Routing does not settle selected-row incidence

10 October 2026. Assessment 205. Dedicated primary-proof review. Native theorem 67 remains the strongest construction; no new native theorem or root admission.

Later developments: review 206 tests the clipped-cone schedule, review 207 supplies learned shared contexts, and review 208 identifies the remaining late-context union interface.

Outcome

Removing only the fixed global spine bound from theorem 67 admits arbitrary binary row-to-target incidence, even when each occupied spine has just one terminal depth and satisfies the fixed mixed-budget parameter two. A unique content route identifies the target; it does not answer whether any selected row contains that target. This exact scope diagnostic redirects the next construction toward restrictions on incidence as well as routing.

The two primary arguments below supply a positive routing mechanism and a negative-observation comparison. Neither supplies a resource-preserving transfer to the Lab’s native model. The ordinary-input semantic bridge from entry 129 is retained, not repeated or counted as a new advance.

Two primary proofs

Bille, Gawrychowski, Gørtz, Landau and Weimann, Top Tree Compression of Tries, full proof version, 20 September 2019. All 20 PDF pages were studied; the ISAAC 2019 version identifies the conference publication. Theorem 1 gives a static comparison/pointer structure for total string length $n$ over alphabet size $\sigma$, using $O(n/\log_\sigma n)$ space and $O(\min\{m\log\sigma,m+\log n\})$ query time. Sections 4–5 telescope cluster-height charges along vertical extraction. Section 6, Lemmas 8–9, replaces startup height cost by online path extraction: a static top/bottom partition stores root-path lists, and emitted top-path nodes pay for collecting the bottom path. Output has constant delay from a supplied node. Section 7 uses weighted ancestors and gapped grammars for horizontal access; Section 8 combines horizontal and vertical traversals. Theorem 2’s comparison lower bound distinguishes constant-answer boxes using parity. The preliminary randomized fingerprint method is not the final deterministic construction.

Alstrup, Husfeldt and Rauhe, Marked Ancestor Problems, BRICS RS-98-7, April 1998. The complete 39-page PDF was studied independently, including its appendix; this is the preliminary report, not an assertion that another edition’s proof was read. In Section 3.3, Theorem 3 proves for existential as well as nearest marked ancestor

$$t_q=\Omega\!\left(\frac{\log n}{\log(t_u b\log n)}\right).$$

It already covers insertion-only worst-case operations. Age-separated memory generations and sparse markings force negative queries to inspect many generations. Section 3.4’s amortized fully dynamic extension uses refreshes and must be distinguished from that statement. Section 4’s RAM upper bounds use microtrees with $O(\log n)$ branching nodes but unbounded total size or boundary count, compressed paths, ancestor masks and predecessor structures; existential marking takes constant time, querying $O(\log n/\log\log n)$, and unmarking $O(\log\log n)$. Section 5 gives amortized constant leaf insertion. Colored queries in Section 2.5 and the appendix specify one color, not a selected set of colors.

Resource comparison

For a binary alphabet, the compressed-trie query bound is $O(m)$. Thus growing content diversity alone need not force a scan of every spine. This is a useful component comparison, not an ordinary-input persistent construction: ready handles, static preparation and the logarithmic-bit primitives must still be replaced or paid. Constant-delay path extraction is stronger than a total-time claim, but does not implement selected-row incidence updates.

The marked-ancestor comparison does not fail merely because reuse is insertion-only. Its worst-case lower bound already includes that case. The quantitative obstruction to the direct transfer is different. With $b=\Theta(\log n)$ and polylogarithmic update cost, the imported lower bound is $\Omega(\log n/\log\log n)$. A conventional ordinary binary operation identifier takes $\Theta(\log n)$ symbols and hence permits $\Theta(\log n)$ native actions at fixed per-symbol cost. Those bounds are compatible. An encoding-sensitive reduction would need stronger separation, including name routing, all preprocessing, storage observations and output. No native language lower bound follows from this comparison.

Marking nodes semantically embeds into row selection by placing in a node’s row the sibling pairs below its descendant-leaf parents. A leaf query tests intersection with its ancestor set. This is only a semantic control: marking one node is not automatically a paid update to every explicit row incidence. Nor does a single-color ancestor dictionary answer intersection with an arbitrary selected-color population.

Exact incidence diagnostic

Let $B\subseteq[q]\times[h]$ be arbitrary. Choose distinct binary parents $p_1,\ldots,p_h$ of one length $d\geq1$, and put $L=d+1$. Define literal rows

$$S_i=\{p_j0,p_j1:(i,j)\in B\}.$$

Every row is sibling-closed. The parents are incomparable, so each observed parent determines one distinct maximal guide. In each row, each occupied guide has the singleton depth set $\{L\}$. Write $\ell=\lfloor\log_2 L\rfloor+1$ for its band index and $a=2^{\lfloor\log_2\ell\rfloor}$ for the start of that index block. A singleton in the canonical band tree has exactly $\ell-1$ mixed ancestors. Since $a\leq\ell<2a$, its mixed count is less than $2a$. All other bands are empty. Thus every per-slot mixed budget passes with fixed parameter $r=2$, in every clause order.

For selected row set $A$, the query at either sibling has answer

$$p_jb\in\bigcup_{i\in A}S_i \quad\Longleftrightarrow\quad A\cap V_j\ne\varnothing, \qquad V_j=\{i:(i,j)\in B\}.$$

The forward direction takes the row witnessing literal membership; the reverse takes any index in the intersection. The two siblings have the same incidence by construction. If a common prepared geometry is desired across all matrices, add one never-selected row containing all pairs. This fixes the guide universe without changing any queried selected union, and also passes the same budget.

This encoding has polynomial ordinary length. With the maintained syntax, the right records contribute $\sum_{(i,j)\in B}2(9+|v_i|+L)$ symbols; each selection record contributes $10+|u|+|v_i|$ and each endpoint wrapper $11+|u|+L$. Binary names and $d=O(\log(h+1))$ suffice, with polynomial extra input for the common-geometry row. This is an exact scope consequence of the existing relation encoding and theorem 67’s budget, not a new semantic bridge or a hardness conclusion from information size.

Consequently, deterministic routing plus the old per-slot budget does not remove arbitrary selected incidence. The failed route is the attempted inference that a unique next trie edge makes the pending union locally readable. No impossibility for a stronger persistent readout is established.

A restricted next test, and why two summaries fail

Test rows whose parent set is one complete prefix cone clipped by one depth interval:

$$P_v=\{p:q_v\text{ is a prefix of }p,\ a_v\leq |p|+1\leq b_v\}, \qquad S_v=\{p0,p1:p\in P_v\}.$$

Take finite endpoints with $|q_v|+1\leq a_v\leq b_v$. These parameters describe a candidate subclass; they are not supplied decompositions or free input metadata. A native construction must learn and check the property from the literal row stream. Empty rows need their own constant representation.

For $w=pc$ and selected set $A$, literal membership is exactly the joint predicate

$$\exists v\in A:\ q_v\preceq p\ \text{and}\ a_v\leq |w|\leq b_v.$$

This identity follows directly from the row definition; it is not a scheduling theorem. The restriction forbids arbitrary subsets at one depth in one row while allowing unbounded prefix diversity. Empty-prefix rows recover complete depth intervals. An ancestor-existence specialization is another semantic control. Neither separately solves their conjunction.

For example, select rows $(q,a,b)=(0,3,3)$ and $(1,2,2)$ and query 00. Its parent 0 matches the first prefix; depth two matches the second interval. No single selected row matches both. AND-ing separate prefix-existence and occupied-depth summaries therefore returns a false positive. This counterexample identifies a necessary joint readout operation, including exact negative answers.

The next decisive test is an ordinary-input producer and persistent consumer for these complete clipped cones, with long shared prefixes, staggered branch depths, unequal intervals and late short-name selections. Charge learning cone completeness, allocating/routing names and pointers, retained old-storage observations, interval preparation, and every selection’s disturbance of pending ancestor-and-depth work. The producer must finish or expose a deadline-safe pending representation using mandatory later input; the consumer may not scan all selected ancestors, use a free LCP oracle or defer work to EOF. A counterexample to that schedule is useful only with its exact failed invariant. Unrestricted radial profiles and independent-store joins remain separate frontiers.

Finite evidence and independent review

The primary audit exhausts 512 incidence matrices and 4,096 selections, checking 24,576 sibling queries and 1,024 singleton budgets. It also checks 1,984 clipped-cone queries across 32 selections, including 1,264 negative answers, and the correlation counterexample.

The independent audit, with no imports from the primary or producer models, exhausts 4,096 matrices, 262,144 literal/intersection checks, 393,216 insert-only retained-prefix checks, 512 ancestor-incidence checks and 4,095 singleton budgets. It checks eight terminal-incidence vectors on one geometry and the same independently derived correlation failure. Both passed. Python sets, strings and integers are semantic oracles; these are finite experiments, not native transition implementations or lower-bound proofs.

Independent adversarial review confirmed the arbitrary-incidence construction and its order-independent singleton budget, corrected the possible deletion-based dismissal of the imported lower bound, and supplied the joint-summary counterexample. The full primary-proof comparison changes the target rather than claiming a new mathematical admission.

Failure ledger and integration

Source custody: primary PDFs and extracted text remain private scratch; the maintained review and two audit sources live in GitLab. Claim mapping: this is a primary-proof comparison and exact semantic scope diagnostic; theorem 67 and root CLAIMS.yaml are unchanged. Evidence and failures are above. The surviving problem is the paid joint prefix/depth mechanism. The existing foundations entry and checkpoint carry the reader synthesis; generated HTML/KaTeX notes and highlighted code retain line anchors and byte-exact raw downloads. Build, scoped rendered-link checks and exact GitLab-to-Site receipts are recorded in integration state. Browser viewport inspection is unavailable under the supported workflow; responsive layout and equation clipping remain unverified.

Replay slices still charge each branch’s service separately. Affine-fragment joins give no independent-store conversion. Persistent causal algebra preserves histories but does not pay access. Unary coherent cubical algebra gives no automatic growing-branch sewing closure. Native SAT controls keep ordinary late pins, right-first orientation and the final-symbol complement answer; native recognition only yields the stated PEG reversal consequence. Entry 129’s imported growing-alphabet state/port results and conditional diagram transfers remain intact.

Scheduled count 109 to 110 once; mathematical count 130, window two and reassessment 128 unchanged. Next 111 is research, primary-proof review 115 and editorial 114. Installed reverse-writer retirement remains separate and unverified.