Conditional κ = 0.00311325989 (+1.2194e-7 over #5756): second rank-four block

research/p104-second-low-rank-block/NOTICE

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+Second rank-four contact block on the complete PR5756 word.
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+No subagents. These are author-side checks.
+
+Joel Pulikkan (GamingPuzzled), with Claude Opus 5.5 assistance, supplied the
+R22 reframe mechanism and the different block in PR5739/5756.
+This package adds a different 102-operation block; it does not claim that mechanism.
+
+The source PR5756 notice 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.

research/p104-second-low-rank-block/PLAN.json

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+{
+  "basis": [
+    11,
+    21,
+    38,
+    448
+  ],
+  "old_frame": 267,
+  "roles": [
+    57,
+    81,
+    92,
+    224,
+    225,
+    227,
+    229,
+    234,
+    742
+  ],
+  "initial_helper": 742,
+  "final_helper": 225,
+  "runs": [
+    {
+      "first_scalar": 9858,
+      "length": 90,
+      "old_sha256": "2c4d8332eed5d21830748b694d88a830f98f50773e7faddfb65b654800d0e296"
+    },
+    {
+      "first_scalar": 11957,
+      "length": 12,
+      "old_sha256": "bcb7d071dbe4b830053892decd7c8b6aef5a26b06a06e010148882871cf07ee2"
+    }
+  ]
+}

research/p104-second-low-rank-block/PRICING-EDITS.json

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+[
+  [
+    "KAPPA, TESTED = Q('.00311301600'), Q('.003113016005')",
+    "KAPPA, TESTED = Q('0.00311325989'), Q('0.003113259895')"
+  ],
+  [
+    "PREVIOUS = Q('.00311024570')",
+    "PREVIOUS = Q('.00311313795')"
+  ],
+  [
+    "sum(paid.values()) == 66834",
+    "sum(paid.values()) == 66870"
+  ],
+  [
+    "'kappa_decimal':'0.00311301600'",
+    "'kappa_decimal':'0.00311325989'"
+  ],
+  [
+    "'previous_pr5718':PREVIOUS",
+    "'previous_pr5756':PREVIOUS"
+  ],
+  [
+    "Paid intermediate meetings: Ali Romman with substantial OpenAI Codex assistance.",
+    "Paid intermediate meetings: Ali Romman with substantial OpenAI Codex assistance.\nThe source R22 block is by Joel Pulikkan (GamingPuzzled).\nThe second rank-four block is by Ali Romman with OpenAI Codex assistance."
+  ]
+]

research/p104-second-low-rank-block/PROOF.md

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+# Proof and full cost
+
+## Exact local change
+
+Let C(F) be the retained native chart map for binary subspace F. A MOVE from F
+to G implements `C(G) C(F)^-1`. The source checker factors each map into its
+recursive calls and ordinary operations. It checks the signed Pauli action
+and the exact global phase. The frame induction then applies at every positive
+tensor width. A scalar instruction acts on registers in one common chart.
+Changing that chart preserves its abstract scalar action.
+
+The constructor uses the two exact runs in PLAN.json. It keeps their scalar
+instructions unchanged and assigns the rank-four chart. It rebuilds all MOVE
+instructions from the complete scalar sequence. The only changed public
+boundary charts belong to helpers 742 and 225. The source contract permits
+these free helper endpoints and charges their resulting target transforms.
+No main endpoint changes. No helper input is assumed to be zero.
+
+`verify.py` checks every scalar coefficient on all 767 public inputs and nine
+owned coordinates. It separately checks every transpose column in reverse
+chronological order. The native replay checks all maps and phases, every main
+adapter and every helper normalizer. Thus this check includes the boundary
+cost, not just the interior rank-four operations.
+
+The local paid profile is:
+
+```text
+rank:   1     2    3   4    5  6  7
+count: 2221  454  492  94  173  6 15
+```
+
+The local productive helper profile is:
+
+```text
+rank:   1  2   4  5  7   8    9
+count: 23  5  93  8  6  36  388
+```
+
+Compared with the frozen source, the paid delta is `H1+H4-H5` and the helper
+delta is `-H1+H4+H5-H8`. Paid rank mass stays 5,987. The scalar allowance stays
+224,138. There are eleven owned lifetimes and at most two live owned fields.
+
+## Physical caller and complete price
+
+The unchanged productive controller makes eighteen full local calls. It also
+pays fifteen thin queries, each of cost `104 H1`, and fifteen whole idles.
+The idle ranks are `8L`: two instances of each L from two through eight and
+one instance of L=9. Each instance costs 208 such jobs.
+
+Its main productive targets are `208 H145` and the fifteen idles with rank
+increased by one. The full target profile is this main profile plus eighteen
+copies of the local helper profile. The full paid profile is eighteen copies
+of the local paid profile plus all thin queries and all whole idles.
+The physical replay checks each contact and each central-square column.
+The full profiles appear literally in RESULT.json. They contain 13,390 targets
+and 66,870 paid calls, with rank-mass surplus 760.
+
+For a saving k, set `t=1-k`. The raw condition compares the complete moments
+`sum(T[r] r^t)` and `sum(P[r] r^t)`. Both exact rational interval engines give
+a positive margin at the tested saving `0.003113259895`. They put the raw zero
+in `(0.00311325991155, 0.00311325991156)`. The retained derivative bound is
+negative throughout `[0,0.0034]`, which gives the fixed-profile ceiling.
+The old complete source profile has a strictly negative upper margin at the
+new claimed kappa. This proves a new profile gain, not extra decimal places
+from the old profile.
+
+The completion proof uses cutoff 16, floor 17, tail offset 1/7, decay 125/128,
+power 1024 and reserve 1e-12. It bounds the whole tail, productive stock and
+normalized paid cost. The compiler checks all 54 strict inequalities and
+four functional conditions. It includes ordinary scalar operations,
+COPY/CLEAR, restoration, native adapters, thin queries, idles and routes.
+The inherited BIT saving is 0.0034. The complex cost binds the resulting
+conditional multiplication saving `0.00311325989`.
+
+`PRICING-EDITS.json` declares all edits to the archived PR #5751 pricing code:
+the claim and tested value, previous claim, full paid-call count, result label
+and attribution. The verifier requires those exact edits. It does not weaken
+any mathematical price predicate.
+
+## Failure controls and limits
+
+The replay rejects an old helper entry, an old block chart, a missing native
+MOVE, a wrong scalar coefficient, an unpaid query, an extra target credit,
+a damaged source byte and a missing central translation. The native checker
+also rejects the wrong global phase. Explicit exceptions preserve these checks
+under Python `-O`.
+
+The finite receipts and inherited frame induction remain conditional on the
+five framework assumptions listed in README.md. They do not materialize a
+uniform word or rounded multiplier. These are author checks. Independent
+review and any selected-frontier admission are separate obligations.

research/p104-second-low-rank-block/README.md

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+# Second rank-four block: conditional kappa 0.00311325989
+
+This package gives a complete conditional kappa of **0.00311325989**. It improves
+[PR #5756](https://github.com/CrocSwap/integer-mult-bounds/pull/5756)'s claim by
+**0.00000012194**. The exact raw crossing lies in
+**(0.00311325991155, 0.00311325991156)**. This is a small gain. It does not meet
+the donor's request for a much larger improvement.
+
+[PROOF.md](PROOF.md) gives the argument. [RESULT.json](RESULT.json) records the
+complete bill and checks. [SOURCES.json](SOURCES.json) pins all retained bytes.
+The base is canonical main `de5ec2e7a6074068452f45b1a191b5c72832df4a`, priority
+revision 8. The selected frontier is unchanged.
+
+Later [PR #5786](https://github.com/CrocSwap/integer-mult-bounds/pull/5786)
+claims the larger bound 0.00312265044 through retained copies and asynchronous
+hubs. This package does not supersede that result. Its new block is released
+for the separate [source-composition investigation](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19202093); no gain is obtained by adding the published decimal values.
+
+## Change and scope
+
+The exact source is PR #5756 at
+`a1b338e5d9271b541f2a198f4fc19cefc565feed`. It contains the paid-meeting changes
+from [PR #5751](https://github.com/CrocSwap/integer-mult-bounds/pull/5751) and
+Joel Pulikkan's R22 block from
+[PR #5739](https://github.com/CrocSwap/integer-mult-bounds/pull/5739).
+This package changes a different block on roles
+57, 81, 92, 224, 225, 227, 229, 234 and 742.
+
+[PLAN.json](PLAN.json) pins two scalar runs by index, length and SHA256. All 102
+operations move from chart 267 of rank eight to `span[11,21,38,448]` of rank
+four. Helper 742 starts there; helper 225 ends there. Every scalar coefficient,
+scalar order, owned event, main endpoint and port is unchanged. The constructor
+rebuilds every native MOVE, including entry and final routes.
+
+Per local call, the paid change is `H1 + H4 - H5`. The helper target change is
+`-H1 + H4 + H5 - H8`. Here `Hr` denotes a recursive job of rank r, not an
+array-volume exponent. The complete caller has 13,390 productive fields,
+66,870 paid calls and rank-mass surplus 760. A count reduction alone is not the
+source of the gain: the complete signed moment improves.
+
+The exact raw ceiling applies only to this fixed full target/paid profile.
+It leaves between `2.155e-11` and `2.156e-11` above the claim. Further useful
+progress must change that profile. No optimum over other words is claimed.
+
+## Replay
+
+Python 3.12 and the standard library are sufficient. From a checkout:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/p104-second-low-rank-block
+```
+
+The registered commands use ordinary Python and `-O`, in sequence. Each ends:
+
+```text
+PASS_P104_SECOND_RANK4_BLOCK_KAPPA_00311325989
+```
+
+A source export needs no Git, network or solver. In its package directory:
+
+```sh
+python3.12 -I -B verify.py
+python3.12 -I -O -B verify.py
+```
+
+The source word is frozen, not regenerated from the PR #5756 search. The new
+word is regenerated from that frozen word. All 602,176 scalar coefficients and
+a separate dense transpose replay are checked. The replay also checks 2,934
+native maps with their absolute phases, 208 main adapters, 1,118 helper
+normalizers, 7,488 physical contacts and 53,248 central-square columns. Two
+rational moment engines check completion and the full bill. All ordinary
+COPY/CLEAR, scalar, restoration and route charges remain in the inherited
+compiler proof. See [VALIDATION.md](VALIDATION.md) for the run record.
+
+## Conditions and limits
+
+The five inherited framework assumptions remain: `native_all_columns`,
+`owned_erasure`, `compiler`, `precision_tape` and `analytic_recovery`.
+This is a complete conditional price in that framework. The uniform word and
+rounded integer multiplier are not materialized. No independent human review,
+proof-kernel acceptance or selected-frontier change is claimed.
+
+## Open next questions
+
+[TASK.json](TASK.json) gives the next questions and their first checks. A larger
+gain needs a new full profile. Independent review must still assess this exact
+source closure, word and price. Check current Discussions before taking a new
+scalar, helper or caller scope.
+
+## Credit
+
+Ali Romman with substantial OpenAI Codex assistance supplied this second block,
+its constructor and this replay package. No subagents were used. Apache-2.0.
+Joel Pulikkan (GamingPuzzled), with Claude Opus 5.5 assistance, supplied the R22
+reframe mechanism and the different block retained from PR #5756. We do not
+claim that mechanism as new.
+
+The retained PR #5751 source supplies the exact local, native, caller and price
+checks. Its source chain includes flyingviper's distinct-boundary work in
+[PR #5718](https://github.com/CrocSwap/integer-mult-bounds/pull/5718), the earlier
+P104 helper work, and the controller, compiler and analytic work.
+[NOTICE](NOTICE) preserves their notices. Exact source commits, blob IDs and
+SHA256 hashes are in SOURCES.json. The nested archives are unchanged.