[zxv]

Conditional candidate with a complete local replay; awaiting external mathematical review.

Result

Reference point	κ	Gain from this PR
Repository-selected conditional result (certificates/selected-result.json)	0.003125804	+5.803%
#6716 · exact caller baseline	0.00330420	+0.09079%
This PR	0.00330720

The absolute increase over #6716 is 3.000 × 10⁻⁶. Separately, an integral transfer of the same caller raises the ordinary BIT saving to 0.00376669, up 0.00002212 from the retained 0.00374457. That BIT value is a dependency bound, not the overall multiplication κ. The selected result is unchanged by this PR.

Both claims are conditional on the five retained framework assumptions: native_all_columns, owned_erasure, compiler, precision_tape, and analytic_recovery. The package supplies exact caller identities, physical-map checks, complete invoices, finite compiler certificates, and outer inequalities. It does not discharge those assumptions.

The key idea: pay to split shared idle work

The usual instinct is to fuse shared idle work as far as possible, because every extra paid job costs. Here, splitting one combined idle job into two paid pieces exposes a smaller prefix that can be retired as an independent output. When the output gained is worth more than the extra payment, the complete bill improves, and strict concavity of H(r) = r^(1-κ) makes that happen at these ranks.

Concretely, each fragment turns one rank-41 target into independent targets of ranks 32 and 9, and one paid rank-40 idle into paid jobs of ranks 32 and 8. Relative to the same unfragmented root and allocation, the combined change from both fragments across 104 ports is

Δtargets = 208 (H32 + H9 - H41)
Δpaid    = 208 (H32 + H8 - H40)
Δgap     = 208 ((H9 - H8) - (H41 - H40)) > 0

The headline comparison with #6716 uses the two full invoices, including the changed root and allocation; it does not follow from this local calculation alone.

Construction and complete invoice

The root has eight fields and nine queries, with six full sites of ranks (2,4,4,4,4,2). Its two shared idle banks admit integral determinant-one rotations. In each bank, the split exposes a genuine prefix/suffix dual pair. An ordinary basis change and handoff then retire an independent output and introduce a fresh arbitrary input. The second cut also needs an ordinary phase normalization, which is included.

The child sizes are (4,4,1,1,4,4). The final caller has 34 arbitrary independent inputs, 606 instructions, 18 unchanged native invocations, and 15 thin queries. Every helper input, paid fragment, binary idle, normalization, and ordinary route is charged:

Invoice	Productive fields	Paid jobs	Rank-mass surplus	Raw crossing
Complex	13,670	66,852	760	(0.0033072446848, 0.0033072446849)
BIT	11,640	56,673	753	(0.0037667234775, 0.0037667234776)

Two independent rational interval engines enclose each raw crossing. The claimed κ = 0.00330720 sits just below the complex crossing, leaving margin for the finite compiler and outer transfer.

Integral BIT transfer

The caller's root and handoff bases are integral and unimodular, so the same construction transfers to BIT without dividing by two. First specialize the central relation to q²=1, which removes the complex λ normalizations. The remaining identity has integer coefficients, so it can then be reduced modulo two. The retained BIT native source is hash-bound, and its scalar endpoint and native charts are checked.

The binary caller has 34 independent inputs and 356 instructions; both coefficient engines check all 1,156 entries. Because the new covered saving exceeds the old wrapper atom 0.00375, the atom is retuned to 0.004. Finite admission and two strict ordinary-wrapper steps, 0.00374998 → 0.00376663 → 0.00376669, then establish the claim. Storage, routing, discarded targets, and ordinary work are all included.

Findings and limits

• Allocation-only extensions reach this root's raw ceiling. Exact interval dynamic programming excludes every positive six-child allocation with 6–49 total leaves at κ = 0.0033072446849. A concave envelope has negative value and derivative at 50, which excludes the entire larger tail. The exclusion persists through κ = 0.004. It covers this root, these two fragment orientations, and ordinary binary supplier trees, but not changed roots, sources, or additional retirements.
• The scalar header budget is saturated. After setting body variables to one, 32 targets carry q and two carry the identity. The existing 33 queries are minimal under the same central scalar contract with q-independent invertible ordinary controls. The argument covers the complex and formal BIT involutions; it does not exclude operator-valued port coupling.
• Internal fragmentation has a scoped obstruction. At the completed-left-child cut, a pure L or SL prefix on the parent's auxiliary bank cannot have the required unit pairing, even after ordinary rotations of that bank. Exact matrix and phase fixtures support the algebra. Broader changed histories remain outside the proof.
• Native-rewrite and helper-promotion pilots found no further gain. The retained notes separate local frame neighborhoods, contact-response tests, and finite retirement searches from general impossibility claims. The late promotion receipts cover all 47 owned lifetimes and their tested cleanup models. This is local exploratory evidence, not an independently reviewed source-wide closure theorem.

The findings and coverage note gives exact scopes, evidence paths, and reopening conditions. Allocation-only extensions above the proved raw ceiling, within this exact root, the same two fragment orientations, and ordinary binary supplier trees, are retired over the certified κ interval through 0.004. The unsuccessful rewrite, internal-fragment, and promotion branches are parked. No supported next construction action is currently recommended on those branches without changed source, target, or legal unit-pair evidence.

Evidence and reproduction

The proposed package is research/p104-paid-idle-fragmentation/. The exact publication commit is 435bbe91f; all 236 committed package files passed a working-byte comparison, including the unchanged frozen export. Its frozen/ directory preserves the complete source export, original notices, and a 215-file byte manifest. Start with the complex proof, the two retirement witnesses, the complete complex certificate, and the BIT proof.

From the package directory, using only the Python standard library:

python3 -I -B frozen/verify_bundle.py
python3 -I -B frozen/allocation_tail.py
python3 -I -B frozen/internal_prefix_lemma.py

The bundle runner authenticates the retained files and replays them in a temporary copy. For the complex caller it checks both 1,156-entry coefficient engines, both handoffs, all 7,488 physical contacts and absolute phases, the complete invoice, the finite compiler, and the 54+4 outer predicates. It also replays both new BIT admissions and rejects substantive missing-gate and unearned-credit controls. The decompressed BIT caller and every mathematical receipt field must match exactly; elapsed time and gzip container differences are handled separately.

The unchanged native proof is reused from the authenticated full source replay. To repeat it as well:

python3 -I -B frozen/verify_bundle.py --full-native

What was run, and where:

• Fresh during packaging (Python 3.11.11): the default bundle and both supplementary proof checks.
• Retained earlier logs: the full-native replay and the Python 3.14.5 cross-version run.
• Not rerun: exploratory sweeps, Lean, and Docker.

The enormous finite uniform multiplier is specified, not materialized. The package uses direct replay commands and does not register a new CI manifest.

Significance for researchers

Paying for fragmentation can improve the complete concave moment when it exposes a smaller independently productive prefix. The gained output increment exceeds the extra paid increment in this construction. The integral witnesses make the same mechanism available to the actual BIT dependency, while the allocation ceiling identifies where further price optimization of this exact root stops paying and a changed construction becomes necessary.

Open next questions

• Independent application review. Check both literal dual-pair cuts, the second cut's absolute phase, all arbitrary fresh inputs, and the paid-fragment chronology against the retained native interface. Begin with the witnesses and the separately written coefficient engine, then run the portable verifier. Existing reviews of the predecessor do not cover this changed construction.
• Framework assumptions. The five retained all-size assumptions remain separate obligations. A successful finite replay does not discharge them or establish an unconditional multiplication theorem.
• A different construction beyond the ceiling. A changed root, source schedule, or additional legal retirement could improve the bound. Reopening requires an actual source-bound witness and a complete profitable invoice; a larger allocation sweep or an optimistic stage-rank budget is not enough. This package nominates no successor search.

Attribution

PR	Submitter	Used here
#6716	THU New Media Research Center· Prof. Shen Yang	Eight-field common-idle caller, fresh-input cut and full signed invoice; exact baseline at a467457692bc9e11113d9b2e2e34ea8ffb553b58
#6702	@DreamingOfClouds	Paid-job merges and split public helper register in the native word; retained source at edd6a37a9ff87b254fb7466184348d1432d0c8c0
#6551	Ali Romman	Retirement-aware tree rebalancing
#6545	Ali Romman	Independent Gaussian retirement and exact dual pairs, the method extended here
#6529	Ali Romman	Scaled transport fusion and the complete native word
#6348, #6398	Ali Romman, with Cinna Davis	Binary query suppliers, common-projector root and normalization
#5599, #5602, #5611, #5615	Ali Romman	Causal handoff, independent helper splits and dyadic helper bases
#6399, #6417	Rohan Arun	Clean donor-transport and native merges with the mixed caller
#6453	Erwan L.	Third group-1091 anchor on the clean mixed root
#6375	@DreamingOfClouds	Shared-carrier input lifts
#4268	Jacob Alan Sussman	Mixed-target (productive) recursion

The nested notices also retain native work by Cinna Davis, David Arias, Gareth Davidson and their predecessors; price and interval work by zxv, Cao Yifan, ywz65 and Nick Petalas; and contributions from Mason Hale, Pierre-Loic Doulcet and Chris R. Later predecessor audit additions do not change the pinned construction bytes. The frozen notices preserve the complete native, productive-helper, compiler, interval, and analytic credits.

The new paid-fragment witnesses, integral BIT transfer, and associated findings were developed with substantial OpenAI Codex assistance for zxv. All original source bytes, attribution, and licenses are retained; new contributions are Apache-2.0.