Task-Sufficient Memory under Stochastic Clocks: Coarse-graining operators, fixed points, and finite-policy certification

What information about environmental memory must a quantum controller retain to guarantee a specified computational output? This preprint formulates that question in terms of task-sufficient process representations, intervention-valid predictions, stochastic clock laws, and explicit resource accounting. The central requirement is to preserve the outcomes and costs of the permitted decisions, with a stated accuracy contract and a finite-sample certificate. The paper connects two complementary lines of analysis. An operational construction studies Gaussian dephasing and controlled qubit tasks. A separate stochastic-clock analysis studies direct and inverse time changes, fractional relaxation, and the information that temporal coarse graining must transport. These constructions address related modelling questions but are not identified as a single microscopic noise model. The main elements are: Identical uncontrolled channels, different controlled responses. A continuum of Gaussian phase processes has the same uncontrolled coherence at every time and zero Breuer-Laine-Piilo backflow, yet different spin-echo responses. The construction shows why an uncontrolled channel family alone can fail to determine an intervention-dependent control decision. Task-sufficient equivalence and resource contracts. The manuscript defines a contract functional collecting accepted-output moments, acceptance probabilities, and physical costs over a declared policy family. Its finite linear task quotient identifies distinctions relevant to those predictions. Approximate sufficiency is accompanied by stability bounds that expose the sensitivity of conditional guarantees to low acceptance probabilities and small feasibility margins. Simultaneous finite-policy certification. Conservative confidence bounds certify conditional task bias and expected cost per accepted outcome after selection among a fixed, finite set of candidate policies. The assumptions include independent reset attempts, bounded observations and costs, and a trusted ideal reference. Failure to certify a candidate is explicitly distinguished from proof that no feasible physical policy exists. An explicit coarse-graining framework for stochastic clocks. The proposed decorated phase space records the process, permitted policy family, observable set, and clock law. The clock component transports stable laws under their natural scaling and relates domains of attraction to regular variation of waiting-time tails. Within the stated two-parameter clock-and-coupling family, the relaxation-rate direction scales with eigenvalue c^alpha. These clock results are placed in the established probabilistic renormalization literature; they do not establish hyperbolicity or convergence for the full decorated operator. A reset-sensitive fractional-relaxation calculation. Direct subordination of a quantum dynamical semigroup is distinguished from inverse-stable time change. For the latter, the paper compares uninterrupted evolution with independent renewal of the hidden clock and relevant environment. At half order and dimensionless rate one, the exact erfcx representation gives the short-time defect Delta(h) ~ 0.660989212585294 sqrt(h). The difference concerns the declared reset protocols, rather than a general assertion that every physical gate resets its environment. The reported classical benchmark uses ten candidate policies with explicit pulse depolarisation and separate X- and Z-expectation contracts. At the quasistatic endpoint, midpoint echo increases average identity-gate fidelity from 0.789293 to 0.990050. The fidelity crossover near r = 0.0467883 is distinct from the stricter X-task feasibility threshold r >= 0.9688074. For the selected Z-task input, dephasing is harmless and unnecessary echo pulses reduce the expectation. Conventional calibrated echo matches the best tested control performance; calibration, sampling, rejected attempts, and amortisation remain part of the resource account. Two questions are isolated as conjectures: boundedness of the task-quotient dimension along the coarse-graining orbit, and rigidity of the task quotient as a complete contract invariant. The manuscript specifies falsifiable numerical and experimental tests, including rescaled relaxation-profile collapse and finite tests of quotient-rank growth. It establishes no hardware advantage, scalable task-compression theorem, logical error-rate improvement, or error-correction overhead reduction. This illustrated version 2.0 contains 30 pages, four scientific figures, derivation and reproduction appendices, a methodological comparison with renormalization in one-dimensional dynamics, a glossary, and 58 references. The manuscript is dated 12 November 2026 and supersedes version 1.0 of 10 November 2026. The four figures show the uncontrolled/control-response contrast, task-dependent protection, finite-policy certification alongside dimensionless fractional relaxation, and the exact scalar clock-reset defect.