Improve conditional κ to 0.00310676333 with a four-gate contact rewrite

research/p104-contact-rewrite/NOTICE

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+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.

research/p104-contact-rewrite/PRICING-EDITS.json

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+[
+  [
+    "Q('.00310440546'), Q('.003104405465')",
+    "Q('.00310676333'), Q('.003106763335')"
+  ],
+  [
+    "PREVIOUS = Q('.00308595')",
+    "PREVIOUS = Q('.00310644469')"
+  ],
+  [
+    "== 66546",
+    "== 66474"
+  ],
+  [
+    "'kappa_decimal':'.00310440546'",
+    "'kappa_decimal':'.00310676333'"
+  ],
+  [
+    "previous_open_pr5601",
+    "previous_open_pr5651"
+  ],
+  [
+    "Apache-2.0.",
+    "The paid A2 change is from zxv PR5643. The three helper relations are from\nAli Romman PR5644. The four-gate change is by Ali Romman with Codex assistance.\nApache-2.0."
+  ]
+]

research/p104-contact-rewrite/PROOF.md

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+# A complete conditional contact rewrite
+
+The claimed value is **kappa = 0.00310676333**. The comparison is the open
+PR5651 value 0.00310644469. The increase is 0.00000031864. This construction
+keeps the five selected framework assumptions and the helper endpoint
+contract. It does not change the selected record.
+
+## 1. Exact sources and composition
+
+The retained export is PR5627 at
+`ed4dc88a09a7d201971f6603b8c2074e95dd9c32`. It contains 36 source files and
+one export manifest. Authenticate the archive, its complete inventory and
+each file before import. Also authenticate the retained PR5644 constructor,
+witness, word and the PR5643 word.
+
+First apply the three fixed PR5644 helper relations. Compare the result with
+the complete published word. Each relation changes the local paid profile
+by `-H1-H5+H6`. All public inputs, targets and owned instructions stay fixed.
+
+Then apply zxv's PR5643 A2 change. Check its entire scalar-program difference
+against the published PR5643 word. Locate the same two gates in the PR5644
+word. Replace them by the three published gates and rebuild the nested public
+paths. The complete local paid change is `H2+H6-H3-H5`. Full replay checks the
+composition; numerical headline gains are not added. PR5650 already publishes
+this composition, and PR5651 checks it again. Compare with the retained
+PR5650 word after rebuilding the same public MOVE order. Each changed order
+only swaps consecutive MOVEs on different public fields. The scalar and
+owned chronology stays fixed. No new composition result is claimed here.
+
+## 2. The new four-gate identity
+
+Let `c=165`, `h=586`, `s=733` and `x=21`. The old gates are
+
+```text
+h <- h - 4s       at frame 597
+c <- c + 2s       at frame 597
+c <- c - x/2      at frame 652
+x <- x + h        at frame 652
+```
+
+The first two sites are separated by other operations. The witness checks
+that none of the intervening operations reads or writes c, h or s. The two
+late gates are consecutive in the scalar skeleton. Operations on x before
+the late pair retain their order.
+
+Use these gates instead:
+
+```text
+c <- c + h/2     at frame 3
+h <- h - 4s      at frame 597
+x <- x + h       at frame 652
+c <- c - x/2     at frame 652
+```
+
+Here frame 3 is `{0}`, frame 597 has basis `[82]`, and frame 652 has basis
+`[82,268]`. The new early read uses the old value of h. The new gates are
+placed at the old late pair after the two early updates are removed.
+
+Write the values before these updates as c, h, s and x. Both programs give
+
+```text
+c_out = c + 2s - x/2
+h_out = h - 4s
+s_out = s
+x_out = x + h - 4s
+```
+
+The verifier also compares both full 4 by 4 rational matrices. More generally,
+this identity holds for coefficients alpha, beta, gamma and delta when
+`beta = alpha*gamma*delta`; the early coefficient is `-gamma*delta`.
+The A2 identity and compensated triangle method are prior work. The new
+result is this source-bound four-gate schedule and its paid native path.
+
+The new schedule lets c remain in dimension zero until its dimension-two
+contact. Its old two rank-one requests become one rank-two request. Every
+other paid count has the same value. Thus
+
+```text
+new local paid delta = -2 H1 + H2
+new local target delta = 0
+```
+
+For every `0<t<1`, `2-2^t>0`. The complete caller uses the local word 18 times,
+so this new component improves the signed moment by `18*(2-2^t)`.
+
+For this fixed dimension-zero to dimension-two path of c, the new transport
+also has the minimum cost among strictly nested subdivisions. Such a path
+has either one rank-two step or two rank-one steps. For `0<t<1`, the former
+costs `2^t<2`. This local minimum keeps the neighboring contact charts fixed;
+it does not restrict a changed scalar graph or a different caller.
+
+## 3. Complete native interface
+
+For a frame V, a field contains its canonical native chart applied to its
+logical value. A public MOVE changes between strictly nested frames. Every
+ADD and SCALE uses one common chart. COPY uses a cleared owned field. The
+new program keeps every owned instruction in its original order.
+
+The first interpreter checks all public and dirty-owned input columns. A
+separate dense integer interpreter checks the transposed instruction matrices
+in reverse order. The two checks cover 565,504 endpoint coefficients. All 743
+public inputs keep their original endpoints. The nine owned roles have eleven
+lifetimes and peak occupancy two. They receive no productive target credit.
+
+The native checker binds each paid map, each helper boundary and every main
+adapter, including the global phase. The common-chart induction applies at
+every positive tensor width. Scalar coefficients stay outside the tensor
+power. No tensor power of a sum is replaced by a sum of tensor powers.
+
+The retained physical p16/a7 caller has 18 full forward calls, 15 thin query
+sites and 3,120 whole idle jobs. Its 104 ports, 7,488 boundary contacts and
+53,248 central-square column checks are retained. Each helper target has its
+native normalizers. Every thin query and whole idle remains charged.
+
+## 4. Complete invoice and conditional bound
+
+Relative to PR5627, the local paid change is
+
+```text
+-5 H1 + 2 H2 - H3 - 4 H5 + 4 H6
+```
+
+The final local paid profile is
+
+```text
+{1:2195, 2:458, 3:493, 4:91, 5:159, 6:22, 7:15}
+```
+
+The helper profile remains
+
+```text
+{1:2, 2:4, 4:91, 5:6, 7:5, 8:16, 9:411}
+```
+
+The complete caller has **66,474 paid requests**, **12,958 productive fields**
+and rank-mass surplus **760**. All targets from the consumed source remain.
+The independent path-length count gives the same invoice.
+
+Two rational interval engines check the signed moment. Its unique raw
+crossing lies in `(0.00310676334487,0.00310676334488)`. The derivative bound is
+strictly negative on `[0,0.0034]`. This is a ceiling for this fixed invoice,
+not for other words or callers. The claim leaves between 1.487e-11 and
+1.488e-11 below that crossing. The tested saving is 0.003106763335.
+
+The full retained compiler calculation includes the paid ratio, both tail
+ratios, completion, splitting, rounding, complement capacity, logarithmic
+threshold and all 54 strict plus four functional outer predicates. It keeps
+the ordinary BIT admission 0.0034. `RESULT.json` records the exact constants.
+`PRICING-EDITS.json` lists each change from the authenticated price source;
+no compiler or outer predicate is removed or weakened.
+
+The local ordinary bound includes every scalar instruction, paid-map route,
+helper normalizer, main adapter, boundary contact and owned lifetime. It stays
+below 10^7. The complete ordinary allowance stays below 10^9. The retained
+finite coefficient, operation and native-rank bounds also cover the new word.
+The precision and fixed-tape contracts remain conditional assumptions.
+
+## 5. Scope of the search
+
+The discovery first tried different carriers, paid endpoint extensions,
+MOVE-only scheduling, retained histories and scalar contractions. None of
+those tested domains gave an accepted paid gain. They are not general lower
+bounds and are not needed to verify this construction.
+
+A two-step A2 search found the new identity. The retained `patterns.py`
+checks one specific pattern of public scalar dependencies and coefficients,
+before frame selection. It finds this one pattern in the composed source
+and none in the final word. `verify.py` checks both counts. This census does
+not enumerate all scalar identities or legal frame schedules. Other
+dependency orders, longer sequences, different callers and new native
+providers remain open.
+
+The result is a small complete conditional improvement. It is not evidence
+of a large breakthrough or a global maximum. The enormous uniform compiler
+word and the complete rounded multiplier are not materialized. The author
+checks are not independent scientific acceptance or a proof-kernel theorem.

research/p104-contact-rewrite/README.md

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+# Conditional kappa 0.00310676333 from a four-gate rewrite
+
+This package raises the complete conditional kappa from PR5651's
+**0.00310644469 to 0.00310676333**. The increase is **0.00000031864**.
+It is a small gain. The selected record stays unchanged.
+
+The package reconstructs the PR5644 helper cleanup and zxv's PR5643 paid
+A2 change. PR5650 already publishes this composition; PR5651 checks it again.
+A new four-gate schedule then uses an earlier helper value
+and changes two paid H1 requests into one H2 request. Every independent input,
+productive target and owned instruction is retained.
+
+| Complete quantity | PR5651 | This package |
+| --- | ---: | ---: |
+| Paid requests | 66,492 | 66,474 |
+| Productive fields | 12,958 | 12,958 |
+| Rank-mass surplus | 760 | 760 |
+| Claimed conditional kappa | 0.00310644469 | 0.00310676333 |
+
+The exact raw crossing is `(0.00310676334487,0.00310676334488)`. The new
+four-gate component contributes `18*(2-2^t)` to the signed moment. Relative to PR5644, the
+published A2 component also contributes `18*(3^t+5^t-2^t-6^t)`. Both are strictly
+positive for `0<t<1`. See [PROOF.md](PROOF.md), [WITNESS.json](WITNESS.json)
+and [RESULT.json](RESULT.json).
+
+## Replay
+
+Use Python 3.12 and the standard library:
+
+```sh
+python3 -B scripts/research_package.py --package research/p104-contact-rewrite
+```
+
+In an official Git-free export, run:
+
+```sh
+python3 -I -O -B research/p104-contact-rewrite/verify.py
+```
+
+Expected marker: `PASS_P104_CONTACT_REWRITE_KAPPA_00310676333`.
+The verifier authenticates all retained bytes before source imports. It
+checks the four-column identity, 565,504 complete endpoint coefficients,
+native maps and phases, the full physical caller, two interval engines,
+compiler costs and all 58 outer predicates. Corruption controls test the
+source version, early value, frame, erasure, target credit and paid bill.
+[VALIDATION.md](VALIDATION.md) records the completed checks and their limits.
+
+The five selected assumptions remain: native all-column realization, owned
+erasure, uniform compiler closure, precision and fixed-tape execution, and
+analytic recovery. The helper endpoint contract is also retained. The huge
+uniform word and complete rounded multiplier are not materialized. No global
+optimum, independent human acceptance or proof-kernel theorem is claimed.
+
+## Sources and credit
+
+The exact consumed source is
+[PR5627](https://github.com/CrocSwap/integer-mult-bounds/pull/5627) at
+`ed4dc88a09a7d201971f6603b8c2074e95dd9c32`.
+The released inputs are
+[PR5644](https://github.com/CrocSwap/integer-mult-bounds/pull/5644) at
+`c5871955d89bf73506fd68b0a5bf1cf2b3153c02` and
+[PR5643](https://github.com/CrocSwap/integer-mult-bounds/pull/5643) at
+`a3ff5c628bfc8825553d62d913f3e91f75d6825f`.
+The composition word is also retained from
+[PR5650](https://github.com/CrocSwap/integer-mult-bounds/pull/5650) at
+`7916490124ac946704339a6a332bb76f06e5db5c`. The stronger rounded comparison is
+[PR5651](https://github.com/CrocSwap/integer-mult-bounds/pull/5651) at
+`853192167b27d44aea0d4bc2b0d52d695fadc7c6`. The checker matches the published
+composition after it orders independent public MOVEs in the same way.
+The archive retains its complete source closure. Copied inputs match their
+published Git objects. [SOURCES.json](SOURCES.json) records both locations.
+
+Credit zxv for the paid A2 witness and PR5650 composition; Ali Romman for PR5644 and the prior
+complete P104 caller; Chris R/Clear1nsight and ToikY for the prior A2 method;
+and Cinna Davis for the compensated triangle and scalar compression method.
+Retain flyingviper's relay, Scratch-net's preparation, Pierre-Loic Doulcet's
+cascade, and all David Arias, Gareth Davidson, Cinna Davis, Jacob Alan Sussman,
+Cao Yifan, zxv, geometry, native, BIT, compiler and analytic source notices.
+[NOTICE](NOTICE) retains the detailed source credit.
+
+The new four-gate placement and author checks were prepared by
+Ali Romman with substantial OpenAI Codex assistance. No subagents were used.
+Apache-2.0. The [canonical investigation](https://github.com/CrocSwap/integer-mult-bounds/discussions/5125#discussioncomment-19183208)
+records the scope and the prior failed screens. Those screens are not inputs
+to the positive verifier.
+
+## Open next questions
+
+1. Independently check the four-gate source version and native frame placement,
+   then replay the complete package at its publication commit.
+2. Test a different dependency pattern, a longer sequence or a changed caller.
+   The stated finite search is at a fixed point; it does not exclude those
+   constructions. A new claim must include its complete paid profile.
+3. Resolve the retained framework assumptions. A finite source-bound replay
+   does not supply a complete rounded multiplier or formal multiplication proof.
+
+[TASK.json](TASK.json) states the first step and limit for each question.