Connect current CRT numerical offsets to the physical Beneš row query

research/crt-current-row-query/ATTEMPT.json

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+{
+  "status": "INCOMPLETE_RESOURCE_TIMEOUT",
+  "command": "/opt/local/bin/python3.12 -B research/crt-current-row-query/verify.py --four-axis --write-result",
+  "elapsed_limit_seconds": 180,
+  "exit_code": 1,
+  "completed_receipt": false,
+  "join_sha256": "bcae8074a67fe74ee9979a4608452de6e2decbe7023671624573f362f3caefe5",
+  "verify_sha256": "e0ba8c478eb5054e7f83be1840c42f99472e012e67cc2edf527c84589f80e9bf",
+  "case": "moduli5,7,11,13 widths3,3,4,4 current1,5,0,9 level2 row0 P15; two h4 complete producers",
+  "observation": "No completed producer/query result was emitted before the subprocess timeout killed the child. No progress count or passing h4 result is claimed.",
+  "subsequent_repairs": [
+    "Replace linear active tuple membership with the equivalent constant-time compiled width/node/row predicate.",
+    "Replace fixed2e9 watchdog with the proved T_rom bound computed from compiled P and h."
+  ],
+  "evidence_scope": "The final implementation is tested on the complete h2 actual pipeline. This timeout is not acceptance evidence for either version."
+}

research/crt-current-row-query/PROOF.md

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+# Current numerical offsets to a physical CRT micro-row
+
+JoyboyBrian, with substantial OpenAI Codex assistance. Apache-2.0.
+
+## The exact boundary
+
+Use the admitted `RowEntry` domain of Ali Romman's PR5353: a power-of-two
+axis count d, declared h_i in {ell-1,ell}, ell>=2, pairwise coprime moduli
+2<=s_i<2^h_i, arbitrary padded current coordinates 0<=x_i<2^h_i, one
+compiled tree node [a,a+n) and one global row. The source is its literal
+sequential padded-axis tape, not a host list of numeric values. Eight
+distinct protected numerical words are supplied with the retained legal
+heads. Fixed topology, input ownership and the source numerical-operation
+model remain predecessor premises.
+
+Let b=a+n/2, P=1+sum_{i=a}^{a+n-1}h_i, and A be the non-idle right axes
+of this global row. The result is an owned stream containing, in axis
+order, the h_i bits of the **micro-row output** for every right axis
+b<=i<a+n. Delimiters are L, then one coordinate followed by `|` per axis,
+then R. No active axis means no allocation or tape work and no output;
+the original source already represents the identity row. When some axes
+are idle, their coordinates are copied into the returned stream unchanged.
+
+All borrowed source/dirty words and entry heads return unchanged. The
+numerical outputs, new descriptors, produced ROM and complete unselected
+layer output, producer workspace and query countdown are retired. The
+owned coordinate result remains until `close`, which erases it and closes
+the original Entry. Successful calls leave every new physical head at
+zero; the inherited numerical ports have their retained head-one exit.
+Failure rollback and arbitrary dirty backing acquisition are not claimed.
+
+This theorem closes an actual encoded-numerical-output to physical-query
+boundary. It is not the full three-epoch exceptional key, a full CRT row
+permutation of all payload addresses, or a complete multiplication theorem.
+
+## Numerical input and encoded descriptor compatibility
+
+The unchanged PR5353 RowEntry and PR5053 producer reconstruct the actual
+current left child. On a valid left child they return k_A; an invalid left
+child disables it by k_A=0. For each right axis i, their sequential output
+contains
+
+    c_i = k_A * (product_{j=a}^{i-1}s_j)^(-1) mod s_i,
+
+followed by a whole-rotation updated value. The inverse exists by pairwise
+coprimality. All preceding right moduli are included. We consume the offset
+word and **discard the updated-value word by a charged scan**: a complete
+cyclic rotation is not one Beneš micro-row.
+
+For an active axis, the join reads its original modulus directly from the
+borrowed source, translates each numeric bit to the producer's finite bit
+symbol, and pads with zeros against the actual unary P guide. It similarly
+copies exactly P bits from the actual numerical offset stream, enforcing
+the guide and field delimiters. This creates a singleton offset list
+`L [P bits] | R`; no decoded integer or host-built offset list is used.
+Compiled h_i and the selected local-row constant are written in binary and
+zero-padded through that same guide. The finite topology is the only source
+of those constants. The ROM program itself is precompiled without any
+input values and passed to the join.
+
+The existing PR5092 producer requires W>=h_i+3. A node has at least two
+axes, and every modulus is at least two, hence every admitted h_j>=2.
+Consequently P>=h_i+2+1=h_i+3. Taking W=P meets its width contract; its
+modulus and singleton-offset guards follow from the numerical theorem.
+The local row is g for a long axis, and g or g-2 on the two active pieces
+of a short axis, with global rows ell-1 and ell idle. This is exactly the
+pinned `local_row` function, and lies in [0,2h_i-2].
+
+Thus the unchanged encoded producer emits the exact canonical Beneš
+network for the rotation x->x+c_i mod s_i on x<s_i, fixing padded wires.
+Its selected `rom` is L followed by exactly N_i=2^h_i bits and R. Its
+`layers` output contains all 2h_i-1 layers; the join pays to erase it even
+though the query needs only one selected layer. Both producer outputs are
+consumed on their original backing lists, through local tape operations.
+There is no verifier-supplied row or source table lookup in execution.
+
+## Physical query and the current-coordinate convention
+
+The query copies the actual current h_i-bit coordinate to an owned
+little-endian countdown and a separate output coordinate. The ROM scan
+visits every produced row bit in order. Until selection, a zero test scans
+the countdown, returning its head, and a nonzero countdown is decremented
+by ordinary borrow propagation and head return. At scan position j, while
+selection is enabled, the countdown represents x_i-j. Hence precisely
+position x_i sees zero and selects its row bit. Since 0<=x_i<N_i, selection
+always occurs before R. After selection, the scan continues to R without
+changing the chosen bit. The countdown is then erased.
+
+If the local row is r, its source target bit is min(r,2h_i-2-r). The join
+moves to that compiled bit position in the owned coordinate, XORs the
+selected bit and returns the head. This is exactly `Caller.row`'s table
+action. No check x_i<s_i suppresses a query. A Beneš implementation of a
+padded permutation can pass through padded coordinates at intermediate
+rows; the queried coordinate must be the current one.
+
+At a fixed node, the left child is unchanged throughout its micro-rows.
+Right offsets depend on the successive modulus prefixes, not on the
+current right coordinate values. The numerical producer's current-child
+convention therefore gives the same offsets as the source caller at each
+row. Each active right axis receives its proper current-coordinate query;
+idle axes retain their coordinate. These facts prove the whole returned
+right-node row stream equals the corresponding slice of `Caller.row` for
+every admitted source state, not merely the verifier examples.
+
+## Ownership and a reused producer bank
+
+The numerical producer and RowEntry stay unchanged and retain their full
+63-port lifecycle. New header, coordinate, countdown and output ports are
+distinct from all borrowed roles. The join allocates one fixed PR5092
+service bank, then invokes the same program for each active axis on the
+same bank. Its private tapes return blank with heads zero by the producer
+theorem. The join erases both generated output roles and all four headers
+before the next invocation. These steps establish the next call's entry
+premises without constructing a new ROM or replacing the bank backing.
+Restoring the emulator's initial head configuration represents those
+already returned physical heads, not a free head jump.
+
+The finite producer register configuration is fixed control. Reentering
+its fixed start state has constant cost, included in the conservative
+adapter allowance below; it is not an input-sized store reset. Initial
+blank owned backing remains an explicit storage premise. Erasing cells
+does not release or subtract their retained capacity.
+
+The source and numerical output heads rewind by local movement. Numerical
+results are retired only after every offset has been consumed, so no
+output lifetime is silently shortened. `close` disposes of the coordinate
+result and calls the original Entry close exactly once. All malformed
+or rejected executions remain outside the successful cleanup theorem.
+
+## Complete local time and retained-storage invoice
+
+Let L=2+sum_{i=0}^{d-1}(3h_i+4), N_i=2^h_i, and
+
+    F = L + (n+1)(P+2)
+        + sum_{i in A}[(N_i+1)(h_i+1)
+                      + N_i(2h_i-1) + 2h_i+4].
+
+The inherited Entry, including numerical computation and retirement, costs
+
+    T_entry = 2^20(P+2)^4 + 2^16(P+2)^2 + 128(L+P+n+1).
+
+The complete producer is charged once per active axis at m=1, W=P:
+
+    T_rom(i) = 2*10^6 P^2 (N_i+1)^3 (h_i+1)^4.
+
+The new serialization, current-coordinate query, updates, returned-stream
+emission, every header/ROM/layer erase and every added rewind cost at most
+4096F source-tape/control operations. One direct count is: each additional
+source symbol is scanned a fixed number of times; each header/offset/skip
+word uses O(P) charged reads/writes/moves; each row query visits at most
+N_i positions, with at most two h_i-bit scans plus head returns at a
+position; disposal scans both output lengths N_i and N_i(2h_i-1)+O(h_i).
+The displayed routines use fewer than 64 primitive tape operations per
+unit represented in F. Between these charged accesses there are fewer
+than 64 finite tests, assignments and compiled-loop transitions, including
+fixed-role dispatch and the per-invocation fixed control reset. Thus 4096F
+is a deliberately loose complete bound, not a timing extrapolation.
+Compiled topology/ROM-program creation and supplied-input serialization
+are external setup, explicitly not runtime numeric decoding.
+Activity uses the fixed compiled node/width/row predicate, not a linear
+membership scan of all active axes. The emulator watchdog is the displayed
+T_rom(i), computed from compiled P and h_i; no fixed finite watchdog limits
+the admitted all-size successful execution theorem. External resource
+timeouts remain test limits, not machine rejection.
+
+The full invoice is
+
+    T_join <= T_entry + sum_{i in A} T_rom(i) + 4096F.
+
+This adds compatible fixed-alphabet source-operation bounds. It is not a
+claim that all numerical control was lowered to one binary transition table.
+
+Let J_i=N_i(h_i^2+3h_i+1). Use the predecessor Entry retained-backing bound
+
+    B_entry = 57(P+2)+3(P-2)(2P+4)+6+3P+4n+4,
+
+and, including the four producer inputs of total length 4P+9,
+
+    B_rom(i) = 100(P+2)+5P J_i+N_i(4h_i-1)+10(h_i+1)+4P+9.
+
+The single bank's sum of per-role historical maxima during this finite
+row is at most sum_i B_rom(i). We deliberately use this conservative sum,
+not the unjustified maximum of aggregate bounds. The two coordinate/
+countdown ports and returned stream add at most
+
+    B_extra = 2(max_{b<=i<a+n}h_i+3)
+              + sum_{b<=i<a+n}h_i + n/2 + 3.
+
+Hence total new owned retained backing is bounded by
+`B_entry + sum_i B_rom(i) + B_extra`. Input source L and protected numerical
+backing 8(P+2), plus any supplied exterior blank capacity, remain borrowed
+and separately charged. The verifier sums distinct physical backing
+identities, so ROM/header aliases in observer wrappers are not double
+counted. No arbitrary cross-session persistent-bank theorem is implied.
+
+## Evidence and limits
+
+The first source fixture is (7,3), widths (3,2), current state (1,1).
+Five separately encoded row invocations follow the actual state sequence
+(1,1), (1,1), (1,3), (1,3), (1,3), (1,2). The last query must run on
+padded coordinate 3. Test-harness serialization between invocations is
+explicit; this is not an uninterrupted physical whole-level execution.
+Invalid left padding disables the offset and returns identity. Four
+faults change the supplied offset, selected local row, required padded
+query or complete-layer disposal, and are detected at semantic or ownership
+checks. The source four-axis discriminator has preceding-right-modulus
+offset3 rather than the incorrect7, but the new full h=4 producer/query
+replay exceeded its180-second resource limit and produced no completion
+receipt. It is explicitly uncompleted; the executed new full pipeline
+evidence is h=2. The quantified argument is source-relative and does not
+depend on claiming that uncompleted test passed.
+
+Finite checks corroborate the written source-composition and countdown
+arguments. Initial axis-record production, a uniform binary compiler,
+full three-epoch exceptional key, global ROM reuse, full E(p,ell), native
+and outer multiplication admission remain unresolved. There is no new
+kappa or selected-frontier claim, human acceptance or formal-kernel proof.
+
+## Credit
+
+Ali Romman supplies PR5353's actual row-entry contract; JoyboyBrian supplies
+the retained PR5053/5219 numerical modules and PR5092 control producer.
+Retain Hiroki KUMAZAKI's current-coordinate and mixed-width caller,
+Cinna Davis's CRT/Beneš construction, ToikY's encoded arena/DFS proof,
+Tekalign's original-coordinate stream work, Jose Manuel, Ryanson Jonathan,
+unpaidmoon, Douglas Colkitt/John-Michael Smith and all pinned notices.

research/crt-current-row-query/README.md

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+# Actual CRT offsets to a physical current-row query
+
+The sequential numerical result of Ali Romman's PR5353 now feeds the
+unchanged PR5092 encoded Beneš ROM producer. The join reads the produced
+ROM on its original backing, selects the current padded coordinate with
+an encoded countdown, and returns the actual micro-row coordinate stream.
+It pays for serialization, queries, rewinds and disposal of both the
+selected ROM and the unused complete-layer output.
+
+[PROOF.md](PROOF.md) gives the source-relative quantified interface theorem,
+admission W=P, current-coordinate convention, complete local time bound
+and retained-backing bound. [RESULT.json](RESULT.json) records the small
+actual producer/query cases and four falsifying controls. The h=2 checks
+exercise the complete encoded producer, not a verifier-supplied row table.
+
+```sh
+python3.12 -B scripts/research_package.py --package research/crt-current-row-query
+```
+
+The registered normal and optimized runs authenticate every source input
+and compare typed deterministic receipts. The separate, expensive h=4
+four-axis test is explicitly opt-in:
+
+```sh
+python3.12 -B research/crt-current-row-query/verify.py --four-axis
+```
+
+This optional whole-ROM test is distinct from the small executed h=2 ABI
+checks. The author attempt exceeded180seconds without completion; no
+ACTUAL4 receipt exists and no h=4 joined result is claimed. See
+[ATTEMPT.json](ATTEMPT.json) for exact attempted bytes and the subsequent
+all-size repairs. A future completed optional run checks the caller and
+invoices even without a frozen receipt; `--write-result` records one.
+
+The small mixed-width case reaches padded coordinate3 before its last
+query. Suppressing that query gives the wrong endpoint. Mutating the
+offset, selected row or unused-layer disposal also fails. All source and
+protected words are restored; all63 numerical ports and every joined
+owned backing are retired. Separate row invocations use test-harness
+input serialization; no uninterrupted physical level execution is claimed.
+
+## Open next questions
+
+- Physically update the canonical axis input with the returned row stream,
+  so successive rows need no host serialization.
+- Compose the current query into the complete three-epoch exceptional key,
+  paying for all original/current coordinate conversions and restorations.
+- Resolve cross-session bank ownership and a complete binary numerical
+  lowering before claiming a global machine or outer multiplication bound.
+
+No new kappa, selected frontier, human acceptance or formal-kernel proof is
+claimed. Source models and initial blank owned backing remain explicit
+premises. This is one actual caller boundary, not full total repair.
+
+## Provenance and credit
+
+Base32890bc5de034ab8fc93ac88ead7279a6a416d04, priority7. The scope takes
+PR5353's explicit physical-query handoff, Discussion4709 comment19152738;
+the new claim is comment19173182. Fresh overlap covered352 open PRs and
+4747 fully paginated Discussion records through2026-10-25T04:14:44Z,
+plus exact local source packages. Unpublished work was not searched.
+[SOURCES.json](SOURCES.json) freezes the complete input closure.
+
+JoyboyBrian, with substantial OpenAI Codex assistance. Apache-2.0.
+Retain Ali Romman's actual RowEntry, Hiroki KUMAZAKI's mixed-width caller,
+JoyboyBrian's numerical entry and ROM producer, Cinna Davis's CRT/Beneš
+construction, ToikY's arena/DFS proof and all predecessor notices and
+credits listed in PROOF.md. No predecessor file is modified.