[kumagi]

Change and mathematical scope

PR5188 explicitly left retained tape backing open: fifteen fixed private roles do not bound the arrays retained across varying calls. This package derives their exact per-role extents from the unchanged 4,589-instruction interchange and proves the history identity

K(k) = sum_r max(entry_capacity_r, c_r(call1), ..., c_r(callk)).

The order of maxima matters. Three actual successful calls with H=1 and V=256, placing factor 64 in A, G and S respectively, each retain 811 private cells in a fresh owner. One reused owner retains 811 → 823 → 835, although every call erases private contents and returns heads. This falsifies max-of-standalone-totals accounting, not the predecessor program.

For maximum historical volume Vmax, a fresh bank needs at most 4Vmax + 8 bit_length(Vmax) + 23 retained private cells, independently of call count. Arbitrary blank entry backing K0 has the sufficient grant K0 + 4Vmax + 8 bit_length(Vmax) + 8. A separate conservative visited-span allowance adds 15 to the fresh grant. The exact actual H2/A1/G1024/S12 private capacity is 786497 cells; this is derived from the theorem, not a new execution of that 196608-bit caller.

Scope is successful canonical calls in the existing five-symbol model. Borrowed/old caller arrays, descriptor supply, binary representation, global recursive storage, failure rollback, Python heap capacity and full raw/repair composition remain outside the local bank theorem. Endpoint correctness/time bounds remain inherited. No selected frontier or κ changes.

Task and overlap

Card uniform-transfer-audit, refined to PR5188 open question2. Family decision: narrow and stop, one source-capacity proof and finite accounting audit.

• Canonical round base: ff3cc58e370d0f678e529246f8c73e8094e18fcf, priority 7. Six live predecessor files are pinned by Git blob and SHA256 in SOURCES.json.
• Exact predecessor PR5188: bffbe6f4b31c102bbf53d12ebbefde6c1a86337a. Its unchanged program, interpreter, proof and receipt are consumed directly; its heavy full verifier is not run.
• Canonical claim/result; hub pointer.
• The numerical-arena and packet retained-backing audits already establish historical accounting for different owners/allocators. This contribution does not claim that general principle as new; it binds the literal interchange's actual roles and constructs its mixed-history discriminator.
• Pay the two-target CRT stencil producer and join its repair mask #4990's upstream descriptor-production claim, fixed-block binary lowering and whole raw/repair composition are distinct and not taken here. Expansion accuracy was considered but not claimed.

TASK.json records the initial overlap and refreshed upstream checkpoint. The final refresh completed at 2026-10-23 15:46:39 UTC: all 7,091 base-tracked note paths plus two new package notes, 342 open/draft PR bodies (15:35:35 UTC), and 4,540 Discussion records (15:46:39 UTC), with complete inventory/pagination and no warnings. No competing successor was found. A targeted 15:48:33 UTC recheck confirmed the unchanged merged #5188 head and our claim as the only returned recent #4709 activity. #5274 concerns a distinct BIT continuation-stack model and expressly excludes erased backing. Searches cover PR bodies, Discussion text and tracked package notes, not arbitrary diffs or unpublished work.

Submitted head: b1577bd405f5c7ade3616486b5fe44478e0ba5cb; the final canonical-main checkpoint remains ff3cc58e370d0f678e529246f8c73e8094e18fcf. Eight new package files; no predecessor, shared guidance or selected record edits.

Validation

Python 3.12.13, standard library only:

python3.12 -B scripts/research_package.py --package research/interchange-retained-backing --check-only
python3.12 -B scripts/research_package.py --package research/interchange-retained-backing
python3.12 -B scripts/check_research_commit.py --package research/interchange-retained-backing --check-source-history

• Ordinary and optimized registered replay pass. All six consumed sources authenticate before captured-byte compilation; frozen RESULT.json comparison is read-only and type-sensitive.
• 110 fresh geometries, all six orders of the fixed-volume witness, shrinking-volume calls and preexisting blank backing. Actual endpoints, protected inputs, cleanup, heads and every per-role allocation are checked; guards intercept each source allocation.
• Independent sparse interpreter: all 16 fresh/warm entry comparisons agree with the original machine on complete allocated contents, lengths, heads, status and exact instruction/read/write/move counts. It imports no geometry/cap formula or source implementation.
• Registered primitive probe separates nonallocating reads/moves from allocating blank writes. Actual-source accounting mutants reject allocating reads and shrinking erased arrays. Other controls reject max-of-totals, erased-content-only, current-volume-only and discarded-entry-backing grants. Removing OUTPUT's marker capacity rejects the actual H0/V1 execution.
• Committed-byte/source-history check passes for all 14 package-plus-source files. Two portable exports are byte-identical; export validation passes. A fresh Git-free extraction passes python3.12 -I -O -W error -B research/interchange-retained-backing/verify.py in 1.716 seconds without file changes. Deliberate source-byte and frozen-receipt drift are rejected and then restored.
• The written proof establishes the quantified bounds; finite replays corroborate it. Separate agent review is not independent human or proof-kernel acceptance.

No heavy source-sized caller, historical suite, native/Lean/Docker, full CRT layer or selected scientific replay was run. Publication-byte/history and portable export checks are separate from mathematical acceptance; hosted CI is separate from local validation.

Credit

JoyboyBrian/#5188 supplies the program and open question; #5162, Jose Manuel's physical ordinary wrapper and the PR2809/PR2753 stream/cardinality lineage supply its source context. Cinna Davis, Hiroki KUMAZAKI, Douglas Colkitt/OpenAI and archived notices retain their CRT/physical-source credit. Ross Richardson's separate numerical/packet-backing work retains its accounting-method credit. No predecessor source is edited.

Prepared by Hiroki KUMAZAKI (@kumagi) with substantial OpenAI Codex assistance in proof, implementation, separate interpreter/review, verification and coordination. Apache-2.0.

Open next questions

README.md and TASK.json specify the next bounded tasks: combine the private grant with one actual caller's simultaneously live borrowed, descriptor, ROM and old-input objects; or transfer it through an independently admitted fixed-block binary translation. Both require their own concrete ownership/representation contract and fresh overlap decision. This local result does not establish a global reserve or automatically start another implementation campaign.