A feasible schedule is not yet an accessible policy

13 October 2026. Dedicated primary-proof comparison. At assessment 235, construction 80 was the latest native result. This assessment makes no new native theorem, lower-bound or priority claim.

The useful distinction is between scheduling known mandatory work and choosing work whose relevance is still hidden. The first admits workload tests. The second needs a single policy that respects all shared selector histories. Neither statement, by itself, pays to locate the next job in a growing persistent representation.

Primary proof 1: mandatory jobs and deadline service

C. L. Liu and James W. Layland, Scheduling Algorithms for Multiprogramming in a Hard-Real-Time Environment, JACM 20(1), 46-61, January 1973, DOI. Review scope: the primary paper’s environment, critical-instant and rate-monotonic arguments, and especially the complete deadline-driven proof in section 7; section 8 supplies the residual-service comparison.

Tasks have periodic requests, deadlines at the next request, independent requests and fixed execution costs. On one preemptive processor, Theorem 7 characterizes deadline-driven feasibility by $\sum_i C_i/T_i\le1$. Necessity counts processor demand. Sufficiency assumes a missed deadline and isolates an interval containing only work due by that deadline; integer request counts then force utilization above one. The residual-availability argument repeats this comparison after higher-priority work consumes capacity. Critical-instant shifting and adjacent-priority exchanges support the earlier fixed-priority result, whose utilization guarantee is weaker.

The important hypothesis is that the jobs and their obligations are already specified. The paper allows overhead to be included in execution costs; it does not prove that discovering and selecting a job has bounded cost on an immutable bounded-radius graph. Its periodic utilization theorem is not an arbitrary-release or optional-work theorem.

Primary proof 2: one decision for each input history

Amir Pnueli and Roni Rosner, On the Synthesis of a Reactive Module, POPL 1989, 179-190. Author-uploaded full text. Review scope: sections 3-5, including the complete history-tree argument of Lemmas 1-2 and Theorem 1; Theorem 2 and Proposition 3 were examined for the finite-state construction’s resource boundary.

A synchronous semantic program maps each nonempty input history to an output. Separate successful output sequences for separate complete inputs need not give such a program. Theorem 1 instead uses a full input tree with one output labeling satisfying the specification on every branch. Its proof transfers a labeling to any other tree by following the unique input history; conversely, a history function labels the full tree. Shared prefixes therefore receive the same decision.

For finite domains, Theorem 2 constructs a finite transducer through a word automaton, deterministic Rabin representation and tree nonemptiness. Theorem 2 gives a doubly exponential upper bound on construction time and transducer size in specification length. Proposition 3 tracks acceptance progress in a finite product and extracts a regular witness. This effective synthesis result supplies neither input-paid preparation nor bounded local access for a growing input-derived policy.

Where the two mechanisms meet the current frontier

The existing traversal game has states $(p,I)$: observed selector prefix $p$ and the ideal $I$ of completed task-tree edges. One input bit is observed before at most $K$ service jobs execute. Its recurrence has the order

$$\forall a\text{ with }pa\in Q,\quad\exists J\in\mathcal A_K(pa,I):\quad V(pa,J).$$

Here $Q$ is the selector trie, and $\mathcal A_K(pa,I)$ contains exactly the admissible completed ideals: they extend $I$ by at most $K$ edges, and new edges lie on candidate paths still possible after $pa$. Ancestors must be completed before descendants.

The existential choice can depend on the new bit and the actual old state, but not on later bits. At a leaf, the selected path must be complete. This is a finite service-specific version of causal strategy choice, already proved in assessment 234. The present literature comparison does not count it again.

Three attempted reductions expose the implementation boundary:

The first two failures are witnessed by existing controls, without a new experiment. For a full binary task tree of depth $n$, a selector of length $n$ and $K=1$, following the revealed edge succeeds. Requiring all $2^{n+1}-2$ edges within $n$ jobs is unnecessary. Conversely, four disjoint two-edge paths selected by 000, 001, 010 and 011 each fit individually into three jobs. After the first bit only one path can have received its first job. The second bit can select the other pair; one job cannot prepare both of their first edges, so the final bit can demand an unprepared two-job path. There is no compatible policy. This counterexample concerns the explicit traversal contract, not unrestricted native recognition.

The third route is a genuine specification method but not the desired resource theorem. For a finite tree with $M$ edges, a direct ideal encoding has up to $2^M$ subsets before reachability or downward closure removes states. Enumerating them is an upper-bound implementation choice, not a lower bound. A small quotient or structured policy could exist. Conversely, calling the resulting controller finite does not make it one fixed native machine for all growing input instances. Recompiling its transition table for each instance would require a paid uniform producer or would be a nonuniform family.

What a successful persistent implementation must expose

The next candidate should identify a policy whose next action is available at the current local root. Give the policy finite instructions such as follow a labeled edge, retain a cursor, restore a paid continuation or report a terminal observation. Then account for the instruction stream and its representation together:

1. Ordinary records must build and check the selector/task correspondence. The policy is not an input annotation or supplied decomposition.
2. Every service job must use the captured current activation version. A cursor prepared under an older activation snapshot does not automatically answer the current query.
3. Retaining, finding and restoring a cursor consumes actual access, allocation and routing work. The abstract game’s free choice among ready edges is not a native operation.
4. Total per-symbol work includes policy selection, worker service, persistent queue or stack operations, and output. Service constants cannot silently multiply by a growing frontier width.
5. Preparation and all dependent work must fit the actual encoded input and its cuts. No extra selector padding, delayed end marker or EOF work funds the policy.

These are obligations for the next construction, not a sufficient structural theorem in disguise. A promising concrete test is a shared traversal with a locally maintained continuation stack and an input-built route from selector prefixes to the correct retained cursor. Its hard part is the cross-index: independently building the traversal history and the selector directory does not construct the links between them for free. The test must expose when each such link is learned, how its old-storage target is reached, and which actual symbols pay for it. A failed cross-index construction should retain its exact release deadline, rather than reverting to an information-size argument.

The older controls remain separate. Replay slices charge each descendant’s replay. Affine joins retain their coordinate-routing premises. Persistent causal algebra preserves histories without free current observations. Unary coherent cubical controls do not establish arbitrary branching or feedback closure. Native SAT constructions preserve late ordinary-name pins and final-symbol negative output. A native recognizer would still yield a PEG for the reversal unless a separate closure argument applies. Entry 129’s relation quotient and diagram bounds keep their matching and positive-readout hypotheses.

Later work tests selector-index routing and constructs shared activation indexes for restricted learned views. The current checkpoint retains the open general problem.

Evidence, disposition and next test

This is a primary-proof synthesis and failed-transfer analysis. The imported theorems are established results; the comparison applies their hypotheses to the existing Lab model. No native subclass extension or unrestricted lower bound has been obtained. The two small controls and finite game are reused from assessment 234 and its readable checker, rather than claimed as fresh mathematical results.

Next test at assessment 235: construct an ordinary-record producer for a variable-width task/selector correspondence with a locally exposed service policy. Charge shared current-version traversal, discovery and retrieval of the selected cursor at every actual-input cut. Test the selector-to-cursor cross-index explicitly. Preserve arbitrary whole-suffix permutations, arbitrary-name routing, cyclic sharing and heterogeneous service as open.

Independent adversarial review and scoped publication checks are recorded in integration state. Primary-source text remains outside the public repository. This review has one maintained source and a generated styled reading; existing source-code links retain highlighted pages, line anchors and exact downloads. No homepage recency bump or new paper is warranted.

Scheduled completion 140 is the dedicated two-paper review, counted once after completion. Mathematical 143, window three, reassessment 140 and native 80 remain unchanged. Next 141 is research, next primary review 145, next editorial 144. Browser viewport inspection is unavailable; publication receipts distinguish build/render checks from that limit.