Conditional κ = 0.00310626335 (+1.85789e-6 over #5627): three helper cleanups

research/p104-neutral-helper-cleanup/NOTICE

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+P104 neutral helper cleanup
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+
+The retained PR5627 archive contains the full source notices. The source
+pricing module retains its predecessor notices. Read README.md for links
+and the contribution map. No source notice was removed.

research/p104-neutral-helper-cleanup/PROOF.md

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+# Three neutral helper relations remove paid calls
+
+This construction starts from the exact PR5627 word at
+`ed4dc88a09a7d201971f6603b8c2074e95dd9c32`. It keeps all 743 independent
+inputs, all 535 helper targets and the 104 main pairs. The selected frontier
+is unchanged. The five source framework assumptions and the helper endpoint
+contract remain.
+
+## 1. The boundary identity
+
+Write the scalar state in native normalized coordinates. Let h be a helper
+with initial chart S and final chart F. At a chronological cut, suppose h
+and each cancellation supplier are at F. Suppose the exact prefix rows give
+
+```math
+r_h-\sum_j b_j r_j=c x_h+\sum_{j\ne h} a_j x_j.
+```
+
+Every input on the right has initial chart S, and c is a dyadic unit. Suppose
+there is no later source read of h. Before the word, apply
+`h += -a_j/c * j` for every j other than h. At the cut, apply
+`h -= b_j * j` and then `h *= 1/c`. The resulting h is exactly the original
+independent input x_h. Delete the later scalar operations whose destination
+is h. They have no effect on another output.
+
+The source returns every helper unchanged and has no helper contribution
+to a main output. Thus the input basis change can affect only the old h
+output. The cut repairs that output exactly. No helper is set to zero,
+no new input is created and no productive target is removed or added.
+All basis coefficients are dyadic, and every SCALE has a dyadic inverse.
+
+The constructor checks the complete prefix relation, common charts, the
+last-read condition, the exact removed operations and the full arbitrary-input
+endpoint after each step. It also checks that the complete chronological
+subsequence of owned operations is unchanged.
+
+## 2. Exact witnesses
+
+The cuts below are zero-based and refer to each successive word after
+transport coalescing. `WITNESS.json` retains the full event anchors and vectors.
+Here `e_j` denotes the j-th independent input.
+
+| Helper | Cut | Final chart | Prefix row after cancellation |
+| ---: | ---: | ---: | --- |
+| 291 | 10627 | 234 | -2 e291 - e293 + 2 e39 + 2 e303 |
+| 321 | 11072 | 237 | e321 - 2 e320 - 2 e34 + 2 e330 |
+| 369 | 11718 | 243 | e369 + 2 e383 - 2 e372 - 2 e3 - 2 e382 - 2 e378 |
+
+Their cancellation vectors are respectively
+`{293:-1}`, `{22:2,320:-3/2,330:2}`, and
+`{368:-3/2,26:2,378:-2}`. All initial basis operations use chart 3.
+The three steps remove six, five and five old scalar cleanup events.
+The inserted cancellation and input operations are included in the cost.
+
+## 3. The paid change
+
+The removed cleanup contacts let main roles 55, 49 and 80 bypass one
+intermediate chart each. Their old paths are
+
+```text
+role 55: 722 -> 234 -> 127
+role 49: 734 -> 237 -> 127
+role 80: 749 -> 243 -> 127
+```
+
+Each path has ranks 1 and 5. The new direct path has rank 6. For chart
+operators C_A, C_B and C_D, the exact product is
+
+```math
+(C_D C_B^{-1})(C_B C_A^{-1})=C_D C_A^{-1}.
+```
+
+No scalar operation uses that role between these transports. The chronological
+replay checks strict chart containment, and the native verifier checks the
+exact signed map and global phase. This identity applies at every positive
+tensor width under the retained native implementation assumption.
+
+The local paid delta is `-3 H1 -3 H5 +3 H6`. Every target stays fixed.
+For 0<t<1, the complete target-minus-paid moment therefore increases by
+
+```math
+54[1+5^t-6^t]>0.
+```
+
+This is a changed physical invoice. It does not add predecessor kappa gains.
+The alternative PR5613 inventory has 72 additional H5 paid calls and 72
+additional H5 targets. It has the same signed source moment, but this package
+uses only the smaller PR5627 inventory.
+
+## 4. Complete cost and verification
+
+The final local word has 25,448 events, 743 independent inputs, 535 helpers,
+nine owned roles, eleven owned lifetimes and peak owned occupancy two.
+Its paid profile is
+
+```text
+{1:2197, 2:456, 3:494, 4:91, 5:160, 6:21, 7:15}.
+```
+
+The complete caller retains 18 full forward calls, 15 thin query sites,
+3,120 whole idle jobs and 3,328 main targets. Its complete invoice has
+66,492 paid calls and 12,958 productive targets. The rank-mass surplus is 760.
+Helpers, native normalizers, main adapters, basis operations, COPY/CLEAR,
+all owned lifetimes, routes, thin queries and idles are charged.
+
+A separate dense integer transpose checks all 565,504 matrix entries.
+The native check verifies 2,916 distinct maps. The physical caller verifies
+all 7,488 contacts, 53,248 central-square columns and complete main phases.
+Two rational interval engines check the same complete moment.
+
+The exact raw crossing lies in
+`(0.00310626336369, 0.00310626336370)`. The claim is `0.00310626335`, tested
+at `0.003106263355`. The gain over PR5627's claim is `0.00000185789`,
+about 0.05985%. This is a small gain.
+The raw ceiling is proved only for this fixed invoice on `[0,0.0034]`.
+Its headroom above the claim is between `1.369e-11` and `1.370e-11`.
+It is not an optimum over constructions.
+
+The source strictification parameters remain: power 1024, cutoff 17,
+target floor 18, tail offset 1/5, reserve 10^-12 and tail decay 39/40.
+The compiler checks its finite integer threshold, the preceding integer,
+initial range, shifted exits and complement capacity. All 54 strict outer
+conditions and four functional conditions pass. The ordinary BIT contract
+remains 0.0034.
+
+The local scalar allowance is 220,208 and the local route allowance is
+5,172,000, below the retained 10^7 bound. The complete ordinary-work bound
+is 184,647,552, below 10^9. Fewer than 26,000 scalar events and the checked
+coefficient bound retain the source growth bound. The paid native rank sum
+is 5,986 per local word. No growth or route cost is omitted.
+
+The huge uniform word and complete rounded multiplier are not materialized.
+These are author checks by the same AI-assisted author. Independent scientific
+acceptance and a proof-kernel theorem remain absent. Source native, owned
+erasure, uniform compiler, precision/tape and analytic recovery assumptions
+remain. The package makes no large-gain or global-optimum claim.

research/p104-neutral-helper-cleanup/README.md

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+# Conditional kappa 0.00310626335 from three helper cleanups
+
+This construction raises complete conditional kappa from
+**0.00310440546 to 0.00310626335**, a gain of **0.00000185789**
+(about **0.05985%**) over
+[PR #5627](https://github.com/CrocSwap/integer-mult-bounds/pull/5627).
+This is a small gain. The requested large improvement is not established.
+The selected frontier is not changed by this package.
+
+Three exact helper relations remove scalar contacts on main roles 55, 49
+and 80. Each role can then replace an H1+H5 path with H6. The local paid
+delta is `-3 H1 -3 H5 +3 H6`. All independent inputs, productive targets
+and owned operations stay fixed. The complete caller saves 54 paid calls.
+
+Read [PROOF.md](PROOF.md), [WITNESS.json](WITNESS.json),
+[RESULT.json](RESULT.json), [SOURCES.json](SOURCES.json), and
+[VALIDATION.md](VALIDATION.md). The constructor checks every relation before
+it emits the new [word](WORD.json.gz).
+
+| Complete quantity | Value |
+| --- | ---: |
+| Paid calls | 66,492 |
+| Productive targets | 12,958 |
+| Rank-mass surplus | 760 |
+| Public inputs / helpers per full call | 743 / 535 |
+| Owned roles / lifetimes / peak | 9 / 11 / 2 |
+| Local events | 25,448 |
+| Raw crossing | (0.00310626336369, 0.00310626336370) |
+| Tested saving | 0.003106263355 |
+| Claimed conditional kappa | 0.00310626335 |
+
+The complete cost includes helpers, native maps, input/output normalizers,
+main adapters, scalar bases, COPY/CLEAR, allocation, routes, thin queries,
+whole idles, compiler completion and rounding. Two rational interval engines,
+all compiler conditions and all 58 outer conditions pass. The ordinary BIT
+contract stays at 0.0034. The fixed-invoice ceiling is the raw crossing above;
+it is not a ceiling over constructions.
+
+## Reproduce
+
+Use Python 3.12 and its standard library, from the repository root:
+
+```sh
+python3 -B scripts/research_package.py --package research/p104-neutral-helper-cleanup --check-only
+python3 -B scripts/research_package.py --package research/p104-neutral-helper-cleanup
+```
+
+The registered command runs normally and with `-O`. Expected markers are
+`PASS_P104_NEUTRAL_HELPER_KAPPA_00310626335` and
+`PASS_REGISTERED_RESEARCH_PACKAGE`. Direct replay needs no Git or network:
+
+```sh
+python3 -I -W error -B research/p104-neutral-helper-cleanup/verify.py
+python3 -I -W error -B -O research/p104-neutral-helper-cleanup/verify.py
+```
+
+Normal replay is read-only. `--write` regenerates the word and receipt.
+The verifier authenticates the exact source archive and all 36 payload files
+before it imports source code. It checks all 565,504 endpoint coefficients,
+2,916 signed native maps, the complete physical caller and the full cost.
+Controls reject a false cancellation, wrong cut, missing input basis,
+missing owned erasure, extra target credit and unpaid query. The native,
+controller and pricing checks retain their own controls.
+
+## Source and credit
+
+The consumed source is [PR #5627](https://github.com/CrocSwap/integer-mult-bounds/pull/5627)
+at `ed4dc88a09a7d201971f6603b8c2074e95dd9c32`:
+[source proof](https://github.com/aliromman/integer-mult-bounds/blob/ed4dc88a09a7d201971f6603b8c2074e95dd9c32/research/p104-helper-subspace-splits/PROOF.md),
+[source word](https://github.com/aliromman/integer-mult-bounds/blob/ed4dc88a09a7d201971f6603b8c2074e95dd9c32/research/p104-helper-subspace-splits/WORD.json.gz),
+and [source notices](https://github.com/aliromman/integer-mult-bounds/blob/ed4dc88a09a7d201971f6603b8c2074e95dd9c32/research/p104-helper-subspace-splits/NOTICE).
+Its official self-contained export is retained. Reproduction does not require
+a cached unmerged Git object. The exact word, caller and all shared proof
+engines are unchanged source inputs. The derived pricing file changes only
+the declared claim, comparison and paid count; its byte-level derivation is checked.
+
+Ali Romman supplied the new three relations with substantial OpenAI Codex
+assistance. The same assistant performed the author checks; no subagents or
+independent scientific review. The helper-boundary method extends
+[PR #5602](https://github.com/CrocSwap/integer-mult-bounds/pull/5602).
+The source package supplies the complete composition of
+[PR #5601](https://github.com/CrocSwap/integer-mult-bounds/pull/5601),
+[PR #5605](https://github.com/CrocSwap/integer-mult-bounds/pull/5605),
+[PR #5612](https://github.com/CrocSwap/integer-mult-bounds/pull/5612),
+[PR #5606](https://github.com/CrocSwap/integer-mult-bounds/pull/5606)
+and [PR #5599](https://github.com/CrocSwap/integer-mult-bounds/pull/5599).
+Its notices credit flyingviper's relays, Scratch-net's prepared helper,
+Pierre-Loic Doulcet's cascade, Joel Pulikkan, DreamingOfClouds, thu-nmrc,
+Gareth Davidson, David Arias, Cinna Davis, Jacob Alan Sussman, zxv and
+Cao Yifan, with the earlier geometry, BIT and analytic sources.
+All notices remain in the retained export and pricing module. Apache-2.0.
+
+The discovery first passed on
+[PR #5613](https://github.com/CrocSwap/integer-mult-bounds/pull/5613) at
+`32f23af8790cc3c435bb1f495d861332b4f000e3`. That source has the same signed
+moment but 72 more H5 paid calls and 72 more H5 targets. It is comparison
+evidence, not the consumed input. Its counts are not combined with this receipt.
+
+## Coordination and limits
+
+The frozen research base is canonical main
+`6cf7a683f447b9f1e2333791907fd92be5009a63`, priority revision 7.
+The [canonical claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/5125#discussioncomment-19154639)
+records the source pins, live overlap checks and released finite screens.
+The separate Discussion5600 main-port retirement campaign is excluded.
+The final publication checkpoint is recorded in `VALIDATION.md`.
+
+The five selected assumptions remain: native implementation at all widths,
+owned erasure, uniform compiler closure, precision/tape cost and analytic
+recovery. The helper endpoint contract also remains. The huge uniform word
+and complete rounded multiplier are not materialized. This package does not
+claim independent acceptance, a proof-kernel theorem or global optimality.
+
+## Open next questions
+
+An independent reviewer can check the three prefix relations and transport
+products, then replay the full package. Further coordinated bases may remove
+more scalar contacts, but the larger exploratory screens have not produced
+a new complete gain. A new construction must first check current claims and
+must include every changed target and paid operation. [TASK.json](TASK.json)
+records the next bounded question and its blockers.