Fixed-point amplification as a stability result.

Standard Grover-style reflections can help when accepted evidence is rare, but they can overshoot when the belief becomes concentrated. A one-off experiment can avoid that regime with a guard. A reusable service faces it repeatedly because every observation changes the next prior.

The paper tests fixed-point amplitude amplification with softer phase rotations at every listen step. The point is not simply a larger accepted-event amplitude. It is whether the full posterior remains readable by the planner after the quantum step is reused across a trajectory.

The reported 8-step high-shot trajectory reaches a maximum Hellinger distance of 0.009, while the 12-step primary run reaches 0.021. These are posterior-distance results after sequential feedback, not isolated amplitude benchmarks.

The 20-step and 32-step rows remain supporting controls. They extend the observed operating envelope and show that the loop does not immediately diverge, but the paper deliberately keeps the primary claim anchored to the 8-step and 12-step Tiger runs.

The 0.009 headline uses 32,768 shots per step. The matched-shot FPAA control uses 10,000 shots per step and reports a maximum Hellinger distance of about 0.033. The reduction in fidelity under the lower budget is central to the interpretation, not a detail to hide.

The guarded Grover row used an adaptive allocation between 8,000 and 16,000 shots per step, averaging about 10,000. That makes it budget-aligned rather than strictly constant-shot. Without a constant-shot Grover rerun, the table does not establish equal-budget FPAA superiority.

The paper does not stop at Hellinger distance. It applies the same Tiger immediate-reward rule to the hardware-derived and exact Bayes posteriors. Across the reported matched-shot and longer-horizon checks, the selected immediate action is the same.

Action agreement does not erase posterior error. It says the error stayed on the same side of the tested decision threshold. A different reward model or a posterior closer to a threshold could produce a different operational result.

Primary evidence covers the sequential Tiger posterior on Heron, including fidelity, matched-shot context, and decision checks. Fez repeats, longer horizons, mitigation A/B, Heron R3 transfer, and rare-event sweeps support the operating envelope.

The four-state corridor and UCGate/QSD pilots are exploratory. They identify where larger encodings and deeper circuits become difficult; they do not validate realistic autonomous-system scale.

A responsible evidence record stores shots per step, backend, trajectory, amplification policy, circuit counts, mitigation state, and the exact reference beside the posterior metric. A single best number without that context invites the wrong conclusion.

For QANTIS, the honest summary is useful: high-shot all-step FPAA produced the best reported fidelity; the matched-shot run remained action-consistent but did not prove equal-budget superiority. That wording preserves both the result and the remaining work.

The high-shot FPAA trajectory answers a fidelity question: can fixed-point amplification be applied at every listen step while keeping the returned posterior close to exact Bayes? On the reported 8-step run, the maximum Hellinger distance is 0.009; the 12-step primary result reports 0.021. Those are useful hardware-stability observations.

The matched-budget question is stricter. At roughly 10,000 shots per step, the FPAA control reaches maximum Hellinger about 0.033. That result does not support an equal-budget accuracy win. Keeping it beside the 32,768-shot headline protects readers from attributing a budget effect entirely to the algorithm.

For a sequential service, the broader lesson is methodological: compare algorithms on the same path, expose the resource allocation, and judge whether error propagates into a different decision. Stability is valuable on its own; it does not need an unsupported superiority claim.

arXiv:2607.06760, submitted 7 July 2026, reports the high-shot FPAA result and a matched-shot hardware control explicitly so readers can separate best observed fidelity from budget-relevant behavior. The paper itself says the comparison is not a fully budget-matched proof because the guarded Grover run uses an adaptive allocation rather than a strictly constant shot count.

Accordingly, this article uses 'sequential stability' for the supported result. It does not describe FPAA as universally better, faster, or proven to hold an advantage at equal cost.