Assessment 226: routing large rows and materializing bounded exceptions
12 October 2026. Research assessment and six-result reassessment. Construction 77 supplies a native sufficient criterion for overlapping full-row stores. No root admission or external priority claim.
Contents
Outcome and payoff
The graph-activation test now has a positive mixed interface. One distinguished owner is exposed by the target’s actual prefix; each store has at most a fixed number of foreign targets. Unbounded primary rows remain shared through one activation bit. Only the foreign packets are materialized per source, with the last packet kept directly readable. There is no bound on how many stores a source selects.
A star illustrates why this is more than a bounded-degree control. The center can own arbitrarily many edges in its namespace; each leaf has one foreign edge. The two stores share exactly the same literal target. Activating the center changes all incident answers without enumerating them. Activating a leaf creates one bounded pending exception.
This is not the unrestricted graph test: actual target names supply a primary-owner route. The common positive store-name width and the checked bound on target length pay packet service. Target lengths themselves may vary. No free relabeling, supplied graph, final-store backpatch or deferred end-of-input work is used.
Argument and adversarial review
The proof separates static primary support from source activation. A query reads the primary-support dictionary from its first bit, while its encoded prefix retrieves the source’s activation bit. Their conjunction excludes unrecorded words in an active namespace. Committed foreign membership and at most a fixed number of pending word comparisons run alongside those readers.
The old exception packet is flushed during every next selection for the same source, including a selection of absent alias 1. Alias 1 never retracts an earlier activation. Each packet contains at most words of length at most ; its bit/close steps fit a fixed service rate over the next actual decoded store bits. Retained versions keep coherent activation, aggregate and pending roots.
Independent adversarial review checked the candidate against the inherited serial dictionary, strict FIFO and bounded-record compiler. It approved variable target lengths, exact endpoint exhaustion, source interleaving and foreign namespaces naming no prepared store. The manuscript identifies every checked gate and the exact formula serialization. The all-size claim rests on this written schedule; finite container operations are not a native implementation.
The independent executable audit passed 218 right instances, 7,841 histories, 18,336 selections, 395,574 queries, 170,862 direct Horn checks and 922,925 retained-snapshot queries. Seven star families reach 64 leaves. Three invalid-gate cases are rejected. Abstract service-token checks include packet completion and do not assert machine timing measurements.
Failed broader routes and reopen gates
| Attempt | Exact limit | Reopen gate |
|---|---|---|
| Apply fractional cascading directly | Supplied atomic keys, mutable bridges and amortized repairs do not meet the native contract | An input-built immutable continuation with a strict service proof |
| Materialize every selected row | Unbounded row incidence can be released by one short selection | A charged compressed foreign predicate or bounded scatter budget |
| Use arbitrary next store names as service | A later short name may not finish an older long packet | Paid length-band scheduling or another exact pending representation |
| Replace arbitrary targets by endpoint-prefixed targets | Changes literal identity, encoding length and the recognized language | A separately proved resource-preserving reduction |
The construction solves the second route only under bounded foreign scatter and the explicit service gate. It does not revive the factored-reader representation ruled out in assessment 224. It prepares a different source-dependent aggregate.
Six-result reassessment at mathematical 140
The six substantive results since 134 form a progression across three distinct obligations:
| Result | What became affordable | Restriction retained |
|---|---|---|
| 135, construction 72 | Late support growth without changing old component meanings | Fixed component count and mixed-region budget |
| 136, construction 73 | Unbounded births within learned stable coordinates | Fixed envelope and ambient mixed budget |
| 137, construction 74 | Sparse growing support via exact row classes | At most one omission and its original length gate |
| 138, construction 75 | Every fixed omission bound via paid scans and short codes | Grouped local preparation and common local widths |
| 139, construction 76 | Independent stores with separate service streams | Bounded independent readout or disjoint prefix routing |
| 140, construction 77 | Shared targets through routed activation and bounded scatter | Literal primary-owner route, bounded foreign packet and common service width |
The useful explanatory direction is now clearer: preserving old row meanings, paying pending composition, and exposing relevant source state are separate obligations. Cheap local rows alone do not imply cheap joins. A successful interface can divide a large relation into shared query-routed contributions and bounded contributions whose construction is paid by later ordinary input. Neither semantic overlap nor small information size alone supplies that interface.
This is a collection of explicit sufficient criteria, not a complete characterization. Construction 77 restricts rows to the full-row case; it does not subsume construction 76’s omission machinery. The common-directory and factored-reader obstructions retain their representation-specific scope. Entry 129’s matching and positive-path bounds retain the growing alphabet and their exact readout contracts. Nothing here gives a native-language lower bound.
Replay slices still charge every traversal; affine joins retain their coordinate requirements. Causal sharing preserves history without servicing a pending join. Unary coherent algebra does not automatically supply a branching access index. The native SAT and scaffold controls, and the PEG reversal orientation, are unchanged.
Next decisive test and disposition
Keep actual primary-owner prefixes and the common store service width, then remove bounded foreign scatter. Start with two owners per target and full rows. Construct shared foreign predicates that can be installed by a short selection and read alongside the routed primary contribution by the target deadline. Charge their right-input preparation, persistent activation, pointer routing and any replay. Arbitrary-name routing and heterogeneous service windows remain separate open fronts.
Source custody is this maintained proof and independent checker. The claim is written native construction 77, with no addition to CLAIMS.yaml. Proof and finite evidence are linked above; failed routes and reopen gates are explicit; the surviving frontier is the preceding paragraph. Reader synthesis belongs to the existing foundations guide, and styled HTML/KaTeX plus exact-download source readers are generated projections. This is a research note, not a new paper or a publication-rank change.
Scheduled completion 130 to 131 and mathematical 139 to 140 count once. The six-result window resets to zero at 140. Next 132 is editorial and preserves this target; the next dedicated primary review is 135. Deployment, scoped checks, any broad verification failures and exact source parity are recorded in integration state. Installed reverse-writer retirement remains separate and unverified.