Conditional κ = 0.00310815849 (+1.39516e-6 over #5660): paid interior bases

research/p104-paid-interior-bases/NOTICE

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+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.

research/p104-paid-interior-bases/PRICING-EDITS.json

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+[
+  [
+    "Q('.00310676333'), Q('.003106763335')",
+    "Q('.00310815849'), Q('.003108158495')"
+  ],
+  [
+    "PREVIOUS = Q('.00310644469')",
+    "PREVIOUS = Q('.00310676333')"
+  ],
+  [
+    "== 66474",
+    "== 66402"
+  ],
+  [
+    "'kappa_decimal':'.00310676333'",
+    "'kappa_decimal':'.00310815849'"
+  ],
+  [
+    "previous_open_pr5651",
+    "previous_open_pr5660"
+  ],
+  [
+    "Ali Romman PR5644. The four-gate change is by Ali Romman with Codex assistance.",
+    "Ali Romman PR5644. The four-gate change is by Ali Romman with Codex assistance.\nThe four interior basis changes are by Ali Romman with Codex assistance."
+  ]
+]

research/p104-paid-interior-bases/PROOF.md

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+# Exact construction and complete price
+
+## 1. One temporary basis
+
+Let `a` and `b` be distinct public registers. Set `R = I + c E_ab` for a
+nonzero dyadic scalar `c`. Then `R^-1 = I - c E_ab`.
+
+Choose two internal cuts where both registers can reach one common chart.
+Insert `R` at the first cut. Between the cuts, replace each scalar block
+`M_j` by `R M_j R^-1`. Insert `R^-1` at the second cut. The complete interval
+is unchanged:
+
+```text
+R^-1 (R M_k R^-1) ... (R M_1 R^-1) R = M_k ... M_1.
+```
+
+A block that does not use either register commutes with `R`. The same is
+true for each owned initialization and erasure, since `a` and `b` are public.
+The checker requires one certificate for every affected block. It checks
+the exact old event list and every rational matrix coefficient. It also
+rejects extra certificates. Thus this is an identity for arbitrary inputs,
+not a test on selected vectors.
+
+Each scalar block has one physical chart. The compiler moves each operand
+to that chart before it is used. At every step, its physical value is the
+declared native chart applied to its logical value. A scalar basis change
+at a common chart therefore implements the same rational matrix `R`.
+Native transport changes the chart and preserves the logical value.
+The final basis removal restores the original logical interval.
+
+## 2. Fixed witnesses
+
+The four certificates are applied in the order in WITNESS.json. Block
+indices refer to the complete word produced by the preceding step.
+
+| Public pair | Coefficient | Local paid delta |
+| --- | ---: | --- |
+| 438, 733 | -4 | `-H1-H2+H3` |
+| 119, 125 | -1 | `-2H1+H2` |
+| 125, 128 | -1 | `-2H1+H2` |
+| 128, 139 | -1 | `-2H1+H2` |
+
+There are eight changed scalar blocks. Each certificate gives the full
+before and after lists. The new contact graph permits four fewer local
+native calls. The transport builder includes both meeting charts and all
+remaining scalar contacts. It does not assign free movement to a basis
+operation. All public paths remain nested.
+
+The separate endpoint checks cover 743 arbitrary public inputs and nine
+owned register slots. There are eleven owned lifetimes and peak occupancy
+two. Every owned instruction and transport stays in the same order. All
+535 productive helper targets and all public boundary charts stay fixed.
+
+## 3. Native and caller binding
+
+The retained compiler lowers each chart change to a signed Gaussian-dyadic
+recipe. A separate interpreter checks every signed Pauli image and the
+global phase. It checks 2,918 distinct maps and 208 main endpoint adapters.
+The helper normalizers retain the exact productive target profile.
+
+The p16/a7 caller is unchanged. Its physical binding checks all 7,488
+contacts and all 53,248 central-square columns. It has 18 full forward
+calls, 15 thin query shears and 3,120 paid whole-idle jobs. The 3,328 main
+targets and 9,630 helper targets give 12,958 productive fields.
+
+These exact finite identities use the retained factorwise tensor rule.
+They do not add a new analytic or all-size admission assumption. The five
+existing framework assumptions and helper endpoint contract remain.
+
+## 4. Complete cost
+
+For `H_r(t) = r^t` and `0 < t < 1`, the paid change per local call is
+
+```text
+delta P = -7 H1 + 2 H2 + H3,
+delta T = 0.
+```
+
+Strict concavity gives `1+2^t>3^t` and `2>2^t`. Thus the complete signed
+moment increases by `18 * (7 - 2*2^t - 3^t) > 0` throughout that interval.
+The new bill has 66,402 paid calls and rank surplus 760. The local paid rank
+mass remains 5,986; the helper rank mass remains 4,266. Their difference is
+the 1,664 main progress plus 56 owned descent debt.
+
+Two rational interval engines verify the full moment and put its unique
+raw crossing in `(0.00310815850980, 0.00310815850981)`. The claim is
+`0.00310815849`, and the tested saving is `0.003108158495`.
+
+All scalar changes are priced. The local scalar allowance is 220,290 and
+the route allowance is 5,167,000. The retained complete raw ordinary bound
+is 184,647,552, below its 1,000,000,000 budget. The power-1024 compiler,
+cutoff 17, floor 18, tail parameter 1/5, reserve 10^-12 and decay 39/40 are
+unchanged. All 54 strict and four functional outer conditions pass.
+
+## 5. Verification limits
+
+The full source archive is authenticated before any retained module is
+loaded. The nested predecessor archive is authenticated again before its
+local or native checker is loaded. The new checker verifies the source
+word and every intermediate complete word. Its separate dense transpose
+engine checks 565,504 entries.
+
+False basis signs, false block coefficients, a missing block certificate,
+a missing native transport, a missing owned erasure, an unpaid query and
+extra target credit are rejected. The inherited phase, caller and price
+controls also run.
+
+The huge uniform word and rounded multiplier are algorithm-defined and
+are not materialized. These are author checks, not independent scientific
+acceptance or a proof-kernel theorem. The numerical improvement is small.
+No large gain or global optimum is established.

research/p104-paid-interior-bases/README.md

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+# Paid interior bases for the P104 word
+
+Complete conditional kappa is **0.00310815849**, a gain of **0.00000139516**
+(about **0.04491%**) over
+[PR #5660](https://github.com/CrocSwap/integer-mult-bounds/pull/5660).
+This is a small gain. A large improvement and a global optimum remain open.
+
+Four temporary changes of the public register basis remove paid contacts.
+The basis changes occur at internal meeting charts. All transports to those
+charts are included. The source and output interfaces stay fixed.
+The complete local delta is `-7 H1 + 2 H2 + H3`. Across 18 full calls, this
+removes 72 paid requests. The complete bill has **66,402 paid calls**,
+**12,958 productive targets**, and rank surplus **760**.
+
+[PROOF.md](PROOF.md) states the identity and the complete cost.
+[WITNESS.json](WITNESS.json) contains all eight scalar block certificates.
+[RESULT.json](RESULT.json) is the full receipt.
+[SOURCES.json](SOURCES.json) pins every consumed source.
+
+## Replay
+
+Python 3.12 and the standard library suffice. From the repository root:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/p104-paid-interior-bases
+```
+
+The hook runs normal and optimized checks. The expected final marker is:
+
+```text
+PASS_P104_PAID_INTERIOR_BASES_KAPPA_00310815849
+```
+
+The direct commands also work in a source export with no Git or network:
+
+```sh
+python3.12 -I -B research/p104-paid-interior-bases/verify.py
+python3.12 -I -O -B research/p104-paid-interior-bases/verify.py
+```
+
+The checks authenticate the 23-file PR5660 export and its 36-file predecessor
+export before source imports. They check every exact block identity, all
+565,504 transpose coefficients, 2,918 signed native maps, the physical caller,
+both rational moment engines, all compiler costs, and all 58 outer conditions.
+See [VALIDATION.md](VALIDATION.md) for the exact checks that ran.
+
+## Scope and limits
+
+The five inherited framework assumptions and the helper endpoint contract
+remain. The huge uniform program and complete rounded multiplier are not
+materialized. Author checks are not independent scientific acceptance.
+The selected frontier is unchanged.
+
+The source is PR5660 at `3b0b4f37d5005f381ed9cfcccebfbe1ab35c71aa`.
+The branch starts from canonical main `df0137efd35e11b4445734bf612bdd4c7ebc330c`,
+priority revision 7. The complete source is retained, including all notices.
+
+## Open next questions
+
+1. Can an independent reviewer confirm the block identities and full bill?
+2. Can a larger temporary basis or a different scalar schedule give a large
+   gain after all meeting transports and ordinary work are paid?
+3. Can one of the inherited framework assumptions be resolved?
+
+The finite discovery searches do not prove a general obstruction. Further
+work must check live Discussions and pin its complete input word.
+
+## Credit
+
+[PR #5660](https://github.com/CrocSwap/integer-mult-bounds/pull/5660) supplies
+the complete input and the four-gate contact rewrite. It retains zxv's
+[PR #5643](https://github.com/CrocSwap/integer-mult-bounds/pull/5643) A2 change,
+the [PR #5644](https://github.com/CrocSwap/integer-mult-bounds/pull/5644) helper
+relations, and the [PR #5650](https://github.com/CrocSwap/integer-mult-bounds/pull/5650)
+composition. The source notices retain the P104, side-correction, helper,
+native, caller, productive-recursion, interval and compiler authors.
+The elementary change-of-basis identity is not claimed as a new theorem.
+
+The paid interior construction and this verification package are by
+Ali Romman with substantial OpenAI Codex assistance. No subagents.
+Apache-2.0. See [NOTICE](NOTICE) for the full retained attribution.