An unbounded component directory needs its own budget

12 October 2026. Written representation obstruction and conditional repair contract. Native construction 72 is unchanged; no root admission or external priority claim.

The current investigation guide places this result alongside its successors. The two-omission review identifies the current paid strict-extension target; it does not remove this note’s hypotheses or close its remaining frontier.

Why the next extension is difficult

Construction 72 stores a fixed number of support components. A row may contain a whole component or none of it, at zero internal mixed cost. With unboundedly many components, these individually simple choices can form a complicated global pattern. A directory must represent that pattern as well as route a later target.

For example, label eight singleton components by the binary strings 000 through 111. Consider the three rows that select, respectively, the strings whose first, second or third bit is 1. Every row is uniform inside each singleton. Nevertheless, every binary directory joining the eight components has an internal cluster that is nonuniform for one of these rows. Across its seven internal clusters there must be at least seven such incidences. Rearranging the directory cannot make all three rows cheap.

The argument below makes this simultaneous obstruction quantitative. It then identifies the exact extra quantity that pays the inherited union algorithm. It rules out a transfer of construction 72’s internal-only budget, not bounded-local-cost recognition in general.

1. Common-tree obstruction

Let $k=2^r$, with $r\ge1$, and index singleton components by $b\in\{0,1\}^r$. For each $i\in\{1,\ldots,r\}$, let

$$X_i=\{b:b_i=1\}.$$

A common binary directory is any rooted full binary tree $T$ with these $k$ components as its leaves. The topology and leaf order may depend on every row. At an internal node, a row is mixed when its descendant leaves meet both the row and its complement. Let $M_T(X)$ count these nodes. Unary routing chains, if present, are suppressed for this count.

Proposition. Every such directory satisfies

$$\sum_{i=1}^rM_T(X_i)\ge k-1,\qquad \max_iM_T(X_i)\ge\left\lceil\frac{k-1}{r}\right\rceil.$$

Proof. Every internal node has at least two distinct descendant bit strings. Some coordinate differs between them, so at least one coordinate row is mixed there. A full binary tree on $k$ leaves has $k-1$ internal nodes. Count pairs consisting of an internal node and a coordinate that varies below it, first by node and then by coordinate. The average gives the second inequality. This proof allows an offline, row-aware choice of the entire common tree. It does not assume creation order, a balanced tree or literal-prefix order.

Complementing a row preserves its mixed nodes. Consequently adding all $\overline X_i$ does not weaken the obstruction. Uniform component flags therefore cannot generally be converted to common-tree views with a mixed budget proportional only to their zero internal mixed counts and their short alias lengths.

2. Realization by ordinary records

This family is legal under construction 72’s deterministic birth procedure with only its fixed component cap removed. Use two distinct birth-owner aliases and $2r$ further distinct aliases for $X_i$ and $\overline X_i$. All have common width

$$\ell=\lceil\log_2(2r+2)\rceil.$$

Choose any fixed literal prefix $P$, possibly long. The targets are the actual names $Pb$, all of width $n=|P|+r$. Insert each target once, alternating the two owner aliases. Every owner change freezes the preceding component; the next target is new, so it births a singleton component. After all births, append ordinary records for the coordinate rows and their complements, using only those existing targets. The first coordinate alias freezes the last active component. Every subsequent target belongs to a frozen singleton, so no further birth occurs.

Within a singleton, every row restriction is EMPTY or FULL. Hence every row, including both birth owners, has total internal mixed count zero. Duplicate insertions do not matter. This uses ordinary named Horn edges, not supplied component handles or a promised decomposition. The common prefix is part of each actual target, not free preparation or padding inserted into a reduction.

There are $k$ birth records and exactly $rk$ coordinate/complement records. Their actual right-factor length is

$$N_R=(r+1)k(9+\ell+n),$$

apart from the fixed phase delimiters of the inherited encoding. Large preparation is explicitly paid; it is not an access lower bound. Since $(k-1)/(r\ell)$ is unbounded, no fixed constant converts this family’s internal-only gate into $M_T(X_v)=O(\ell)$ for all rows on one common tree.

More generally, after the singleton births, arbitrary row-to-target incidences can be appended without increasing any internal mixed count. This recovers the arbitrary-incidence diagnostic already present in assessment 205; it is not a new general hardness claim. The new conclusion is that choosing a better shared hierarchical directory cannot by itself repair the mixed-budget transfer.

3. Actual union work, not just representation size

Consider precisely the inherited aligned asymmetric splice: return the other operand for EMPTY, return FULL when either operand is FULL, otherwise recurse into both children and combine the results. Count a call even when it terminates immediately. This contract has no special complement test or arbitrary pointer-identity shortcut.

For baseline $\overline X$ and incoming $X$ on the same tree,

$$\operatorname{calls}(\overline X,X)=1+2M_T(X).$$

Indeed both operands are mixed at exactly the mixed nodes of $X$, so precisely those calls branch into two recursive calls. At every other reached node a uniform case terminates. Counting edges in this recursion tree gives the identity. The result may be FULL, but the specified procedure discovers that result only after its recursive work. Hash-consing repeated subtrees, an independently prepared complement certificate or a different Boolean representation are different interfaces.

Thus one coordinate row in the family forces at least $1+2\lceil(k-1)/r\rceil$ calls, although its alias length is only $\ell$. This defeats every fixed per-alias completion rate for that splice. It remains a statement about a concrete worker, not all possible ways to compute union.

To make the schedule witness independent of the chosen common tree, use one short source name for each coordinate. For that source select $\overline X_i$, then $X_i$, then any short alias. The second selection can install the former pending complement into an EMPTY baseline in constant work. The third selection must complete the complement/coordinate splice under construction 72’s inherited next-alias contract. Include these triples for every coordinate. Source names may have width $\lceil\log_2(r+1)\rceil$, so even crediting the entire next record gives only $O(\log r)$ symbols. Some triple exceeds every fixed service rate as $r$ grows. No total-input padding argument pays this particular deadline.

An immediate pin after the first two selections still has an exact answer from committed and pending views. It is not a counterexample to that query invariant. The failed step is completion before the next selection replaces the pending slot. An off-support negative query can also be easy. These distinctions prevent a worker-cost failure from being mislabeled a native-language lower bound.

4. The exact missing charge

There is a useful conditional replacement for the discarded internal-only bound. Take a fixed binary directory $D$ whose leaves are pairwise disjoint nonempty supports $U_j$. Graft each component’s own binary support tree at its directory leaf, obtaining one aligned tree $G$.

For any row $X$, let $\mu_D(X)$ count directory internal nodes whose entire descendant support meets both $X$ and its complement relative to that support. Partial component rows contribute to this test as partial sets, not as a single activation bit. Then

$$M_G(X)=\mu_D(X)+\sum_jM_{U_j}(X\cap U_j).$$

Proof. The internal nodes of $G$ are the disjoint union of directory internal nodes and internal nodes of the grafted component trees. The row is mixed at a node exactly under the corresponding definition. Add the two disjoint counts.

The asymmetric splice on $G$ consequently has at most

$$1+2\mu_D(X)+2\sum_jM_{U_j}(X\cap U_j)$$

calls. Only incoming mixed nodes can branch; any baseline terminal case can reduce the count. A checked bound on this combined quantity can replace the fixed root-call term in construction 72, provided the following implementation obligations are discharged:

• Construction and every revision of $D$ and its row views are paid by actual input. Growing $D$ must not silently change old budgets or predicates.
• Child views, frames, return steps and pending descriptors have bounded local cost in one fixed finite-control, fixed-degree, fixed-radius old-top machine.
• Ordinary target symbols route through the aligned directory and component coordinates with timely branch events and literal absence validation. A nonliteral component ordering does not supply this for free.
• Every alias’s actual service window completes the previous charged job, including return work, while current lookup and new receipt construction proceed. Query output remains exact from a bounded number of committed/pending readers without an EOF flush.

This is an exact cost identity plus a conditional transfer contract, not a native construction. In particular the new global budget is not proved checkable during component births. Replacing an old directory leaf by a branch can change many old rows from FULL to mixed even when all component interiors stay uniform. The earlier one-branch diagnostic applies again at this higher level.

5. What survives and what to test next

Three attempted extensions now have separate dispositions. The unchanged growing-vector reader batch violates bounded active readers. A common tree with the old internal-only budget fails by the universal coordinate-row argument and actual splice witness. A common tree with the combined budget has a correct conditional union charge, but lacks a charged online producer and query router. None excludes row-specific geometries, shared Boolean circuits, certified complement operations, or a more selective pending interface.

Early unique routing remains a positive control: selection and query names pay for the same unique current local kernel, so no global activation union is required. The present family allows each short alias to touch many target components, which violates that control’s same-route premise. Frozen support 71 and bounded components 72 remain valid. Replay, affine joins, causal sharing and unary coherent algebra do not independently supply this missing branching-directory operation. Imported entry 129 retains its growing alphabet and conditional state/port bounds; no matching-diagram extraction or native lower bound is claimed here.

Next decisive target: construct a producer and literal query router for a combined directory-plus-component mixed budget. Begin with singleton components born in alternating owner blocks under one long common prefix. Account for budget changes to old rows at every directory refinement, and test complement-pair selections followed immediately by another short alias. If row-specific or DAG sharing avoids the common-tree obstruction, specify and charge its cross-row union operation and exact negative readout. Keep the early unique-routing subclass as a control; independent-store joins remain separate.

The independent finite audit and review record accompany this written argument. Finite enumeration corroborates the identities; it does not prove native scheduling. The contribution is a supporting representation obstruction and precise repair contract within the Lab’s existing mixed-region program, not a new native theorem or an external originality claim.