Entry 129: persistent composition and negative readout

Current boundary: late contraction receipts need current routing

Assessment 244 separates service from current destination access. A bounded global pending overlay works under a checked successive-service condition; a descending family of arbitrarily strong contractions defeats its unrestricted explicit-insertion version. An all-shorter-class sweep has enough total service, but a late selected bucket still needs a charged current-version route. This is a completed failed extension; construction 85 remains the latest native theorem. Review and evidence.

Primary comparison and editorial review 245 distinguishes version-sensitive predecessor lookup from constant repair overhead after a node is located. Neither imported mechanism supplies the missing native current selector. The construction 85 opening example and recent notation are corrected without changing its theorem.

Assessment 246 rules out eager owner-by-owner current shadows under an exact immutable-tree contract. The same alias family has a small suffix quotient, so multiplicity alone is not an obstruction. The next concrete test is an input-built quotient with paid current source-specific routes.

Next: Construct a paid current-version selector for late contraction receipts, or a shared aggregate that bypasses bucket insertion. Ordinary preparation and activation must make the selected shorter answer accessible by its deadline, without supplied routes, stale handles or final traversal. Test both service-admissible sequences and descending contractions. Preserve arbitrary whole-suffix permutations, arbitrary-name routing, cyclic sharing and heterogeneous service as open.

Scheduled 151 completed once after assessment 246. Mathematical 148, native 85, window 2 and last reassessment 146 unchanged. Assessment 245 completes the primary review and editorial duties; the mathematical target is preserved. Installed-writer retirement remains separate and unverified.

Preserved construction 85: local length classes preserve current reuse

Construction 85 removes global owner width. A class-local pending overlay and width-tagged shared index admit arbitrary interleavings of lengths without distant scans. Queries route to their class during the namespace spelling and pay pending parsing before late key production. Local owner/namespace width matching remains required. Review and evidence.

Next at assessment 243: Test cross-scale reuse where a much shorter ordinary target selects a much longer activated owner, with unbounded image and no supplied route. Seek a paid shared current predicate avoiding long query-time traversal. Preserve arbitrary whole-suffix permutations, arbitrary-name routing, cyclic sharing and heterogeneous service as open.

Scheduled 148, mathematical 148 and native 85 completed once. Window 2; last reassessment 146. Next 149 research; 150 both primary and editorial. Installed-writer retirement remains separate and unverified.

Preserved construction 84: unbounded filler fibres and exact pending readout

Construction 84 removes the fixed occupied-fibre bound. A shared injective key routes the learned filler early; a bounded pending overlay keeps the newest activation readable while the next actual same-source record pays installation. Two roots and one pending receipt suffice, with no per-namespace update or final flush. Common width, local seeds and synchronous short maps remain explicit restrictions. Review and independent finite audit.

Next at assessment 242: Remove the common owner-width hypothesis using charged local length classes, while preserving paid namespace routing, current pending selection and late transformed-key readout. Test section 15 class-window service against the factored query deadline. Preserve arbitrary whole-suffix permutations, arbitrary-name routing, cyclic sharing and heterogeneous service as open.

Scheduled 147, mathematical 147 and native 84 completed once. Mathematical window one; last reassessment 146. Next 148 research; primary and editorial both due 150. Installed-writer retirement remains separate and unverified.

Preserved construction 83: learned fillers and occupied owner fibres

Construction 83 admits independently learned binary fillers through a shared short-key activation mask. Ordinary preparation checks fixed occupied owner-fibre capacity and assigns stable slots; actual activations update one bit, and supported queries select the correct current bit after a paid short lookup. This is a sufficient subclass, with seed, finite-capacity and synchronous-map restrictions. Review and six-result reassessment.

Next at assessment 241: Allow unbounded occupied owner fibres through a checked shared inner representation. Charge selector-route construction, literal activation updates and timely current readout without per-namespace updates or a final traversal initiated by a late selector. Preserve arbitrary whole-suffix permutations, arbitrary-name routing, cyclic sharing and heterogeneous service as open.

Scheduled 146, mathematical 146 and native 83 completed once. Reassessment 146 resets the mathematical window to zero. Next scheduled 147 is research; primary and editorial duties both due at 150. Last primary 240, editorial 239. Installed-writer retirement remains separate and unverified.

Preserved checkpoint after review 240

Primary review 240 separates recognition, report dispatch and incremental matcher construction. Terminal match bursts and literal table changes block those specific transfer routes, not shared aggregates or unrestricted native methods. The next learned-filler mechanism must expose a bounded shared activation record and paid selected-current-answer access. Construction 82 was the latest native result at that review.

Editorial assessment 239 reviewed the recent service, routing and normalization readings; scope and checks. Construction 82 and the mathematical next target below are unchanged.

Latest primary review 240; latest research construction assessment 238; latest editorial assessment 239; then-latest native construction 82. Updated 13 October 2026. Research active; installed-writer retirement remains separate and unverified.

Editorial review 233 improves notation and successor navigation. At that checkpoint, scheduled 138 was complete; mathematical 143 and native 80 were unchanged. Assessment 234 below continues its preserved target.

Frontier at assessment 238: a shared basis for learned views

Construction 82 replaces the common exceptional cut length by independently learned namespace lengths. Three source-shared roots suffice: original owners, leading-normalized owners and trailing-normalized owners. A saved persistent path endpoint after the last 1 updates the trailing chart without a zero-run flush. Fixed-width injectivity and exact support keep answers faithful; the normalized producer meets final-symbol deadlines. The local first-foreign-seed restriction is explicit. The written factorization criterion requires both paid updates and paid normalized readout; semantic factorization alone is insufficient. Review and evidence.

Next: Replace common zero filler by independently learned filler words. Seek a bounded shared normalization basis, charge pattern discovery and literal activation updates without scanning the growing pattern family, and prove timely current readout. Preserve arbitrary whole-suffix permutations, arbitrary-name routing, cyclic sharing and heterogeneous service as open. Preserve this mathematical target through the next editorial pass.

Scheduled 145 completed with primary review 240. Mathematical 145, window five, native 82 and last reassessment 140 are unchanged. Next 146 is research; both primary and editorial duties are due at 150. Both-side trimming loses the deleted-position coordinate even at fixed width. Section 8 preserves that coordinate in the trailing chart and traverses it before the late key, completing the zero-padded interior-position case. Independently learned nonzero filler words remain open.

Construction 81 remains the earlier common-shape control. Construction 82 includes its per-namespace-seeded cases and the sparse envelope-width witness, not every encoding it accepted. The prior service-game and cross-index failures remain valid for their specified representations.

Preserved construction 80: growing delay with a checked release envelope

Construction 80 characterizes feasible non-speculative ordered suffix output at fixed service rate by a common-prefix envelope. Actual ordinary records build and validate its bounded chunks, without computing longest common prefixes or accepting a supplied schedule. The resulting native subclass shares one current activation dictionary across growing namespaces even with growing output delay. First-half suffix reversal separates it from every fixed-delay and fixed-prefix-ambiguity contract, with the exact scoped witnesses in the proof. Its bounded-width extension builds finite branch slots from ordinary records, forks immutable workers, and pays all candidates before the final input selects a unique owner.

Target at assessment 232, sharpened above: Replace fixed simultaneous width by an input-derived speculative frontier with paid local service. Charge discovery, traversal and elimination at their release times, including stored but temporarily unserved candidates; a total-work bound alone does not suffice. Seek a useful variable-width subclass without frontier scans or supplied decompositions. Arbitrary whole-suffix permutations, arbitrary-name routing, cyclic state sharing and heterogeneous service remain open. Preserve this target through the next editorial pass.

Review and independent evidence. Scheduled 137, mathematical 143, window three and native 80 completed once. The subsequent editorial assessment 233 leaves these mathematical counts unchanged; last reassessment 140. No root admission.

Preserved fixed-delay construction

Construction 79 and its review retain the fixed-delay producer and bounded final burst. Construction 80 includes these contracts by choosing service parameter C=d+1. Its necessity statement is only for the specified ordered non-speculative reader, not arbitrary native representations.

Previous primary comparison

Assessment 230. The primary-proof comparison separates fixed-consumer summaries from paid growing-dictionary readout. An exact serial release-tail test detects work released too late; neither imported theorem supplies a native current-activation reader for free.

Next: Construct and validate an input-built bounded-delay suffix transducer after the learned first-half replacement. Charge transition discovery, consistency checks, replay and current activation traversal against actual symbols. Then test unbounded delay with the release-tail criterion. Arbitrary whole-suffix permutations, arbitrary-name routing and heterogeneous service remain open.

Scheduled 135 completed as primary review; mathematical 141 and native 78 unchanged. Next 136 research, primary review 140, editorial 138.

Preserved construction 78 and its starting target

Construction 78 permits growing independently prepared namespaces to share one current source activation dictionary. Each target has a routed primary owner and a foreign owner obtained by replacing its first suffix half through an arbitrary learned map. The unchanged second half pays the delayed activation lookup and replay, with a ready final-symbol answer. Actual ordinary records build and check the maps; no namespace-wide activation updates are needed.

The criterion is a paid release window: an input-built lookup can finish late if meaningful remaining input pays its complete dependent work. It admits partial and noninjective maps with actual support guards. This is a sufficient subclass construction, not arbitrary whole-suffix permutation access or a general SAT/scaffold characterization.

Construction 77 retains routed activation plus bounded foreign packets; construction 76 retains separate store service and bounded or disjoint readout. Assessment 228 retains the arbitrary independent-reindexing frontier. Prefix-tree automorphisms are an earlier section 16 control, not a new advance. The primary comparison and restricted factored-reader failure keep their qualifications.

Next: Remove the unchanged suffix condition: test input-built suffix transformations whose released lookups and consumed-prefix replay are paid by the actual remaining query symbols. Preserve arbitrary whole-suffix permutations, arbitrary-name routing and heterogeneous service as open. Next primary review should compare delayed-output transduction with persistent multi-index access under strict release deadlines.

Scheduled 134 and mathematical 141 completed once; window one, latest reassessment 140, native construction 78, no root admission. Review and evidence. Last editorial assessment 227 remains. Next 135 is a primary-proof review; editorial 138.

Reading the current investigation

The goal is to combine completed computations while keeping later answers accessible with bounded local work. The recent notes separate preparation, deferred work and query readout:

The pending-receipt construction removes the residue selector for every fixed final number of balls. The preceding histogram construction bypasses stale bucket references by routing through the current histogram during each changing word. The earlier notes retain the late-repair obstruction, shared-gate escape and conditional completion budget for more general profiles; that general worker remains open.

Claim and evidence map

These are local statement labels, not new CLAIMS.yaml admissions. No novelty claim is made. Finite checks corroborate examples, not general proofs.

Complete readout argument

Sections 8–11 of the maintained written argument contain N1 (exact observation certificates), N2 (chronological raw-branching obstruction), N3 (sharp preprocessing/probe tradeoff), and N4 (failed native transfers and the precise remaining target). This checkpoint owns the Horn proof below; the reader note owns the readout proof, avoiding a second editable copy.

Foreground mathematical packet, reconciled without recounting

The following proof sections preserve the 7 October foreground Horn argument and its exact contracts. They supplement, rather than replace, run121’s 3CNF encoding and independent-store theorem.

2. Full relation composition in Horn 2CNF

Let $H=[h]$, $h\geq2$. For a relation $R\subseteq H\times H$, use distinct input variables $x_1,\ldots,x_h$ and output variables $y_1,\ldots,y_h$ and put

$$F_R(x,y)=\bigwedge_{(i,j)\in R}(\neg x_i\lor y_j).$$

An empty conjunction is true. Products are in path order: $(i,j)\in RS$ exactly when some $k$ satisfies $(i,k)\in R$ and $(k,j)\in S$.

Written adaptation A: exact composition. For disjoint vectors $x,z,y$,

$$\exists z\,[F_R(x,z)\land F_S(z,y)]\quad\Longleftrightarrow\quad F_{RS}(x,y).$$

For the forward implication, a true $x_i$ forces $z_k$ along every edge of $R$, and that $z_k$ forces $y_j$ along every edge of $S$. For the converse, choose

$$z_k=\bigvee_{i:(i,k)\in R}x_i.$$

Every clause of $F_R$ then holds. If this choice makes $z_k$ true and $(k,j)\in S$, some true $x_i$ has $(i,j)\in RS$; $F_{RS}$ forces $y_j$. Thus every clause of $F_S$ holds too.

Consequently, existentially hiding the internal layers of any nonempty chain $F_{R_1}\land\cdots\land F_{R_t}$ gives $F_{R_1\cdots R_t}$ on its endpoint vectors. The carrier $\{F_R:R\in\mathcal R_H\}$ is closed under this operation, and $F_{I_H}$ is its internal identity. This identity assertion is relative to this carrier; the full category of Boolean solution relations has equality as its identity. A zero-layer chain identifies its input and output variables and has implication-path relation $I_H$ as well.

Written adaptation B: every relation entry is distinguishable. Append the unit clauses $x_i$ and $\neg y_j$. Then

$$F_R\land x_i\land\neg y_j\text{ is satisfiable} \quad\Longleftrightarrow\quad (i,j)\notin R.$$

An edge $(i,j)$ directly contradicts those pins. When it is absent, set only $x_i$ true among the inputs and only $y_j$ false among the outputs. This satisfies every implication. The composition proof extends that witness through all hidden layers. Hence different relation products remain distinct under ordinary future SAT tests.

For a single Boolean answer convenient for the algebraic transfer, define $C(w)$ for $w=R_1\cdots R_t$ by its implication layers, positive units on every initial variable and negative units on every terminal variable. Then

$$C(w)\text{ is satisfiable}\quad\Longleftrightarrow\quad R_1\cdots R_t=\varnothing.$$

For the empty word the two endpoint vectors are identified, so $C(\varepsilon)$ is unsatisfiable, agreeing with the nonempty identity relation. For singleton identities $I_{\{i\}}$ and $I_{\{j\}}$,

$$C(I_{\{i\}}wI_{\{j\}})\text{ is satisfiable} \quad\Longleftrightarrow\quad (i,j)\notin R_1\cdots R_t.$$

Encoding and cost. This is ordinary signed 2CNF over a fixed alphabet: clause delimiters, signs, binary digits and variable-name delimiters suffice. Assign the layer-$\ell$ variable with index $i$ the ordinary integer name $\ell h+i$. All names, including the endpoint units, are present in the encoded formula and charged. For $t\geq1$, there are at most $(t+1)h$ variables and $\sum_\ell|R_\ell|+2h$ clauses, giving

$$N=O\!\left((\textstyle\sum_\ell|R_\ell|+h) \log((t+1)h+2)\right) \subseteq O(th^2\log((t+1)h+2)).$$

The exponentially large set of possible relations does not require an exponentially long encoding of one relation. Each relation has at most $h^2$ implication clauses. The construction supplies a controlled stress family; it supplies no free decomposition or handles to a native recognizer. Abstract layer composition below is not being claimed as a finite-state transduction of the globally named formula encoding.

3. Exact consequence for matching interfaces

Conditional consequence C. Suppose a compositional summary of relation layers is a monoid homomorphism

$$\tau:\mathcal R_H^*\longrightarrow D_k,$$

where $D_k$ is entry 129’s Brauer matching monoid, and the SAT answer for $C(w)$ depends only on $\tau(w)$. The final answer function may be arbitrary. Then

$$k\geq2^{\lfloor(h-2)/31\rfloor}.$$

Proof: let $P_w$ be the relation product of $w$. If $\tau(u)=\tau(v)$ but $P_u\ne P_v$, choose an entry $(i,j)$ on which they differ. Multiplicativity makes the summaries of $I_{\{i\}}uI_{\{j\}}$ and $I_{\{i\}}vI_{\{j\}}$ equal, but section 2 makes their SAT answers different. This is impossible. Therefore $\phi(\tau(w))=P_w$ is well-defined on the identity-containing image submonoid. It preserves products and identity and is onto $\mathcal R_H$, since every relation is a layer. Apply the liveness paper’s Corollary 3.2.

This consequence imports the algebraic theorem directly. It does not require that each layer be one physical input symbol or that the answer itself be a matching-connectivity test. To turn it into a persistent-resource lower bound, one must derive this summary contract and bound $k$ in terms of the actual resources of the representation.

4. Exact consequence for positive path readout

The companion paper permits branching and repeated crossing. Its finite path-diagram monoid $T_m$ consists of four relations: forward, backward, left return and right return, each on $m$ boundary labels. Products join boundaries and retain finite traversals. The componentwise edge order $\preceq$ is preserved by multiplication.

Conditional consequence D. Suppose $\tau:\mathcal R_H^*\longrightarrow T_m$ is a monoid homomorphism and a predicate $A$ on $T_m$ gives exactly the SAT answer for $C(w)$. If $A$ is increasing under edge inclusion, then

$$2m\geq2^{\lfloor(h-2)/127\rfloor}.$$

An existential test for an accepting boundary path is one such predicate. It remains increasing even when paths revisit the interfaces.

The equality argument from section 3 again defines a surjective multiplicative map $\phi(\tau(w))=P_w$. If $\tau(u)\preceq\tau(v)$, the same inequality holds after any singleton-identity sandwich. An accepted sandwich for $u$ must remain accepted for $v$. Thus every entry absent from $P_u$ is absent from $P_v$:

$$\tau(u)\preceq\tau(v)\quad\Longrightarrow\quad P_v\subseteq P_u.$$

The map reverses inclusion. The companion paper’s Theorem 2.1 gives the bound. Its hypothesis does not additionally require identity preservation.

This establishes a useful distinction. A branching representation can retain the implication relation compactly, but an exact positive path test for the absence of every source-to-target implication has the exponential interface obstruction above. Native finite control can inspect missing pointers or labels and use negative tests; that stronger readout is outside this positive predicate contract.

5. Persistent branching escapes the matching representation

Written construction E. All relation layers have a compositional chronological backward-DAG realization with $h$ input and $h$ output ports, outdegree at most 2, a fixed finite label set, and at most $h+|R|$ new nodes per layer. A chain retains $O(h+th+\sum_\ell|R_\ell|)$ nodes.

Start with $h$ terminal input nodes. For each output index $j$, list the input ports $i$ with $(i,j)\in R$. Build a binary tree whose leaves point to those already existing ports; every internal node points to at most two older nodes. Create a fresh output root. If the list is empty it is a dead terminal; if it has one member it points to that input; otherwise the root points to the tree. A list of size $d_j>0$ uses at most $d_j$ new nodes, and an empty list uses one. Summing gives at most $h+|R|$.

An output root $j$ reaches input port $i$ exactly when $(i,j)\in R$. To append the next layer, use the preceding output roots as its input ports, without changing any old node. New-to-old reachability is exactly relational composition. Labels can be terminal, dead or branch; input indices are port designations, not an unbounded node-label alphabet. Pointer sharing and unbounded indegree are allowed, as in a scaffold.

This graph has the relevant immutable chronological shape. It has not yet provided a native per-symbol construction schedule or a bounded local answer procedure from ordinary names. The port vector is the explicit boundary of the representation, not an uncharged vector granted to the automaton.

The construction explains why a matching-only extraction cannot cover all such persistent graphs with polynomially many ports. A matching represents one paired continuation at each port. A branch node retains several possible continuations, with shared older endpoints. A graph can preserve those choices without immediately resolving them. Its compact storage does not give the SAT answer required in section 4: the answer tests nonreachability.

Primary sources

1. OpenAI, An exponential two-way deterministic state lower bound for one-way liveness, 25 September 2026. Sections2-3; Corollary 3.2, page 12; Theorem 4.1 and Lemma 5.2.
2. OpenAI, An exponential state lower bound for two-way nondeterministic complementation, 25 September 2026. Section 2.1; Theorem 2.1, page 4; Lemma 2.2 and Proposition 2.3; Sections3-4 for recurrent-class transport and amplification.

The source theorems concern growing relation alphabets and state/interface parameters. The Horn encoding in section 2 removes an exponential formula-size objection to using the algebra, while leaving the native resource translation as the explicit frontier. No uniform logarithmic-space separation follows.

Review, validation and durable handoff

Assessment 123 derives the pointer-valued dictionary extension, nested-key commit schedule, duplicate-checked frozen-row composition, late query mechanism and reuse consequence in one argument. Claim mapping is reader section 12’s written native theorem; no new CLAIMS.yaml identifier or formal proof admission. Mathematical 88->89/window 2->3, last reassessment 86 retained. Scheduled 28->29 exactly once; next 30 dedicated primary review. Earlier foreground reconciliation and publication retries do not increment either counter.

Proof audit by the writing agent: arbitrary pointer payloads are copied through fixed fields, not followed recursively; FIFO schedulers never inspect their contents. An outer capsule can remain suspended without a restoration walk. Both inner and outer tips remain explicit. Duplicate middle names are checked using Seen, not equality of row roots. Empty rows, missing query names, prefix-related names, long-then-short names, empty blocks, a rejected degree overflow, malformed/stopped input and the final delimiter are covered. The selected d is a machine constant, not an input integer or nonuniform advice. No fixed-width promise, free root, uncharged replay or EOF work is used. No independent reviewer or compiler replay is claimed.

Finite evidence: check_frozen_row_composition.py freshly passed 1600 exhaustive Horn truth-table queries, 2097 retained versions, 169857 retained-version queries with duplicates, prefix names, missing names and names exceeding 250 bits, and a stale-snapshot counterexample, plus 3600 fixed-depth product queries through up to four bounded factors. Capsule tries and a separate Boolean-clause/set oracle agree. Tuple queues are a semantic model of the separately proved strict FIFO; Python timings are not native resource evidence. The written schedule establishes the symbolic cost.

Contribution comparison: Demaine–Langerman–Price, Confluently Persistent Tries, primary author PDF §1 and §2/Theorem 1 proof were inspected, alongside the Lab’s existing primary comparison and unary, FIFO and exact batch-compiler proofs. The container ideas and bounded union are established techniques. The additional written result is the checked ordinary-Horn input and native pointer-capture schedule. This focused single-paper comparison does not count as the dedicated multi-paper review due next. Primary PDFs and extracted text remain outside public source.

Source, claims, evidence, failure and reader dispositions: one maintained reader proof in section 12, this checkpoint as continuation, one semantic checker, no duplicated proof or new paper. Later right updates invalidate the frozen-snapshot implementation; unbounded selected rows invalidate its bounded cursor schedule. These are failure gates for this implementation, not universal nonrecognition. All earlier support-omission, equality, affine, replay, causal and unary-cubical results/frontiers remain retained. Homepage and program summarize the actual native subclass without changing paper order or coordinated editions. Existing Site identity and public audience are preserved.

Validation: finite semantic checker and git diff –check passed. Declared full make verify was attempted with the existing declared-dependency environment: 1490 passed, 1 skipped, 5 failed; four unchanged custody-provenance failures and unavailable lake, then later targets were not reached. No full-suite or compiler replay success is claimed. Site build and final reading regeneration passed: 349 styled HTML readings, raw Markdown excluded. Six structural checks and three HTML routes passed, with 309 rendered formulas and 41 in-scope anchors, including the new theorem/corollary. Source publication is prepared; final version/deployment/parity receipt is in state.json research_run.publication. Base GitLab 960c52ce641f7ee0312e2b686c04e2356cfdcfd9, freshly opened Site 520/source 55a853eab382c7bc7fa228a35003210824c46167. The prior full handoff remains available at that GitLab commit. Lease foundations-entry129-20261007-123 acquired conditionally at 67daca53919db4ff882a3e687c6f9fe40c3db24a. No old reverse backup is used or claimed retired.

Publication receipt, assessment 123

Published as existing public Site version 521. Reviewed GitLab content: 1e2bbd181a799e52f5ae073aadca0dd697e03c2f. Sites source: 5c54c4551116dcf9103096f2999a97e567a25669. Deployment appgdep_6ac690b4e4848191bcb57b9cde0473a9 succeeded. The reopened current source matches all 1,837 paths and 2,131 tree entries of the pinned GitLab site/ projection, with no exclusions. Tree: 5f6fff8a0984e054bb4e261aabf8da2bc2dd745b. Full version, archive and parity receipt is in docs/integration/state.json under research_run.publication. Verified 2026-10-07 18:35 UTC; lease released. Next scheduled assessment 30 is the dedicated primary-proof review described above.

Assessment 124 validation and handoff

The written invariant and finite checker are separate: check_finite_image_composition.py uses persistent capsule operations against direct set products, passing 3312 exhaustive queries, 1920 retained versions and 69120 retained multifactor queries, with duplicate, missing, prefix-related, long names, checked overflow and empty-factor coverage. Python tuple queues model the established strict FIFO and supply no timing evidence. No independent reviewer or formal replay is claimed.

Primary PDFs and extracted text remain private scratch. Source attribution, claim/non-admission mapping, proof/evidence, implementation failures/reopen gates, surviving target, reader synthesis and Site disposition are recorded in review 124. Public section 13 is the single maintained proof; no new paper or duplicated theorem body. Final validation and publication receipts are recorded under state.json research_run. The existing public Site and prior history remain intact. No old reverse backup or installed retirement is claimed.

Publication and validation receipt, assessment 124

Existing public Site version 522 deployed successfully as appgdep_6ac6983e82bc8191b9a1324f59df81b9. GitLab content 8ae07359f9d936d68d2d4be494decfc360af3bdd; Sites source 2da26598d0b48a9ae5ef9226b76687aecdc9d4e3. Reopened actual current source has exact path/blob/mode/type parity over 1, 838 paths and 2, 132 tree entries, no exclusions; tree 2f05a42030369c3a19ae198e0d4c980facff7cb6. Full archive/version/parity receipt is state.json research_run.publication. Verified 2026-10-07 19:07 UTC; lease released.

Finite checker and diff checks passed. Fresh build/final reading regeneration passed: 349 styled readings, raw Markdown excluded. Scoped checks: 6 structural tests, 3 HTML routes, 362 rendered formulas, 47 anchor references. Full make verify: 1490 passed, 1 skipped, 5 failed, retaining the four known custody-provenance failures and unavailable lake; subsequent targets not reached. No full-suite or compiler success is claimed. Next 31 is research, next dedicated 35. No mathematical counter advance.

Assessment 125 validation and handoff

The two-sided native routing theorem is written in reader section 14; its actual-draft audit and seven-surface dispositions are in run-125-review.md. Forward finite evidence: 189 right-mask checks, 2304 exhaustive queries, 500 retained versions/42120 retained queries, 126 growing-image queries. Backward evidence: 540 checked right pairs, 600 retained versions/48600 retained queries. All passed against separate whole-string/set-product oracles. The semantic model’s tuple queues provide no timing evidence. No independent reviewer, compiler replay or formal admission is claimed.

Fresh dependency installation and final Site build passed: 349 styled HTML readings, raw Markdown excluded. All 6 structural checks and 3 changed HTML routes passed, with 471 rendered formulas and 55 in-scope anchor references. Diff checks passed. Full make verify was attempted: 1490 passed, 1 skipped, 5 failed in 230.13s, retaining the four custody-provenance failures and unavailable lake from the preceding receipt; subsequent make targets were not reached. No full-suite success is claimed.

Base GitLab 461ac13ab76c3f56f29b5c88c5db0a6597ae7093 and freshly opened existing Site 522/source 2da26598d0b48a9ae5ef9226b76687aecdc9d4e3 had identical Site subtree 2f05a42030369c3a19ae198e0d4c980facff7cb6. Conditional lease acquired at 90a6cd14876904e8bfc7f830c31e946a454e122c. GitLab-first publication stages and final receipt belong to state.json research_run.publication. Installed reverse-writer retirement remains unverified and nonblocking; no old backup is used. Next 32 research: reverse-name index under actual deadlines; dedicated review 35. Scheduled 31/mathematical 90/window 4 counted once for this durable packet, not again for publication.

Publication receipt, assessment 125

Published on the existing public Site as version 523; deployment appgdep_6ac6a993b9288191969d94519575e448 succeeded. Reviewed GitLab content 061227bd0dcb14bb27a65e89480953a591c7941e; Sites source d1c3c00d5a03751ecaabeb8bb79bdcbf27fcd0d1. Reopened actual current source matches all 1, 839 paths and 2, 133 tree entries of the pinned GitLab site/ projection, with no exclusions; tree 2beb5eea388ce5bb632bfa9a2b9736a450bb3dd0. Final archive SHA-256 b52550b475420a25f4b9fce119a80f228494fa60afe00f8e10a3a6933e76e6a7, independently verified after atomic placement. Earlier truncated archives were discarded before any save; there was one version save and one deployment. Full archive/version/parity receipt is state.json research_run.publication.

Verified 2026-10-07 20:21 UTC; lease released. Next 32 research remains the reverse-name persistent-index test, broader arbitrary functional-right substitution open; dedicated review 35. Counters scheduled 31/mathematical 90/window 4 are final and must not be incremented by a retry. All earlier results and frontiers are retained.

Assessment 126 validation and handoff

The maintained written theorem is reader section 15; the actual-draft writing-agent audit and seven-surface dispositions are run-126-review.md. The checker check_pending_substitution.py passed 2304 exhaustive product queries, 540 right-row degree/span checks, 504 retained versions/30240 queries and 62 arbitrary-permutation queries. It exercised 322 serviced batches, 2 discarded partial jobs, 1047 pending-only answers, 69427 reader promotions and pending lists up to 3 entries. The Python directory, clock and tuple queues are semantic models, not native instructions or timing evidence. No independent review or formal replay is claimed.

Final Site build passed with 349 styled HTML readings and raw Markdown excluded. All 6 structural checks and 3 changed HTML routes passed, with 576 rendered formulas and 64 in-scope anchors. Diff checks passed. Declared full make verify was attempted: 1490 passed, 1 skipped, 5 failed in 231.43s, with the same four custody-provenance failures and unavailable lake as the preceding receipt; subsequent make targets were not reached. No full-suite success is claimed.

Base GitLab 86b295f097dde002345c30ca2a97247e2f081d11 and freshly opened existing public Site 523/source d1c3c00d5a03751ecaabeb8bb79bdcbf27fcd0d1. Conditional lease6e9e0d25e9d94a00463b2eb8e7388818883051c8. Final deployment/source-parity stages belong to state.json research_run.publication. Use the bundled packager with a closed archive atomically placed into the final workspace path and independently integrity-checked, because the preceding run observed archives truncated after direct workspace writes. Do not resave an already saved version on retry.

Next 33 research: cross-scale functional substitution with unbounded selected image; dedicated review 35. Scheduled 32/mathematical 91/window 5 counted once. The next substantive mathematical result reaches 92 and requires the preserved six-result reassessment from86. Publication retries and administrative work do not advance these counters. Installed reverse-writer retirement remains unverified/nonblocking; no old reverse backup is used.

Publication receipt, assessment 126

Published on the existing public Site as version 524; deployment appgdep_6ac6b6b222a48191a02b54e2ddb03049 succeeded. Reviewed GitLab content d23400d836fca6d913e1a1f0e99c4360d2ab177a; Sites source fa5f701a882195671e694e6a24f94692d2cd5283. Reopened actual current source matches all 1, 840 paths and 2, 134 tree entries of the pinned GitLab site/ projection, with no exclusions; tree 47087e50bcc36f12a8a96d11e0333d284a923991. Bundled workflow archive was atomically placed and passed independent integrity checks in separate calls, SHA-256 b073fc2af245b31a9489fc63e7ed5ad2d6be22ef47c082d9a41757929ffaee57. One source push, one version save and one deployment; no replacement Site or audience change. Full archive/version/parity receipt is state.json research_run.publication.

Verified 2026-10-07 21:17 UTC; lease released. Next 33 research is cross-scale functional substitution with unbounded selected image; dedicated review 35. Scheduled 32/mathematical 91/window 5 are final. Next substantive result 92 requires the preserved six-result reassessment. Do not recount this packet on publication or source retries.

Assessment 127 validation and handoff

The cross-scale note and checker are canonical original research artifacts; no new native endpoint or root admission. The finite checker passed 276 offline table cases, 2048 retained versions and 10240 membership comparisons, plus the explicitly restricted arithmetic/alias/code-instability checks. Immutable tries and a separate set-image oracle agree. Python arithmetic and timing are not native evidence. Writing-agent audit is review 127; no independent review or formal replay is claimed.

Fresh make verify: 1490 passed, 1 skipped, 5 failed in 231.50s. The same four custody-provenance failures and missing lake remain; subsequent make targets were not reached. Diff, Python syntax, seven local evidence links and cadence checks passed. No build/deployment was performed because only research continuation documents and a root checker changed, with no public page, homepage entry or reader theorem edit. Existing public Site version 524 was freshly observed; its previously verified source remains the historical publication baseline. New canonical continuation files are not claimed deployed or source-parity verified.

Base GitLab 616465d93cb7187681acdd209c3d8764e489adb0; conditional lease committed at ed468fbe127ec339f6666d7d27b2c8f1c38f3108 (actual commit time 2026-10-07T22: 05: 01Z). Final state releases the lease and records this no-native-advance assessment once: scheduled 33, next 34 research/review 35; mathematical 91/window 5/last reassessment 86 unchanged. The next substantive 92 requires reassessment. Next target is the target-directed current-version reader for arbitrary injective expansion tables. No old reverse backup, retirement claim or automation change.

Fresh full make verify: 1490 passed, 1 skipped, 5 failed in 224.88s. The retained failures are four custody-provenance checks and unavailable lake; later make targets were not reached. Site build passed with 349 styled readings and raw Markdown excluded. Six structural checks and three changed HTML routes passed, covering 680 rendered formulas and 74 anchors. Python syntax, cadence consistency and diff checks passed. Reopened hosting source matched the previous GitLab site subtree exactly. Publication requires the separate root receipt.

Assessment 129 final publication and handoff

Content commit 9f4b056c7d1f9e042756747e837fca97648ac9f3 contains the two-paper proof review, mixed-condition argument, finite checker and coordinated continuation. The supporting combination earns no additional mathematical count. Dedicated completion 35 is complete; next 36 research, next dedicated 40, substantive 92 and reassessment 92/window 0 retained.

Existing public Site 526 deployed successfully as appgdep_6ac6e1489a288191b0b434f2130437dc from source 78441ff14470b521928ab09359c72213926b1ec5. At 2026-10-08T00: 18: 54Z, freshly reopened current hosting source and current GitLab site/ exactly matched the pinned source: 1843 paths, 2137 tree entries, path/blob/mode/type parity, no exclusions. Tree de599361dc6cf3fbac6dd12f87f4b0c5eac88b2f. Root state retains prepared, committed, source-pushed, version-saved, deployed and parity-verified receipts through Git history. Site audience remains public and the shared lease is released. The active research task continues with the far-expansion/pending-context test above.

Validation: 4096 contract/order cases, 6114 retained versions, 189234 query comparisons and 512 rectangle-equivalence cases passed; seven expanded-reversal width witnesses. Python syntax, counters and diff checks passed. Full make verify retained 1490 passes, 1 skip and 5 failures in 227.04s: four custody-provenance checks and unavailable lake; later make targets were not reached. The final Site build produced 349 styled HTML readings with raw Markdown excluded. Six structural checks and three changed routes passed, with 774 rendered formulas and 81 anchors. Publication success and source parity are recorded separately in root state.

Assessment 130 final publication and handoff

Content commit 993e0cc72829bf29d3ced1e190316ba3d923dc8f contains the finite-fibre normalization theorem, far-expanded reversal consequence, exact rectangle prerequisite for the next test, finite checker and coordinated continuation. Counted once as mathematical 93/window 1 and scheduled 36; next 37 research, dedicated 40, last reassessment 92. No new root admission or novelty claim.

Existing public Site 527 deployed successfully as appgdep_6ac6efdf953c8191a90a2b59be6473a3 from source 71d13f1748ce2258839bd7affcea9d0f164f2693. At 2026-10-08T01: 21: 29Z, freshly reopened current hosting source and current GitLab site/ exactly matched the pinned tree fa3497bcbca9a73839be6f93e9ba3d81fe94613c: 1844 paths, 2138 tree entries, identical path/blob/mode/type with no exclusions. Root state and its commit history preserve every publication stage, archive and deployment receipt. Audience remains public. Shared lease released; active research continues with charged complete-rectangle validation, not another catalog inventory.

Assessment 135 final publication and handoff

The written supporting theorem is maintained in reader section 22: fixed bounded final answer diversity admits unambiguous nondeterministic native validation and current-answer reuse under the stated normalization, degree and length contract. Canonical first-occurrence predictions, lane rectangles and revisable distinction counters force a unique successful classification. Deterministic discovery remains open. The common-prefix family realizes 2^(m-1) possible final partitions and excludes early immutable final labels, not shared/revisable selectors or native recognition generally. No root admission, novelty claim or major-result increment.

Finite checker passed 4794 sequences, 13219 assignments, 69809 prefix checks, 50999 retained snapshots, 115894 product queries, 80 revisable-distinction steps and 255 extendable prefix partitions. Python syntax, counters and diff checks passed. Full make verify: 1490 passed, 1 skipped, 5 failed in 238.52s; four retained custody-provenance failures and unavailable lake, later targets not reached. Site build passed with 349 styled readings and raw Markdown excluded. All 6 structural checks and 3 changed HTML routes passed, covering 1067 formulas and 113 anchors. The finite model corroborates semantics; the written schedule carries native timing.

Reviewed GitLab content 757b7552d9dd981c9ed74e01790f068c37efbf88 was published as existing public Site 531, source c72899a545ae019f272e574daa9c775d96bf3316. Deployment appgdep_6ac736f24aa08191a496f59a3031fe1a succeeded. At 2026-10-08T06: 24: 50Z, freshly reopened current source, current GitLab site/ and the pinned projection matched exactly: 1849 paths, 2143 tree entries, path/blob/mode/type parity, zero exclusions or mismatches. Tree 66a4d17bebd4066e490791f8501c636de21be9c7. Root state and Git history retain every prepared/committed/source-pushed/version-saved/deployed/parity-verified stage, exact archive receipt and saved operation IDs. Hosting source was clean. The known 134 continuation-only metadata was intentionally included. Existing identity, public audience and editions are preserved; no old reverse backup or retirement claim.

Scheduled 40->41 counted once; next 42 research, next dedicated 45. Mathematical 95/window 3/last reassessment 92 unchanged. The next target is a charged shared or revisable deterministic selector for two types, including late resolution for earlier long targets and current-row queries. The unique validator does not find the labels. All prior frontiers remain; research continues. Shared lease released.

Assessment 136 final publication and handoff

Reader 23 now gives a written supporting deterministic native criterion: fixed finite-state predecessor typing, checked per-key/color rectangles, locally revised target masks and exact final-type population counters. Both row and target groups may be unbounded, right order is arbitrary, and the normalized degree/length conditions remain. Reader 23.8 carries the dependent input-derived literal-row palette through the same push. The palette handles two-row instances beyond every fixed finite menu of the colored-rectangle criteria. These do not solve arbitrary latent row-profile discovery or a free independent-store join. No new root admission, novelty claim or major-result increment.

Fresh finite evidence: 3549 sequences, 22412 prefix checks, 25952 retained prefixes, 7652 retained source versions, 130293 product queries, 720 cell permutations, 510 deterministic prefix continuations and 512 learned-palette cases. Type counts 1, 2, 2, 3, 2, 1 confirm temporary excess classes can collapse before the final gate. Finite checks corroborate semantics; the written schedule carries native costs. Python syntax, JSON/cadence and diff checks passed. Full make verify: 1490 passed, 1 skipped, 5 failed in 242.10s, with the four retained custody-provenance failures and unavailable lake; later targets not reached. Final build passed: 349 styled readings, raw Markdown excluded. Six structural checks and three changed routes passed, covering 1182 rendered formulas and 122 anchors.

GitLab content 14e23a894030dcdd1bfc6ade1b793f78aadc2f2d was published as existing public Site 532 from source acf825c7929117c6eac3549b1d6627f422f4b887. Deployment appgdep_6ac7451e5e4481918711f97b651e7b0a succeeded. At 2026-10-08T07: 25: 20Z, freshly reopened hosting source, current GitLab site/ and the pinned projection matched 1850 paths and 2144 tree entries exactly, including path/blob/mode/type, with zero exclusions or mismatches. Tree 0599c58abce4a1f8931e662f437424e661813986. Hosting source was clean. Root state/history retains every publication stage and saved operation ID. A credential-expiry refusal occurred before source mutation; renewal produced one successful source push, save and deployment. No Site replacement, audience/edition change or reverse backup was used.

Scheduled 41->42 counted once; next 43 research, next dedicated 45. Mathematical 95/window 3/last reassessment 92 unchanged. Next: learn types with unboundedly many literal predecessors while charging row-action revisions, earlier normalized receipts and target-mask populations. The bounded literal palette is already done; do not repeat it as the next advance. All prior frontiers remain and research continues. Shared lease released.

Assessment 137 publication receipt

The written streamed-row-key extension and finite audit are committed at 8b26ae3d19a503bb9caa2e01ea36ac1095fb360b. Existing public Site 533 deployed successfully from source 8c91a45f7f068bfdd4c9977ea4b2aa0dfe4de14d; deployment appgdep_6ac7519c421c81919c94beca5a8ed734. At 2026-10-08T08: 18: 31Z the actual reopened hosting source, pinned projection and current GitLab site tree matched exactly: 1851 files, 2145 path/blob/mode/type entries, zero exclusions or mismatches, tree 72cc0e5edadf096fce5042785cf92fb28e4dee13. Hosting source was clean. State retains the archive, version and staged receipts.

Build and scoped HTML/KaTeX checks passed. Full make verify had 1490 passes, 1 skip and the same five retained failures (four custody-provenance, unavailable lake); later targets were not reached. No compiler replay or all-gates success is claimed. Scheduled 43 is complete; next 44 research, dedicated 45. Mathematical 95/window 3 unchanged. Lease released. Next work is the relation-dependent split stated at the top, not another fixed-key control.

Assessment 138 publication receipt

The written anchor-refinement/deferred-dispatch construction and finite audit are committed at 263506aec91ab4990aee04fcf3dcf4c1d7e176a9. Existing public Site 534 deployed successfully from source a6a835b9da6589d863ec1d468292ec9dda154f82; deployment appgdep_6ac7608db8b0819182bd16b8d8a2e8b8. At 2026-10-08T09: 22: 27Z the actual reopened hosting source, pinned projection and current GitLab site tree matched exactly: 1852 files, 2146 path/blob/mode/type entries, zero exclusions or mismatches, tree 300020df8177a9c0bb8ce20f50faec0b3861dab3. Hosting source was clean. State retains archive, version and staged receipts.

Finite checks passed 1958 sequences and 48530 product queries. Build and scoped HTML/KaTeX checks passed. Full make verify had 1490 passes, 1 skip and five retained failures (four custody-provenance, unavailable lake); later targets were not reached. No compiler replay or all-gates success is claimed. Scheduled 44 complete; next 45 dedicated primary-proof review, at least two full papers. Mathematical 95/window 3 unchanged. Lease released. The next target is shared correction after late anchor incidences, as specified at the top.