noise levels | 5 |
| Normalized observation-noise cases | 30 |
| Resource accounts | 30 |
| Target/readout records | 1650 |
| Singular-value records | 420 |
| Feature-scale records | 420 |
| Signal/SNR diagnostic records | 420 |
| Paired target differences | 825 |
| Cross-seed replication outcomes | 275 |
SQLite reports integrity: ok and zero foreign-key violations. Derived JSON, CSV, and Markdown
exports are produced from the same in-memory suite. Selected SHA-256 hashes from the evidence run
are:
| Artifact | SHA-256 |
|---|---|
results.sqlite |
4A1A403E23737CCFBA169B1F15198865BC37AB288102A5A8427F3D43749F6315 |
noise-suite.json |
E2FFD54D9ACDBA49EF066E3E2D4E60FE6B85822A48D098AFA8E7922D237B6D98 |
noise-replication.csv |
454A6CF56A701F292D4EECE7D781559B08051B14AD1D1C25A267307EB40BC732 |
Deutsch–Popperian critique
The local conjecture under test was not that Kerr memory is generally robust. It was the narrower claim that the Ledger 13 lag-three/four quadrature advantage would survive a predeclared normalized independent observation-noise floor at one frozen parameter point. The experiment exposed that claim to failure and it survived.
That survival identifies new errors to attack rather than supplying confirmation:
- The noise has no detector physics. It is independent additive Gaussian noise in normalized quadrature coordinates. Correlation, phase dependence, local-oscillator limits, finite bandwidth, quantization, drift, and shot noise remain unmodeled.
- Raw readout amplification remains severe. Gate-target raw-equivalent norms are between (1.24\times10^6) and (4.94\times10^6). The measured noise gains remain finite in this model, but no actuator, estimator, or electronic dynamic-range constraint has been imposed.
- Any nonzero noise changes the standardized control design. Ten exactly constant Kerr-disabled features become pure-noise columns and are standardized to unit variance. Its observed standardized rank therefore jumps from 4 to 14 even though noiseless and noise-aware ranks correctly remain 4. The sharp capacity change between zero and (10^{-10}) is consequently an estimator/readout effect, not a physical enhancement. The paired design remains fair, but the magnitude of the noisy Kerr advantage is not monotone evidence of stronger physics.
- The observable-dimension rule is declared, not derived from an instrument. A one-standard- deviation principal-axis floor is an auditable robustness criterion, not a universal definition of observability.
- Replication is still local. Three seeds test sampling variation at one dynamical point. They do not establish a connected parameter region, fabrication tolerance, or comparison against a resource-matched linear cascade.
- Only one level was inferentially decisive. The (10^{-8}) gate was fixed in advance. The remaining levels map severity and must not be mined post hoc for a more favorable threshold.
The next severe test should therefore seek a connected region in normalized pump, detuning, Kerr strength, and symbol duration while retaining the same paired noise and replication rules. It should add a declared readout-gain constraint and a resource-matched linear or cascaded control. The siege should try to destroy the lead, not optimize a single successful point.
Verification
The implementation passed:
cargo fmt --check;cargo check --all-targets;cargo test --all-targetswith 31 passing tests;- direct/pseudospectral full-RHS cross-check error (5.389158\times10^{-16}), below (10^{-9});
- the release-mode frozen noise-suite command;
- SQLite integrity and foreign-key checks;
git diff --check.
Decision
The predeclared detector-noise gate is passed. The Kerr-associated lag-three/four coherent linear memory survives normalized detector noise at (10^{-8}) across all three frozen seeds, while the noise-aware Kerr observation dimension falls from 14 to 12.
Advance the lead to a bounded categorical parameter siege, with readout-gain and matched-linear controls carried forward. Do not claim physical detector viability. Do not formulate Conjecture 5: the required replicated nonlinear delayed capacity remains zero.