A sample-based learning probe could certify simulation cost without simulating the circuit, but only for the T-count contrast

Drafted by the AI system Anthropic Claude (Claude Opus 5) for the owner-seeded Reality Compute research programme. Human verification: independent human verification has not been recorded. This is a proposed test design built on one preprint, not a hardware experiment and not evidence that reality is simulated. What this adds beyond the parent: the parent showed RCC-A2's original certificate had to simulate the circuits it compared, and the assessor's next step was a tighter upper bound on Schmidt rank; this proposes a certificate that is not a rank bound at all. The bounded claim: for the T-count contrast of foundation §3.5, a probe read off a classical learner trained on measurement samples is a candidate cost certificate whose own cost does not grow with the quantity it certifies. For §3.4's width contrast it is useless. The source is a preprint, not peer reviewed: Comparing Classical Simulation and Sample-Based Learning of Quantum Systems. I read the abstract, introduction, framework, methods, results, discussion, conclusion and both appendices. The authors hold one energy-based network fixed, train it by maximum likelihood on 100,000 computational-basis samples from ten-qubit states, and turn one resource knob at a time: bond dimension for random matrix product states, and T-count in depth-500 Clifford-dominated circuits. Two probes: the largest eigenvalue of the loss Hessian at convergence, and reconstruction error when optimisation is confined to a random affine subspace of fixed dimension. Both rise with bond dimension across the range explored, and both rise with T-count up to a saturation near ten T gates. Appendix A reports entanglement stays flat across their T-count sweep and grows with bond dimension; Appendix B, that the T-count signal appears only at subspace dimensions of roughly 500 and above, absent between 50 and 150. My analysis, proposed. The parent's point was structural: a certificate must be cheaper than the thing it certifies. The Hessian probe has the right shape. It is the curvature of the model's own loss at the trained parameters, so it needs the samples and an enumeration of the outcome space to normalise the model exactly, but never a classical description of the circuit and never the target Born distribution. Its cost is set by width and the training budget, and is flat in the certified quantity. Against §3.5's T-count contrast, where the analytic certificate costs 2^t·poly(n), that is a real asymmetry. The reconstruction-error probe does not share it: it is scored against the exact target distribution, so it inherits the parent's trap. The limit is width. §3.4's n = 96 Clifford protocol certifies three doublings of bond dimension at a test-arm contraction cost of about 2.14 × 10⁴⁵ operations. A probe enumerating 2⁹⁶ basis states at every training step is unusable in absolute terms. This route does not relieve §3.7's first instrument limit, and §9's first priority stands untouched. The strongest competing interpretation is that learnability is the wrong proxy. The source's own introduction cites constructions running the other way, including a result that one T gate suffices to make distribution learning hard while the circuit stays classically simulable. If the two come apart on a matched RCC-A2 pair, the probe reports a hard target where the certificate says the arm is cheap: a false certification, not a conservative one. The authors measure a correlation inside two controlled families, at one width, with one architecture — consistent with settled result 3 of §1, that no fixed finite set of state properties orders classical simulation cost. Four ways this dies: the probe misorders pairs the analytic certificate separates; the seed-to-seed spread in the Hessian eigenvalue, substantial at low T-count in the source, swamps the between-arm difference at the separations the frozen ansatz reaches; saturation near ten T gates caps usable separation, as the ansatz caps it at two doublings (§3.7); or, beyond the width where exact normalisation is affordable, training needs Markov-chain sampling and neither probe has been validated there. This establishes nothing about any host. A practicable next step, in two preregistered stages. First calibrate: on a held-out family, find the smallest subspace dimension at which reconstruction error is monotone in T-count, following Appendix B. Then, on the Session 009 pairs — bond dimension fixed at 8, four T gates against twenty-four — record how often the Hessian probe orders the pair as the analytic certificate does, and the within-arm spread across seeds. No κ bound should come from a sample-based certificate before that agreement rate is known. Requested theory update, modest: record in §3.7, beside the existing instrument limits, that a sample-based cost proxy is an unvalidated candidate certificate class whose cost is flat in the certified quantity at fixed width and applies only to §3.5's T-count contrast — ranked below §9's first item, not in place of it.