Finite bit construction: fixed-prefix helper470 retirement and five cut obstructions (no new kappa)

research/bit-early-retirement-cut/CLAIM.md

@@ -0,0 +1,21 @@
+# Bit early-retirement investigation
+
+- Track: bit supplier; OS-random assignment seed **8**, drawn 2026-10-13 20:44:34 UTC.
+- Base: `49b3157d1d89adcef16df8be98445451672be5ba`.
+- Selected record: PR1094, kappa `836330098538459/500000000000000000`; bit supplier PR589/555; complex binds.
+- Three candidates: semantic helper-star H2/K repair (overlap PR1194); local R439 gate synthesis (overlap PR1188/1208/1221); early retirement of PR1128's six necessary-condition survivors (chosen).
+- Scope: fixed PR589 D/A/scatter prefix; helpers 465,466,468,469,470,476; retire one helper at its rank-seven common frame using exact dirty restoration and erasure. Test whether promotion destroys required target access. Seed 8 rotates the six-helper order to begin at 468; it is not a novelty claim.
+- Predecessor: Ali Romman's PR1128 at `df29b159239b79d7e451d6ddda826a73e0e50a67`; leaves these six early-cut candidates open. The present test changes the after-B fixed-prefix restriction by retiring before B, but bounds its explicit common-frame mechanism.
+- Method: exact F2 boundary inversion, rational frame containment, constructive restore/clear stars, and target dependency cuts. Check optimistic complete expense before attempting any full supplier.
+- Expected deliverable: one exact finite construction or a scoped obstruction with machine-checkable witnesses, negative controls, source pins and remaining obligations. No new kappa without complete admission.
+- Budget: at most 60 minutes; orientation at most 12 minutes; final 15 minutes reserved for verification and handoff. Expected finish by 2026-10-13 21:44 UTC; earlier if the decisive test closes the mechanism.
+- Overlap: full 838-open-PR/body and 12-discussion snapshot plus 322 local records; PR1128 is the only precise boundary lead. No pinned sprint issue. Local claim follows the donor's no-pinned-issue fallback; no separate reservation issue.
+- Attribution: Ross Richardson, with substantial OpenAI Codex assistance; Apache-2.0. Predecessor PR589/555 and PR1128 authors retained.
+
+## Result and handoff
+
+Completed the six-helper domain: all six fail a single rank-seven retirement block; five also fail their least possible first restoration frame. Helper470 admits a lower-frame restore/clear schedule with the original D/A/scatter/B unchanged, all661 symbolic columns correct, and local rank mass5129 ->5128 for two additional ADDs. This is finite construction/interface evidence, not a new supplier bound or kappa. PR1043's earlier source-deletion result is explicitly credited and is not improved upon.
+
+Exact implementation and evidence: [source-pinned artifact](https://github.com/RossRichardson/integer-mult-bounds/blob/2a6d342645ba919392c121f94f3345430afe0240/research/bit-early-retirement-cut/README.md). The Python3.12 registered replay, Python3.9.6 standalone replay, both independently represented checkers and all eleven test groups pass. The CPython3.14.6 receipt regenerates byte-identically. Native hosting, changed H4 admission, packed precision/tape and an outer bound remain unresolved.
+
+Submission refresh: main `dadfea6e8a78ff6ba901705f87532a71c6f5bbb8`, same selected certificate and identical input blob; 845 open PRs and17 discussions checked with complete pagination. Nearest work PR1043/1079/1128/1179 is distinguished in the README. No pinned coordination issue existed. This is the one local claim update, with the immutable artifact link; the focused PR is the publication handoff. No upstream or selected-frontier changes, merge, or closure is authorized or performed.

research/bit-early-retirement-cut/README.md

@@ -0,0 +1,255 @@
+# Early retirement in the fixed PR589 bit prefix
+
+**Finite construction and scoped negative result; no new kappa.** After the
+unchanged PR589 `D/A/scatter` prefix, five of PR1128's six width-seven candidates
+cannot even be restored by a first single-frame block without losing a required
+target input. Helper **470** is the exception: restore it at rank two, clear its
+dirt at rank seven, and change two cleanup instructions. The complete finite
+word preserves all 661 input columns and reduces that helper's terminal width
+from eight to seven.
+
+This fills a concrete early-cut opening in Ali Romman's
+[PR1128](https://github.com/CrocSwap/integer-mult-bounds/pull/1128), pinned at
+`df29b159239b79d7e451d6ddda826a73e0e50a67`. It does **not** improve on his
+[PR1043](https://github.com/CrocSwap/integer-mult-bounds/pull/1043), pinned at
+`e309e17f83f5759b1ea1fa24a76f24838784cb4d`, which already deletes helper470 by
+changing the earlier source boundary and supplies a complete hosted invoice.
+Here `D`, `A`, retained copies, scatter, and the original `B` are byte-for-byte
+unchanged; a three-gate block is inserted between scatter and B. No savings
+from the two constructions are combined.
+
+The base is `49b3157d1d89adcef16df8be98445451672be5ba`. Its selected conditional
+kappa is `836330098538459/500000000000000000` (PR1094), with the complex side
+binding. The selected record and `upstream/` are unchanged. Assignment seed 8
+selected the bit-supplier track and rotated the six-case inspection order to
+start at helper468. This is a complete six-case calculation, not a randomized
+optimization or a universal bit-supplier obstruction.
+
+## Exact finite construction
+
+Use the original register/frame identifiers in the pinned PR589 word. Insert:
+
+| Frame | Operation | Purpose |
+|---:|---|---|
+| 186 (rank 2) | `470 ^= 77` | Restore helper470 to its original dirty input |
+| 513 (rank 7) | `444 ^= 470` | Remove that dirty input from register444 |
+| 513 (rank 7) | `475 ^= 470` | Remove it from the retained-total register475 |
+
+The earlier scatter already consumed the unchanged snapshots; it is not rerun.
+After these three gates, helper470 holds its initial input and no other register
+contains its dirty column. The original B does not touch register470.
+In C, replace zero-based gate 580, `444 ^= 470`, by `444 ^= 77` at the same
+full frame. Omit zero-based gate 2343, `470 ^= 77`. End helper470 at frame513;
+all other final frames remain unchanged. These indices are checked against the
+pinned operands, not accepted as unverified positions.
+
+The complete patch is recorded in [CERTIFICATE.json](CERTIFICATE.json).
+`patched_sections` in [verify.py](verify.py) reconstructs its literal stream
+without copying or editing the frozen source word. Both checkers replay the
+whole stream and its literal scalar inverse, including every dirty column.
+Forward frame legality is checked over Q. The inverse replay is scalar only;
+it does not claim that reversed frame climbs are a legal physical program.
+
+| Complete local quantity | PR589 | Early retirement |
+|---|---:|---:|
+| ADDs, including 527 scatter ADDs | 24,305 | 24,307 |
+| MOVEs | 2,911 | 2,912 |
+| COPY / ERASE | 9 / 9 | 9 / 9 |
+| COPY ranks | nine rank-seven copies | unchanged |
+| Records | 27,234 | 27,237 |
+| Paid children, including COPY charges | 2,920 | 2,921 |
+| Paid rank mass | 5,129 | 5,128 |
+| Helper470 frame dimensions | `0 -> 1 -> 8` | `0 -> 1 -> 2 -> 7` |
+| Other 660 frame itineraries | baseline | identical |
+
+The exact local histogram changes by `+H1 + H5 - H7`. Both words have scalar
+positive-majorant row-sum peak 49,347 (16 bits), calculated by replacing every
+XOR by addition on initial unit row sums. This is a finite scalar coefficient
+bound, **not** the packed/native precision invoice or the nine-stage prefix
+bound. All nine existing COPY/ERASE charges are retained; this package does not
+recompile their native implementations.
+
+## Why the other eager restorations fail
+
+The cut is the finite 661-register scalar contract of PR1128: `w = M u` just
+after D/A/scatter, with its original current frames. Retained scatter-copy
+channels are unavailable for new suffix reads. M is invertible over F2.
+Allowed suffix operations are invertible scalar operations at a common rational
+frame and monotone frame enlargements, with no new COPY, descent, reset or other
+high-to-low transport. Extra workspace starts with independent dirty inputs.
+
+For a helper h, let `a_h = e_h^T M^-1` be its restoration row and `d_h` the
+support of its dirty column in M. A **first single-frame restoration block**
+performs all scalar work at one frame F, with no earlier lower-frame preparation.
+It returns h to its original input. Every boundary coordinate in `support(a_h)`
+must participate: otherwise the block cannot distinguish two independent
+boundary states differing only in that coordinate. Participating registers
+must reach F. This argument allows arbitrary block length, register mixing,
+target feedback and any number of independent dirty auxiliaries at F.
+
+If the block also isolates h's dirty input from every other register, every
+member of `d_h` must participate too. Hence its forced footprint contains
+`P_h = support(a_h) union d_h union {h}`. A rational-span computation gives a
+unique rank-seven join U for these six footprints, reproducing the specific
+necessary-condition survivors named in PR1128. We do not rerun that PR's
+separate exclusion of the other 433 helpers.
+
+For target t the desired final value is `y_t + x_(t-111)`, whose row in boundary
+coordinates is `b_t = (e_t + e_(t-111))^T M^-1`. Suppose F is not contained in
+that target's final frame V_t. No dependency path from a register at F can ever
+reach t, even through later scalar mixing: every subsequent frame contains F.
+If `b_t` has a nonzero coefficient on a boundary coordinate forced into the
+block, then two inputs differing only in that coordinate have identical
+remaining accessible information but different required target values. No
+allowed completion exists. Moving additional registers to F cannot repair it.
+
+This is an information-access obstruction, even if the block, all remaining
+scalar work, and every permitted frame change are optimistically priced at
+zero. An improved invoice cannot fix this failed identity. It does not exclude
+lower-frame preprocessing, additional transported information, delayed
+restoration after target reads, different copies, or a changed prefix.
+
+| Helper | Full-isolation U | Lost target obligations at U | First restoration frame J | Rank J | Lost targets already at J |
+|---:|---:|---:|---:|---:|---:|
+| 465 | 512 | 10 | 507 | 4 | 1 (target122) |
+| 466 | 514 | 10 | 508 | 4 | 1 (target208) |
+| 468 | 513 | 10 | 509 | 4 | 1 (target144) |
+| 469 | 517 | 10 | 517 | 7 | 10 |
+| 470 | 513 | 10 | 186 | 2 | 0 |
+| 476 | 514 | 9 | 514 | 7 | 9 |
+
+J is the rational span of the restoration donors' current frames. Thus the
+five exclusions at J also exclude every larger first restoration frame,
+including unlisted rational frames. Helper470 instead uses J before U; its
+source77 stays available at frame186. This is the evidence-driven refinement
+of the initially tested one-common-frame mechanism, not a rerun with another
+random seed.
+
+The certificate contains all 666 target checks, the 59 nonzero U-cut rows,
+exact rational separating normals, and six pairs of original inputs witnessing
+failure. The ranks of the six restricted missing-dependency matrices are
+`2,2,2,1,1,2`. These are scalar dependency ranks, not recursive COPY prices.
+The common-frame restore/clear stars have 4,4,4,3,3,4 ADDs respectively and do
+isolate their dirty inputs; their failure is geometric, not a scalar mismatch.
+For helpers465/466/468 they would additionally force targets163/198/208 into
+frames that cannot reach those targets' required final frames.
+
+## Favorable complete slot relaxation, and its limit
+
+A local rank reduction alone is not a supplier improvement. As a planning test,
+move the width-seven residual from its old width-eight allocation into one
+formerly idle rank-seven slot. In the old host, return the full eight freed
+coordinates to an existing idle remainder r. For the incumbent occupied-slot
+remainders `r in {2,4,6}`, all local children, freed idle rank and consumed idle
+rank give the **two-slot relaxation**
+
+`Delta H = e1 + e5 - 2 e7 + e_(r+8) - e_r`.
+
+One `-e7` is the changed local path and the other is the consumed idle slot.
+Both child count and rank mass are unchanged. The inherited child-count
+surcharge would therefore be unchanged under this exact relaxation. At the
+pinned bit exponent `a = 1733427039589844234749/10^24`, an exact rational
+atanh/Taylor enclosure certifies a strictly negative change in
+`sum H_r * r * (64/r)^a` for all three cases (approximately -0.0042490,
+-0.0088265 and -0.0121653 per local conversion). The complete rational intervals
+and profile deltas are in the certificate. No root search or new exponent is
+claimed. This accounts for every child in the stated slot model; it does not
+establish that these slots can be occupied in the real chronology.
+
+**Unresolved:** an actual owner/window assignment for this endpoint, native
+five/nine/eleven-stage compilation, packed precision and tape invoices, and a
+replacement/adaptation of H4 (which currently requires literal full-frame
+cleanup of every helper). The source still has the original H1/H2/H3/H5 scalar
+shape; the changed endpoint is outside the selected H4 contract. The two-slot
+calculation assumes successful admission and unchanged allowances and supplies
+no proof of either. The complex supplier also remains limiting. None of this
+selects a new frontier, a physical supplier, a practical speedup or a complete
+multiplication theorem.
+
+## Reproduce and verification coverage
+
+Python standard library only. From the repository root:
+
+```sh
+python3 -I -O -B research/bit-early-retirement-cut/verify.py
+python3 -I -O -B research/bit-early-retirement-cut/independent.py
+python3 -I -O -B research/bit-early-retirement-cut/test_package.py
+```
+
+The first command ends with:
+
+```text
+PASS_BIT_EARLY_RETIREMENT_CUT helpers=6 lost=59 least_frame_survivor=470 width=8->7 ADDs=24305->24307 mass=5129->5128 no_new_kappa
+```
+
+`verify.py` uses packed binary rows and Fraction geometry; `independent.py`
+uses set-valued rows, separate Gaussian inversion, and fraction-free integer
+geometry. They share the authenticated source and declarative receipt, and
+neither imports the other or predecessor executable code. The second checker
+independently verifies the cut classification, input pairs, full construction,
+frame paths and local invoice; the analytic slot intervals are certified by
+the first checker. Separate implementations written with the same AI assistance
+do not constitute independent human review.
+
+Eleven test groups include all 22 star-gate omissions, three retirement-gate
+omissions, both cleanup mistakes, premature source promotion, invalid final
+frame, altered input/geometry witnesses, 60 small invertible-matrix fixtures,
+the returned-idle accounting check, four receipt corruptions rejected by both
+checkers, source corruption, and read-only offline relocation. Decimal values
+are only a diagnostic cross-check of the rational intervals. Full-frame target
+and smaller-first-frame controls explicitly guard against overstating the
+negative result.
+
+`--emit NEW_PATH` regenerates the complete canonical receipt and refuses to
+overwrite a file. The registered hook in [VERIFY.json](VERIFY.json) runs the
+three commands on Python3.12. Standalone replay also works from a source-only
+export containing this package and its one declared input. Metadata provenance
+checking additionally needs the pinned Git history.
+
+No full `make verify`, selected cold replay, native supplier replay or Lean
+compilation is claimed. This package changes no generator or selected input;
+its focused checks cover the new evidence. Hosted CI status is separate from
+local validation. The registered Python3.12 replay and all eleven test groups
+passed locally; both checkers and all eleven groups also passed on CPython3.9.6.
+Receipt regeneration on CPython3.14.6 was byte-identical. The selected-record
+integrity check and all five selected-result tests passed. The final overlap
+refresh covered 845 open PRs and 17 discussions, with complete pagination; it
+found the explicitly credited neighboring work above. Refreshed main
+`dadfea6e8a78ff6ba901705f87532a71c6f5bbb8` retains the exact input blob and the
+same selected result.
+
+## Open next questions
+
+1. Can the rank-seven endpoint pass the native recursive and owner/window
+   contracts? Start by emitting helper470's `0,79,186,513` path and checking an
+   exact rank-seven idle window, including pending source access. A positive
+   slot moment is useful only after the H4 and physical admission gaps close.
+   Compare with PR1043 before investing: its helper deletion may remain better.
+2. Can helpers465/466/468 be restored after their single blocking target read
+   while retaining a favorable complete profile? Start with targets122/208/144
+   respectively, recompute the inverse-row cut at the corresponding B boundary,
+   then check every dirty column and subsequent target dependency. The present
+   fixed-after-scatter theorem does not cover those later cuts. No search there
+   was performed.
+
+Structured next steps and the deliberate H2/K-to-H4 scope change are recorded
+in [TASK.json](TASK.json).
+
+## Attribution and source status
+
+Ross Richardson prepared this contribution with substantial OpenAI Codex
+assistance in mathematical reasoning, code, tests and writing. New material is
+Apache-2.0, the repository license.
+
+Credit DreamingOfClouds (PR589/545), eumemic (PR555/543 and the interleaved
+interface), their retained E8/native supplier lineage, and Ali Romman
+(PR1128's two-sided boundary conditions and PR1043's earlier helper470
+source-deletion construction). Sagar Thakrar's PR1023 motivates width-seven
+hosting. Hanson Sun's PR1079 independently identifies helper470 in the distinct
+source-substitution class. Adam Gall's PR1179 changes a kernel entrance;
+that mechanism is not imported or combined here. Frozen sources and their
+original notices remain untouched.
+
+Exact Git/SHA256 input and historical context pins are in [SOURCES.json](SOURCES.json).
+Pending PRs are research predecessors, not accepted replacements for the
+selected source. [CLAIM.md](CLAIM.md) records task choice and overlap coverage.