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When Does Interference Help Learning? Kernel Geometry as a Pre-Experimental Test for Photonic Reservoir Computing

Oishik Kar, Aswath Babu H

arXiv:2610.11360v1quant-phcs.CE

TL;DR

Whether photon interference contributes to learning remains an empirical question, and task-dependent reservoir results lack a task-independent characterization of feature-map geometry. This paper compares reservoir kernels through geometry and task alignment, finding that task-level advantages in the examined record coincide with aligned tasks.

  • Problem

    Whether interference contributes to learning remains an empirical question, and the reservoir setting lacks a task-independent characterization of feature-map geometry under partial distinguishability.

    Source: §I · p. 1

  • Method

    The paper compares indistinguishable- and distinguishable-photon reservoir kernels, then evaluates kernel geometry and task alignment before hardware commitment.

    Source: §I · p. 1, §II · p. 2, §VIII · p. 5

  • Results

    The geometric difference identifies quantum-kernel directions absent from the classical map, and task-level advantages in the examined record occur on aligned tasks.

    Source: §VIII · p. 5

  • Takeaways & Limitations

    Interference merits experimental cost only when both kernel geometric separation and task alignment are favorable.

    Source: §VIII · p. 5

    Results are numerical at classically simulable system sizes and bracket, but do not replace, hardware validation.

    Source: §VIII · p. 6

Abstract

from arXiv · show

Multi-photon interference is the resource most native to photonic quantum machine learning, yet experiments on photonic reservoirs have reported advantages, null results, and train-only effects. These outcomes cannot be compared: each measures task-dependent accuracy on different tasks and platforms. Here we resolve the question task-independently through the kernel geometry of boson-sampling feature maps. Interference redistributes feature variance across roughly twice as many effective dimensions as distinguishable-particle statistics, generating a large geometric separation between the corresponding kernels. The separation grows superlinearly with indistinguishability, is linear in Hong--Ou--Mandel visibility at small visibility for two to four photons, and retains approximately 80% of its magnitude at the source quality of current quantum processors. Under experimental sampling budgets the separation is learnable on adversarially constructed tasks (accuracy advantage +0.22 plus or minus 0.05) and an order of magnitude smaller on natural tasks, gated by kernel--task alignment. The results reconcile the existing experimental record and supply a pre-experimental protocol: kernel geometry establishes that an interference-specific learning resource exists, and task alignment determines whether a given task can access it; both are computable before committing significant hardware resources.

I. INTRODUCTION · II. BOSON-SAMPLING RESERVOIRS AND THEIR KERNELS

The paper argues that interference creates a task-independent kernel-geometry separation, while kernel–task alignment determines whether that separation improves learning. It proposes computing geometry and alignment before substantial hardware investment.

I. INTRODUCTION

  • Prior experiments found an indistinguishability advantage on nonlinear temporal tasks but not linear memory, no resolved two-photon effect despite higher feature-space rank, and higher training but unchanged test accuracy.The paper argues these results are not directly comparable because they use different tasks, readouts, platforms, and task-dependent metrics.Source: §I · p. 1
  • Because reservoir readouts are regularized linear models, their learning behavior is governed by the induced kernel, whose geometry can be computed before a task and bounds linear-readout performance.The magnitude and partial-distinguishability behavior of this geometry, and its practical sufficiency for natural tasks, had not been characterized in this setting.Source: §I · p. 1
  • +0.222 ± 0.046 accuracy advantage was measured on adversarially designed tasks; it was an order of magnitude smaller on natural tasks and zero when neither kernel aligned with the task.The results show that geometric separation alone does not ensure a learning advantage; task alignment gates whether it is accessible.Source: §I · p. 1
  • Interference redistributes feature variance across roughly twice as many effective dimensions, with kernel separation growing superlinearly with indistinguishability and persisting under hardware-typical imperfections and experimental sampling budgets.The paper presents this separation as a property of kernel geometry, rather than a task-specific accuracy result.Source: §I · p. 1
  • The proposed protocol reconciles the apparent disagreement among experiments by computing kernel geometry and task alignment before committing significant hardware resources.Geometry establishes whether an interference-specific resource exists; alignment indicates whether a task can access it.Source: §I · pp. 1–2

II. BOSON-SAMPLING RESERVOIRS AND THEIR KERNELS

  • The reservoir encodes classical inputs into interferometer phases, injects N photons into the first N modes, and uses post-selected, renormalized output measurements as features.The architecture uses independent Haar-random interferometers.Source: §II · p. 2
  • Indistinguishable and distinguishable photon statistics differ in the placement of the modulus, but both are degree-2N nonlinear functions, so their learning difference reflects interference-induced sign structure.Source: §II · p. 2
  • Partial distinguishability is parameterized by photon mode overlaps; for uniform overlaps, the Hong–Ou–Mandel visibility is V = I^2.Source: §II · p. 2

III. INTERFERENCE BROADENS THE FEATURE SPECTRUM · IV. A GEOMETRIC DIAL CONTROLLED BY INDISTINGUISHABILITY

At the flagship configuration, interference spreads the same total kernel variance across roughly twice as many effective dimensions as distinguishable-photon statistics. The geometric separation is a smooth, strongly nonlinear function of indistinguishability and remains substantial at reported hardware source qualities.

III. INTERFERENCE BROADENS THE FEATURE SPECTRUM

  • Interference cancellations spread feature variance: nonnegative permanents cannot cancel, whereas signed amplitudes decorrelate outcome fluctuations.The passage relates this mechanism to expressivity arguments, noting hardware evidence of higher feature-space rank at two photons and an account based on richer transformations.Source: §III · p. 2

IV. A GEOMETRIC DIAL CONTROLLED BY INDISTINGUISHABILITY

  • The geometric difference g upper-bounds the ratio of kernel model complexities; g > 1 certifies tasks favoring indistinguishable photons, but guarantees nothing for any particular task.Here model complexity is sK(y) = y⊤Ky.Source: §IV · p. 2

III. INTERFERENCE BROADENS THE FEATURE SPECTRUM

  • At the flagship configuration, indistinguishable-photon variance reaches 90% of its trace in 61 dimensions, versus 33 for distinguishable photons.Both Gram matrices have identical trace; in a representative realization, their leading eigenvalues are 5.7 for KQ and 10.0 for KC.Source: §III · p. 2, Figure 2 · p. 3

IV. A GEOMETRIC DIAL CONTROLLED BY INDISTINGUISHABILITY

  • Across the full visibility dial, the effective exponent rises from 1.16 at N = 2 to 1.22 at N = 3.The excess above unity is the convex correction; the N = 4 small-visibility fit uses a restricted window.Source: §IV · pp. 2–3
  • At N = 3, separation retention is 80.3% ± 0.4% at ref. source quality (V = 0.864) and 95.8% ± 0.1% at ref. source quality (V = 0.972).Retention is the fraction of excess separation retained at visibility V.Source: §IV · p. 3
  • At V = 0.864, retention decreases from 83.1% at N = 2 to 80.4% at N = 3.Source: §IV · p. 3
  • At full indistinguishability, the geometric separation is g = 10.3 ± 1.1 across ten interferometer realizations (M = 12, N = 3, n = 100).The separation varies smoothly and strongly nonlinearly with visibility.Source: §IV · p. 2, Figure 3 · p. 3
  • At small visibility, g − 1 is nearly linear for N = 2, 3, 4, with fitted exponents kV = 1.06, 1.09, 1.12, respectively.For N = 2, g(V) − 1 = (s/2)V + O(V^2) exactly, where s is the largest eigenvalue of the cross-statistics matrix.Source: §IV · p. 2, Figure 3 · p. 3

V. ROBUSTNESS OF THE SEPARATION · A. Sampling Budgets · B. Conditioning and the Meaning of g

At S = 3 × 10^4 shots, the designed-task advantage matches the exact-arithmetic result, while matched-sample comparisons show that task advantage persists despite conditioning-sensitive changes in g. Uniform loss cancels in renormalized post-selected distributions, but multiphoton-emission noise remains unmodeled.

A. Sampling Budgets

  • +0.244 ± 0.112 designed-task accuracy advantage at S = 3 × 10^4 shots is indistinguishable from exact arithmetic; the advantage is stable from 10^3 to 10^6 shots.At each budget, outcome distributions were resampled and the kernels, adversarial task labels, and readout reconstructed from sampled data.Source: A. Sampling Budgets · pp. 3–4, Figure 4 · p. 4

B. Conditioning and the Meaning of g

  • At matched n = 180, g falls 5.6-fold between rank-limited and unconstrained configurations, while designed-task advantages (+0.248 ± 0.075 vs. +0.207 ± 0.044) are unchanged within error.The comparison uses M = 12, D = 220 for the rank-limited configuration and M = 20, D = 1140 for the unconstrained configuration.Source: B. Conditioning and the Meaning of g · p. 4, Figure 4 · p. 4
  • Across n = 60–180, g · λmin(KC) stays approximately 4 × 10^-3 across a 27-fold range of g, because the geometric difference lies in the empirical classical kernel’s tail directions.Thus, the absolute value of g depends on how deeply the finite-sample Gram matrix resolves the tail, and comparisons are made at matched n.Source: B. Conditioning and the Meaning of g · p. 4
  • Uniform per-mode efficiency loss cancels exactly in renormalized post-selected distributions, while mode-dependent loss is statistically absorbed into an effective interferometer.Uniform loss rescales each collision-free N-photon outcome by η^N before renormalization.Source: B. Conditioning and the Meaning of g · p. 4
  • Multiphoton-emission noise (g(2)) is unmodeled and is identified as the imperfection most likely to alter geometric quantities at fixed V.Its characterization is prioritized for hardware follow-up because it perturbs the input state rather than the channel.Source: B. Conditioning and the Meaning of g · p. 4

VI. LEARNABILITY IS GATED BY KERNEL–TASK ALIGNMENT · VII. RECONCILING THE EXPERIMENTAL RECORD

Learning gains from the quantum photonic kernel depend on kernel–task alignment, not feature dimension alone. This alignment-based account explains contrasting experimental outcomes and predicts a published null result within its variability.

VI. LEARNABILITY IS GATED BY KERNEL–TASK ALIGNMENT

  • +0.222 ± 0.046 accuracy advantage for the indistinguishable- over distinguishable-photon kernel on designed tasks contrasts with small, graded natural-task results.Natural-task advantages range from +0.041±0.027 on the radial family to −0.006 ± 0.024 on the checkerboard family; the real digits task gives +0.036 ± 0.054. The pooled designed–natural separation is +0.187 (paired t = 8.69, p < 10−4).Source: Figure 5 · p. 4, §VI · p. 5
  • The advantage appears at alignment extremes: the designed task realizes the full gain, while zero- or negligible-alignment tasks and random labels show no advantage.The random-label control produces +0.009, indistinguishable from zero. Fine-grained ordering among natural tasks remains unresolved: in an expanded battery, alignment predicted the sign in 9 of 13 cases, while rank correlation was not significant (ρ = 0.027, p = 0.93).Source: §VI · p. 5
  • At matched M = 20 and n, designed-task advantages are +0.305 ± 0.152, +0.207 ± 0.044, and +0.165 ± 0.059 for N = 2, 3, and 4, respectively.These gains are constant within error bars across a 25-fold range of feature dimension, consistent with alignment rather than Hilbert-space size governing learnability.Source: §VI · p. 5
  • Dimension-matched random Fourier features do not recover the designed-task advantage, while both photonic kernels outperform them on tasks designed against the RFF map.Against the classical-kernel-designed task, the quantum kernel attains 0.761 accuracy versus RFF at 0.583; against the RFF-designed task, quantum and classical attain 0.883 and 0.878, respectively, versus RFF at 0.439, while the photonic kernels remain indistinguishable from each other.Source: §VI · p. 5

VII. RECONCILING THE EXPERIMENTAL RECORD

  • The four experiments in Table II sample different regions of one landscape: task structure differentiates opposite N = 2 outcomes, and generalization follows alignment rather than trainability alone.Within Di Bartolo et al., the advantage varies with task demand and vanishes on linear memory recall; Rambach et al. show improved trainability on all tasks but an alignment-dependent train–test gap.Source: §VII · p. 5
  • For ref., the framework predicted a +0.015 indistinguishability effect against ±0.017 interferometer spread, matching the reported null structure and effect magnitude within 10%.The prediction used ref. ’s photon number, detector count, task, and sample size without reference to its results; predicted geometric difference was g = 4.4±0.5, but task alignment gap was ∆A = +0.008. The published effect was +0.014 against ±0.042 variability.Source: §VII · p. 5
  • The kernel-space account complements ref. ’s Fisher-information interpretation: both describe variance spread across output-space directions, framed respectively in kernel space and parameter space.In kernel terms, sensitivity of output probabilities to input variation corresponds to variance spread across directions of the outcome space.Source: §VII · p. 5

VIII. DISCUSSION

The discussion frames interference as worthwhile only when both kernel geometry and task alignment are favorable. It presents a pre-experimental protocol but identifies prospective validation before measurement as the definitive test.

Pre-experimental protocol

  • The protocol recommends interference only when the geometric difference g is large and quantum-kernel alignment AQ exceeds classical-kernel alignment AC for the candidate task family.It estimates both kernels from a pilot set of unlabeled inputs; about 100 inputs and a modest shot budget of S ≈ 3 × 10^4 are reported as sufficient.Source: §VIII · pp. 5–6

Validation and follow-up

  • The framework reproduces the null result of ref. when applied blind to that configuration, while the pre-registered task battery was specified before its results existed.Prospective testing g and task alignment before measurement is identified as the definitive validation and immediate follow-up.Source: §VIII · pp. 5–6

Interior-optimum analysis

  • The cited passage characterizes the result as a readout-level effect or as within sampling variability, and explicitly rules out a geometric explanation.Source: §VIII · p. 6

Partial distinguishability

  • Partial distinguishability acts as a continuous resource, with high retention at hardware-typical visibility, while optimal task performance need not occur at full indistinguishability.Interior optima in distinguishability are non-geometric; whether readout-level mechanisms can produce them, as suggested by ref., remains open.Source: §VIII · p. 6

Broader interpretation

  • The framework links photonic-reservoir studies to quantum-kernel theory, framing expressivity as geometry and generalization as alignment.It interprets concentration warnings for large quantum kernels as conditioning sensitivity of g.Source: §VIII · p. 6

Limitations

  • The results are numerical at classically simulable sizes, and the synthetic adversarial tasks and modeled shot noise and visibility do not replace hardware validation.Natural-task alignment ordering remains unresolved at current sample sizes; g is conditioning-sensitive and sample-size dependent, so comparisons are made at matched configuration.Source: §VIII · pp. 6–7

Computational scope

  • Because estimating KQ requires #P-hard matrix permanents at scale, the protocol applies through the small-to-midscale engineering regime, not large systems.For N ≥ 50, classical evaluation is unavailable; predicting utility beyond that frontier requires alignment-heuristic extrapolation, which remains open.Source: §VIII · p. 7

METHODS … Numerical Integrity

The methods define boson-sampling features and kernels, construct adversarial and natural tasks, and evaluate readouts across repeated splits and shot-noise samples. They also document numerical checks for the simulation pipeline.

Feature Maps and Kernels

  • Features are post-selected, renormalized collision-free outcome distributions, with exact probabilities from Ryser permanents and raw kernels analyzed alongside standardized-kernel cross-checks.The flagship geometry configuration is M = 12, N = 3, d = 12, n = 100 (D = 220); the accuracy configuration is M = 20, N = 3, d = 12, n = 180 (D = 1140). Visibility is V = I2.Source: Feature Maps and Kernels · p. 7

Geometric Difference and Alignment

  • Geometric difference follows the square-root convention of ref., retention is computed per interferometer realization, and kernel–target alignment follows Eq. (8).Regularization ε is at least three orders of magnitude below the smallest eigenvalue of KC; convergence and the conditioning-pinning relation g · λmin(KC) ≈const. are documented in Supplementary Note 1.Source: Geometric Difference and Alignment · p. 7

Task Construction

  • Adversarial labels follow ref., using a leading generalized eigenvector and median binarization; out-of-sample labels derive the direction from training data and use a kernel expansion.The reverse construction exchanges the roles of KQ and KC.Source: Task Construction · p. 7
  • Natural tasks comprise threshold, checkerboard, radial, and linear-combination label families on the input manifold, plus 3-vs-8 digit classification.Exact task generators are given in Supplementary Note 3.Source: Task Construction · p. 7

Evaluation Protocol

  • Evaluation uses cross-validated kernel ridge classifiers, 10 Haar realizations and 10 stratified splits, paired t-tests for gating, and matched-D random Fourier feature baselines.Shot-noise experiments resample each outcome distribution at budget S, then reconstruct kernels, labels, and readouts from sampled data alone; outcomes below 1/(2S) are trimmed.Source: Evaluation Protocol · p. 7

Numerical Integrity

  • Numerical checks validate permanents against brute force, confirm unit-norm statistics including collisions and N = 1 agreement, reproduce both partial-distinguishability endpoints, and recover ref. ’s overlap-kernel setting.All results are regenerated from a single version-controlled pipeline.Source: Numerical Integrity · p. 7

Supplementary Information · Conditioning-pinning: · Kernel-convention cross-check:

Supplementary checks show that the geometric difference converges under regularization, varies with kernel conditioning, and retains its spectral contrast under standardized feature kernels. The conditioning dependence is characterized explicitly, while task-level results are reported as insensitive to it at matched configurations.

Supplementary Information

GEOMETRIC DIFFERENCE

Conditioning-pinning:

  • Across n = 60–180, g · λmin(KC) stays approximately constant at 4 × 10^-3 in the squared convention, despite a 27-fold range of g.Table IV documents sample-size dependence for the flagship configuration and a representative realization.Source: Conditioning-pinning · p. 10, Table IV · p. 10
  • The generalized-eigenvalue characterization makes g a conditioning readout: g^2 scales as ⟨vmin, KQvmin⟩/λmin(KC), with an approximately n-independent numerator.The maximum is attained on the eigenvector of KC with its smallest eigenvalue.Source: Conditioning-pinning · p. 10
  • Main-text comparisons use matched n and configuration, and direct measurement found task-level results insensitive to conditioning in the dissociation experiment.The passage identifies the measurement as main-text Fig. 3b.Source: Conditioning-pinning · p. 10

Kernel-convention cross-check:

  • Under standardized feature kernels from refs., the 90%-variance effective dimension is 61 for indistinguishable particles versus 33 for distinguishable particles, with the leading-eigenvalue ratio in the same direction.Source: Kernel-convention cross-check · p. 10
  • Z-scoring suppresses g relative to its raw value, while both kernel conventions remain monotone in V and preserve identical retention orderings.The z-scored comparison is reported at ε = 10^-3.Source: Kernel-convention cross-check · p. 10

NOTE 2: EXPONENT ANALYSIS OF THE DIAL … Shot-noise protocol:

The notes explain why small-visibility kernel separation is nearly linear, describe the natural and adversarial task constructions, and specify how sampling-limited experiments are simulated. Natural-task alignment varies with task structure, while adversarial labels are derived from the kernel pair.

Small visibility:

  • At small V, g(V)−1 is approximately linear for N=2–4 because transpositions with weight I2=V dominate; for N=2, the 1.057 fitted exponent reflects window bias.The N=2 outcome probabilities are exactly affine in V, with quadratic corrections to g−1; longer permutation cycles are suppressed by higher powers.Source: Small visibility · p. 10

NOTE 3: NATURAL TASK FAMILIES AND GENERATORS

  • Natural tasks use labels on [0, 1]^d, with median binarization where required, and the task domain has d=12.The tasks use the same uniform input sets as the kernels, except for the digits task; Table VI gives the generators as implemented in the frozen pipeline.Source: NOTE 3: NATURAL TASK FAMILIES AND… · p. 11, Table VI · p. 11
  • The natural families span coordinate, high-frequency, smooth geometric, and generic linear structure; the checkerboard has the lowest absolute alignment, AQ=0.049.The radial boundary is a radius-0.5 sphere centered at the manifold center in the first three coordinates; the linear boundary uses weights (0.4, 0.3, 0.2, 0.1) over the first four coordinates and threshold 0.5. Checkerboard frequency is two oscillations per unit interval in each of the first two coordinates.Source: NOTE 3: NATURAL TASK FAMILIES AND… · p. 11

Adversarial tasks:

  • Adversarial labels follow ref.: a leading generalized eigenvector of the kernel pair is mapped and median-binarized, and the reverse construction exchanges KQ and KC.This construction designs labels from the kernels rather than specifying an independent natural task.Source: Adversarial tasks · p. 11
  • In the out-of-sample adversarial variant, the eigenvector uses only the training split, and test labels are assigned by the sign of the training-point kernel expansion.The expansion uses training coefficients αi and KQ(x, xi).Source: Adversarial tasks · p. 11

Shot-noise protocol:

  • At budget S, each exact collision-free outcome distribution is replaced by a multinomial sample, and the pipeline reconstructs kernels, adversarial labels, and kernel-ridge classification from sampled data.Outcomes with exact probability below 1/(2S) are trimmed before empirical distributions are renormalized; adversarial labels are re-derived from the noisy kernels, while the out-of-sample protocol is unchanged.Source: Shot-noise protocol · p. 11
  • The z-scored cross-check kernels use a standardization floor of 10^-3 of outcome-space standard deviation, and each budget is evaluated over the full realization-by-split ensemble.Source: Shot-noise protocol · p. 11

Random Fourier feature baseline: … Status:

This block documents the random-feature control and evaluation protocol, numerical validation checks, and a post-hoc weighted-alignment observation whose ordering claim was not supported by an expanded battery.

Random Fourier feature baseline:

  • The classical random-feature control uses RBFSampler with D-matched components and the same random-state seeding as the interferometer realizations.Inputs were standardized, bandwidth fixed to γ = 1.0 after preliminary cross-validation, and the baseline used the photonic maps’ trimming, standardization, readout, and split pipeline.Source: Random Fourier feature baseline · p. 11

Evaluation:

  • Evaluation uses kernel ridge classification, 3-fold cross-validation for regularization, stratified 80/20 splits, and 10 interferometer realizations × 10 data splits.Gating is assessed by a paired t-test on per-realization designed-minus-natural gap differences.Source: Evaluation · p. 11

NOTE 4: NUMERICAL INTEGRITY

  • Numerical checks reproduce Hong–Ou–Mandel bunching and distinguishable-photon binomial splitting to 10^-12, and verify normalization for both outcome statistics.Permanents for n = 2, 3, 4 agree with brute-force enumeration to 10^-9; normalization includes collision terms with factorial denominators.Source: NOTE 4: NUMERICAL INTEGRITY · p. 11
  • At N=1 the two statistics coincide exactly, while the partial-distinguishability construction matches KQ at V =1 and KC at V =0 below 10^-15 entrywise.The endpoint check holds at every configuration used; calibration in the overlap-kernel setting of ref. reproduces a designed-task advantage gap of +0.067.Source: NOTE 4: NUMERICAL INTEGRITY · p. 12

Statistical protocol:

  • Uncertainties are standard deviations over the stated ensembles, and paired testing preserves shared interferometer realizations and data splits across both arms.An unpaired comparison would underestimate the gating separation.Source: Statistical protocol · p. 12

NOTE 5: WEIGHTED ALIGNMENT READING

  • The post-hoc quantity P = AQ·∆A orders the settings consistently with the accuracy gap, including the checkerboard exception, but remains hypothesis-generating.The checkerboard family has sizeable ∆A but no advantage and the lowest absolute alignment; Table VII reports the weighted-alignment ordering.Source: NOTE 5: WEIGHTED ALIGNMENT READING · p. 12, Table VII · p. 12

Status:

  • The expanded pre-registered battery does not support the weighted reading’s ordering claim, so the paper makes no established criterion claim for P.The observation was identified after seeing the checkerboard exception and evaluated on the same settings; validation on held-out task families is left to future work.Source: Status · p. 12

NOTE 6: HELD-OUT PREDICTION FOR THE CONFIGURATION … Affine mixture:

For the two-photon configuration, predicted and published accuracy gains from indistinguishability are both smaller than their realization variability, matching the reported null. The analysis also predicts a geometric and task-alignment gap, while the two-photon visibility dial makes each raw Gram-matrix entry quadratic in visibility.

NOTE 6 — Protocol:

  • For ’s two-photon, 22-detector, 8 × 8 MNIST 0-vs-1 configuration, predictions were made without using its reported results.The model used independent complex-Gaussian transmission matrices in two layers (64 SLM pixels → 290 fibre modes → 22 detectors); predictions averaged 100 draws at 105 coincidences per input, using crossvalidated kernel ridge regression on standardized features.Source: Protocol · p. 12

NOTE 6 — Results:

  • At full indistinguishability, predicted geometric difference was g = 4.4 ± 0.5; task alignment gap was ∆A = +0.008 (AQ = 0.334 vs. AC = 0.326), with AX = 0.071.AX is below both AQ = 0.080 and AC = 0.072; the alignment gap rises monotonically with visibility as accuracy does.Source: Results · p. 12

NOTE 6 — Caveats:

  • Absolute predicted accuracies differ by +0.04 across the visibility dial, attributed to readout differences and unmodeled detector noise, while the visibility dependence is unaffected.The predicted curve is monotone rather than showing the published interior optimum near α ≈ 0.73; that optimum’s edge (0.001) lies within reported variability.Source: Caveats · p. 12
  • The predicted transmission-matrix spread is narrower than reported error bars, which also include dataset resampling; 356 of 360 images are available in the standard distribution.The α-mixing convention was checked against [2]’s Eq. (1).Source: Caveats · p. 12

NOTE 6 — Results:

  • +0.015 predicted accuracy effect (TM-to-TM spread ±0.017) and +0.014 published effect (variability ±0.042) both fall below their spreads, reproducing the reported null.Table VIII reports mean ± s.d. over 100 transmission-matrix draws at 105 coincidences.Source: Table VIII · p. 12, Results · p. 12

NOTE 7 — Affine mixture:

  • For N = 2, partial distinguishability is affine in visibility, P(V) = V PQ + (1 − V)PC, so every raw Gram-matrix entry is quadratic in V.The quadratic includes distinguishable, indistinguishable, and cross-statistics Gram terms.Source: Affine mixture · p. 12

C ). Proposition 1 (slope at the origin):

Near zero visibility, the geometric separation has a linear leading term despite a degenerate, flat spectrum at the origin. For every labeling, model complexity is convex in visibility, so a nonconstant complexity reaches its maximum at an endpoint.

C ). Proposition 1 (slope at the origin):

  • The geometric difference is 2V + O(V^2) near the origin.The derivation uses the exact identity for the largest eigenvalue at the flat-spectrum point V = 0, rather than eigenvalue differentiability at a degeneracy.Source: C ). Proposition 1 (slope at the origin) · pp. 12–13
  • The largest-eigenvalue quantity is bounded by 1 + Vs ± V^2ρ(C), yielding g2(V) = 1 + Vs + O(V^2).The derivation also gives an exact 1 + Vs expression for V ≥ 0; the closed-form slope matches finite differences at V = 0.01 to approximately 1% across the frozen seeds.Source: C ). Proposition 1 (slope at the origin) · p. 13

Proposition 2 (convexity; interior optima are impossible):

  • For every labeling, model complexity is convex in V and, unless constant, strictly convex with its maximum on [0, 1] attained at V ∈ {0, 1}.It is constant only when sQ = sC; the stated convexity argument does not require KX to be symmetric or positive semidefinite.Source: Proposition 2 (convexity; interior… · p. 13
  • Across 2000 random labelings, minimum curvature was +0.105 and minimum Cauchy–Schwarz slack was +0.052, with no interior maxima.The reported interior optimum of was not reproducible at the geometry level; at that configuration, AX = 0.071 was below AQ = 0.080 and AC = 0.072.Source: Proposition 2 (convexity; interior… · p. 13

Power analysis of the reported interior optimum: … Interpretation:

The reported interior optimum is too small to resolve with the experiment’s sample size, although the monotone rise is supported. Preregistered controls passed, but the alignment–advantage relationship and fine-grained ordering among natural tasks were not resolved.

Power analysis of the reported interior optimum:

  • 0.0021 separates the peak accuracy (0.9140 at α = 0.775) from the α = 1 endpoint, below the ±0.042 spread; resolving it at 1σ needs ~800 matrix draws versus 100 used.The monotone rise of 0.0127 requires about 22 draws and is supported; the printed spread exceeds plausible digitization error on the edge by an order of magnitude.Source: Power analysis of the reported… · p. 13

Registration:

  • 13 task families were preregistered with prediction rules and success criteria, but the file states 14 while enumerating 13; registered thresholds were evaluated unchanged.The criteria tested alignment–advantage correlation, sign agreement, a designed-task positive control, and a random-label null control.Source: Registration · p. 13
  • +0.225 on the designed task passed the positive-control criterion, while +0.009 for random labels was indistinguishable from zero and passed the null-control criterion.These outcomes validate the pipeline’s two controls.Source: Registration · p. 13
  • The results report that H1 failed, with ρ = 0.027 and p = 0.93; they do not state the specific preregistered criterion used to judge this result.Source: Registration · p. 13
  • 9 of 13 sign agreements failed H2’s preregistered threshold of 10 of 14.Source: Registration · p. 13

Interpretation:

  • At the extremes, geometry without alignment yielded no advantage, but natural-task alignment gaps (∆A = 0.003–0.037) fell within accuracy-difference realization noise (±0.02–0.045).The gating claim holds at the extremes; fine-grained ordering within the natural-task regime remains a resolution limit.Source: Interpretation · p. 13
  • Approximately five times the realizations, or larger alignment gaps, are needed to resolve natural-task ordering and bound the sample sizes for experimental tests.Source: Interpretation · p. 13
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