Improve P104 helper split with distinct input and output bases

research/p104-two-boundary-helper-split/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.
+
+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-two-boundary-helper-split/PRICING-EDITS.json

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+[
+  [
+    "sum(targets.values()) == 12958 and sum(paid.values()) == 66348",
+    "sum(targets.values()) == 12976 and sum(paid.values()) == 66348"
+  ],
+  [
+    "KAPPA, TESTED = Q('155457611/50000000000'), Q('124366089/40000000000')",
+    "KAPPA, TESTED = Q('31102457/10000000000'), Q('622049141/200000000000')"
+  ],
+  [
+    "PREVIOUS = Q('.00310871312')",
+    "PREVIOUS = Q('.00310915222')"
+  ],
+  [
+    "'kappa_decimal':'0.00310915222'",
+    "'kappa_decimal':'0.00311024570'"
+  ],
+  [
+    "'previous_open_pr5660'",
+    "'previous_open_pr5713'"
+  ]
+]

research/p104-two-boundary-helper-split/PROOF.md

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+# Two boundary functionals for helper708
+
+## Source and scope
+
+This construction starts from PR5713 head
+`a5348f7143f95fe0046b92b96172e30763c7d731`, package
+`p104-temporal-updates`, canonical word SHA256
+`32adce6656e49aa020d3e6317f1c45c11fccadeeb67a8d506172d101703d2138`.
+Its source lineage and authors remain credited. The new construction changes
+scalar dependencies and adds one independent public helper while retaining
+every paid MOVE. It retains nine owned labels, eleven lifetimes and peak two.
+All owned identifiers increase by one to make room for the new public field743.
+
+The resulting word has744 independent public fields and nine dirty owned input
+columns. Its canonical SHA256 is
+`a1c8277189a811546bdc7b147d72e6a594504ff9585c9bf59a5681aa873abb01`.
+The exact constructor and independent checks below are finite checks; the
+publication verifier separately binds native maps, the complete caller,
+ordinary allowances and the numerical pricing claim.
+
+## Initial basis and retirement
+
+Use the cut immediately after source event14330. Roles708 and154 share frame602,
+of rank2. Let A be the scalar prefix, B its suffix, T=BA, e=e708, and define
+
+```
+a = e708^T A,        c = e154^T A / 2,
+h = a+c,            d = B e708 = -2e126+2e188+e708,
+k = h-e708^T.
+```
+
+Exact replay verifies that T has identity input columns and output rows for
+helpers126,188,708. It also gives h708=d708=1, hd=1 and k708=kd=0. Every term of h has its
+initial value at the empty frame. Its only main-input terms are
+`(x154+x217)/4`. The covector c has zero coefficients on126,188,708.
+
+Apply the ordinary initial basis change
+
+```
+Q = (I+(d-e)e^T)(I-e k) = I+(d-e)e^T-d k.
+```
+
+The first factor is implemented after the second:87 initial ADDs
+`708 -= k_r*r`, then `126 -=2*708` and `188 +=2*708`. All operands share the
+empty frame, and the main fields are read but unchanged. The identities
+`hQ=e^T`, `Qe=d` and `cQ=c` follow directly.
+
+At the cut, `708 +=154/2` restores exactly x708. Retire original708 at rank2.
+A new independent helper starts in frame602 with arbitrary input z and replaces
+every subsequent occurrence of original708. No subtraction of154/2 is applied
+to this new helper, and no MOVE is inserted or removed.
+
+## Output normalization at an existing meeting
+
+Before retirement the transformed continuation value was `aQ=e^T-c`. Replacing
+it by z injects `d*(z-e^T+c)` into the suffix. Combining this with TQ gives the
+following three dirty full-frame values, with every other endpoint unchanged:
+
+```
+126 = x126 - 2*t,
+188 = x188 + 2*t,
+new = t = z-k+c.
+```
+
+The output nuisance is `g=k-c=a-e708^T`. It differs from the input nuisance k.
+An independent backwards calculation expresses g using118 full-frame endpoint
+fields. This provides a direct exact endpoint check, but the final program
+uses the shorter current-version identity below.
+
+Original708 has only one destination update after the cut:
+`708 +=146/2` at source event17433, at frame127 of rank9. Because the original
+endpoint is x708, the exact value of146 at that event satisfies
+`g=-value(146)/2`. In the candidate the renamed update is event17523,
+`new +=146/2`. Immediately after it, insert these three ordinary operations:
+
+```
+126 += 2*new
+188 -= 2*new
+new -= 146/2
+```
+
+The first two remove the126/188 defects at this point; later ADDs still build
+their final values. The third restores new to z. Their order is
+necessary: applying the inverse146 update first leaves errors at126 and188.
+The literal suffix has no later source reads from126,188 or new, and no later
+write to new. Later writes to126 and188 are only ADDs at frame127. Consequently
+the first two normalizations commute with that suffix, and the source146 is
+read at precisely its required live version. A separate covector check verifies
+that this146 value is unchanged by the initial basis and raw-z replacement: its
+126/188/708 input coefficients and its response to the cut perturbation vanish.
+
+The pure constructor first builds the equivalent terminal normalization and
+then performs this checked substitution: three early operations replace the
+two terminal restorations plus118 final correction operations. No algebraic
+search, historical-value assumption, or new transport is used.
+
+## Complete signed invoice
+
+The paid histogram is unchanged. Original708's H9 productive target is replaced
+by its retired H2 target and the new independent helper's H7 target:
+
+```
+Delta paid = 0
+Delta targets = H2+H7-H9
+Delta(targets-paid) = H2+H7-H9 > 0.
+```
+
+For `H_r=r^t` and0<t<1, strict subadditivity gives `2^t+7^t>9^t`. Rank mass
+is unchanged. The eighteen local calls multiply this delta by18. The complete
+paid occurrence count stays66348 and the productive count increases to12976;
+the complete rank surplus remains760. The extra independent helper's ordinary
+routing and allocation costs are still charged.
+
+The final local word has25546 events and scalar allowance221474. Its paid
+histogram is `{1:2184,2:459,3:495,4:92,5:159,6:22,7:15}` and its productive
+histogram is `{1:2,2:5,4:91,5:6,7:6,8:15,9:411}`.
+
+## Finite verification
+
+The pure constructor accepts only the pinned canonical source and returns
+the exact final candidate plus a witness. The independent literal checker
+verifies all753 arbitrary input columns, frames, owned initialization/use/clear
+rules, both complete local histograms and the ordinary scalar count. A separate
+backwards implementation verifies all753 transpose columns.
+
+Ten actual mutations are rejected under ordinary Python and `python -O`:
+missing retirement; either omitted early restoration; omitted or wrong-sign146
+inverse; incorrect cleanup order; reinstated obsolete decoder; omitted owned
+identifier shift; overstated old helper target; and wrong new helper boundary.
+
+Independent agents separately derived the88-term h and118-term output
+functional and checked the temporal simplification. These are independent
+agent calculations, not independent human review or proof-kernel acceptance.
+Prepared for flyingviper with substantial OpenAI Codex assistance. Apache-2.0.

research/p104-two-boundary-helper-split/README.md

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+# A P104 helper split using different input and output bases
+
+This package improves the complete conditional exponent saving from
+**0.00310915222 to 0.00311024570**, building on Ali Romman's
+[PR5713](https://github.com/CrocSwap/integer-mult-bounds/pull/5713), head
+`a5348f7143f95fe0046b92b96172e30763c7d731`.
+The increase is 0.00000109348, approximately 0.0352% over that source.
+This is a modest improvement; the selected frontier is unchanged.
+
+The initial helper basis can use main inputs at zero-dimensional frames.
+Those inputs have arbitrary values: they are not initialized zeros. An actual
+coframed relation later restores the old helper and permits a new independent
+helper to take its place. A different, legal output combination restores the
+remaining fields. Requiring the same combination at both boundaries would
+miss this split. The construction adds no paid recursive transport.
+
+## Complete cost
+
+For `H_r = r^(1-kappa)`, one local call changes the signed invoice by
+
+```
+paid:          unchanged
+targets:       +H2 +H7 -H9
+targets-paid:  H2 +H7 -H9 > 0.
+```
+
+Strict subadditivity proves the positive gain for `0 < kappa < 1`. All added
+ordinary work is separately funded. The complete caller uses eighteen LOCALs.
+
+| Quantity | Verified value |
+|---|---:|
+| Conditional kappa | 0.00311024570 |
+| Strict raw crossing enclosure | (0.00311024572012, 0.00311024572013) |
+| Complete paid jobs | 66,348 |
+| Complete productive fields | 12,976 |
+| Complete rank surplus | 760 |
+| Local independent inputs / owned labels | 744 / 9 |
+| Local owned lifetimes / peak simultaneous | 11 / 2 |
+| Distinct native maps checked | 2,917 |
+| Local scalar / routing allowance | 221,474 / 5,166,000 |
+
+Original helper708 retires at rank two. New helper743 starts there and follows
+the original continuation to rank nine. All nine owned identifiers shift by
+one, without a new lifetime. The 25,546-instruction word contains 89 initial
+basis additions, one retirement addition and three cleanup additions. At the
+cleanup contact, the exact current146 value replaces a 118-term terminal
+functional; the proof verifies both its version and the required order.
+
+## Reproduction and checks
+
+```sh
+python3 -B scripts/research_package.py --package research/p104-two-boundary-helper-split
+python3 -B scripts/check_research_commit.py --package research/p104-two-boundary-helper-split --check-source-history
+```
+
+Expected marker: `PASS_P104_TWO_BOUNDARY_SPLIT_KAPPA_00311024570`.
+The registered normal and optimized commands check the exact construction,
+all 753 arbitrary-input columns, separate dense transpose, actual binary
+coframes and owned intervals, every native map and absolute phase, and the
+complete physical caller. The price replay uses two exact moment engines,
+compiler completion and all 54 strict plus four functional outer predicates.
+Complete WORD, WITNESS and RESULT payloads are compared; compression container
+bytes are not used as mathematical equality.
+
+Ten actual instruction/boundary mutations are rejected, covering missing or
+wrong retirement/cleanup, reversed cleanup order, the obsolete fresh decoder,
+wrong helper boundaries and an omitted owned-identifier shift. Two further
+invoice controls reject an unpaid query and an invented target. Inherited
+caller, phase, cutoff and BIT controls remain active. See [PROOF.md](PROOF.md).
+
+Separate agents derived the input basis and output correction, verified the
+exact intermediate146 value, and replayed all scalar columns and ownership
+without importing the constructor. Compact receipts are retained in `evidence/`.
+The final caller-audit and publication checks are reported in
+[VALIDATION.md](VALIDATION.md). Independent AI checks are not independent human
+or formal mathematical acceptance.
+
+## Source custody, credit and limits
+
+Main provenance is `df0137efd35e11b4445734bf612bdd4c7ebc330c`.
+The exact 21-file PR5713 export is authenticated before source imports; all
+nested source checks remain active. Git-history metadata binds the archive
+retained in this branch; `archived_files` separately preserves its exact
+origin-head and payload pins. Replay needs no cached pending-PR Git objects. The complete price adapter differs only
+by the explicit substitutions in `PRICING-EDITS.json`. Frozen predecessors and
+shared framework files are unchanged.
+
+Credit Ali Romman's PR5713 temporal update and predecessor proofs, Joel
+Pulikkan/GamingPuzzled's PR5684, the retained helper-split constructions, and
+the caller, native and pricing authors listed in NOTICE. Chris R/Clear1nsight's
+PR5550 and the maintainer's PR5599 review supply the separate caller lineage.
+The elementary linear basis identities are standard; the contribution is this
+source-specific construction, legal schedule and complete invoice. Prepared
+for flyingviper with substantial OpenAI Codex assistance, including separate
+verification and literature agents. Apache-2.0.
+
+The native all-width, owned erasure, compiler, precision/tape and
+analytic/recovery assumptions remain. COPY is an ADD into an ideally initialized
+owned zero with numerical residuals retained. No complete rounded multiplier
+or huge uniform word is materialized. This package claims neither a large gain,
+a general optimality result, nor admission to the selected frontier.
+
+## Open next questions
+
+- Independently inspect the two boundary functionals and early cleanup order;
+  both matter to the endpoint identity.
+- Find another pair of legal boundary functionals at an actual coframed cut,
+  with the complete native, helper and ordinary bill included.
+- Audit one remaining source-specific framework assumption before promoting
+  a complete supplier into the selected result.
+
+Concrete first steps and blockers are recorded in `TASK.json`.