Width-ready registration without old-mask rewriting

Assessment164, 9 October2026. Written supporting argument and conditional charged interface. No new native-language endpoint or root admission. This extends the prepared reader and repair, using reader41’s live coordinate transport. The dedicated review164 compares the primary minimal-realization and persistent-scheduling mechanisms.

1. Exact claim and payoff

Fix r once for one machine. A width-ready registration can replace a learned row at width n, preserve every other row’s logical mask, and repair one shared cross-width parity reader in O_r(n) work and new storage. It needs neither one reader per width nor a traversal of old row keys. Unbounded width diversity is allowed. Its gate is hereditary bounded future-prefix rank of the registered family, not final-only rank.

The qualifier width-ready is substantive: before the paying n-bit target word starts, the current row, its prepared coefficient streams, the width registry, and the current reader root must already be locally available. This theorem charges the transaction after that point. It does not grant these handles to an ordinary-input native recognizer. Establishing that routing and release time is the next construction problem.

2. Representation and invariant

At each width n keep an independent local coefficient registry and basis B_n, padded with zero rows to r rows, and a matrix C_n in GL(r,2). The externally visible syndrome is A_n x = C_n B_n x. A registered row v has an immutable Boolean truth table g_(v,n) on the full r-bit syndrome space. Its actual target set is {x : g_(v,n)(A_n x)=1}. Original row coefficient streams and their local truth tables are retained as in reader41. The registry reference-counts these actual coefficient vectors, using a bounded distinct list under the rank gate; there are at most 2^r-1 nonzero distinct vectors. It never walks the population of referencing keys.

Use reader41’s full ambient Fourier-support invariant: each character in an old mask’s support pulls back under C_n B_n to its registered actual coefficient, with the corresponding full-space mask representation retained. Equality only on the current attainable syndromes would not justify recycling a direction later. The shared compiled reader realizes all A_n, with minimal cut state and future-prefix rank rho_t at most r. All published versions carry compatible reader roots and row-profile roots together.

The width directory, row capsule and depth-mask profiles are interfaces, not unit-cost random access. Their paths must have been paid by the caller. A finite current root bundle contains only a fixed number of these handles, regardless of the number of stored widths or keys.

3. Atomic adoption and repair proof

Consider replacing row v at width n. During the transaction the old version remains current. Compute the surviving registered coefficient space S and a new padded basis B’_n for the replacement registry, by the paid bounded-list operations of reader41. If its rank exceeds r, reject the checked contract. Let c_old(a), c_new(a) be the padded coordinates of a in the respective bases. The maps from S into the two coordinate spaces are injective. Extend their induced isomorphism to an invertible r by r matrix Q, so

c_old(a) Q = c_new(a)  for every a in S.

Set C’_n = C_n Q. For each Fourier character ell of a surviving mask, its old actual coefficient a satisfies ell C_n = c_old(a), by the full ambient invariant. Consequently

ell C'_n B'_n = c_old(a) Q B'_n = a.

Every Fourier character of every surviving mask therefore retains its meaning; linearity of the Fourier expansion proves the entire Boolean predicate is unchanged. The mask itself is not rewritten. The replacement row’s new mask is its truth table in B’_n coordinates composed with (C’_n)^(-1). The same invariant holds for the new version, including when registered rank decreases.

Now replace only A_n by A’_n=C’_n B’_n in the shared reader. The vector-output repair theorem163 applies with q=r. It scans the affected prefix, computes reachable augmented states and their future-observation quotient, builds the new immutable prefix, and joins the unchanged strict suffix through its bridge. If the proposed rho bound fails, publish rejection instead of an invalid version. For every other width m, the physical output A_m is unchanged. Thus all its row masks also retain their meaning even though internal state bases in the compiled prefix may have changed. Internal state coordinates and the externally observed width syndrome must not be conflated.

Invertible C’_n changes no row space, hence the future-rank test depends on the actual coefficient spaces, not the chosen coordinate adoption. This makes the gate compatible with all choices of Q satisfying the surviving constraints. Publish the new registry, current row payload, physical basis and compiled root in one fixed-size version bundle. Existing versions keep their old bundle. Induction gives the same guarantees for arbitrarily many completed updates. This is a conditional interface theorem, not a general dynamic join theorem.

4. Resource contract

For fixed r, coefficient comparisons, basis recovery, Q selection and mask conversion cost O_r(n); finite truth tables and r-dimensional matrix operations have machine-dependent constant size. Preparation of the candidate before registration remains a separate charged learner obligation. The prefix repair cost is O_r(n) including old-node reads, reachable-space construction, backward quotient calculation, bridge construction and allocation. Combined storage allocation and old-storage observations are O_r(n). Each elementary worker step uses a fixed number of pointer traversals and finite labels/control. Old nodes are immutable; continuations and new nodes refer only to previously available nodes. The inherited reader41/163 stream and capsule implementations supply the fixed-degree, fixed-radius old-top operations, rather than Python’s arrays.

There is a constant K_r such that servicing K_r elementary worker steps per actual symbol of the paying n-bit word completes the sequential transaction by its delimiter. Forward scans and backward construction are scheduled worker steps, not additional passes over the input. A cursor into already retained coefficient or reader storage is paid for each traversal. The word’s bits still perform their ordinary membership/learning work at a separate constant rate. No other registration is released within this atomic record. There is no padding, EOF flush, supplied decomposition or free old-version handle.

This schedule is valid only because all transaction inputs are ready at word start and the O_r(n) constant is uniform over widths and histories. A job released at the delimiter cannot retrospectively charge that word. The model allows old current answers during construction and only switches at commit; it does not promise that an incompletely learned row already has its eventual final semantics. Final row certification remains the learner’s responsibility. For n beyond the old maximum width, a prospective implementation can append zero-output columns before applying repair, spending O_r(n) work. That extension and its native bootstrap are not asserted by the finite checker below.

5. Minimal realization explains the state, not the producer

For a finite family y_n=A_n x_[1:n], form a linear behavior whose event order is x_1,y_1,x_2,y_2,…,x_D,y_D. Cut just after x_t and before y_t. Forney’s state quotient identifies pasts with identical possible future behavior modulo past-only trajectories. In this functional causal behavior, two input prefixes are equivalent precisely when every future coefficient-prefix difference vanishes. The quotient dimension is rho_t. Cutting after y_t would remove the width-t observation and can yield a smaller dimension; this event convention is essential to the strict stopping query.

This is an original application to the prepared reader contract, not an input reduction inserting output letters. Actual input length N receives no extra credit from the semantic event expansion. The classical minimality theorem explains why future dependence is the correct invariant. It does not supply learning, persistent repair, named routing or the registration deadline.

6. Why a real-time queue is not a registration scheduler

Hood and Melville’s recopy proof has a specific reserve: an old front of length n remains readable while at most 2n+1 copy steps execute. Two copy steps per update finish before that front can be exhausted. The old structure answers queries throughout. A queue of arbitrary repair jobs has no analogous reserve merely because enqueue/dequeue are constant time.

In particular, if a width-n context becomes selectable only at its target word’s delimiter, queuing its O_r(n) repair then gives no service before an immediate required answer. Total linear work is insufficient to prove this record deadline. This refutes only the proposed late-release scheduling argument; pending semantic overlays or earlier dispatch can escape it. It is not a lower bound for native recognition. Likewise an old compiled reader may be structurally readable while assigning stale meanings to new masks; section3 solves that second problem, conditional on the first problem’s ready inputs.

7. Next decisive test: checked width blocks

Investigate the following narrower ordinary syntax. Right incidences occur in contiguous blocks of one actual target width, each width in at most one block. There are no width headers, decomposition certificates or extra work symbols. Within a block after its first record, the context from the preceding record can dispatch the width-n learner before the next target begins. This is a potential source of the ready handles in section1, with unbounded blocks.

The sustained bootstrap attempt separates four obligations:

1. A first target builds a raw singleton and a unary width path as its bits arrive. At its delimiter, compare with the active width and reject reentry to a finished width using a paid seen-width path. Do not assume that this late comparison can itself prepare every row profile or compiled column.
2. During an apparent same-width record, service the active learner and its pending registration speculatively. If actual width differs, discard this transaction and start the raw first-record context. Ordinary validation of the new word still has to have been done in a fixed additional lane.
3. For subsequent records, combine reader40’s determining-prefix margin with section4 to ensure the last candidate receives an own registration word before block completion. The global hereditary future-rank gate must hold for published provisional versions; final-only rank is not enough.
4. Before leaving the right phase, install the ordinary-key depth-mask profiles needed by reader36’s fixed dyadic span. First-record profile allocation, closure of the current key capsule and new-width reader extension must all meet the same delimiter. No free width-to-key transposition is allowed.

The first two lanes explain why arbitrary width speculation might be avoidable, but they do not yet implement obligation4. A width revealed only at the first word’s end cannot be used at that word’s start to select an existing nested profile. A fresh raw singleton may defer this, provided an explicit capsule invariant proves where its row key and width cursor live and which later actual symbols pay conversion. This is the precise next proof target. We have not proved either impossibility or a full native width-blocked construction.

The useful test family has arbitrary learned parity coefficients across unbounded widths while the entire family has bounded future-prefix rank. Fixed finite-state routes and reader42’s fixed width palette remain controls, not evidence for this new scheduling obligation. Independent-store joins, final-only rank, unrestricted width order, arbitrary-key heterogeneous widths and the SAT/scaffold boundary remain open. Any future native recognition result transfers to a PEG for the reversal of that language only.

8. Finite evidence and scope

python tools/research/check_cross_width_adoption.py combines vector-output prefix repair with independent per-width mask transport. Seed164, capacities 1 and2, maximum width7,903 proposals:707 accepted transactions,42 local-rank and154 global future-rank rejections,10 nonidentity transports. All861 repair rank gates match complete independent recompilation. Literal sets verify 671632 current-row queries,51934 retained-row queries,193548 union queries and179578 syndrome queries across55 retained versions. The forced rank-drop case exercises a surviving mask under a nontrivial coordinate change.

The checker uses full recompilation only as an oracle. Its Python dictionaries, integer coordinates and object identity audit of shared suffixes are not native primitives or communication channels. It corroborates algebra and retained semantics; section4 supplies the conditional work argument. It does not simulate ordinary-input dispatch, bootstrap, native allocation clocks or a PEG.