[petalas]

Summary

Scoped accounting proof with finite continuation certificates. Lowering one or more of PR6030's 62 market-helper births into the associated hub cannot improve the recursive moment while preserving every original paid MOVE and all other productive endpoints, even with arbitrary scalar resynthesis. Every changed helper loses at least 18*(4^t+3^t-7^t)>0 for every 0<t<1. A single three-rank ascent attains this loss in the inherited recursive bill.

old helper: G(dim 5) -> unchanged continuation -> full frame(dim 9)
new helper: B <= G intersect G^perp -> paid ascent to G -> unchanged continuation
credit gain: 18*((4+d)^t-4^t)
extra paid: at least 18*d^t
net change: strictly negative, d >= 3

This addresses the birth-lowering subfamily of the bottom-to-market opening in DreamingOfClouds's PR6170 Lemma 11. It does not settle all coupled words. The next useful candidate must delete an old paid transition or obtain another justified productive-boundary gain. No new kappa, supplier admission or selected-frontier change is claimed.

Package: research/p104-prefix-birth-obstruction/. This contribution was produced by Nick Petalas. Classical concavity is background. Credit DreamingOfClouds for the source word, geometry, ledger and question, Ali Romman, Cinna Davis and the retained P104/native/helper/caller/compiler/BIT lineage. No predecessor result was audited or replayed.

Advance CLAIM. The delivered scope narrows the proposed coupled search to a sharp negative preparation screen; the wider coupling question remains open.

Evidence

Before, the fixed-slot separation explicitly left bottom-to-market coupling untested. After, the written proof rejects all scalar resyntheses within the stated unchanged-paid-inventory and productive-boundary contract, for the whole open power interval.

The new exact checker constructs 310 arbitrary-input continuation fixtures over fourteen markets and 62 eligible roles. Added ranks are three in 62 fixtures, four in 186 and five in 62. It checks literal unchanged suffixes, materialized new frame bases, exact entry-frame restoration, 28 integer path compositions, seven rejecting controls and a paid-deletion escape control. Generality rests on the proof; the finite tests do not enumerate scalar programs. Source identity and pricing are inherited premises.

From the repository root, using the interpreter used locally:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/p104-prefix-birth-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/p104-prefix-birth-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p104-prefix-birth-obstruction
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p104-prefix-birth-obstruction --check-source-history --check-modes

Elsewhere substitute a CPython 3.12 interpreter. Standard library only. Direct marker: PASS_P104_BIRTH_OBSTRUCTION splices=310 compositions=28 rejections=7 no_new_kappa.

All commands passed locally on CPython 3.12.15. Each registered command was also timed directly, at 0.097 and 0.098 seconds. Both registered replay modes passed. Committed bytes, execute modes and source ancestry passed for nine package/input files at b967838c246dc4b2a5bed2f506608dcbb62c4532. Official export and input-coverage checks passed. Both direct modes passed in a read-only Git-free export with an empty PATH; all ten exported files retained their hashes and modes. Generated receipts are emitted to stdout and bound to file SHA256 hashes.

Scientific base and submission refresh: canonical main a5bff3bf133e482541c09eba4e773884bdb2b1b7, priority revision 8. The selected binding-only orientation passed. Bounded live PR/Discussion, main README/formula and merged-PR searches found no matching birth-lowering result; sparse notes, arbitrary historical diffs and unpublished work remain limitations.

Limitations and unrun obligations

This is an optimistic recursive-invoice obstruction, not a universal routing barrier or complete multiplication theorem. It retains every old paid transition and the other productive endpoints. Replacing or deleting MOVEs, changing helper finals, obtaining new independent targets or changing the native interface requires another comparison. Sharpness concerns the recursive bill, not the complete physical cost.

The source arithmetic was cited rather than replayed. No new native/caller/compiler/precision/stock/outer cold replay, physical multiplier, Lean or Docker run was needed or performed. The five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed.

Open next questions

Can one bottom-to-market coupling delete a named old paid transition while preserving every arbitrary helper input? Start with one bound market role and solve its changed scalar continuation with the complete signed bill. That word, native realization and consumer bill remain missing. No follow-up search is reserved.

Related open work

• Conditional κ 0.00309871 from a prepared P104 split helper #5606 changes a prepared P104 helper to owned storage and removes paid queries while debiting its productive target. It changes hypotheses retained here and is not excluded by this result.

• Two successor hand-offs on the P104 word: conditional κ 0.0030428242 #5553 splits helper lifetimes into fresh independent successors at intermediate frames. It changes the productive target boundary, outside this result's retained-endpoint contract.

• Price the p19 coded-reverse chain: conditional κ = 0.0033999964 #5440 studies a p19 coded-reverse caller and physical lowering, rather than market-birth transport. No common numerical result is computed here.

• Certify conditional kappa 0.0033999964 via the p19 word without the coded overlay (complex 0.007578535, BIT 0.0034, zero field bill) #5270 studies the uncoded p19 caller and its inherited producer contract. It does not instantiate this birth-lowering family; no claimed kappa is compared or rechecked.

• Prove strict traps for exact rank-one projection cuts (no new κ) #6194 proves strict traps for rank-one projection cuts in an abstract paid frame graph. It shares concave accounting, but changes a different contract and computes no market-birth deficit or common-instance number.

• Prove coordinate charts suffice for coordinate-terminal cache trees (no κ) #6202 restricts intermediate charts in coordinate-terminal cache trees. Its Lagrangian distance model supplies no market-birth continuation or competing value for this family.

The completion refresh inspected all twelve other account PRs still open and created since the CLAIM at 2026-10-29T16:17:21Z: #6192, #6193, #6194, #6195, #6196, #6198, #6199, #6200, #6201, #6202, #6203 and #6204. None studies this exact market-birth family or reports a conflicting same-instance number. No additional predecessor computation was run. Related-work coverage is bounded to the live open-PR list and targeted searches.

CI notes

Both hosted checks passed at exact head b967838c246dc4b2a5bed2f506608dcbb62c4532 in run 40911824813: plan in 4m28s and replay (research/p104-prefix-birth-obstruction) in 2m5s. The completed check list was read once and both entries report pass.

Hosted replay ran the registered normal and optimized package checks at this head. Local publication-byte, mode, source-history and read-only Git-free export checks are separate passing evidence. No CI failure remains to report. Independent human and formal proof-kernel review remain unclaimed.

Merge Danger

Door: two-way.

Blast Radius: isolated research package. No shared executable, frozen source, existing credit or selected record changes.