Indexed preparation and readable unfinished work

Assessment 154, 8 October 2026. Written conditional scheduling argument and failed transfer, following the primary-proof review in review 154. No new native recognizer or root admission.

Exact conclusion and payoff

Linear preparation can be spread across actual input symbols while retaining only one unfinished job in each dyadic length class. This does not make that job, or its completed index, readable by a late ordinary-name query. For the width-first prefix-counter candidate, selecting an exact-width root only after the query delimiter leaves the word traversal unpaid. The useful next target is a joint prefix/width representation with direct access to both committed and unfinished coverage, including complementary partial coverage at union.

This separates two obligations that a generic persistent container or a worst-case queue does not discharge together. It does not exclude either mechanism as a component of a successful construction.

1. A charged abstract scheduling lemma

Fix an integer A>=1. Each record exposes n>=1 actual target bits and releases one job only after its endpoint. Its class is j=floor(log2 n). The job has at most A*2^j bounded primitive steps. All dependencies of each step must already be available; forcing an unbounded recursive thunk is not a primitive step. An unfinished job has an explicit immutable continuation. Service preserves its accumulated work. There is one slot per class, with a separate committed root. Assume the slot selected below and the input-position ruler are locally available. Their construction and routing are not proved by this lemma.

During every subsequent target word, position 1 services slot 0 for A steps. For j>=1, positions 2^(j-1)<p<=2j service slot j for 2A steps each, stopping if it is empty or complete. Completed work publishes its class root. At the record endpoint, install its new job in its class slot.

Proposition. The installation never replaces an unfinished job. Service uses at most 2A steps per actual target bit, with at most one unfinished job per class. There is no final flush.

Proof. A new class-j record has n>=2^j. Before its endpoint it has supplied A steps to slot 0 if j=0, or 2^(j-1)2A=A2j steps to slot j otherwise. The previous job in that slot required no more than that, even without any service from intervening shorter words. No new job in the slot appears before this endpoint. Hence its predecessor is finished. Induct over endpoints. Intervals partition the positive positions, so no bit serves two slots. Unused service is discarded, not saved as uncharged credit. QED.

Empty target names, if legal, require a separate fixed-size class serviced at record punctuation; this statement concerns nonempty names. With fixed length span s between ordinary v and w, O(|v|+|w|) isolated preparation fits A_s*2^j. This is a conditional work-scheduling statement for one fixed A_s, not a nonuniform machine allowed to acquire more finite control as n grows.

It allows O(log max n) unfinished jobs globally, which is essential. It grants neither a scan of those jobs at a pin nor free class roots. Pending queries and immutable publication into the current version remain separate obligations. The existing native class-window constructions supply stronger, specialized routing/readout guarantees; this lemma does not replace their proofs.

2. Why one global FIFO is a different contract

Let a FIFO service at most q>=1 steps per subsequent target bit, and release a job of size n after a record of length n. Choose B=4q+4 and record lengths B^m,B(m-1),…,1. Before the first job completes, all later bits supply at most q*(B^m-1)/(B-1)<Bm/4 steps. Thus all m+1 jobs can be pending simultaneously, even though each job’s size equals its own record length and total preparation is linear in total input. This is a FIFO counterexample, not a lower bound against a class scheduler or a client that ignores irrelevant widths. It also does not charge a fixed record delimiter as zero work: any fixed extra service per record is dominated by choosing the initial scale larger. The displayed exact formula uses service on target bits only, as stated.

3. Width-first preparation and its unresolved readout

For a row X and exact width n, store a prefix trie. At prefix p of depth d<=n, let c(p,n) count distinct stored words of width n extending p. Fullness means c(p,n)=2^(n-d). Raw ordinary (v,w) incidence lookup detects duplicates before a preparation job is released. The word and its capacity ruler were retained while its actual bits arrived.

Once n is known, an isolated worker can walk an exact-width unary directory, then the v directory and the w prefix path. Every genuinely new word increments one counter per visited prefix. The immutable run counter of reader 35 makes an increment constant-local, even when an ordinary binary counter would carry far. New zero counters share tails of the same n-ruler: at depth d the tail has length n-d. One must not independently construct n-d cells at every prefix. The worker can retain path capsules and rebuild the changed trie path in O(n) steps. This accounts for O(n+|v|) work and allocation for an isolated job in a pointer-worker model with available input handles and explicit continuations. It is not a complete old-top native implementation.

Class-partitioned compiled roots can avoid one stale-publication failure: only the one worker in a class may change that class’s compiled root, while raw incidence input uses a separate representation. A later worker in that class starts after its predecessor publishes. A global class directory still needs paid immutable routing and publication; a ready handle is not supplied for free. Concurrent workers cannot each rebuild an old global root and then silently overwrite one another’s updates.

Now consider the late target pin carrying ordinary b. In the restricted width-first reader, no b-path cursor starts until the exact-width root has been selected from n=|b| at the delimiter. Even granting that root for free at that instant, walking its arbitrary b-path takes n dependent edge steps. There are no remaining name symbols to pay for it. Fixed-radius final work cannot perform this traversal for unbounded n. Remembering b as a list does not change the deadline. Granting a leading width header or a second replay changes the encoding and is not a transfer for the stipulated input.

This is a failure of that reader, not of every width-first data structure. Maintaining several speculative cursors, storing a transposed index, jumping through prepared joint summaries, or directly querying pending records could escape it, but their construction, fan-out and final selection must be paid. The raw incidence trie already answers single-row membership in input order; that control does not implement prefix fullness or union of partial coverage.

Moreover counters do not by themselves compose: c(A union B)=c(A)+c(B)-c(A intersection B). Equal counts may conceal different holes. The one-hole witnesses in the preceding profile-access note remain a direct test against an overlap-free union proposal. No ordinary balanced sequence operation supplies these intersection counts for free.

4. Resource ledger and decisive test

One fixed finite-control machine must supply every missing operation with constants depending only on fixed gates such as s. Preparation, old-storage observations, allocation and output count against actual encoded N; binary pointer descriptions may cost O(N log N) bits even when node count is O(N). No decomposition, preprocessing advice, padded service, identity/metadata or timing side channel is granted. Neither paper reviewed here proves the missing operations. Entry 129 retains its growing alphabet and state/port scope.

Next: test a prefix-first index whose exact-width capacity information is updated by shared residual actions, or prove a restricted obstruction to that action. It must answer both committed and pending states in the streamed name order and handle complementary holes on union. Test on the diagonal family from assessment 153 and on complete levels (reader 35); also preserve bounded spectra, aggregate class regions, replay, affine-fragment joins, persistent causal algebra and unary coherent cubical algebra as controls. There is no new native-language lower bound and no new PEG transfer. A future native L construction yields a PEG for reversal(L) under the existing equivalence, without silently reversing the input contract.

Python scheduling checker provides the finite experiment associated with this note. Its execution is evidence about finite cases, not a native implementation.