Conditional κ = 0.003128645 (+2.59e-7 over #6144): retain an operand to remove a root stop

research/p104-retained-operand-descent/NOTICE

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+Retained-operand descent: Ali Romman, with substantial OpenAI Codex assistance.
+Apache-2.0.
+
+The exact source word and price are from DreamingOfClouds, PR6144 and PR6143,
+with Claude Opus 5.5 (Anthropic), via Claude Code. They extend PR6030 and PR5907.
+The source reader is adapted from that work. The gzip portability controls
+retain Douglas Colkitt and Maxime Fleury credit.
+
+The pinned PR5827 checker and all earlier source notices are retained below.
+
+Unread-register path resynthesis
+Copyright 2026 Ali Romman. Apache-2.0.
+
+New construction, code, calculations and prose: Ali Romman with substantial
+OpenAI Codex assistance. No subagents. All new checks are author-side.
+
+Mason Hale supplied PR5811, the complete two-block composition and the exact
+role92 remapping and fourth accumulation. The earlier elementary accumulation,
+retained-copy and second-block work is by Ali Romman. Joel Pulikkan supplied
+R22. Dugongue supplied the concurrent baseline composition proof. Their source
+notice and proof are retained byte for byte under inputs/pr5811-*.
+
+The source closure retains flyingviper, Cinna Davis, David Arias, Gareth
+Davidson, Jacob Alan Sussman, zxv, Cao Yifan, Pierre-Loic Doulcet, thu-nmrc,
+Scratch-net and all native/helper/controller/compiler/geometry/analytic
+predecessors. The exact PR5786 source archive and its nested notices remain.
+This package does not claim any predecessor construction as new.
+
+Exact PR5811 vendor/accumulated/NOTICE follows. Source head:
+33d7760d0a6fea6394aefd014eea68f0af02a92e
+
+Accumulated native updates
+New work: Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+No subagents. The same assistant made the construction and the author checks.
+
+Concurrent scope: Discussion5125 now tests retained copies in the transposed
+supplier. That route uses the same source/destination duality. This package
+claims the three fixed forward-supplier groups and their complete result,
+not the elementary accumulation identity or the broader transposed search.
+
+Exact source: PR5786 at 6b936acd68a7e0f9062037900f0fe9bc4b6fdd25.
+The complete predecessor notice follows unchanged.
+
+Asynchronous helper hubs
+New work: Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+
+Retained native copies from PR5774 at 7d6ec0cdef31d0c37b7d2563e709979caa0d5eae.
+All prior credit follows unchanged.
+
+Paid intermediate meetings and collective helper splits.
+Ali Romman, with substantial OpenAI Codex assistance. Apache-2.0.
+No subagents. All checks are author-side calculations.
+
+The complete PR5718 notice follows unchanged.
+
+Four paid interior public-basis changes.
+Ali Romman, with substantial OpenAI Codex assistance. Apache-2.0.
+No subagents.
+
+The following exact notice is retained from PR5660.
+
+New helper-subspace construction, proof and replay:
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+The same assistant performed the construction and author-side checks.
+
+Retained contributions:
+flyingviper: PR5605 shared relay, PR5612 delayed relay, and the PR5608/PR5613
+relay/split compositions.
+Scratch-net: PR5606 prepared-helper identities and target debit.
+Pierre-Loic Doulcet (hexapode): PR5565 commuting-contact construction, used as the exact source.
+Ali Romman: PR5602 fixed boundary relations, PR5601 three-contact frame
+change, PR5580 contact step,
+and PR5544/5545 complete P104 caller.
+Joel Pulikkan (GamingPuzzled): PR5569 priced helper handoff.
+DreamingOfClouds: PR5561 free helper endpoints and PR5568 search tools.
+thu-nmrc: PR5551 contact improvements.
+Gareth Davidson (bitplane): PR2305 source composition.
+David Arias: PR2152 coalesced P104 word.
+Cinna Davis: PR2219 private reuse and the controller lineage.
+Jacob Alan Sussman: productive mixed-target recursion.
+zxv and Cao Yifan: exact intervals, compiler and native/pricing lineage.
+GamingPuzzled: PR5472 successor handoff, a predecessor to the split concept.
+The retained source notices preserve earlier geometry, BIT and analytic credit.
+
+The elementary rank-one basis identity is not claimed as a new general
+linear-algebra theorem. The contribution is its explicit independent-target
+use on this P104 word and the complete conditional kappa improvement.
+
+New contribution: Ali Romman, with substantial OpenAI Codex assistance.
+The four-gate source-version rewrite composes zxv PR5643 and Ali Romman PR5644.
+The A2 method retains Chris R/Clear1nsight PR3998 and ToikY PR4019 credit.
+The compensated triangle method retains Cinna Davis PR4692 credit.
+No subagents were used for this contribution. Apache-2.0.
+
+Concurrent PR5650 by zxv publishes the PR5643/PR5644 composition. PR5651 by
+Ali Romman corroborates it. This package credits that released composition
+and contributes the further four-gate placement.
+
+New work: distributed scalar updates on an unobserved public field interval, the exact witness and complete conditional price. Prepared for Ali Romman with substantial OpenAI Codex assistance. No subagents. Apache-2.0. The source archive retains all predecessor notices.
+
+Two-boundary main-assisted helper split and early cleanup: flyingviper with substantial
+OpenAI Codex assistance, including independent AI review agents. Apache-2.0.
+
+Retained native-copy construction
+Prepared by Ali Romman with substantial OpenAI Codex assistance. No subagents.
+The paid-meeting source is PR5751 by Ali Romman with OpenAI Codex assistance.
+The dimension-four span [265,194,164,75] is from Joel Pulikkan / GamingPuzzled
+PR5739, with Claude Opus 5.5 assistance. The R22 block edits are not included.
+The retained-copy mechanism, exact constructor, source-specific verification,
+new price and proof are the contribution of this package. Apache-2.0.
+
+Exact PR5811 vendor/blocks/NOTICE follows. Source head:
+33d7760d0a6fea6394aefd014eea68f0af02a92e
+
+The second block is PR5796 at4b30391e633ba0c7c674f03586ae97d443e9bc9c, by Ali Romman with substantial OpenAI Codex assistance, following Joel Pulikkan/GamingPuzzled R22. Exact notice, plan, proof and source words are retained in source/SECOND-* unchanged.
+
+The final immediate source is PR5786 asynchronous helper hubs by Ali Romman with substantial OpenAI Codex assistance. Its three temporal helper splits and complete completion parameters are retained and credited. The earlier PR5774-only composition is a superseded checkpoint, not a separate contribution.
+
+New complete retained-copy and R22 composition: Ali Romman with substantial OpenAI Codex assistance. No subagents. Apache-2.0.
+
+The thirteen retained native conversions and changed helper targets are by Ali Romman with OpenAI Codex assistance in PR5774. The R22 block and rank-four basis are by Joel Pulikkan (GamingPuzzled), with Claude Opus5.5 assistance, in PR5739/5756. The complete source archive and R22 notice remain unchanged.
+
+The construction retains Ali Romman PR5751 paid intermediate meetings, flyingviper PR5718 boundary splits, Cinna Davis's productive controller, Jacob Alan Sussman's productive recursion, zxv and Cao Yifan's interval/compiler line, and every P104/native/BIT predecessor in the source notices. The source engines are reused under exact byte bindings, not claimed as new implementations.

research/p104-retained-operand-descent/PRICING-EDITS.json

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+[
+  [
+    "KAPPA, TESTED = Q('0.003128386'), Q('0.0031283910')",
+    "KAPPA, TESTED = Q('0.003128645'), Q('0.003128650')"
+  ],
+  [
+    "PREVIOUS = Q('0.003125804')",
+    "PREVIOUS = Q('0.003128386')"
+  ],
+  [
+    "'kappa_decimal':'0.003128386','previous_pr6030':PREVIOUS",
+    "'kappa_decimal':'0.003128645','previous_pr6144':PREVIOUS"
+  ]
+]

research/p104-retained-operand-descent/PROOF.md

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+# Retain an operand to remove a root stop
+
+Prepared by Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+The source construction is PR6144 by DreamingOfClouds. See NOTICE for all
+retained source credit.
+
+## Exact source and claim
+
+Use PR6144 at `5ec9b6dafc1ec54e2c0ca0771e8ed032f7888417`. Under the same five
+framework assumptions as the selected P104 result, the changed finite word and
+complete caller support **κ = 0.003128645**. The proof has two parts: equality
+of the changed local operation and a complete cost and compiler check.
+
+Event positions below are zero-based positions in the immutable source word.
+The native frames have these dimensions and containments:
+
+```text
+dim(3)=0, dim(595)=1, dim(737)=4, dim(520)=6;
+3 ⊂ 595 ⊂ 737 ⊂ 520 ⊂ 529.
+```
+
+The checker proves these containments and the ranks through the unchanged
+source replay and the exact native-map checker. No new frame is introduced.
+
+## Equality at the cut
+
+Let `a` be the decoded value of reader 451 before event 12918. Let `b` be the
+decoded value of helper 228 at event 12918. Let `q` be the decoded root value
+in register 772 at event 13485. The source performs:
+
+1. Move reader 451 from frame 3 to frame 737 and add `b` at event 12918.
+2. Move root 772 from frame 520 to frame 737 at event 13486.
+3. Add `-4q` to reader 451 at event 13487.
+4. Move root 772 from frame 737 to frame 595 at event 13488.
+
+At this cut, reader 451 has value `a+b-4q` in frame 737. Root 772 has value `q`
+in frame 595. No other register reads reader 451 between events 12918 and
+13488. Its next external read is event 14524.
+
+The new word performs:
+
+1. Move reader 451 from frame 3 to frame 595 at event 12913.
+2. At event 12918, initialize owned register 770 in frame 737 and copy `b` into
+   it. This uses the source value before helper 228 leaves that frame.
+3. At event 13488, move root 772 directly from frame 520 to frame 595. Add
+   `-4q` to reader 451 in that frame.
+4. Move reader 451 from frame 595 to frame 737, add the saved `b`, and clear
+   register 770.
+
+The value is now `a-4q+b = a+b-4q`. The two schedules give the same values and
+frames for all public and live owned registers at the cut. The saved value is
+zero on exit. Register 770 has no earlier live lifetime; its source lifetime
+starts at event 13658. The new lifetime ends before that event.
+
+Each scalar operation acts in a common native chart. Each new MOVE has the
+proper nested direction: public registers move to a containing frame and owned
+registers move to a contained frame. The native checker verifies every signed
+Pauli image and the exact global phase. Full arbitrary-input and
+reverse-chronology transpose replays also check the complete word, including
+all later reads and public restoration.
+
+## Complete bill
+
+The old three changed routes have ranks `4, 2, 3`. The new three routes have
+ranks `1, 5, 3`. The rank-3 terms cancel. For `H_r=r^t` and `0<t<1`, the saving
+is
+
+```text
+H_2 + H_4 - H_1 - H_5
+  = (H_2 - H_1) - (H_5 - H_4) > 0.
+```
+
+Strict concavity gives the last inequality. The number of paid calls and rank
+mass stay the same. All productive targets and all 104 main ports stay the
+same. The complete caller has 18 local calls, so its paid change is
+`18(H_1-H_2-H_4+H_5)`. The 15 thin-query groups, 3,120 whole idle jobs, main
+adapters and helper normalizers retain their full charges.
+
+The full word has 22,821 events, 770 public registers, 29 owned register labels,
+32 owned lifetimes and peak owned use 13. The new cache adds one INIT, one COPY
+and one CLEAR. The local scalar allowance rises from 189,658 to 189,670. The
+route bound is 5,325,000 and includes four route allowances for each owned
+lifetime. Their sum is below the retained local budget of `10^7`.
+The full nonrecursive bound is 184,647,552, below `10^9`.
+The scalar-growth bound uses 26,000 events; the word remains below that bound.
+
+The complete invoice has 66,762 paid calls, 13,444 productive fields and rank
+surplus 760. A separate census from the two caller path lengths agrees with
+the physical-event census. No target is assigned to the new cache.
+
+## Price, completion and transfer
+
+The two exact rational moment engines give a unique raw crossing in
+
+```text
+(0.00312865201590, 0.00312865201591).
+```
+
+The derivative bound is negative on the retained interval up to the BIT cap
+0.0034. This ceiling is for this fixed invoice. The old PR6144 invoice has a
+negative upper margin at the new claim.
+
+Use tested saving 0.003128650, cutoff 16, floor 17, tensor power 1024, tail
+offset `1/7`, reserve `10^-12`, decay `125/128` and tail block 128. The paid,
+offset and completion ratios have upper bounds below, respectively,
+`1-10^-12`, `125/128` and `125/128`. Splitting and rounding charges are included.
+The compiler gap is `1/1400000000`. The finite threshold is
+`N = 2^65807000000000`; its integer threshold test rejects the preceding value.
+All 54 strict outer inequalities and four functional inequalities pass.
+
+These checks retain the native-all-columns, owned-erasure, uniform-compiler,
+precision/tape and analytic-recovery assumptions. They do not materialize the
+uniform word or the rounded integer multiplier. Independent scientific review
+remains open. The selected frontier is not changed.

research/p104-retained-operand-descent/README.md

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+# Retained operand: conditional κ = 0.003128645
+
+This package raises the conditional claim of
+[PR6144](https://github.com/CrocSwap/integer-mult-bounds/pull/6144)
+from **0.003128386 to 0.003128645**. The new gain is **0.000000259**
+(about 0.00828%). This is a small gain from a changed paid profile.
+The source is fixed at `5ec9b6dafc1ec54e2c0ca0771e8ed032f7888417`.
+The branch starts at canonical main `3f0152a32eeebac0df11946dbd50cabe5a7f7839`.
+
+The new operation order removes one stop from root register 772. Save helper
+228 at frame 737 in register 770. Reader 451 can then receive the root at frame
+595 before it adds that saved operand at frame 737. Clear the saved operand
+before the existing use of register 770. All public outputs and productive
+targets stay the same. The declared register count stays 29 and peak use stays
+13. The extra lifetime and 12 scalar operations are charged.
+
+For `H_r = r^(1-κ)`, the local paid change is
+`H_1 - H_2 - H_4 + H_5`. It is negative for `0 < κ < 1`.
+The complete caller uses this change 18 times. It still has 66,762 paid calls,
+13,444 productive fields and rank surplus 760.
+
+The raw crossing is strictly between **0.00312865201590** and
+**0.00312865201591**. The claim leaves more than `7.01590e-9` of raw headroom.
+The old PR6144 profile fails at the new claimed κ. This excludes an improvement
+that comes only from a smaller rounding gap.
+
+## Evidence and scope
+
+[construct.py](construct.py) applies five guarded replacements to the pinned
+source word. [PROOF.md](PROOF.md) proves the changed schedule and price.
+[RESULT.json](RESULT.json) records the complete check. [VERIFY.json](VERIFY.json)
+registers the replay. [VALIDATION.md](VALIDATION.md) records the environment and
+the publication checks.
+
+The verifier checks the full regenerated payload, all 799 input columns,
+638,401 endpoint coefficients, a separate transpose calculation, native maps
+and exact phases, main adapters, helper normalization and the physical caller.
+Two rational interval engines check the price. The completion, compiler and
+all 54 strict plus four functional outer inequalities are checked. Controls
+remove or change the cache, the later addition, restoration and cleanup.
+
+The five inherited assumptions remain: native operators at all column widths,
+owned erasure, uniform compiler closure, precision and tape movement, and
+analytic integer recovery. The uniform word at the stated threshold and the
+rounded multiplier are not materialized. These are author checks. Independent
+scientific acceptance remains open. The selected record stays at 0.003125804.
+
+## Reproduce
+
+Python 3.12 and its standard library are sufficient. From the repository root:
+
+```sh
+python3.12 -I -B research/p104-retained-operand-descent/verify.py
+python3.12 -I -O -B research/p104-retained-operand-descent/verify.py
+python3.12 scripts/research_package.py --package research/p104-retained-operand-descent
+```
+
+Use `--build-only` on `verify.py` for the exact endpoint and inventory checks.
+That option does not check the full native and compiler chain. The full command
+does not need Git or network access. It authenticates every file in the retained
+PR5827 source archive before use. Its local checker is unmodified. The price
+equals PR6144's price with the three declared edits in
+[PRICING-EDITS.json](PRICING-EDITS.json).
+
+Immutable input hashes stay exact. Regenerated gzip output is compared by every
+decompressed payload byte, so different container headers are accepted.
+
+## Coordination and credit
+
+The canonical investigation is
+[Discussion 5645](https://github.com/CrocSwap/integer-mult-bounds/discussions/5645#discussioncomment-19205459).
+PR6144 explicitly left the reader 451/497 root-2 stop question open. This work
+completes its reader 451 branch. The changed premise is a retained historical
+operand and a changed addition order. The protected R22 and 102-contact event
+sequences stay byte-exact. The overlap coverage and its limits are in
+[TASK.json](TASK.json).
+
+New construction, proof and checks: Ali Romman, with substantial OpenAI Codex
+assistance. PR6144 and PR6143 are by DreamingOfClouds with Claude Opus 5.5
+(Anthropic), via Claude Code. PR6030 and PR5907 retain that author's credit.
+PR5827, PR5811, PR5802, PR5786 and their source authors retain all earlier credit.
+The source reader follows DreamingOfClouds and Ali Romman. The portability
+controls retain Douglas Colkitt and Maxime Fleury credit. [NOTICE](NOTICE)
+retains the full source notices, including native, caller, compiler and BIT work.
+All new work is Apache-2.0.
+
+## Open next questions
+
+- Can an independent reviewer confirm the changed chronology, full invoice and
+  complete replay? Start with the five replacements and the cache controls.
+  Author checks and hosted CI do not establish this acceptance.
+- Can reader 497 avoid the remaining stop at frame 520? Its earlier operand at
+  frame 519 is an extra constraint: frame 595 is not a subspace of frame 519.
+  Start with an explicit paid schedule for both operands. A rank estimate or
+  an available historical span alone does not satisfy this task.
+
+The structured questions and their blockers are in [TASK.json](TASK.json).