Conditional κ 0.00309871 from a prepared P104 split helper

research/p104-prepared-split-reuse/PRICING-EDITS.json

@@ -0,0 +1,34 @@
+[
+  [
+    "Q('.00309839'), Q('.0030983905')",
+    "Q('.00309871'), Q('.0030987101')"
+  ],
+  [
+    "PREVIOUS = Q('.00308189')",
+    "PREVIOUS = Q('.00309839')"
+  ],
+  [
+    "13084",
+    "13066"
+  ],
+  [
+    "66852",
+    "66816"
+  ],
+  [
+    "'kappa_decimal':'.00309839'",
+    "'kappa_decimal':'.00309871'"
+  ],
+  [
+    "previous_open_pr5580",
+    "previous_open_pr5599"
+  ],
+  [
+    "'peak_owned_fields':1",
+    "'peak_owned_fields':local['peak_owned_fields']"
+  ],
+  [
+    "'local_paid_rank_mass':local['paid_rank_mass']",
+    "'local_paid_rank_mass':sum(r*n for r,n in local['paid'].items())"
+  ]
+]

research/p104-prepared-split-reuse/PROOF.md

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+# Two paid queries replaced by a prepared split helper
+
+Use PR5599's exact complete word at
+`0644592d421b375c221045954fdf13932575ffa8`.
+Its fifteen helper splits, scalar bases, phase conventions and caller remain
+credited to that source. This construction changes only the preparation
+and two queries described below.
+
+## Exact source-version identities
+
+Write `h_r` for original input r and `x_r` for its current normalized value.
+The source has 750 independent productive inputs and eight owned roles.
+Fresh exact replay, before any preparation, proves:
+
+| Source event | Current-value identity | Paid source operation |
+| --- | --- | --- |
+| 13670 | `x573-x572+2*x732 = -2*h573` | `MOVE 573: 3 -> 595`, rank 1 |
+| 13691 | `x573-x572-2*x710 = -2*h573` | `MOVE 573: 595 -> 603`, rank 1 |
+
+The identities include every source input coefficient and current data version.
+They are derived by `verify.py:source_relations`, rather than read from a
+saved relation table. The source scalar basis changes are included in replay.
+
+Source helper 573 starts at frame 3, the zero subspace, and finishes at
+frame 603, a two-dimensional subspace. It is the rank-two prefix target
+created by PR5599's eleventh split. The rank-seven successor has its own
+independent input and remains in the word.
+
+Declare role 573 owned and initialize it to zero before the source's entry
+basis program. Its arbitrary dirty entry is erased. Thus `h573=0` is an
+internal preparation, not a promise on any retained productive input.
+Remove its rank-two target from the productive invoice.
+
+All other source output rows are independent of `h573`. Exact replay of the
+new word checks all 749 retained independent inputs, 208 main outputs,
+541 helper outputs and every owned return. A dense integer transpose checks
+the same complete linear map through a separate arithmetic path.
+
+## Explicit replacements
+
+At source event 13670, roles 572 and 732 are already at frame 595.
+Replace the rank-one MOVE with:
+
+```text
+CLEAR 573 at 3
+INIT 573 at 595
+COPY 573 <- 572
+ADD 573 += -2*732
+```
+
+The first identity proves that this gives the required value in frame 595.
+Keep the following source ADD and every later source instruction.
+
+At source event 13691, roles 572 and 710 are already at frame 603.
+Replace the rank-one MOVE with:
+
+```text
+CLEAR 573 at 595
+INIT 573 at 603
+COPY 573 <- 572
+ADD 573 += 2*710
+```
+
+The second identity proves that this gives the required value in frame 603.
+In particular, the following source SCALE and split basis operations remain
+unchanged. No operation is moved across their reads or writes.
+
+After the source program, clear role 573 at frame 603. Its exact output was
+already zero because its prepared input was zero. The eight original owned
+roles retain their source operations and cleanup.
+
+The word has 758 total roles, nine owned roles, eleven owned lifetimes and
+peak owned occupancy two. All nine original dirty owned columns vanish from
+all outputs. The new instructions use only the source INIT, CLEAR, COPY and
+ADD operations. Neither a zero retag nor a new Gaussian operator is needed.
+
+## Native meaning and every positive width
+
+The normalized row in a frame denotes a linear combination of arbitrary
+original fields under that frame's native operator. Both reconstructed
+values use sources already in the required physical frame.
+The COPY and scalar ADD operations therefore act on the actual physical
+fields, with no omitted transport or phase correction.
+
+For each legal positive tensor width f, substitute the tensor power of every
+source native chart. Keep rational scalar coefficients outside the tensor
+power. Equal charts commute with these whole-field scalar combinations.
+The two source identities therefore prove both replacements at every width,
+under the same native premise as the source.
+
+No retained independent input is copied into another productive input slot.
+Role 573 is owned and receives no target credit. Its old external entrance
+normalizer can be omitted because its input is replaced by zero. Its old
+exit normalizer can be omitted because its output is owned zero.
+All other helper normalizers and both main interfaces remain unchanged.
+
+The verifier checks that every retained paid-map pair occurs in the source.
+The source native evaluator rechecks the signed Pauli images and absolute
+phase anchors of all those maps, all original helper endpoints, and the
+208 main adapters. Checking the discarded helper endpoint is harmless extra
+evidence; its target and two unused normalizers are removed from accounting.
+
+## Complete invoice and conditional bound
+
+The local paid profile is:
+
+```text
+{1:2209, 2:459, 3:494, 4:94, 5:164, 6:18, 7:14}.
+```
+
+The retained productive helper profile is:
+
+```text
+{1:4, 2:4, 4:90, 5:10, 7:5, 8:18, 9:410}.
+```
+
+Relative to PR5599, the local paid delta is `-2 H1`, and the target delta
+is `-H2`. For `0<t<1`, `2^t<2`. The target-minus-paid moment therefore
+increases by `2-2^t>0`. At t=1 the delta is zero.
+
+The unchanged physical controller has 18 local calls, 15 thin query stages
+and 15 whole idle stages. Each call uses its own 541 fresh helper inputs.
+It invokes the complete new supplier, which returns all owned storage zero.
+The global deltas are `-36 H1` paid and `-18 H2` productive.
+There are 66,816 paid jobs and 13,066 independent productive fields.
+The complete rank surplus stays 760.
+
+The new word has 25,510 instructions and scalar allowance 220,432.
+All initialization, erasure, copying, scalar operations and routes are charged.
+The route allowance includes eleven owned lifetimes and peak occupancy two.
+The local cap 10^7 and global cap 10^9 remain sufficient.
+The added owned role replaces a productive role, so the total field count
+stays 758. The native paid-rank sum is recomputed from the changed word.
+
+All scalar coefficients are dyadic and satisfy the source growth bound.
+There are fewer than 26,000 operations, each with growth below 2^16.
+The same precision and tape premise covers this explicit finite template
+with its charged constants. The rounded multiplier is not materialized.
+
+The source pricing code is retained with the exact edits listed in
+`PRICING-EDITS.json`. The target debit is included in both completion tails.
+The source's tail decay 39/40 and compiler logarithm power 1024 are retained.
+Two rational moment engines test saving 0.0030987101 and fund the claim
+**kappa=0.00309871**. They also reproduce PR5599's predecessor crossing.
+The same run checks the compiler threshold, BIT stock, 54 strict outer
+predicates and four functional predicates.
+
+The five inherited framework assumptions remain. This is an author-checked
+conditional construction, without independent human or proof-kernel acceptance.

research/p104-prepared-split-reuse/README.md

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+# Prepared split helper: conditional kappa 0.00309871
+
+This construction raises the complete conditional bound from
+**0.00309839 to 0.00309871**. It composes an owned-helper preparation with
+Ali Romman's fifteen independent helper splits in
+[PR5599](https://github.com/CrocSwap/integer-mult-bounds/pull/5599), pinned at
+`0644592d421b375c221045954fdf13932575ffa8`.
+The selected repository record remains unchanged.
+
+The exact crossing lies in `(0.00309871051439, 0.00309871051440)`.
+It exceeds PR5599's crossing by between `0.00000031699227` and
+`0.00000031699229`.
+
+The rank-two prefix helper 573 becomes owned storage, initialized to zero.
+Two exact source relations then replace its two rank-one queries with
+ordinary copies and additions. Its rank-two output is debited. The independent
+rank-seven successor from PR5599 remains productive.
+For every `0<t<1`, the complete local moment improves by `2-2^t>0`.
+
+The complete caller applies the change in all 18 local calls. It has
+**66,816 paid jobs and 13,066 independent productive fields**, with rank
+surplus 760. The local interface has 749 independent productive inputs and
+nine owned roles. All owned roles return zero; peak occupancy is two.
+There are no new paid native maps or new instruction types.
+
+The word keeps the source register numbers. Its legacy `public` slot limit
+is 750; `independent_public` lists the 749 actual productive inputs.
+`owned_roles` includes role 573 and the eight original owned roles.
+
+These are conditional, author-checked results. The five inherited framework
+assumptions remain: native implementation at every width, owned erasure,
+uniform compilation, precision and tape costs, and analytic recovery.
+The complete uniform word and rounded multiplier are not materialized.
+Independent human review, proof-kernel acceptance and frontier promotion
+are not claimed.
+
+Read [PROOF.md](PROOF.md) for the source identities and physical implementation.
+[construct.py](construct.py) emits the complete word.
+[RESULT.json](RESULT.json) records the exact checks and complete bill.
+[VALIDATION.md](VALIDATION.md) records the executed validation.
+
+Reproduce with Python 3.12 and the standard library:
+
+```sh
+python3 -B scripts/research_package.py --package research/p104-prepared-split-reuse
+```
+
+The verifier checks all 758 endpoint rows and independent transpose columns.
+It derives both relations before specializing the source input.
+It checks every retained native map with absolute phase, the full physical
+caller, two rational moment engines and all 58 outer predicates.
+Seven semantic controls test initialization, reconstruction, chart matching,
+owned cleanup, paid accounting and the productive debit.
+
+[SOURCES.json](SOURCES.json) pins the complete PR5599 word, its local and
+pricing code, its split witnesses, proof and notices. The unchanged PR5545
+export supplies 33 authenticated runtime files. Replay needs no network,
+solver or pending Git objects. The source scalar basis programs remain intact.
+
+This changes the independent-input contract and actual paid word. It falls
+outside the fixed-input passive-history obstruction in
+[PR5598](https://github.com/CrocSwap/integer-mult-bounds/pull/5598).
+No global optimality or worldwide novelty claim is made.
+
+Ali Romman supplied PR5599's fifteen splits, PR5580, and the P104 caller and
+pricing machinery. GamingPuzzled supplied PR5569; DreamingOfClouds supplied
+PR5561; thu-nmrc supplied PR5551. David Arias, Cinna Davis and Gareth Davidson
+supplied the underlying P104 word and private-copy construction.
+The preparation and two query replacements are by Scratch-net, with
+substantial OpenAI Codex assistance. No subagents were used.
+
+The reuse question comes from Dugongue's PR5586 atlas. Cached-relation
+precedents include Ali Romman's PR5383 and Dillon Amburgey's PR5402.
+Retain Chris R/Clear1nsight's review and native interpreter, Cao Yifan's
+native lineage, Jacob Alan Sussman's productive recursion, zxv's compiler
+and outer machinery, and all authors in the source closure and
+[retained PR5599 notice](inputs/pr5599-NOTICE). New work is Apache-2.0.
+
+## Open next questions
+
+Independent review should check the changed owned interface and the two
+reconstruction identities, then replay the full caller and funding checks.
+All source dependencies are present. The five framework assumptions and
+independent scientific acceptance remain open.