Construct actual mixed-width CRT row entry with complete lifecycle

research/crt-mixed-row-entry/PROOF.md

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+# Entry at a mixed-width CRT row
+
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+
+## Domain and result
+
+Use the logical Caller in the pinned mixed-width source. Its fixed topology
+has d>=2 axes, d a power of two, ell>=2, and declared widths h_i in
+{ell-1,ell}. The supplied moduli satisfy 2<=s_i<2^h_i and are pairwise
+coprime. One compiled node has interval [a,a+n), n=2^level, a divisible
+by n, and midpoint b=a+n/2. The compiled global row is in [0,2ell-2].
+The source is one sequential tape. Each axis has the record
+
+    ROOT, ONE^h_i, WORD_END, h_i modulus bits, WORD_END,
+    h_i current coordinate bits, RECORD_END.
+
+Bits are little endian. LEFT is at cell zero; RIGHT ends the entire stream;
+the head is one. The tape is the encoded representation of the actual
+caller's moduli and current row state, not a host integer argument to the
+bridge. Representation of that upstream state is a premise. The bridge
+checks all field lengths, bits and delimiters. Modulus validity and equality
+to the chosen logical Caller are domain premises; the numerical consumer
+also checks its own modulus and coprimality guards.
+
+Let S=sum_{i=a}^{a+n-1} h_i and P=S+1. Eight distinct borrowed numerical
+ports have the retained P-bit ABI, arbitrary contents and legal heads.
+They are disjoint from the source. Their production is a supplied boundary.
+All new ports are fresh and blank. The program uses only local port reads,
+writes and unit moves for runtime data. Topology loops and axis labels are
+compiled finite control; no input modulus, coordinate, product or numerical
+width is decoded into host control.
+
+For an active node, RowEntry produces the canonical paired left/right
+streams and the unary P guide, invokes the unchanged Entry and produce,
+and returns their actual owned result. Its offsets are exactly the actual
+Caller.offsets values on this node. close consumes the result, retires all
+63 owned ports, and preserves every borrowed byte and entry head. A node
+with no active target does no tape work and makes no numerical invocation.
+This is a source-operation entry theorem. It does not implement a Beneš
+key machine, the whole row permutation, native admission or E(p,ell).
+
+## Sequential representation proof
+
+The program scans all d input records once in order. Compile-time position
+decides whether to copy a record to the left stream, the right stream, or
+neither. Every selected width symbol appends one ONE to the guide. One final
+ONE gives P=S+1 without multiplication or binary-to-unary expansion of a
+host integer. Since every selected modulus is at least two, its width is at
+least two, S>=4, and P>=5. Also product(s_i)<2^S=2^(P-1). Thus the exact
+consumer's width and complete-product margin hold. The source's padded
+coordinate fits h_i<P bits.
+
+For each copied h_i-bit field, canonical_end scans its trailing high zeros
+backwards and erases them. If it meets the preceding ROOT or WORD_END, it
+writes one zero. Otherwise it stops at the highest ONE. It then writes
+WORD_END. The resulting number is unchanged and its encoding is minimum
+length, including zero. Every removed cell is blank. A later field can reuse
+these cells by ordinary writes. The two fields and RECORD_END have exactly
+the retained canonical pair ABI. All supplied source bits are read only;
+a local rewind restores head one. The guide and paired streams then satisfy
+every representation premise of the retained Entry proof.
+
+The retained Entry initializes its 60 owned ports, pads the canonical
+streams, executes the unchanged numerical producer, and consumes and retires
+its state. The additional three ports are retired after that call. Each
+canonical body is contiguous through RIGHT and has only blank cells after
+it; clear therefore erases all of its allocated nonblank cells. Its LEFT
+marker is also erased. No borrowed port is retired. Failure rollback is
+outside the successful-domain theorem.
+
+## Binding to the actual caller
+
+The comment in Caller.offsets uses the word original for the unchanged
+left-child slice of a level. Its executable decode_group inverts the
+already completed child transform. It does not interpret that slice as
+original mixed-radix digits. The retained current-coordinate producer has
+that exact decode_group contract. At any micro-row of one level, the left
+child is unchanged. It therefore reconstructs the same k_A as the frozen
+level controls, or zero if a left coordinate is invalid.
+
+For each right axis j, the producer advances the prefix through every
+preceding right modulus. Its output is
+
+    offset_j = k_A * (product_{i=a}^{j-1} s_i)^(-1) mod s_j.
+
+This equals Caller.offsets. The modulus product has an inverse at each
+step by pairwise coprimality. Feeding those offsets to the logical row
+query thus selects the identical retained Beneš table and target bit.
+The bridge itself returns a sequential numerical result; physical transport
+from that stream into a key/ROM query is not supplied by this theorem.
+
+Do not apply the returned new-coordinate field as a micro-row update.
+It is the endpoint of an entire cyclic rotation. In a partial Beneš path,
+a valid endpoint can pass through a padded target coordinate. The actual
+row still queries that coordinate; gating it on x_j<s_j changes the
+permutation. The independent source case (7,3), widths(3,2), input(1,1)
+returns (1,2), while that gate returns (1,3).
+
+For widths(3,3,4,4), ell4, the short axes are idle at global rows3 and4.
+They are active at row0. The active set in the bridge is exactly the
+source local_row criterion; an all-idle node returns before any port
+allocation or source movement.
+
+The main discriminator is moduli(5,7,11,13), state(1,5,0,9), level2.
+The current left child decodes to11, with product35. The right offsets
+are0 and3; the whole-rotation values are0 and12. Ignoring the preceding
+right modulus11 would give7 on modulus13. This is a semantic distinction,
+not a choice of formatting or a new CRT identity.
+
+## Complete invoice
+
+Put L=2+sum_i(3h_i+4), the supplied source extent including end markers.
+Selection, copying, zero trimming, guide construction, rewinds and final
+retirement of the three entry ports take at most
+
+    128(L+P+n+1)
+
+local and finite-control operations. Each input symbol has a fixed number
+of tests and moves. Each selected bit is copied at most once and scanned
+backwards at most once during trimming. Rewinds and retirement each visit
+only the constructed stream lengths. The fixed overhead is absorbed by
+the displayed constant. The code uses no whole-array read, slice, length,
+absolute head assignment or host numerical field conversion for this work.
+
+The complete existing Entry and numerical tariff must be included once:
+
+    T <= 2^20(P+2)^4 + 2^16(P+2)^2 + 128(L+P+n+1).
+
+This is a constructive upper bound, not a measured speedup. The source
+numerical theorem and arithmetic-operation model remain inherited.
+
+The guide retains at most P+1 cells. The two canonical streams together
+retain at most 2(P-1)+4n+4 cells: each copied field can temporarily use all
+h_i bits, even if high zeros are later erased. Thus the three additional
+ports retain at most 3P+4n+4 cells. Include all existing Entry storage:
+
+    K_owned <= 57(P+2) + 3(P-2)(2P+4) + 6 + 3P+4n+4.
+
+Borrowed storage is separate and charged: L source cells and 8(P+2)
+numerical cells, plus any preexisting blank backing. No erased backing is
+subtracted. These are per-invocation fresh-owner bounds; the complete
+trace uses separately retired owners. An arbitrary persistent bank's
+history, global backing, codecs and literal binary lowering need their own
+contracts. The complete kappa calculation is unchanged.

research/crt-mixed-row-entry/README.md

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+# Mixed-width CRT row entry
+
+The actual mixed-width caller can now supply the retained encoded numerical
+entry from one sequential tape of padded axis records. Runtime data use
+local reads, writes and unit moves. The bridge constructs canonical paired
+streams and a unary precision guide, invokes the unchanged numerical
+consumer, and retires all owned ports. Fixed tree topology is compiled data.
+
+[PROOF.md](PROOF.md) gives the all-admitted-width argument, the exact caller
+coordinate convention, and complete source-operation and storage bounds.
+[RESULT.json](RESULT.json) records 27 source-bound cases, including a complete
+four-axis forward caller trace, two literal idle cases, padded states, mixed
+widths, source restoration, 63-port cleanup and six failure controls.
+
+The (5,7,11,13) source state (1,5,0,9) has lower11 and offsets(0,3).
+Whole-rotation outputs must not replace Beneš micro-row updates. The existing
+logical caller needs its table query even at an intermediate padded target.
+
+Run with Python 3.12, standard library only:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/crt-mixed-row-entry
+```
+
+Expected result: `PASS_ACTUAL_CRT_ROW_ENTRY` in both registered modes.
+Frozen sources and the complete deterministic receipt are checked. The
+proof is quantified; finite source runs corroborate it. No independent
+human or proof-kernel acceptance is claimed.
+
+This closes one numerical entry boundary. The encoded upstream state and
+eight dirty numerical ports are supplied. The numerical source-operation
+contract is inherited. Physical key/ROM transport, production geometry,
+whole E, persistent-bank storage, binary arithmetic, native admission and
+outer recovery are separate. No complete kappa improvement follows.
+
+## Open next questions
+
+- Connect the sequential offset stream to an actual row key query. Start
+  with this four-axis source trace and charge query, return and restoration.
+  A logical table lookup does not supply physical transport.
+- Prove one reused bank's backing bound across varying P. Start from the
+  63 role capacities and keep their individual historical maxima. This
+  package proves a fresh-owner per-call bound only.
+
+## Credit and provenance
+
+Base: canonical main 7c472563474e60185e6ca101a17622133d047e56, priority7.
+[SOURCES.json](SOURCES.json) pins every input at that base. The local task
+refines uniform-transfer-audit and takes the released CRT row-entry handoff.
+Live overlap covered304 open PRs,4606 closed PRs and4621 Discussion records;
+relevant canonical-main notes supplement sparse working-tree coverage.
+Arbitrary historical diffs and unpublished work were not searched.
+
+JoyboyBrian supplied PR5053's current-coordinate arithmetic and PR5219's
+encoded Entry. Hiroki KUMAZAKI supplied the mixed-width Caller. Cinna Davis
+supplied the retained CRT construction; Tekalign supplied the separate
+original-coordinate stream work. ToikY, Jose Manuel, Douglas Colkitt,
+John-Michael Smith and the retained word/record/archive sources keep their
+credit. The original source notice is an authenticated input. No predecessor
+file is changed. Ali Romman with substantial OpenAI Codex assistance in
+proof, implementation, review and verification. Apache-2.0.