Learned widths: future rank, a prepared reader, and the missing producer
Assessment163, 9 October2026. Written exact argument and conditional transfer; restricted failed preparation routes. No new native language or root admission. The native endpoint remains reader42, including its fixed learned width palette.
Contents
Result and payoff
There is an exact criterion for compressing all possible stopping lengths into one prepared linear reader. At each cut, retain the rank of the coefficient prefixes of every test whose stopping length has not yet passed. If this future rank is bounded by a fixed r, one immutable column stream and r finite state bits suffice. Coordinates may be dropped or reset as lengths pass. The query need not select a width first or run one reader per width.
Small rank separately at every width does not imply this condition. Conversely, requiring all widths to be restrictions of a fixed tuple of coefficient streams is too strong: even the last-bit classifier would require a growing tuple under that representation. The correct prepared mechanism permits coordinate loss, not only the invertible coordinate transport used for persistent row updates.
This settles the shared readout design at a precisely stated interface. It does not construct that interface from ordinary right incidences before their strict boundary. A dependent prepared-worker theorem repairs one changed width n in O_r(n) work, preserving the already compiled longer suffix. The remaining producer must discover width-indexed tests, certify row masks, and schedule these repairs on actual input time. Two tempting global preparation schedules fail below. The open problem is now a producer with an explicit output contract, rather than an unspecified family of ready readers.
1. Exact observation contract
Fix a finite maximum width D. At each n=1,…,D let A_n be a binary q_n by n matrix, with q_n at most a fixed r. Zero rows and q_n=0 are allowed. A late word x of length n requests its syndrome A_n x. A shared, mask-independent classifier state must support every Boolean mask on that syndrome, not only one accidentally constant answer. A reader specialized to a mask chosen in advance is a different contract and need not satisfy the necessity statement. The actual word order is x_1,…,x_n, and its terminal delimiter announces n. D, the matrices and a width header are not supplied by the ordinary Horn syntax. For the moment they define the completed semantic object to be represented.
At a cut t, put
V_t = span { rows of A_n restricted to coordinates 1,...,t : t<=n<=D },
rho_t = dim V_t.At t=0 the space is zero. At t=D it is the row space of A_D. Widths below t have expired and must not be included. To compare prefixes of length t, use all common suffixes of lengths n-t for all n>=max(1,t), together with their terminal outputs. In particular ending immediately at a positive t is a legal future observation. The record containing the last ordinary bit may compute its output early and retain it until the delimiter.
Exact statement. A deterministic layered reader preserving every stipulated syndrome needs exactly 2^rho_t reachable states at layer t. The minimum is simultaneously achievable at every layer. This is a bound on a prepared layered reader’s states, not on all native persistent representations.
Proof. Prefixes p,p’ are observationally equivalent iff every restricted row above annihilates p+p’. Necessity follows by choosing a distinguishing row, its width n and the all-zero suffix: the corresponding syndrome bits differ. For sufficiency, the difference between A_n(pz) and A_n(p’z) is precisely the prefix contribution, so it vanishes for every suffix z. Thus the equivalence classes are the cosets of V_t’s annihilator and number 2^rho_t. Any deterministic reader merging different classes fails one of those future observations. The linear construction below realizes exactly those classes. QED.
Complementing a selected output bit does not alter the equivalence relation. Negative final observations therefore do not compress this universal interface. If only a fixed row mask is demanded, some syndromes may be observationally irrelevant and this necessity direction need not apply. For example, a parity mask on an identity syndrome can itself be accumulated in one bit even though the full identity syndrome needs all input bits. The shared classifier in the conditional transfer is deliberately independent of source masks.
This is the same residual-separation method as the retained trellis adaptation in the single-width note, with different quantifiers. That note intersects constraints at one terminal width and counts nonempty affine residuals. Here one of many widths will be selected by stopping; all still-possible syndromes, rather than their conjunction, must remain readable. No new imported theorem or literature-priority claim is needed for the proof.
2. A stream of finite linear instructions
Choose a row basis C_t of V_t. After t bits retain sigma_t=C_t p. Every row of C_(t+1), restricted to its first t coordinates, belongs to V_t: it is a linear combination of restrictions of rows at widths at least t+1, all included at t. Consequently there are matrices T_t and columns b_t such that
C_(t+1) = [ T_t C_t | b_t ],
sigma_(t+1) = T_t sigma_t + b_t x_(t+1).There is also a matrix O_t with A_t=O_t C_t. Thus output at width t is O_t sigma_t. Dimensions are bounded by r when rho_t<=r. Pad every vector and matrix to r coordinates with zeros and retain dimensions in finite labels. All T_t, b_t and O_(t+1) then belong to one finite instruction alphabet. The initial state is zero, and induction proves both the state and output identities. All 2^rho_t states are reachable because C_t has independent rows.
Store the instruction for step t+1 in a linked immutable column record. One record observation and one fixed finite transition update the accumulator and the candidate terminal syndrome on the next ordinary bit. A constant end label handles widths beyond D as absent target levels. Output is already in finite control when punctuation arrives. The records have fixed degree, finite labels and constant observation radius for fixed r; no address, pointer equality, metadata, elapsed time or control-flow side channel encodes the answer.
Preparation is distinct. From explicit finite matrices, ordinary elimination can select C_t and solve for T_t and O_t; this is an effective finite compiler, not a native input procedure. Its matrix scans, arithmetic, coefficient routing, allocation and retained versions all have costs. Even once the finite instruction labels are known, a backward-pointing immutable realization needs an O_r(D) scan into a stack and an O_r(D) reverse construction so the root reads columns in forward name order. The resulting stream has O_r(D) records and O_r(D log(D+2)) explicit pointer bits. A ready root must be routed before the first query bit; it is not supplied by a late width delimiter.
The finite experiment uses ordinary Python integers and elimination to prepare these records. It measures semantics, not those construction deadlines.
2.1 Forgetting is essential
Let A_n be the single row e_n for every n<=D. The output is the last input bit. At cut t only the width-t test has a nonzero prefix, so rho_t=1 for t>0. The stream simply resets its accumulator to the arriving bit at every step.
Suppose instead q fixed coefficient streams B_1,…,B_q were required, with A_n in the span of their n-prefixes for every n. Let v_j in F_2^q be their coefficient column at position j. At width n there must be a functional c_n with c_n v_j=0 for j<n and c_n v_n=1. Hence v_n is not in the span of the previous columns. Inductively q>=D. Static shared prefix streams therefore miss even this fixed finite-state control from reader36. More general reset positions give the same stream mechanism with a learned instruction sequence; learning those positions remains a separate obligation.
2.2 Per-width simplicity does not bound simultaneous future rank
Choose m and, at widths n=m+j for j=1,…,m, use A_n=e_j, padded to length n. Other widths may have no tests. Every active width has rank one. At cut m the future prefix rows are e_1,…,e_m, so rho_m=m and a universal layered reader needs 2^m states. No output at an expired width accounts for this bound.
One literal right row can realize the union of the halfspaces x_j=1 at their respective widths. Its target-character encoding already costs sum_(j=1)^m (m+j)2^(m+j-1)=Theta(m 2^(2m)), before charging ordinary intermediate names, syntax, duplicates or left pins. This is not a superlinear storage lower bound in N. Indeed the raw target trie reads that single row in bounded work per arriving bit; bounded-selection composition remains an existing native control. The example rules out a fixed-state universal linear interface for that family, not native recognition or branching/sharing itself.
3. Prefix-local repair of an already prepared minimal reader
The full-restart failure has a constructive repair at the prepared interface. Fix r. Suppose the old stream is the minimal linear reader of section2 with future rank at most r. Replace the entire test matrix at one width n<=D by q<=r new n-bit coefficient rows. Old stream and new coefficient roots must already be routed into the worker capsule; their discovery and routing are not free. All other widths retain their tests.
Prepared persistent repair theorem. A bounded-local worker can check whether the new family has future rank at most r and, if so, construct its minimal linear reader using O_r(n+1) old-record observations, work and allocation. It changes only the first n columns and one bridge column n+1, sharing the unchanged strict suffix after that bridge. Every old reader remains intact. The worker has an explicit immutable continuation; completion is not an instantaneous join and requires a separately paid service interval.
Forward pass. Simultaneously read the first n old column records and the q new coefficient streams. Augment the old accumulator sigma_t with q partial new dot products zeta_t. This uses at most 2r coordinates. Each augmented transition is a finite matrix F_t and column g_t. Starting from zero, compute and retain a basis R_t for the reachable augmented-state subspace:
R_(t+1) = span( F_t R_t, g_t ).Here R_t denotes a subspace, and the displayed image is generated by applying F_t to its retained basis. All matrices have bounded dimensions, so exhaustive finite-table elimination is one machine-constant operation, not unit-cost arithmetic on growing integers. Store each column, output map and R_t in an immutable stack. Read one further old column when n<D to expose T_n, the transition from the old state at n to its state at n+1. This is constant extra work. If n=D there is no strict future.
Backward pass. At n the observations still required from the augmented state are the new terminal tests zeta_n and T_n sigma_n. The latter determine the input-independent contribution to the next old state. Because the old reader is minimal, its states at n+1 are separated by outputs at widths>=n+1; thus these functionals represent exactly the strict-future distinctions at n. Include no old width-n output, since it is being replaced.
Choose functionals K_n forming a basis of those observations restricted to R_n. Recursively at t<n choose K_t from the span of the unchanged width-t output map and K_(t+1) F_t, again modulo the annihilator of R_t. At cut0 there is no output and the reachable subspace is zero. The new state is eta_t=K_t(sigma_t,zeta_t). Induction on the backward pass shows that two reachable augmented states have the same eta_t iff every new-family future output agrees. Hence dim eta_t is exactly the new rho_t, and testing all these finite ranks checks the gate. Cuts after n are unchanged. If a rank exceeds r, return a failure flag and retain the old reader, charging the discarded work.
The quotient by R_t is essential. Normalizing functionals on all of the ambient 2r-dimensional space may keep distinctions no actual prefix can realize and incorrectly reject an update. For example, if the new test equals an existing prefix accumulator, the augmented coordinates are correlated, not independent.
Transitions and sharing. On R_t, each row of K_(t+1)F_t is a linear combination of rows of K_t. Solve this fixed finite system to obtain the new transition; the new input column is K_(t+1)g_t. Express the terminal output in K_t coordinates in the same way. At the right boundary of the repaired prefix, express T_n sigma_n in K_n coordinates. That gives one bridge instruction to the unchanged old coordinates at n+1, with the old input column and output map. Share its successor, which is the old column n+2. If n=D, create no bridge. While popping the stack, create these instructions from right to left, so every new pointer goes to an already allocated suffix. The last root is the new forward reader. Only O_r(n+1) records are read or allocated, including the forward stack, and explicit pointer descriptions refer to the charged total retained storage. QED.
This theorem also covers deletion by replacing the output with zero rows. A sequence of updates whose every completed family satisfies the rank gate keeps the same fixed r; constants do not grow with the number of widths or updates. Each worker temporarily uses at most 2r coordinates, independent of D. No pointer identity test, numeric address, variable-degree node or hidden old-node mutation occurs. This is a concrete persistent representation subclass with charged repair and shared readout. It is not an implementation of ordinary incidence learning, and it does not handle arbitrary temporary rank followed only by a final low-rank gate.
3.1 Why this does not finish the native producer
The worker starts after its new coefficient streams are available. Its reverse pass still needs O_r(n) subsequent service. An n-bit coefficient object being present in storage is not n additional input symbols, and an old long record cannot pay a newly released job. Ordinary extensional incidences also do not identify their width until their target spelling ends. Selecting which learner to advance, servicing several outstanding repairs, atomically publishing their current shared version, and recoding existing row masks are all producer work. The theorem removes the need to rescan longer widths; it does not grant these other operations. Concurrent workers cannot publish stale copies over one another’s changes.
4. Conditional composition through depth masks
Here is the exact producer contract sufficient to reuse the shared stream. Use reader36’s ordinary right-first signed Horn syntax, fixed dyadic span s between intermediate and target names, input orientation and late endpoint pins. Fix r and s in one machine. Permit unboundedly many occupied widths.
By the right boundary, a producer must have built, from the actual incidences:
- A frozen column stream as in section2, with rho_t<=r and a finite-label terminal linear map at every occupied width. A fixed local record also identifies the rank of that terminal map, hence rank(A_n).
- For every right row and dyadic class, a ready unary depth profile of finite masks on the r-bit terminal syndrome. It must certify that the actual row at each width is exactly the union of those syndrome cells. Unreachable syndrome labels are canonicalized to zero.
- The existing ordinary-name directories and current-version root capsules, with no supplied source handle. The stream is jointly frozen for all masks.
The producer must start from the fixed initial configuration. It is allowed only a fixed amount of bounded-local old-storage observation and allocation per actual symbol of the original encoding. It must include learning, distinct incidence detection, routing, all restarts and all abandoned work. There is no header, padding, advice, free prephase or boundary/EOF flush. Each retained record has fixed degree and finite labels. Calling this a producer contract does not assert that a producer satisfying it exists for arbitrary learned parity families.
Conditional transfer. If that producer exists, the ordinary left phase and late pins have the same bounded-local persistent composition/readout schedule as reader36, with constants depending only on r,s and the producer.
Proof. At width n, certified rows refer to the same syndrome partition. Union is bitwise OR of their finite masks, irrespective of overlap. A row/class profile has depth at most the upper end of its dyadic class. The immutable profile union, class-window service and committed-plus-one-pending invariant of reader35/36 apply unchanged: their operations inspect masks, not the classifier’s internal structure. Actual intermediate-name bits pay the same O_(r,s)(class width) union jobs. The fixed span is retained here; no claim is made for reader42’s arbitrary-key, finite-palette extension.
During a late target, run the one global instruction stream while advancing the current/ahead mask-profile readers for committed and pending roots. At the terminal symbol, the syndrome and the two masks for the actual depth are already exposed. Their OR gives the product membership and the opposite SAT polarity. Pending profiles use precisely the same frozen classifier, so no pending coefficient conversion or per-width worker is needed at query time. Access to the global root is part of the bounded capsule, while source lookup is paid by the source pin. The immutable stream is shared across all retained source versions. No old version is selected for free.
Together with the hypothesized producer, observations and allocations total O_(r,s)(N), with bounded work at every symbol including the last. The existing fixed-tuple old-top normalization gives N+1 scaffold nodes; explicit pointer bits cost O_(r,s)(N log(N+2)). A native recognizer of L would supply a PEG for reversal(L), not automatically for L. QED.
There is a concrete certification component if the shared stream is available before the incidences that must be classified. At width n its rank d_n<=r is a finite terminal label. Every reachable syndrome cell has size 2^(n-d_n). Reader36’s increment-only coverage counter can use the n-ruler with d_n units removed, at constant local cost. Distinct incidences increment their cell; occupied incomplete cells contribute to the same global deficiency stack. This closes the variable-cell-size issue. It does not justify using a final learned classifier retroactively on earlier incidences without paid replay.
5. Sustained preparation attempts and exact failures
5.1 Separate width workers do not extend a fixed-lane proof
Reader42.6 performs speculative work for every member of a fixed width palette while the next ordinary target is arriving. The fixed palette is essential: with k occupied widths, its specified update performs k bounded worker actions per bit. Letting k grow violates the single fixed machine’s local budget. The same failure applies before queries, during positive-prefix learning and registration. A width directory selected only at the word delimiter does not supply the selected learner’s handle during the preceding bits. The present prepared stream solves late query dispatch; it does not repair this producer. This excludes that literal parallel schedule, not all width-indexed service.
5.2 A whole-stream restart has a short-notice deadline failure
Consider a compiler that restarts a Theta(D)-step full column scan after every right-row registration, even if the changed row concerns only a fixed small width. First complete all long-width rows. Then introduce and complete one small-width row, issuing its last registration with only a bounded number of small ordinary records before the right boundary. Use fixed short intermediate keys for the small row; the old long rows may have their own long keys, so the fixed span condition is respected. The small width is a fixed machine constant, and hence its entire remaining name/syntax budget is bounded independently of D. The final family may even be the last-bit family of2.1, with future rank one. Choose D larger than that remaining budget times the compiler’s fixed service rate. Its full restart misses the boundary. Extra input spent on old long rows occurred before this registration and cannot service the newly restarted scan.
The counterexample concerns the stipulated full restart. It does not show that the short registration changes every column, that any repair must scan D, or that a previously valid shared reader must be discarded. It points instead to repair limited to the affected prefix and preservation of already paid future structure. A local update at width n changes the generating requirements only at cuts t<=n; that semantic locality alone supplies neither a pointer route nor a constant-rate persistent repair algorithm.
5.3 Final low rank is not an online invariant
At widths m+j first register the independent halfspaces of2.2. Later add each missing complementary half until all these levels are full. The final masks are constant and need no parity tests, but the intermediate canonical Walsh support family has future rank m at cut m. Each addition is an ordinary positive incidence. This is the cross-width version of the earlier temporary rank problem: a final-only rank gate does not justify always maintaining the bounded instruction alphabet while learning. Reader42 overcame the common-width problem with a short-key compiler and a directly readable pending descriptor. The new producer needs a corresponding readable remainder or a hereditary gate; one cannot silently import the completed common-width proof.
5.4 What the attempts resolve, and what remains
The future-rank test resolves whether a bounded-state jointly prepared linear reader exists, including negative masks and all stopping widths. Finite depth masks then support the already charged left union and direct pending query. It removes the need for one query reader per width. It does not remove the right-phase learning dispatch, changing coordinate interpretations, mask certification or repair deadlines. These are construction and access costs, not an argument from semantic information size.
The narrower hereditary-rank route now has the O_r(n) prefix-repair worker of section3. Its next missing component is a schedule selecting the right width learner and serializing these repairs before their paid windows expire, with exact source masks in the resulting coordinates. If that route fails, retain the precise conflicting update sequence. Final-only rank and arbitrary intermediate lengths should remain open until this simpler producer is resolved.
6. Evidence, controls and next test
tools/research/check_cross_width_parity.py independently enumerates literal future truth tables, compares their distinct residual counts with Gaussian rank, compiles finite linear records, and compares every streamed prefix output with literal dot products. It also checks profile-mask unions against literal sets, the independent-width witness and the reset control. It also replaces every scalar width test in every rank-at-most-two family through maximum width4, compares acceptance with an independent complete recompilation, checks every output against literal parity, and verifies that the strict suffix records are shared and the old record list unchanged. Exhaustive scalar families through maximum width4 and seeded cases through width7 are finite experiments. They do not establish a native schedule or validate an unbounded producer. The vector-output proof is in sections1-2; the experiment is scalar.
The completed native controls stay intact: replay slices restrict active observations; affine-fragment joins distinguish elimination from late readout; persistent causal and unary cubical algebras keep semantic composition separate from its coordinates; readers35/36 give paid depth-profile union; readers40-42 provide positive-prefix learning and finite-palette completion. Entry129’s matching and positive-path results remain imported theorems under their exact contracts. The present reader uses finite linear updates and negative masks; no matching extraction, native general lower bound or scientific admission follows from it.
Next dedicated review: compare at least two complete primary arguments for minimal time-varying linear realizations and worst-case incremental/persistent preparation. Test them specifically against the now-proved width-n prefix-repair worker, late learner selection, coordinate adoption and final-symbol deadline. Do not count an amortized matrix bound or a prepared trellis as the missing native producer. The next research target is the hereditary-rank producer above, with unbounded learned widths, fixed span and directly readable retained versions.
Continuation164: coordinate adoption
Width-ready registration combines this prepared repair with per-width invertible transport. Surviving row masks need no rewrite; ordinary-input dispatch and first-width bootstrap remain conditional. Review164 compares Forney and Hood-Melville in full.
Native continuation165
A written native construction now supports unbounded input-learned target widths through a fixed active cache, one shared query reader and permanent row depth profiles. With a common input-learned dyadic density, final future rank suffices: each row publishes only its final classifier. Nonaffine three-quarter slices are included. Bounded width revisit distance and fixed right-edge dyadic span remain checked restrictions.
Reader43 owns the producer, including first-word bootstrap, speculative cached dispatch, exact eviction certificates, zero-tail extension and final-symbol query. These are now proved under its gates, rather than supplied ready-handle premises.
Next: Remove the common-density restriction without assuming a globally supplied row-to-density assignment. Test whether independently learned row density alternatives can be reconciled before width retirement while their combined provisional coefficient rank is unbounded. The fixed finite density-by-key rule is a positive control. Preserve unrestricted revisit order, arbitrary name spans, independent-store joins and SAT/scaffold frontiers.
Assessment166 continuation
Reader44 removes common density under the fixed cache/span gates. A completed row reader is a linear state quotient of the global reader; simultaneous instruction scans convert it to masks in linear depth. A readable pending row per source/class and charged prefix replay eliminate population-wide mask rebuilding. The earlier prepared results remain supporting interfaces, with this native producer and reuse schedule now explicit.