Lift four P104 helper inputs: conditional κ 0.003129111

research/p104-native-input-lifting/NOTICE

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+Native helper input lifting
+Copyright 2026 Ali Romman. Apache-2.0.
+
+New construction, code, calculations and prose: Ali Romman with substantial
+OpenAI Codex assistance. No subagents. All new checks are author checks.
+
+The direct source is PR6314 at 36533d4d1b4e2dd8672427461b12de3001b85810,
+by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.
+It compresses the scalar word of PR6167 without a kappa change.
+The four native input lifts and their complete conditional price are new here.
+The basis identity is elementary linear algebra; no new general theorem
+for that identity is claimed.
+
+The PR6167 source notice follows unchanged. Its nested source archive also
+retains the earlier notices.
+
+Retained-operand descent: Ali Romman, with substantial OpenAI Codex assistance.
+Apache-2.0.
+
+The exact source word and price are from DreamingOfClouds, PR6144 and PR6143,
+with Claude Opus 5.5 (Anthropic), via Claude Code. They extend PR6030 and PR5907.
+The source reader is adapted from that work. The gzip portability controls
+retain Douglas Colkitt and Maxime Fleury credit.
+
+The pinned PR5827 checker and all earlier source notices are retained below.
+
+Unread-register path resynthesis
+Copyright 2026 Ali Romman. Apache-2.0.
+
+New construction, code, calculations and prose: Ali Romman with substantial
+OpenAI Codex assistance. No subagents. All new checks are author-side.
+
+Mason Hale supplied PR5811, the complete two-block composition and the exact
+role92 remapping and fourth accumulation. The earlier elementary accumulation,
+retained-copy and second-block work is by Ali Romman. Joel Pulikkan supplied
+R22. Dugongue supplied the concurrent baseline composition proof. Their source
+notice and proof are retained byte for byte under inputs/pr5811-*.
+
+The source closure retains flyingviper, Cinna Davis, David Arias, Gareth
+Davidson, Jacob Alan Sussman, zxv, Cao Yifan, Pierre-Loic Doulcet, thu-nmrc,
+Scratch-net and all native/helper/controller/compiler/geometry/analytic
+predecessors. The exact PR5786 source archive and its nested notices remain.
+This package does not claim any predecessor construction as new.
+
+Exact PR5811 vendor/accumulated/NOTICE follows. Source head:
+33d7760d0a6fea6394aefd014eea68f0af02a92e
+
+Accumulated native updates
+New work: Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+No subagents. The same assistant made the construction and the author checks.
+
+Concurrent scope: Discussion5125 now tests retained copies in the transposed
+supplier. That route uses the same source/destination duality. This package
+claims the three fixed forward-supplier groups and their complete result,
+not the elementary accumulation identity or the broader transposed search.
+
+Exact source: PR5786 at 6b936acd68a7e0f9062037900f0fe9bc4b6fdd25.
+The complete predecessor notice follows unchanged.
+
+Asynchronous helper hubs
+New work: Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+
+Retained native copies from PR5774 at 7d6ec0cdef31d0c37b7d2563e709979caa0d5eae.
+All prior credit 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.
+
+Retained native-copy construction
+Prepared by Ali Romman with substantial OpenAI Codex assistance. No subagents.
+The paid-meeting source is PR5751 by Ali Romman with OpenAI Codex assistance.
+The dimension-four span [265,194,164,75] is from Joel Pulikkan / GamingPuzzled
+PR5739, with Claude Opus 5.5 assistance. The R22 block edits are not included.
+The retained-copy mechanism, exact constructor, source-specific verification,
+new price and proof are the contribution of this package. Apache-2.0.
+
+Exact PR5811 vendor/blocks/NOTICE follows. Source head:
+33d7760d0a6fea6394aefd014eea68f0af02a92e
+
+The second block is PR5796 at4b30391e633ba0c7c674f03586ae97d443e9bc9c, by Ali Romman with substantial OpenAI Codex assistance, following Joel Pulikkan/GamingPuzzled R22. Exact notice, plan, proof and source words are retained in source/SECOND-* unchanged.
+
+The final immediate source is PR5786 asynchronous helper hubs by Ali Romman with substantial OpenAI Codex assistance. Its three temporal helper splits and complete completion parameters are retained and credited. The earlier PR5774-only composition is a superseded checkpoint, not a separate contribution.
+
+New complete retained-copy and R22 composition: Ali Romman with substantial OpenAI Codex assistance. No subagents. Apache-2.0.
+
+The thirteen retained native conversions and changed helper targets are by Ali Romman with OpenAI Codex assistance in PR5774. The R22 block and rank-four basis are by Joel Pulikkan (GamingPuzzled), with Claude Opus5.5 assistance, in PR5739/5756. The complete source archive and R22 notice remain unchanged.
+
+The construction retains Ali Romman PR5751 paid intermediate meetings, flyingviper PR5718 boundary splits, Cinna Davis's productive controller, Jacob Alan Sussman's productive recursion, zxv and Cao Yifan's interval/compiler line, and every P104/native/BIT predecessor in the source notices. The source engines are reused under exact byte bindings, not claimed as new implementations.

research/p104-native-input-lifting/PRICING-EDITS.json

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+[
+  [
+    "KAPPA, TESTED = Q('0.003128645'), Q('0.003128650')",
+    "KAPPA, TESTED = Q('0.003129111'), Q('0.003129116')"
+  ],
+  [
+    "PREVIOUS = Q('0.003128386')",
+    "PREVIOUS = Q('0.003128645')"
+  ],
+  [
+    "sum(paid.values()) == 66762",
+    "sum(paid.values()) == 66780"
+  ],
+  [
+    "'kappa_decimal':'0.003128645','previous_pr6144':PREVIOUS",
+    "'kappa_decimal':'0.003129111','previous_pr6167':PREVIOUS"
+  ]
+]

research/p104-native-input-lifting/PROOF.md

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+# Native helper input lifting
+
+## Claim and source
+
+Under the five inherited framework assumptions, the emitted supplier and
+the unchanged p16/a7 caller give kappa `0.003129111`. The comparison is
+PR6167's complete claim `0.003128645`. The direct word is PR6314's scalar
+compression of PR6167. The source has 17,445 events. The new word has 17,452.
+
+Write `Z` for the zero subspace, frame ID 3, and `Omega` for the full
+nine-dimensional space, frame ID 127. Frame ID 0 is not `Z`.
+All frame labels below are literal IDs in the emitted word.
+Let `H_r = r^(1-kappa)` and `H_0 = 0`.
+
+## Basis identity
+
+Let `A` be the source preparation map from the inputs in `Z` to their scalar
+values at their first departures. A helper direction `h` has support only on
+helpers that start in `Z` and finish in `Omega`. Choose a primary helper `p`
+with `h_p = 1`. In each construction, the preparation never writes to `p`.
+Let `a = A h`. Suppose the first destination frame for each nonzero component
+of `a` contains one subspace `F`.
+
+Define
+
+```text
+B = I + (h - e_p) e_p^T.
+B^-1 = I - (h - e_p) e_p^T.
+```
+
+The square of `(h - e_p) e_p^T` is zero. Thus `B` is invertible and has
+determinant one. It changes only helper inputs. The source endpoint is the
+identity on these helpers, so it commutes with `B`.
+
+Start the existing independent input `x_p` in frame `F`. Omit its uses in the
+initial preparation. This gives the response `A(x - e_p x_p)`. Deliver the
+response `a x_p` at the stated later common frames. The sum is `A B x`.
+Apply `B^-1` at the full output frame. This restores the source endpoint for
+every input. No helper input is set to zero or replaced by a copy.
+
+This identity alone does not establish a legal schedule. The constructor
+states every delivery, move and output correction. The replay checks the
+actual source frame of every operand and all independent input columns.
+
+## Four directions and schedules
+
+The four supports of `h` are disjoint. Their basis matrices commute.
+Every coefficient is dyadic.
+
+| Primary | New initial frame and basis | Input direction `h` | Departure response `a` |
+| --- | --- | --- | --- |
+| 496 | 595, `[52]` | `e496 - e639 - e643/2` | `h` |
+| 649 | 597, `[82]` | `e649 - e648` | `h + e198` |
+| 663 | 105, `[51]` | `e663 - e665` | `h + e179/2` |
+| 508 | 1093, `[30,42,384]` | `e508 - e631 - e635/2` | `h` |
+
+The verifier computes each response from the full scalar chronology.
+It does not accept the displayed responses without a check.
+Frame 1093 is the intersection of frames 118 and 733. It has dimension three.
+Each other listed common frame has dimension one.
+
+For primary 496, roles 639 and 643 first move from `Z` to frame 1027.
+Role 496 first reaches frame 595 in the source. Replace these movements by:
+
+```text
+at Z:     639 -= 2 * 643
+          move 639 from Z to 1027
+          move 643 from Z to 595
+at 595:   643 -= 496 / 2
+          move 643 from 595 to 1027
+at 1027:  639 += 2 * 643
+```
+
+Role 496 now starts at 595, so its old `Z -> 595` move is removed.
+If roles 639 and 643 hold arbitrary values `v` and `z`, the displayed block
+leaves `v - x496` and `z - x496/2`. The subtraction at `Z` removes the
+carrier's old value from the later delivery. That value is not assumed zero.
+
+Use the same block for primary 508 with roles `(631,635)`, destination 118
+and intermediate frame 1093. Role 508 starts at 1093 and then moves to its
+old first destination 733. No new owned field is used.
+
+For primary 649, add its value to role 198 when role 198 reaches frame 597.
+Move role 648 to frame 120 before role 649 arrives there. Move 649 from 597
+to 120 and subtract it from 648. The original later move of 648 is removed.
+The value of 649 is still its independent input at this subtraction.
+
+For primary 663, move role 179 to frame 105 before role 663 leaves it and
+add half of 663 to 179. Move role 665 to frame 1051, then move 663 from 105
+to 1051 and subtract it from 665. Remove the two original later moves.
+All later reads of roles 179 and 665 remain in their source order.
+
+At frame 127, emit the six inverse-basis additions:
+
+```text
+639 += 496; 643 += 496/2; 648 += 649;
+665 += 663; 631 += 508; 635 += 508/2.
+```
+
+`construct.py` guards all source event indices and all ten omitted initial
+uses. It changes four initial frame declarations, appends frame 1093,
+replaces twelve movement sites and appends the six output additions.
+Every other event is copied. Protected contact events and frames 1091/1092
+remain fixed.
+
+## Exact scalar and native checks
+
+The unchanged checker replays all 770 arbitrary public inputs and all 29 owned
+columns. Each helper has its required independent output. Each main output
+has its required input and shear contribution. Every owned lifetime is
+initialized and cleared at its declared frame. Peak ownership is 13.
+
+A separate reverse-chronology checker uses dense integer rows with binary
+denominators. It checks all 799 columns and 638,401 endpoint coefficients.
+The source and dense checks use different representations and directions.
+Both are author checks made with credited predecessor code.
+
+The native checker lowers every distinct chart change to a signed Clifford
+recipe. A separate exact phase representation checks its signed Pauli images
+and global phase. The 2,930 distinct maps include the changed helper
+normalizers. It also checks 208 free main adapters and 1,124 helper
+normalizers. Its tensor argument applies at each positive tensor width.
+No free native transition is inferred from scalar endpoint equality.
+
+Negative controls omit a carrier subtraction, reverse an injection, omit an
+output correction, use the wrong input chart or intersection, or omit a
+carrier move. Each fails the scalar or frame check. Native phase, caller
+translation, missing price, missing target debit and damaged source controls
+also fail. Explicit checks remain active with Python `-O`.
+
+## Paid and productive profiles
+
+Each of the three one-dimensional lifts has paid change `H4 - H5` and
+target change `H8 - H9`. Its signed margin gain is
+
+```text
+H5 + H8 - H4 - H9.
+```
+
+This is positive for `0 < kappa < 1`: strict concavity gives
+`H5 - H4 > H9 - H8`.
+
+The three-dimensional lift has paid change
+`H1 + H2 + H3 - H4 - H5` and target change `H6 - H9`.
+Its signed margin gain is
+
+```text
+H4 + H5 + H6 - H1 - H2 - H3 - H9.
+```
+
+The exact complete price establishes the required total gain at the tested
+kappa. No claim of positivity for this last expression at every kappa is needed.
+
+The combined local paid change is
+`H1 + H2 + H3 + 2 H4 - 4 H5`. The target change is
+`H6 + 3 H8 - 4 H9`. Both rank totals decrease by six.
+The general concave-order preflight is inconclusive for the complete bill:
+its paid-minus-target hinge change is 18 at cap 1 and -108 at cap 5.
+The claim therefore uses the exact power-moment checks below.
+
+| Rank | Local paid jobs | Local productive targets |
+| --- | ---: | ---: |
+| 1 | 2217 | 14 |
+| 2 | 457 | 8 |
+| 3 | 497 | 4 |
+| 4 | 100 | 107 |
+| 5 | 161 | 14 |
+| 6 | 8 | 3 |
+| 7 | 9 | 9 |
+| 8 | 1 | 22 |
+| 9 | 0 | 381 |
+
+The paid rank total is 5946. The scalar allowance is 127088. There are
+770 independent public inputs, 29 owned labels and 32 owned lifetimes.
+
+## Complete caller and price
+
+The source p16/a7 caller is unchanged. Its literal Laurent replay checks all
+boundary programs and all 18 forward local calls. The physical replay checks
+7,488 contacts, all 53,248 central-square columns, main target phases and
+every whole idle. Let `P_local` and `T_local` denote the table above. The
+complete invoice is
+
+```text
+P = 18 P_local + 1560 H1
+    + 208 sum_(l in {7,8}) sum_(j=1..l) H_(8(j+1)),
+T = 18 T_local + 208 H145
+    + 208 sum_(l in {7,8}) sum_(j=1..l) H_(1+8(j+1)).
+```
+
+It has 66,780 paid calls and 13,444 productive fields. Its rank surplus is
+760. The maximum paid and productive ranks are 72 and 145. Two independent
+censuses agree with the emitted caller. The target debit is included in both.
+
+The strict raw crossing is in
+`(0.00312911821812, 0.00312911821813)`. A rational derivative bound gives
+uniqueness below the retained BIT cap `0.0034`. Two rational moment engines
+give compatible intervals and a strict positive margin at
+`TESTED = 0.003129116`. The old complete profile has a negative margin at
+`KAPPA = 0.003129111`.
+
+The inherited strictification uses power 1024, cutoff 16, floor 17, tail
+offset `1/7`, reserve `10^-12` and decay `125/128`. All completion, tail,
+splitting and rounding inequalities pass with the new invoice. The compiler
+gap is `1/1400000000`. The integer threshold exponent is 47005 and
+`H = 65807000000000`, so its declared uniform threshold is `N = 2^H`.
+The predecessor integer threshold fails the same test. No such large word
+is materialized.
+
+The local scalar allowance plus the route and adapter allowance is below
+`10^7`. The complete nonrecursive bound is 184647552, below the funded
+budget `10^9`. It includes native helpers, adapters, routes, owned operations,
+queries and caller boundaries. All 54 strict and four functional outer
+predicates pass with the retained BIT admission.
+
+`pricing.py` is the exact PR6167 price program with four declared text edits:
+the two new kappa constants, the previous claim, the paid call count and the
+result labels. `PRICING-EDITS.json` binds these edits. The verifier rejects
+any other price-program change.
+
+## Remaining obligations
+
+The five framework assumptions remain `native_all_columns`, `owned_erasure`,
+`compiler`, `precision_tape` and `analytic_recovery`. Native replay verifies
+this finite construction inside that framework. It does not remove those
+assumptions. The rounded integer multiplier is not constructed or benchmarked.
+No independent scientific review or selected-frontier change is claimed.
+
+The small gain does not establish that this mechanism scales to a large gain.
+The published word and exact checks are the contribution. Finite discovery
+screens are not lower bounds or proofs that other lifts are impossible.