Early unique routing and the learned-hierarchy boundary

Assessment 208, 12 October 2026. Written closure specialization of learned contexts, theorem 68, with an unsuccessful extension to unrestricted learned hierarchies. No new numbered native theorem, root admission or external priority claim.

Think of each group as a separate collection of stored answers. The selection name and the later query both identify that collection before local processing begins. The machine can therefore leave every other collection untouched. The difficulty returns when a new shorter prefix makes several old collections relevant to the same query.

Exact useful statement

Unboundedly many resident exact reuse kernels do not require unboundedly many active readers when the ordinary selection and query names expose the same unique route before any local kernel work is needed. The route must select the current version of the local predicate, and every suspended kernel must already answer exactly from its committed and pending state. This replaces a global fixed-context bound by a fixed-context bound per explicitly checked region. It does not allow overlapping regions or arbitrary cross-region aliases.

Here is a concrete specialization, rather than an assumed decomposition oracle. Let

$$e(g)=\operatorname{dbl}(g)11,\qquad \operatorname{dbl}(0)=00,\quad\operatorname{dbl}(1)=01.$$

Use the ordinary right-first Horn encoding of theorem 68. Each intermediate binary name has the form $v=e(g)x$ and each target binary name has the form $w=e(g)t$, with $x,t$ nonempty. The empty group name is permitted. The two occurrences of $g$ on every right edge must agree. Within each group, the subsequence of local right records $(x,t)$ must satisfy theorem 68’s greedy learned-context contract with the same fixed constants $K,d,s$. Records from different groups may be arbitrarily interleaved. The numbers of groups, their route lengths, and their learned context lengths are unbounded.

Closure claim. With the persistent dictionary capsules and old-top compiler already used in theorem 68, this checked language has the same fixed-degree, fixed-radius, finite-label/control native recognition contract and linear total charged storage. The constants depend on $K,d,s$, not on the number of groups. Recognition yields a PEG for reversal of the complete checked language, not automatically for its forward orientation.

Charged construction

Parse route pairs in a fixed two-bit control: 00 and 01 emit a group bit, 11 ends the route, and 10 rejects. A missing terminator, incomplete pair or empty local suffix rejects. Validate this syntax even when a lookup has already discovered a missing group. The code set is prefix-free, so a target belongs to exactly one group.

The right registry, updated persistently during preparation and frozen afterward, maps $g$ to its local context tuple and row dictionary. During an intermediate name, its literal route bits open the group’s persistent dictionary capsule; its literal $x$ bits prepare the local row lookup, length clock and prospective rolling-window producer. During the target’s route, a strict FIFO captured from the intermediate route checks exact pairwise equality, including simultaneous termination. The open group and row capsules remain suspended. The encoded route terminator pays the fixed local initialization before suffix processing; the intermediate route terminator likewise initializes the local alias clock and counter, including its fixed extra cell. The literal $t$ bits execute theorem 68’s existing-prefix candidates, residual clocks, duplicate comparisons and prospective new-context capture. The record delimiter makes only the fixed number of local commitments and capsule closures already supplied by the inherited interface. A new group begins with an empty fixed-size kernel.

For every source $u$, its current source descriptor owns a group dictionary $A_u$. On a left record, the literal $u$ selects that current descriptor. The route bits of $v=e(g)x$ open both the frozen registry and the current $A_u(g)$ in parallel. Then $x$ alone funds all $K$ local service windows. The delimiter appends at most $d$ residual receipts to already serviced local slots and closes the fixed nesting of source, group and local interfaces. Row receipts contain immutable residual information; they never substitute a frozen selected-state pointer for the current source-group root.

On pins $(a,b)$, the positive source name $a$ opens its current source descriptor. Parse $b=e(g)t$. Its route bits find the unique group and current $A_a(g)$. Only that group’s fixed $K$ prefix comparators and residual readers run during $t$. Their terminal disjunction is the exact reachability bit, whose complement is the Horn SAT answer. A well-formed missing group denotes the empty predicate. No traversal or flush occurs after the final symbol.

The active dictionary nesting is fixed: source, group, local kernel. A long route is an incremental dictionary traversal paid by its own bits, not a stack of suspended native frames to be closed at the delimiter. The same fixed capsule interface used for ordinary names in theorem 68 is essential; a conventional path-copying implementation with a linear delimiter-time rebuild would not meet this claim.

An inactive group needs no background service. At every record boundary its committed-plus-pending predicate is already exact. Its pending batch remains queryable while suspended; a later update to the same group uses that later local alias’s service window. Updates to other groups do not spend its service credit or mutate its version. This is the suspension invariant that makes arbitrary interleaving safe.

Resource and semantic proof

For each fixed group, erase the intervening records of other groups. The remaining operations are exactly a theorem 68 producer followed by its serial selection operations and query reader. Suspension preserves the entire local immutable state. Induction on the global operation sequence therefore gives the theorem 68 invariant for every group in every retained source version.

All right edges preserve the parsed group. Thus a selected intermediate in another group cannot reach the queried target. Prefix-free routing and the per-group invariant prove the one-group readout, including absence. This is stronger than a positive witness argument: it establishes that discarded groups contribute no hidden positive answer.

The actual encoded length remains

$$N=\sum_R(9+|v|+|w|)+\sum_L(10+|u|+|v|)+11+|a|+|b|,$$

with the inherited fixed phase delimiters if present. The route spelling is included in these lengths. There is no reduction that adds route headers to an arbitrary instance while preserving its deadline. This is a checked subclass of names already spelled this way.

Each actual route bit performs a bounded number of dictionary/FIFO operations; each local bit runs the fixed inherited kernel. All prospective and discarded work counts. Fixed initialization, parsing, old-storage observations, routing, allocations, pending work, output and retained storage are charged. Hence the logical work/storage is $O_{K,d,s}(N)$. The inherited fixed old-top compilation gives one physical transition per symbol and its fixed initialization convention. Pointer-bit accounting remains $O(N\log N)$ when addresses are externally serialized. No numeric pointer inspection, identity metadata, timing channel, advice, random access or supplied ready handle enters the proof. The construction is one fixed machine for each fixed parameter tuple, not a separately chosen machine for each number of groups.

This is serial persistent reuse within one prepared store. It does not prove that independently allocated stores can be joined at bounded cost. Entry 129’s matching and positive-path bounds retain their original conditional interface contracts and growing alphabet; no such diagram summary has been extracted here.

Why the attempted unrestricted hierarchy does not follow

Three dependent extensions were examined.

1. Cross-group aliases. Drop the same-group gate and put $e(0)0\to e(1)0$ in the right relation. Selecting the intermediate stores a receipt under source group 0, but a query for the target opens source group 1 and misses it. A receipt tagged with target group 1 does not provide paid access to that group’s current dictionary entry during the short local suffix. The wrapper is semantically false without either the gate or an additional charged cross-group operation.
2. Overlapping contexts. Replace the prefix-free route set by contexts $0,00,\ldots,0^h$. A target beginning with $0^h$ may be relevant to every resident context. One chosen local reader is incomplete; activating all of them violates this construction’s fixed active multiplicity. A trie alone locates these contexts but does not supply their exact current selected union. This is a failure of this wrapper, not a native lower bound: shared aggregation might succeed.
3. Late shorter contexts and reparenting. A newly learned shorter prefix can contain arbitrarily many earlier regions. Leaving old receipts in their original regions requires an exact union reader over those regions. Moving them eagerly can disturb arbitrarily many retained local predicates at the release of the new context. Merely linking the old roots below a new parent preserves storage but postpones the missing union readout. Residual coordinates also change: stripping a different prefix changes which literal suffix bits are available to start and pay each inherited reader. The routed proof provides neither paid migration nor a shared queryable aggregate for this operation.

Consequently the route trie supplies unbounded branch count and depth, but only one learned contraction per selected region, with a fixed tuple of local contexts. Calling this a recursive learned hierarchy would conceal the unresolved composition step. No cardinality argument here gives an unrestricted persistent-language lower bound.

Evidence, disposition and next decisive test

The independent finite audit checks routed relation membership against unrestricted direct relational membership across retained selection versions, absent groups and negative queries. It also checks the cross-group and overlap counterexamples. These are semantic experiments, not a native transition simulation; the timing argument is the written reduction to the maintained capsule and theorem 68 contracts. Independent adversarial review approved the closure only with the same-route gate, full syntax checks, current roots, fixed interface nesting and suspended exactness stated above.

This assessment is a useful closure specialization and a precise failed route, not a new mathematical-result count. Source custody remains in this repository. The root claim ledger and native theorem number 68 are unchanged. The failure record is this section; the reader synthesis is the assessment review. This note and its code audit are included in the editorial reader edition. Theorem 68 remains unchanged.

Next decisive test: construct or refute the charged late-shorter-context union interface. At a new prefix release, retain old child kernels without traversing or migrating all children, yet answer the next ordinary-name query from the current source version with bounded active readers and no lost residual-start budget. First isolate a nested chain with arbitrary short aliases and selected-state collisions. A semantic union tree or a root handle is not enough. The routed subclass is a positive control. Complete clipped-cone intervals and independent-store joins remain separate frontiers.