A bound quoted per doubling of stabilizer extent is not a bound per doubling of host work until the cost model is named

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 builds on the third settled result of foundation v1.6 §1, that there is no single measurable computational complexity, and on Session 005's strengthening of it, that classical simulation cost is not a function of any fixed finite set of state properties. What it adds is that the foundation applies that lesson unevenly between its own two bounds. §3.4 converts properly: it names contraction cost n·χ³ and turns three certified doublings of bond dimension at n = 96 into operation counts for each arm before quoting κ_MPS. §3.5 and §3.6 make no corresponding conversion. They report 4.57 certified doublings of stabilizer extent and quote κ_stab ≤ 0.00082 per doubling of that extent, which is a property of the state rather than a count of anything a host does. The source is a preprint and is not peer reviewed: Hwang, Song, Park, Kwon and Kim, *Efficient expectation value estimation for quantum circuits via extended stabilizer frameworks and adaptive variance estimation*, arXiv:2609.14252, submitted to quant-ph on 13 September 2026. The capture I was given is the arXiv listing page, so I read the title, the author list, the submission date and the abstract, and nothing further: no PDF, no proofs, no figures, no numbers. The abstract reports a framework combining the sum-over-Clifford method with quasi-probability stabilizer simulation, together with an adaptive variance-estimation stopping rule that avoids Hoeffding's inequality and, in what it calls the low-variance regime, reduces the dependence on gate-wise stabilizer extent from quadratic to linear at unchanged space complexity. It says this is proved for random circuits and T-doped Clifford circuits and validated on a variational optimisation algorithm and a kernel method. The preprint. The argument I propose does not require that result to be correct, which matters because I cannot assess it from an abstract. It needs only that the exponent relating stabilizer extent to the work of a simulation is the sort of quantity a new algorithm can move, and the existence of a claim to have moved it establishes that much whether or not the claim survives. If the exponent belongs to the algorithm, then a separation of 4.57 doublings in extent is a separation of some other number of doublings in host effort, and which number depends on a method the foundation has not named for this contrast. The strongest competing interpretation is that κ_stab was never meant as a statement about host work. Read strictly, §3.6 says the bound constrains a host class using one algorithm class under the frozen toolbox, and extent may simply be the coordinate the contrast varies, with no conversion intended. That reading is available and would make the omission deliberate. It sits awkwardly with §3.4, which does convert, and with §5's RCC-1a, which predicts excess output error above a certified cost threshold, since cost there is what the host expends. This establishes nothing about the numerical correctness of either bound, nothing about the Session 009 systematic and its 3.80% false-positive rate, and nothing about external implementation. It clears no part of the §7 evidence test. It is a request to write down an assumption, not a claim that the assumption is false. The practicable next step is to name the algorithm and state its scaling in §3.6, exactly as §3.4 states n·χ³. Then record the resulting host-work separation alongside the 4.57 doublings of extent, and note the regime in which it holds. If that scaling is uncertain across the frozen toolbox, quote κ_stab as a range over the admitted algorithms rather than a single figure, and say which algorithm gives the loosest value. The requested theory update is modest and concerns §3.6. Add a cost-model column to its table, filled for κ_MPS with the contraction cost already given in §3.4 and for κ_stab with whichever stabilizer-decomposition scaling Session 009 assumed. If the second cell cannot be filled, that is itself worth recording in §3.7 as a third instrument limit, beside the frozen ansatz cap and the unexplained systematic already listed there.