Growing speculation: service cuts and the unresolved native scheduler
13 October 2026. Supporting written argument and failed extension of construction 80. No new native construction, general lower bound, root admission or priority claim.
Contents
Outcome
The fixed-width proof cannot be extended by giving each candidate a feasible release schedule separately. Before late input selects a candidate, those schedules compete for the same actual service slots. The relevant necessary demand is a union of mandatory traversal prefixes, counting common work once. Even granting free candidate discovery, retained cursors and selection, a growing frontier can fail this demand test.
Conversely, width alone is not an obstruction: many candidates can require no speculative work when their remaining paths fit the remaining input. The missing sufficient mechanism is an input-built representation that shares completed traversal, routes to the selected state and services pending work locally at every release. This note gives an exact finite game for the optimistic traversal model and tests three attempted extensions. It does not claim to implement that game natively.
At assessment 234, construction 80 was the latest native result. Its finite arrays perform bounded discovery, copying, elimination and service; none of those operations becomes constant when its machine parameter W is replaced by the current input width.
Subsequent work charges selector-index routing and develops shared activation contraction and boundary-normalized indexes. These sufficient subclasses escape the explicit-path contract; they do not invalidate its service cuts. See the current checkpoint for the surviving target.
1. Restricted traversal contract
Fix a finite rooted task tree T. An edge is one unit traversal/observation job and can be completed only after its ancestors. A candidate answer requires the complete root-to-candidate path, including observation of its terminal mark. Shared initial paths are the same jobs, not copies. A retained service state is a downward-closed edge set I. It represents already completed work against one captured current activation version. No computed current answers or candidate handles are supplied before this service interval: I is initially empty unless earlier paid service is explicitly included.
An ordinary binary selector arrives over t actual symbols. Its finite input trie Q maps each supported leaf to one candidate. After each symbol, at most K new task edges may be serviced. One fixed K is used across the family. The output is due after the service batch of the last actual symbol; there is no EOF batch. Candidate paths are all visible in this optimistic model. Choosing a ready task, retaining every cursor and routing from the selector to a retained cursor are free. Thus it is a relaxation of the explicit traversal workers, not a native implementation. Shape checks, initialization, program discovery, queue operations and current-version access can only add costs in an implementation.
The contract requires traversal of the selected path even when two marks happen to agree. It excludes a different prepared predicate that answers without that traversal. Consequently its impossibility statements concern this contract only. They do not establish a cell-probe, information-size or native-language lower bound. There are no numerical address, identity, metadata, timing or control-flow channels in the task model. The selected candidate is specified by ordinary binary symbols, not a growing alphabet letter.
2. Mandatory-prefix cut
Consider a supported selector prefix p after its service batch. Suppose every supported continuation has H symbols remaining. Let L(p) be its still possible candidate nodes, d(v) their depths and P(v,k) the first k edges on the path to v. Define
Necessary cut condition. Every successful strategy has R(p,H) contained in its completed ideal I at this cut.
Proof. On a continuation selecting v, at most KH further edges can be serviced. If fewer than d(v)-KH initial edges on its path have been completed, its mandatory remaining path is too long. Downward closure means that completing d(v)-KH path edges entails exactly that initial prefix. Apply this argument to every still possible continuation of the same history. Their shared history has a single I, so it must contain the union. This counts a shared ancestor once, not once per candidate. If I0 is the paid completed ideal at an earlier cut j0 and j is the current time, a further necessary capacity test is
These inequalities deliberately ignore release and access costs, and are only necessary in the online selector game.
For a simpler two-stage relaxation, they are sharp: if the candidate is revealed in full immediately after this cut, with H service symbols still available and free access to each retained cursor, I containing R is sufficient. Each selected residual has at most KH edges; follow it. Starting from I0, the minimum additional preselection work in that relaxation is exactly |R minus I0|: it is necessary as above and achievable in ancestor order because the union with I0 is an ideal. This is not sufficiency when the selector is revealed gradually, nor a native routing theorem.
3. A family defeating isolated path schedules
Let M=2^n candidates have all n-bit prefixes as addresses, followed by disjoint unary tails of length L. Thus each candidate has depth n+L and the shared tree is binary before the tails. Use selectors x s, with x an arbitrary binary word of length A and s the n-bit candidate address. Every stem has all M continuations. The selector is genuinely binary; the task bound does not assume that its full value is revealed in one step.
At the cut after x, H=n. If L>Kn, R contains at least M(L-Kn) tail edges (the full binary prefix contributes another 2M-2 edges). It follows that
is necessary from an empty initial ideal. Take L=2Kn and A=3n. Each individually selected path has n+2Kn edges, no more than K(A+n)=4Kn for K>=1. Every offline single-path schedule can finish. But the common cut requires at least 2^n Kn work before a budget of 3Kn, impossible for n>=2. No allocation of the fixed total service among candidates repairs this family. Even the optimistic model fails, so paying candidate discovery and routing cannot rescue this traversal route.
This family is a task-system diagnostic, not a new Horn input reduction. The stem is not being offered as padding to obtain a native recognition theorem. A valid transfer would have to recover these task paths, their release times, the initial paid ideal and the actual service budget from the ordinary relation encoding. Work during the source pin, E(c), a and earlier input must be included rather than silently discarded. Existing aggregate predicates or shortcuts may violate the traversal contract entirely.
The converse control matters. With a complete binary candidate tree of depth n, n remaining selector bits and K>=1, R is empty regardless of M=2^n. Reading the selector while following its matching task edge finishes with no speculation. Hence a lower bound based solely on the number of possible candidates would be false.
4. Exact finite viability game
For a supported selector prefix p, let V(p,I) say that all supported continuations can be answered from the completed ideal I after the service at p. At a terminal p with candidate v, V holds exactly when the entire path to v lies in I. At a nonterminal prefix,
Here A_K(pa,I) consists of ideals J extending I with at most K additional edges, all new edges lying on paths to candidates still possible at pa. This restriction loses no useful work: eliminated candidates never return along this selector history. Idling is allowed. Ready edges may be added sequentially within the symbol’s batch. Old completed edges from eliminated candidates remain in I and need no deletion operation.
The conjunction comes first: the next input bit is adversarial. The service decision may depend on that bit, but cannot depend on later bits. Backward induction proves the recurrence exact for the stipulated game: a winning policy chooses the witnessing batch after each observed symbol; conversely every successful strategy supplies such a batch on each child. At the root, test V(empty, empty) with no pre-input service. The finite graph is acyclic in selector depth, so exhaustive evaluation terminates.
This criterion accounts for shared traversal, dormant candidates and elimination. It is not a cheap producer: enumerating ideals, transitions and policies can be exponential, and an unbounded policy table cannot be installed as finite control. It also grants free task access and selection. Finite success therefore supplies no native recognizer; finite failure is an exact failure only of the stated task contract. Per-path offline feasibility is a relaxation with a different quantifier order.
Aggregate cuts still do not supply a consistent policy
There is a further obstruction even when the scalar root-budget cut passes at every input prefix. Set K=1 and t=3. Supported selectors are 000, 001, 010 and 011. The four candidates have disjoint two-edge paths from one root, using fixed degree four and fixed edge labels; the selector alphabet remains binary. Each candidate corresponds to one selector. At depth one, two symbols remain and R is empty. At depth two, the selected pair has two mandatory first edges, so |R|=2 fits the two elapsed service slots. At depth three, the selected two-edge path also fits the three elapsed slots. The root cut also passes.
Nevertheless no online policy wins. After the first 0, at most one candidate’s first edge is complete. The second bit can select the pair containing neither that edge nor any completed work. Only one job is then available to prepare that pair’s two mandatory first edges. The final bit selects its unprepared member, whose two-edge path cannot finish in one job. If the first batch idles, either pair works. Hence every scalar cut computed against an empty initial ideal passes, but the required ideals at distinct future cuts cannot all be reached from one shared earlier state. The exact game fails. The state-dependent cut in section 2 detects the deficit once that earlier state is fixed; this example does not refute that necessary statement.
A positive extension therefore needs more than small mandatory unions at isolated prefixes: its paid local policy must reach compatible service states across all shared histories. This is the concrete reason to retain the conjunction-before-disjunction order in the viability recurrence.
5. Three attempted native extensions and their failure gates
Replace the finite slot array by a persistent list. A list stores unbounded candidate tuples with fixed degree and finite labels, but discovery of every child and elimination of unsupported parents now requires a traversal. A round-robin cursor can avoid restarting the scan, yet cannot turn K total jobs into K jobs for each of M candidates. The family in section 3 defeats all such allocations under the explicit-path contract, independently of list implementation. Reopen only with a checked collective demand bound and a locally available policy, not separate path guarantees.
Retain dormant candidates and wake only the selected one. This saves service on losing branches but releases an entire residual path late. The cut condition measures precisely how much must already have been completed for every possible survivor. Dormant storage is harmless only when its unserved residual fits its remaining actual symbols. Knowing that the total work on the eventual winner is linear does not meet this requirement. The shallow binary control passes; long private tails fail.
Traverse the shared task tree once and save every completed cursor. Shared ancestry genuinely reduces demand from a sum of path lengths to union size. A stack can enumerate a bounded-degree tree with linear total navigation, but neither linear total work nor a completed cursor collection gives the selector a paid route to the correct cursor. The tasks depend on the captured current activation version, so static namespace preparation alone does not compute their current marks. Preparing a source-specific cross-index, routing through it, discovering task nodes, and placing service before each cut remain uncharged in this proposal. No native positive is claimed even on families with small union demand.
These failures are complementary: the first is collective service, the second is late release, and the third is current-version access and routing. An arbitrary binary tree can itself be traversed with local stack operations; the missing operation is maintaining its correspondence to the input-derived selector and activation continuations without a frontier scan or free handles.
6. Controls, evidence and exact next target
The fixed-W proof of construction 80 pays all candidates explicitly and therefore survives. The W=1 envelope and fixed-delay controls retain their exact hypotheses. Replay slices charge replay; affine joins require their coordinate structure; persistent causal and unary coherent algebras supply no free cross-index. Entry 129’s imported bounds still concern their matching/positive-path contracts and growing alphabet, not this task game or unrestricted native recognition. Any future native recognizer yields a PEG for the reversed complete checked language only.
The standalone finite checker compares the viability recurrence with a separately written chronological set-of-strategies evaluation, checks cuts against winning states, checks exact two-stage minimum work by enumeration, and contrasts offline paths with late selection. It is a semantic scheduling experiment, not native timing simulation. The assessment record gives counts and independent review; the checker is maintained once and projected into a readable source page.
Next decisive test at assessment 234: build an ordinary-record producer for a variable-width task/selector correspondence with a locally exposed service policy. Charge both the union of required current-version traversals and retrieval of the selected completed cursor at every cut. Start with a bounded-degree shared task tree whose mandatory-prefix union fits the real input budget; determine whether the correspondence can be maintained without arbitrary cross-index updates. Keep growing namespaces, arbitrary whole-suffix permutations, arbitrary-name routing, cyclic sharing and heterogeneous service open.
This is failure evidence and a finite diagnostic, not a new native construction or a replacement for construction 80. The checkpoint and agenda preserve the native target; generated readings expose the maintained note and checker. No primary PDF or private scratch is included.