Many interval components with scale-sparse boundaries

10 October 2026. Assessment 199. Written native theorem 65, independently reviewed. This extends the charged band criterion of theorem 64; root admission and publication significance remain separate.

Current continuation: Theorem 66 replaces the one-interval and partial-band population gates by a checked mixed-coordinate budget, with completion-funded ordinary-input preparation. This dated predecessor remains valid.

Statement and payoff

A row can contain many separated intervals without making its early answers expensive. What matters here is where its partially covered bands occur, rather than how many connected components the whole row has.

Write

$$I_j=[2^{j-1},2^j-1],\qquad J_\ell=[2^{\ell-1},2^\ell-1].$$

The first bands partition positive target lengths. The second partition their band indices. Fix an integer $r\ge0$. In the ordinary right-first Horn encoding, require each frozen right row to satisfy:

1. At every occupied length, it contains every binary word of that length.
2. Its occupied lengths inside each $I_j$ are empty or one interval.
3. In each $J_\ell$, at most $r$ bands $I_j$ are partial: nonempty but not completely occupied.

One fixed native machine checks these conditions from ordinary input and recognizes pinned SAT with bounded work and allocation per actual symbol, including the last. Names, word lengths, row populations, interleaving, duplicates, the number of full bands, the number of partial bands in a row, and retained versions are unrestricted. One fixed machine is used for each fixed $r$.

For $r=2$, this class contains every interval row of theorem 64, since an interval has at most two partial bands in total. The inclusion is strict. Any finite selection of whole bands is allowed, with arbitrarily many gaps. Partial bands can also grow without bound: choose the singleton length $2^{j-1}$ in bands $j=2,4,8,16,\ldots$, stopping anywhere. These partial bands lie in different index blocks $J_\ell$ and already satisfy $r=1$.

This remains checked serial reuse. It neither joins independently supplied stores nor admits arbitrary radial profiles. The PEG consequence concerns the reversal of the checked native language.

1. Why the total component count is the wrong charge

A frozen band record contains EMPTY, FULL or INTERVAL, with ordinary prepared endpoints in the last case. A row cursor visits records in order. EMPTY and FULL each cost bounded work, while an interval in band $j$ has the resumable insertion cost $16j+16$ from theorem 64.

For a cutoff $k\ge1$, put $L=\lfloor\log_2 k\rfloor+1$. The checked condition implies

$$\sum_{\substack{j\le k\\I_j\text{ partial}}}(j+1) \le r\sum_{\ell=1}^{L}2^\ell =r(2^{L+1}-2)\le4rk.$$

The last inequality is strict for $r>0$. For $r=0$, the sum is zero. Thus all partial insertions through band $k$ cost at most $64rk$, even with unbounded total partial-band count. Visiting the intervening records, promoting jobs, and restoring a finished prefix contribute only $O(k)$ more work. This is the structural reason for grouping band indices by doubling ranges.

Independent cost $O(j)$ for each partial band without this sparsity can instead accumulate as $\Theta(k^2)$. Likewise, releasing every component as a separately funded job after reading just one row name would invent input service. The construction releases one row cursor and charges all of its components together.

2. Build both directories during the actual target word

2.1 Inherited primitives and native clocks

Use the maintained ordinary-name and target dictionaries, complete-level validator, unary prefix rulers, bounded-local numeral counter and its power-of-two/all-ones flags from the suffix producer. Retain the exact prepared endpoint-numeral contract used by theorem 64. Use only strict persistent deque end views, end updates and catenation from review 195. These are paid operations on internally constructed roots, not external ready handles.

While a target word is spelled, its ordinary length counter $n$ advances once per bit. At a power-of-two length, advance a second copy of the same native counter for $j$. Its value is the current length-band index. At a power-of-two value of $j$, advance the index-block cursor. Both tests use the inherited bounded-local counter flags; neither scans a carry chain or computes a logarithm at a delimiter. Indices $j,k,\ell$ below are analysis notation, not unbounded labels.

2.2 A dense band deque with paid focusing

Each row retains a dense deque of band records from 1 through its largest visited band. Empty records between occupied bands remain in place. During a target word, keep a speculative finished prefix $P$ and remaining suffix $S$, with the current band at the front of $S$.

At length one, expose band 1. At each later power-of-two length, move the current unchanged band to the back of $P$ and expose the next band of $S$. Create an empty band if no record exists there. These are a fixed number of strict deque operations. A word prefix is not a committed occurrence of its shorter lengths: records passed by the speculative cursor remain unchanged.

When a new band $j$ is created at actual position $a=2^{j-1}$, the target’s prefix unit stack already has exactly $a$ cells. Retain that root as the new band’s missing-length capacity. No stack of size $a$ is constructed at that instant. It was built one unit per actual input symbol. Empty intermediate band records created during this word use their respective earlier prefix roots in the same way.

At the word delimiter, replace the current front record and catenate $P$ with the updated $S$. This is bounded work even when an arbitrarily distant suffix remains. There is no return walk through the prefix and no reconstruction from an obsolete snapshot. Both directories below are committed together in the current row header.

2.3 A second directory for partial-band populations

A row also retains a dense deque of index-block records. Each holds a unary stack counting its current partial bands. Expose block 1 at $j=1$. Begin its speculative prefix/suffix zipper there. When the band-index counter reaches a later power of two, move to the next block record, creating a zero-count record if needed. This event occurs within the actual word and requires only bounded deque work.

At every word delimiter, the two current records are exactly the band containing $n$ and the block containing its band index. A change from EMPTY to PARTIAL increments that block’s count; PARTIAL to FULL decrements it. EMPTY to FULL, unchanged occupancy, and duplicate records leave it unchanged. Restore the block directory by one bounded end update and catenation, alongside the band directory.

A global unary counter counts block records whose partial-band population exceeds $r$. A population update changes it only when crossing $r$. Looking through at most $r+1$ stack cells detects that crossing, a fixed-radius observation for fixed $r$. Intermediate violations are allowed: later completed bands can reduce the population and repair the gate. No scan of all rows or blocks occurs at the right boundary.

3. Discover exact band shape and endpoints

3.1 Occupancy, capacity and complete levels

The inherited unary width dictionary identifies whether this is the first target at length $n$ in the row. Only a first occupied width removes one unit from the current band’s missing-capacity stack. Its initial capacity is $|I_j|=2^{j-1}$. Since every occupied length in that band is distinct, this stack cannot underflow. A nonempty band is FULL exactly when the stack becomes empty; otherwise it is PARTIAL. A finite occupancy flag distinguishes an untouched empty band from a partial one.

Separately, the complete-level validator counts distinct target words against the prepared capacity $2^n$. Its global incomplete-level counter must vanish at the right boundary. A completed band in the length directory is therefore not mistaken for a band containing all binary words: the two gates jointly establish that statement. Duplicate words affect neither distinct-width capacity nor distinct-word completeness.

3.2 One interval within each nonempty band

Keep a unary occupied-run counter $C$ in each band. At a first occupancy of width $n$, observe occupancy at $n-1$ and $n+1$ using the inherited final width-path frame and bounded child view. Clamp the predecessor to zero when $n$ is a power of two, and clamp the successor to zero when $n$ is all ones in binary. These are exactly the lower and upper edges of $I_j$.

For the clamped flags $p,s\in\{0,1\}$, update

$$C\gets C+1-p-s.$$

An isolated position creates a run, an extension preserves the count, and joining two neighboring runs removes one. In the decrement case two distinct runs existed, so a unary pop is legal. A global unary counter tracks bands with at least two runs using the fixed one/two threshold. At the right boundary, its being zero ensures every nonempty band is one interval. Adjacent lengths on different sides of a band boundary do not merge these local run counts.

Together, the complete-level, multiple-run and excessive-partial-population counters are exact finite-root rejection gates. Their being zero is equivalent to the three stated row conditions. Empty right input and absent rows use finite empty flags.

3.3 Endpoint export without a last-symbol comparison

Absolute endpoint comparisons cannot begin only when a band is finally selected at the word delimiter. Instead store, with each old endpoint in band $I_j$, its unary offset ruler from $a=2^{j-1}$, as well as its inherited ordinary numeral.

When the word enters its current band at position $a$, reset a fresh offset ruler to empty and initialize two cursors from the band’s old minimum and maximum offset rulers. On every later target bit within the same band, add one unit to the fresh offset ruler and advance each old cursor by one while nonempty. At a new band, discard the speculative cursors and initialize the next band’s pair. There are only two live cursors.

At the delimiter, the fresh offset is exactly $n-a$. If the minimum cursor has emptied, the old minimum is at most $n$ and remains; otherwise retain $n$ and its prepared roots as the new minimum. If the maximum cursor has emptied, $n$ is at least the old maximum and its roots may replace it; otherwise keep the old maximum. Empty bands initialize both endpoints. Equality may retain either equivalent prepared payload. No large-number comparison occurs at the delimiter.

This also covers a word ending immediately on entry to a band. Its offset is zero. Any positive old minimum offset is then recognized as larger by one nonempty-cursor test; an old zero offset is already empty. A huge old maximum does not delay the answer or endpoint export.

All changes apply to the final target length only. The lower empty band records were already allocated during the word; the delimiter commits their finished-prefix root. Complete-level updates, row dictionary updates, global flags, both restored directories and endpoint payloads require a fixed number of local records at the delimiter. The inherited simultaneous-record compiler pays that fixed batch.

4. Reuse one frozen row cursor

Remove the whole-row interval and maximum-diversity gates of theorem 64. Retain its per-source ordinary-name deduplication, persistent band worker, per-band insertion ownership and sealed output stream.

A distinct nonempty row selection releases one cursor at the front of its frozen band deque. At destination band $j$, read that row’s next band record. EMPTY contributes nothing; FULL sets the destination coverage to FULL; INTERVAL starts the existing resumable range insertion with that band’s prepared endpoints. When the action finishes, advance the row cursor and promote it to band $j+1$. An exhausted cursor finishes its job. Endpoint width is already tied to the band’s position; no cross-band endpoint comparison or search is needed.

An insertion owns its band’s committed root until completion. New jobs join its waiting stack, including jobs promoted from lower bands. Releasing a new row restores the current finished prefix by strict catenation and resumes at band 1. The current records and suspended insertions survive. No later completion overwrites a newer directory root.

For every cutoff $k$, the partial-insertion sum from Section 1, bounded per-record operations and at most $k$ renewed prefix passes give

$$C_k=c_r k,\qquad c_r=256(r+1),$$

as a conservative increase in remaining work. As in theorem 64, additional FULL coverage can only shorten pre-existing waiting insertions along their own endpoint paths. Active insertions keep their unchanged base tree; old jobs retain their relative order. Thus the bound covers an interrupted worker, not merely a fresh empty destination.

Apply the charged band criterion with $c=c_r,d=16$: service $64c_r$ microsteps per actual binary selection-name bit, $1024c_r$ per target-query bit, and 1024 reader steps per query bit. Distinct nonempty names provide the same busy-period bound. A query fork admits no new releases. Its sealed bands are enumerated in order; exhaustion has a local EMPTY-tail certificate. Every emitted Boolean is exact union membership for every target word of that length, including negative gaps.

The proof of theorem 64 therefore supplies every query-prefix deadline and the immediate final pinned-SAT answer. It does not use the number of components as a hidden machine parameter or distribute one name’s service among uncharged fictitious releases.

5. Complete native resource statement

The actual input has length $N$. Initialization has empty directory roots, finite flags and the inherited counter state. Each input symbol performs a fixed number of counter/deque/dictionary/cursor operations, or the fixed selection/query batches above. Directory focusing occurs during the target word at exposed power events. All capacities and endpoint offsets are made from actual input-built unit roots. All routing, old-storage observations, allocation, incomplete work, immutable snapshots, suspensions and output are charged.

For fixed $r$, all records have bounded degree and finite labels/control. Observations have fixed radius, including the $r+1$ population prefix. Updates refer to old roots or the bounded simultaneous-record batch. The inherited compiler gives one fixed native automaton, $O_r(N)$ logical work and storage, $N+1$ packed scaffold nodes, and $O_r(N\log(N+2))$ explicit pointer bits. No supplied interval decomposition, band annotation, class name, padding, advice, address/identity/metadata channel or EOF drain is used.

Imported ingredients are the strict persistent sequence primitives and the existing native counters/dictionaries/compiler. The written extension is the ordinary-input two-directory producer, within-band offset export, checked scale sparsity and its coupled reuse bound. It is a proof extension of those methods, with no claim of external priority. Finite models below are semantic and scheduling evidence; they do not measure native Python running time.

6. Failure boundary, evidence and continuation

An eager component list scanned in full before a short query is not justified by a small endpoint description. A fixed cap on total components would unnecessarily exclude the families above. Counting partial bands alone without weighting their indices would miss the actual insertion work. These failures concern those accounting contracts, not all native representations.

The executable audit separates a per-bit producer from direct set/interval oracles, checks both directory populations after arbitrary insertions and retained forks, and exercises the generalized consumer against direct union membership. Numerals created by the model at a delimiter stand for inherited already-prepared roots; their per-bit construction is not re-audited. Complete-level cases check count/set semantics, not the inherited capacity-counter implementation. It passed 13,020 producer updates, 1,855 retained producer/query forks, 64 complete-level cases, 540 releases and envelope checks, 100,800 query prefixes and 2,036 exact deadline inequalities. The separately designed independent audit, which imports no primary implementation, checks 49,152 occupancy transitions, 196,608 global threshold transitions, 43,392 offset decisions and 127,500 weighted cutoffs. Full adversarial review found no remaining structural blocker after clarifying the zero-sparsity bound, initial block focusing and during-word allocation. Assessment 199 records scope and reassessment; integration state records publication receipts.

Next decisive test: allow several separated occupied runs inside a single partial band, with a checked cumulative mixed-region budget replacing its single-interval payload. The difficult step is exporting and accessing that budgeted shape during ordinary input, without a supplied decomposition or a burst of unpaid work after selection. Unrestricted radial profiles and independent-store composition remain open. Replay, affine-fragment joins, persistent causal algebra and unary coherent controls remain unchanged.