Branching coefficient readers with final-only breadth

Assessment167, 9 October2026. Written persistent representation theorem and conditional producer transfer. This is not yet an unrestricted-width native Horn recognizer.

Outcome

A completed bounded-rank reader need not be maintained in bounded-rank coordinates while its input is still changing. Store its coefficient strings in an ordinary persistent binary prefix trie. Update only the spelled coefficient paths. At a completed query, advance the whole live prefix frontier. Final future rank r bounds that frontier by 2^r, even when earlier versions had arbitrarily large breadth and rank. No final normalization pass is required.

The construction removes the provisional-interface capacity obstacle in reader44. It leaves a specific producer problem: selecting a width/row learner soon enough to process ordinary incidences, and completing its certificates before the right boundary. The fixed revisit gate still supplies that dispatch in the current native theorem.

1. Data and exact scope

Fix constants d>=1 and B>=1. A row v has, at each positive width n, an ordered list A_(v,n) of at most d distinct nonzero binary coefficient words of length n and a finite Boolean table h_(v,n). Its meaning is h_(v,n)((a dot x)(a in A(v,n))) for |x|=n; dot products are over F_2. Use lexicographic coefficient order. An absent row/width has zero output. A full row can have no coefficients and constant-one output. The width set, number of rows, word lengths, update history and provisional ranks are unbounded.

Let C be the union of all currently referenced coefficient words, retaining length as part of the word. Its prefix breadth at depth t is

$$\beta_t(C)=|\{a_{[1:t]}:a\in C,\ |a|\geq t\}|.$$

A completed version is query-ready when every beta_t<=B. Updates need not preserve this condition. The representation checks readiness with a root Boolean maintained by the updates, including deletions. A breadth-failing intermediate version is retained exactly but is not promised a fixed-frontier query.

Written representation theorem. Starting from fixed empty structures, a replacement at width n costs O_(d,B)(n+1) old-record observations, logical work and allocation, including the global reference index, row coefficient trie, row mask profile and final-breadth certificate. Each query-ready version supports one old-root-based frontier step per incoming bit with O_B(1) local operations and a finite Boolean answer at that bit’s delimiter. Old roots retain their exact meanings. A row reader can be converted to global terminal-coordinate masks through depth b in O_(d,B)(b+1) work and allocation. There is no final scan, coefficient-address test, global rebuilding, or intermediate breadth restriction.

Here an update is an explicitly charged worker supplied the old/current roots and the replacement coefficient streams. Root routing, creation of these streams and scheduling that worker are separate obligations of a native producer. This theorem does not declare those inputs free in the ordinary Horn language. The constants determine finite control and a fixed tuple of pointers. All structures have fixed degree; a bounded sequence of record operations has fixed-radius old-top implementation through the Lab’s existing tuple compiler. The O(n) worker is not one native symbol transition.

2. Persistent trie and breadth certificate

The global trie has ordered zero/one child pointers and a terminal reference count. A nonempty trie node is retained exactly when some referenced word ends at or below it. Equal coefficient words at different rows share one terminal whose count records all references. Counts are unary persistent stacks: increment allocates a cell, decrement takes its next pointer, and empty is explicit. A stack cell carries its capped height min(height,B+1), so threshold tests and pops remain local even after unbounded overflow. Node identities never encode data or determine an answer.

Insert or remove one coefficient by scanning its n actual stored bits with a continuation stack, updating its terminal count, and rebuilding that path backwards. Prune nodes that have no terminal reference and no child. Off-path subtries are shared. A nonzero terminal-count change that leaves the count positive does not create or remove prefix nodes. The backward pass records at each depth a delta in {-1,0,1} for whether that depth’s on-path node appeared or disappeared. Reverse this delta stack in a paid pass if necessary. Multiple coefficients in a replacement are processed serially; d is fixed.

Maintain a separate unary depth profile. Its t-th cell contains the exact unary count beta_t and an OR annotation saying whether any cell at depth at least t exceeds B. Apply the delta stream along its first n cells, then rebuild those cells backwards; the unchanged suffix supplies its old OR annotation. A zero-tail sentinel represents absent depths and permits extension. Each changed counter takes constant work; each profile column takes constant work. A deletion at width n cannot alter a prefix deeper than n. No traversal of a longer suffix is needed, even if it remains very wide. Retained zero columns may stay allocated; their cost was paid when created.

The global root’s annotation is therefore exactly [some beta_t>B] by induction. This works through arbitrarily wide intermediate tries, because maintenance follows one coefficient path rather than enumerating a frontier. It is a data certificate constructed and updated at charged cost, not an uncharged metadata channel.

A row has its own coefficient trie and a unary mask profile storing the current h_(v,n) at depth n. Replacing a bounded list updates those tries and that profile in O_d(n+1). Keep the old list in the row/width payload so its references can be removed, and add the new list. The published descriptor changes only after the complete worker finishes; its intermediate private roots are not answers. Publication of a version includes compatible row, global and mask roots. Routing to that payload is still a producer obligation.

The number of logical records allocated is O_(d,B)(1+sum(n_i+1)) over updates, in addition to explicitly charged coefficient/input storage and root routing. Terminal multiplicities do not add a logarithmic arithmetic cost. There is no claim that explicit pointer bits cost only this many bits.

3. Direct branching readout

A nonempty global trie starts with the singleton root frontier and parity zero. At query cut t the frontier is the lexicographically ordered set of nonempty trie nodes at depth t. For its node with coefficient prefix p, retain the bit p dot x_[1:t]. When the next query bit x arrives, replace each frontier node by its existing children in zero/one order, copying its accumulated parity and XORing c*x on child c. Each old node has at most two children. The next frontier has at most B nodes in a query-ready version; a temporary tuple of 2B child slots suffices to construct it.

Induction proves both the frontier and every stored parity. At a terminal depth, select the nodes whose terminal counts are positive. Their parity values are exactly the global coefficient syndromes at that width, in lexicographic order. A row’s own frontier gives its coefficient syndromes in the order expected by h_(v,n), while a synchronized mask-profile cursor supplies that table. Empty frontiers persist as empty, but constant masks remain readable from the separate profile. Virtual zero tails handle longer queries.

Thus branching/sharing supports a fixed number of simultaneous local observations on completed versions. Negative observations are ordinary finite tests of missing children, terminal markers, parity bits and a finite Boolean mask. No global no-path primitive is assumed. The number of local slots depends on B, not on the number of stored widths or the previous maximum breadth. Old versions need no repair when a later version changes.

4. Charged conversion and reuse

Let L be a completed row trie and G its compatible completed global trie. Every coefficient terminal of L occurs in G. Every local frontier prefix consequently has an identical global prefix. Maintain the injection from local frontier slots to global slots as finite state. Begin with roots, and extend the injection using matching zero/one child labels. This requires no pointer identity comparison: the paths and their edge labels establish equality. Each frontier has at most B nodes.

At depth t, local terminal slots inject into the global terminal slots. Compose the local table h_(v,t) with projection onto those slots. This gives a finite table H_(v,t) on the global terminal syndrome vector. Its arity is at most B, so the table size 2^B is a fixed machine constant. The extension to syndrome vectors that no input realizes may be arbitrary; on actual inputs the equality follows directly from the shared coefficient strings.

Scan both tries by these frontiers and the row’s mask profile through depth b, emit the finite tables on a stack, and reverse-build a forward unary profile. This costs O_(d,B)(b+1) observations, work and allocation. It scans neither every coefficient separately nor every complete row key. For a length class j, take b=2^(j+1)-1 and zero the tables before 2^j. The cost is O_(d,B)(2^j).

Consequently reader44’s paid pending-row mechanism transfers without its minimal-linear-coordinate representation. A committed source/class profile is ORed with one directly readable pending row. A later ordinary intermediate name pays conversion and union of the old pending row; its delimiter installs the new row root. The current/ahead replay schedule processes the actual query prefix in the next pending row before promotion. Those existing routing, span and service hypotheses remain required. The new global coordinates are lexicographically ordered terminal coefficients rather than a minimal linear basis. Arbitrary interleaving of completed versions is not granted free historical root selection.

5. Connection to final future rank

For the canonical semantic form, store all nonzero Walsh frequencies of each final row indicator, rather than a noncanonical chosen basis. A rank-at-most-r row has at most d=2^r-1 such frequencies. They and their finite output table can be produced from a learned rank-r basis/table in O_r(n+1): enumerate its constantly many coefficient combinations, compute the finite Walsh table, and retain exactly the nonzero frequencies. The Fourier constant contributes to the finite output table, not a nonzero coefficient word.

Let W_n be the span of all final row frequencies at width n. If

$$\dim\operatorname{span}\{a_{[1:t]}:a\in W_n,\ n\geq t\}\leq r,$$

then the prefix set at t lies inside an F_2 space of dimension at most r and hence beta_t<=2^r. Zero prefixes of nonzero full coefficient words are included in this count. Take B=2^r. This is final-only: a history can insert independent directions at arbitrarily many widths and later remove them, without ever traversing a large frontier while updating.

For a concrete growing history, take widths n_i=m+i for 1<=i<=m. A row at width n_i may first complete the parity half defined by coordinate i and later expand to the full cube. During the first stage its stored nonzero coefficient is e_i padded to n_i. At cut m there are m distinct live prefixes. After all those rows become full, their nonzero support disappears and final rank is zero. Add a separate row at each n_i whose half is defined by the last coordinate n_i. These final coefficients have future rank at most one: at cut t, longer coefficients have zero prefix and only a coefficient ending at t can contribute e_t. Their total zero-padded rank nevertheless grows with m. The breadth certificate recovers without normalizing the old wide versions. This is a semantic history/control, not a newly proved arbitrary-order producer.

At each cut, writing rho_t for the span dimension, the parameters satisfy rho_t<=beta_t<=2^rho_t. The first inequality follows because the distinct prefixes generate that space; the second counts its elements. Thus uniform boundedness of final future rank and final coefficient-prefix breadth are equivalent across a family, with different parameter bounds. This is an equivalence of representation parameters, not a necessity theorem for arbitrary native recognition.

The converse with the same prescribed rank parameter is not asserted. B distinct prefixes can have rank B. The maintained Boolean checks final prefix breadth, not exact final future rank. Thus this representation supplies readout on every final-rank-r version and also some versions outside that semantic class. It does not by itself recognize precisely the final-rank-r gate. A native language theorem must state whether it checks this broader intrinsic Fourier-prefix-breadth gate or separately verifies exact rank.

This identifies a precise stronger operation than a single positive path or matching-only interface: a bounded simultaneous coefficient frontier followed by a finite, possibly nonmonotone, Boolean readout. The entry129 imported bounds retain their hypotheses. Since B=2^r and the finite mask alphabet can have size exponential in B, this is not a uniform polynomial-state escape when r grows with an external relation alphabet. No all-relations lower bound is contradicted.

6. Sustained producer attempt and its unresolved deadline

The cache in reader44 served two purposes: selecting a width’s learner before the next target arrived, and bounding all published provisional coordinates. Sections1-5 remove the second purpose for a prepared updated coefficient store. They do not silently remove the first.

A tested scheduling idea keeps one delayed ordinary right record per dyadic class. A later comparable-length record would replay the earlier record, locate its width/row state and pay its O(n) learning transaction. Fixed serviced class windows suggest a total linear budget. But the last delayed record in each class has no later service. Simply omitting it changes exact validation: a size-m candidate with m-1 observed members plus one outside point has the same final distinct count as a completed candidate, while the final row is different. Padding, an EOF drain, or a free membership test of the delayed word would change the contract.

Nor is the raw coefficient trie automatically a solution during that delayed word. Its current provisional frontier may exceed B. Final breadth gives no bound on the work of reading that earlier version. Replaying the word after selecting its exact-width classifier has to be paid before the boundary. A speculative missing-point dictionary might address this, but constructing it, exposing its current root during ordinary-key routing, and discharging every final delayed record have not been proved.

The next decisive target is therefore a right producer with arbitrary width revisits that supplies exact final local classifiers and committed coefficient updates before the right boundary, using ordinary names and bounded per-symbol work. It may target the broader final Fourier-prefix-breadth gate. Any proposed lag scheduler must explain its final pending record, root freshness and membership validation. The raw trie eliminates global coordinate recompilation from that target. Independent-store joining and removal of the dyadic span remain separate.

7. Evidence and disposition

The dedicated finite checker compares persistent trie updates, exact breadth annotations, direct frontier syndromes, row-to-global conversion and retained versions against literal coefficient sets. It includes wide intermediate versions followed by narrow completed ones and deletion/multiplicity histories. The experiment does not implement ordinary-input learner dispatch or turn the O(n) worker into a one-symbol action.

The mechanism is a written original adaptation of elementary prefix tries to the Lab’s charged coefficient readers. No novelty claim or new root admission is made. It is supporting prepared-interface progress, not a new unrestricted native endpoint. Reader44 remains the latest native language theorem. Assessment167 advances the scheduled assessment once; mathematical count112 and six-result window2 remain unchanged.

Continuation168: ordinary producer under the breadth gate

Reader46 closes the arbitrary-revisit producer target under the broader intrinsic final-breadth gate and the existing dyadic span. Dense-input-paid literal candidate tries keep last-word validation immediate. Class windows service deferred preparation and refresh row lookups before use; the final pending jobs contain no unchecked word. The prepared theorem above remains its representation component. Exact rank-gate recognition and removal of the name span remain distinct.