Deliver child output through the physical continuation boundary

research/continuation-output-return/PROOF.md

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+# Designated child-output return
+
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+
+## Contract and retained stack
+
+Use the seven-symbol tape alphabet and physical push/pop of
+[PR #5356](https://github.com/CrocSwap/integer-mult-bounds/pull/5356), at
+[the exact source](https://github.com/aliromman/integer-mult-bounds/blob/ef1cc26e1e9a08c11cf382de948ee433bebab16d/research/delimited-continuation-stack/stack.py).
+That result restores k=16 payload roles, their presence bits, heads and an
+opaque binary descriptor. It requires every child segment empty at return.
+The unchanged stack and source adapter are retained in vendor/. Their source
+bytes and notices are checked. This PR adds output delivery, not a new stack.
+
+Reserve one occupied empty parent output role before push. Its head is at its
+segment origin. All other parent segments and heads are arbitrary under the
+retained ABI. The child leaves one designated binary output of length L on
+one fixed role, with its head anywhere in that segment. The child may leave
+other binary work on its roles. Slots are compile-time control labels; the
+child and parent labels may be equal. Descriptors are arbitrary finite binary
+words. All calls succeed and are properly nested.
+
+Add one relay tape with a BASE marker and an empty current segment at its
+origin. It is distinct from all payload, control and descriptor tapes. Its
+arbitrary ancestor prefix is preserved. There are nineteen tapes in total,
+independent of nesting depth. The source API's logical counters and lexical
+allocation remain a separate calibration facade; the return routine never
+reads them, a saved origin, a saved length or a parent object.
+
+## Schedule and invariant
+
+First check that the relay head is immediately after BASE and reads blank.
+Rewind the child output to its nearest LEFT marker by unit moves. Copy its
+binary symbols, in order, to the relay until blank. Rewind the relay to BASE.
+This preserves the output in relay cells before any child cell is erased.
+
+For each of the fixed payload roles, rewind to LEFT, erase each current
+binary symbol while moving right, then rewind to LEFT. No ancestor cell is
+read or changed. Invoke the unchanged physical pop. It erases its frame and
+returns each parent payload, head, presence bit and descriptor exactly.
+The relay is outside this bank, so pop cannot erase it.
+
+Check that the reserved target is blank immediately after LEFT. Copy relay
+symbols to that role in order. Erase each relay symbol after copying it.
+Rewind both tapes to their markers. The target now contains exactly the
+child output, with head zero. The relay is empty at its original head.
+All other parent roles and the descriptor have the exact pre-push state.
+
+These assertions prove a matched boundary by loop induction. The copy loop
+preserves the copied prefix and unread suffix. The erase loops affect only
+current segments. The retained pop theorem restores the parent. The final
+copy loop preserves the delivered prefix and the unread relay suffix. Its
+postcondition restores the entry condition of the shared relay. Induction
+on a finite call tree therefore reuses the same relay for arbitrary finite
+proper nesting. At an outer return, a previously delivered inner result is
+an ordinary child output. No recursion frame stores return data in host state.
+
+## Complete bill
+
+Count each current-symbol read, write and unit move as one action. Let Sp be
+the sum of parent extents and Hp the sum of parent head offsets. Let A be the
+number of absent parent roles. Let d and q be the parent and child descriptor
+lengths. Let W and Hc be the sums of child extents and head offsets immediately
+before return. Thus L<=W and Hc<=W. All these symbols are analysis variables;
+the routine does not receive their values.
+
+The retained push costs
+
+```text
+4Hp + 2Sp + 10k + 7d + A + 11.
+```
+
+The retained pop, after payload retirement, costs
+
+```text
+9Hp + 2Sp + 21k + 11d + 5q + A + 28.
+```
+
+Rewinding from offset h to its marker costs 2h+3. Erasing a current segment
+of size s from offset h costs 2h+5s+7. If the original output head is h0,
+checking the relay and copying the output costs 2h0+6L+11. It leaves that
+source at offset L. Retiring all roles then costs
+2(Hc-h0+L)+5W+7k. Checking the target and installing the output while erasing
+the relay costs 9L+11. Thus the complete matched push/return boundary costs
+exactly
+
+```text
+13Hp + 4Sp + 38k + 18d + 5q + 2A + 2Hc + 5W + 17L + 61.
+```
+
+Child computation and its input/descriptor production are separate costs.
+They are not hidden in this formula. Output copying, child cleanup, frame
+cleanup, descriptor return and relay cleanup are included. Since Hp<=Sp,
+Hc<=W, L<=W and A<=k, the boundary is at most
+
+```text
+17Sp + 40k + 18d + 5q + 24W + 61.
+```
+
+There is no ancestor-length or nesting-depth factor. If Sp,W,d,q=O(V), this
+is O(V) and fits the retained per-node allowance. Those actual volume and
+descriptor bounds are explicit premises, not established production facts.
+
+The relay visits L cells in one boundary and is empty at its end. Across
+successful calls its retained backing needs at most the maximum output
+length seen so far, not a sum over nesting depths. The parent output owns L
+cells in its reserved role. During the first copy the child and relay both
+hold the result; neither this duplicate nor the parent output is free stock.
+All inherited continuation frames and parked payload backing remain charged.
+
+## Evidence and scope
+
+The verifier checks 1800 exact small return bills, including zero-length
+outputs and every head position through length three. Twelve nested cases
+use depths 1,2,7,31 and ancestor lengths 16,257,4096. Six actual compact-source
+cases run both retained layouts and deliver all 1024 output bits to the
+parent. An independent direct record permutation checks their values. All
+six returns cost 24388 actions. Six malformed-state controls reject. Errors
+outside the ABI need not restore a partially executed boundary.
+
+The source calibration still uses the predecessor's weak-reference allocator
+and host logical counters. This result does not compile those facilities.
+It does not implement recursive C, packet reorder, a general descriptor
+producer, the full production rectangle, native admission, stock matching,
+precision or outer recovery. It closes the physical output-return boundary
+under a reserved-slot ABI. It does not change kappa or prove the complete
+multiplication theorem. Independent human and proof-kernel review remain open.

research/continuation-output-return/README.md

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+# Physical output return through a continuation
+
+This package closes the designated child-output return question left by
+[PR #5329](https://github.com/CrocSwap/integer-mult-bounds/pull/5329) and
+[PR #5356](https://github.com/CrocSwap/integer-mult-bounds/pull/5356).
+One added relay tape preserves the result through child cleanup and physical
+pop, delivers it into a reserved empty parent role, and returns empty.
+All other parent bytes, heads, presence bits and descriptors are restored.
+The fixed nineteen-tape bank works for arbitrary finite proper nesting.
+
+[PROOF.md](PROOF.md) gives the contract, nesting proof, exact linear bill and
+space charge. [return_output.py](return_output.py) uses only current-symbol
+operations and unit moves. [RESULT.json](RESULT.json) records 1800 exact
+invoice cases, nested returns through depth 31, six unchanged compact-source
+endpoints and six rejecting ABI controls. Both source layouts deliver 1024
+bits in 24388 return actions for dirty ancestor lengths 16,257,4096.
+
+```sh
+python3.12 -B scripts/research_package.py --package research/continuation-output-return
+```
+
+Both registered isolated, warning-strict normal and optimized modes print
+`PASS_CONTINUATION_OUTPUT_RETURN`. Source bytes and the complete receipt are
+checked. No third-party package is needed. The all-size claim rests on the
+written induction, not the finite tests.
+
+The complete kappa calculation is unchanged. A reserved empty parent output
+role is required. Actual packet order, recursive C, physical source allocation,
+logical-counter compilation, full supplier/stock/precision and outer recovery
+remain open. Lean, Docker, selected cold replay and full-history validation
+were not run. The source calibration retains its explicit Python facade.
+
+## Open next questions
+
+Bind this returned stream to the actual FOUR/base packet order and its left
+map. First show where the caller reserves the output role and how its child
+width determines the stream grammar. Separately compile the source's logical
+counters and lexical slots at the repair/stable_radix maximum-live cut.
+Neither obligation is discharged by successful byte delivery alone.
+
+## Credit and coordination
+
+Base 7c472563474e60185e6ca101a17622133d047e56, priority7. The live stack handoff
+was already complete, so this package takes only its remaining output-return
+boundary. Current PR and Discussion bodies and main notes were checked;
+arbitrary historical diffs and unpublished work remain outside coverage.
+
+The byte-exact stack and adapter in vendor/ are consumed from Ali Romman's
+pending PR #5356, head ef1cc26e1e9a08c11cf382de948ee433bebab16d. They are pinned
+in [SOURCES.json](SOURCES.json). Replay pins resolve to the byte-exact copies
+in this PR history. Each upstream_pin retains the original pending source.
+Thus a clean checkout does not need an unmerged predecessor commit. They are
+not claimed as main-branch or independently accepted results. The complete
+linear theorem is conditional on that retained stack contract. JoyboyBrian supplied the segment refinement
+and locality invoice. Jose Manuel/jmbarrancoml supplied the retained sequential
+compact source. Douglas Colkitt supplied guard/repair, Cinna Davis the caller,
+and Matarener the cover/stock lineage. Original notices remain unchanged.
+
+New work: Ali Romman with substantial OpenAI Codex assistance in proof,
+implementation, review and verification. Apache-2.0. No independent human or
+proof-kernel acceptance is claimed.

research/continuation-output-return/RESULT.json

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+{
+  "controls": 6,
+  "matched_boundary": "13Hp+4Sp+38k+18d+5q+2A+2Hc+5W+17L+61",
+  "nested": [
+    {
+      "depth": 1,
+      "steps": 2859
+    },
+    {
+      "depth": 2,
+      "steps": 5911
+    },
+    {
+      "depth": 7,
+      "steps": 22133
+    },
+    {
+      "depth": 31,
+      "steps": 118414
+    }
+  ],
+  "return_sha256": "fbf3167da26c7887b769689075e5dc3afcea2ef668496975bc9e6afec8c85751",
+  "small_cases": 1800,
+  "source": [
+    {
+      "late": false,
+      "output_bits": 1024,
+      "prefix": 16,
+      "records": 128,
+      "return_steps": 24388,
+      "sha256": "4af9929ff7e3d8e72146e260c7e007e975b7c17ef26c0271355333fe64072a48"
+    },
+    {
+      "late": false,
+      "output_bits": 1024,
+      "prefix": 257,
+      "records": 128,
+      "return_steps": 24388,
+      "sha256": "4af9929ff7e3d8e72146e260c7e007e975b7c17ef26c0271355333fe64072a48"
+    },
+    {
+      "late": false,
+      "output_bits": 1024,
+      "prefix": 4096,
+      "records": 128,
+      "return_steps": 24388,
+      "sha256": "4af9929ff7e3d8e72146e260c7e007e975b7c17ef26c0271355333fe64072a48"
+    },
+    {
+      "late": true,
+      "output_bits": 1024,
+      "prefix": 16,
+      "records": 128,
+      "return_steps": 24388,
+      "sha256": "9b6381e2a8cffec1c0d499b50909c409c9dc178584321a9a6e73fafbdbfaf547"
+    },
+    {
+      "late": true,
+      "output_bits": 1024,
+      "prefix": 257,
+      "records": 128,
+      "return_steps": 24388,
+      "sha256": "9b6381e2a8cffec1c0d499b50909c409c9dc178584321a9a6e73fafbdbfaf547"
+    },
+    {
+      "late": true,
+      "output_bits": 1024,
+      "prefix": 4096,
+      "records": 128,
+      "return_steps": 24388,
+      "sha256": "9b6381e2a8cffec1c0d499b50909c409c9dc178584321a9a6e73fafbdbfaf547"
+    }
+  ],
+  "status": "PASS_CONTINUATION_OUTPUT_RETURN"
+}

research/continuation-output-return/SOURCES.json

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