[petalas]

Result

Scoped negative result, finite certificate in the raw-moment relaxation. No new κ.

#6030 notes that removing helper 469's or 471's frame-593 rung is exclusive with #5907's and #6030's removals "unless a new carrier of that content is added". This package prices that route on #6030's published complete profile, using the raw crossing of Σ T_r r^(1-k) − Σ P_r r^(1-k) that this line bisects.

• Granting a free third removal with exactly Conditional κ .003125804 (+5.22e-7 over #5907): remove helper 470's frame-593 rung, the twin of #5907's removal #6030's per-LOCAL delta again (×18 LOCAL invocations) would move the crossing from (.00312581155791, .00312581155792) to (.00312633419236, .00312633419237), a gain of at least 5.2263e-7.
• One added paid call of any rank 1..145 defeats that gain, whether it is added once in the complete invoice or once per LOCAL. All 145 ranks are certified in both settings: the crossing lands below Conditional κ .003125804 (+5.22e-7 over #5907): remove helper 470's frame-593 rung, the twin of #5907's removal #6030's lower endpoint.
• One added rank-1 call alone moves Conditional κ .003125804 (+5.22e-7 over #5907): remove helper 470's frame-593 rung, the twin of #5907's removal #6030's crossing to (.00312167101310, .00312167101311). That loss is at least 7.92 times the free repeat's gain.

A carrier can therefore help only if it adds no paid call, or if it earns productive target credit itself. This package does not price those cases.

Evidence and reproduction

research/p104-rung-carrier-break-even/verify.py uses only the standard library. Signs are certified with correctly rounded Decimal ln/exp at 60 digits and an explicit 10^-50 relative error allowance per moment. Positive controls reproduce #6030's published crossing and, with the stated delta removed once, #5907's (.00312528909884, .00312528909885). They also check the published totals: 13,444 targets, 66,834 paid calls, surplus 760. These controls test this package's pricing; they are not a review of either PR. Three damaged profiles must be rejected, and RESULT.json must equal the recomputation.

~/git/integer-mult-bounds/.venv/bin/python3.12 -B research/p104-rung-carrier-break-even/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p104-rung-carrier-break-even
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p104-rung-carrier-break-even --check-source-history

All three pass locally on CPython 3.12. The verifier takes about 1 s.

Limitations

• The third removal is a hypothesis. No 469/471 removal is constructed, and its real delta may differ.
• Only the raw crossing is evaluated. The tail ratios, compiler and outer checks only add conditions, so they cannot rescue an edit that fails here.
• Carriers that add productive targets are not covered.
• The profile is copied as a literal from Conditional κ .003125804 (+5.22e-7 over #5907): remove helper 470's frame-593 rung, the twin of #5907's removal #6030's RESULT.json at head 772dd10a5859a1d15f4851feb371519789213ad8 (SHA256 84fcbe2a…1924ad). That head is not on main, so it is not a Git pin.
• Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Can a carrier for the content of sources 703/704 itself be a productive helper whose target credit offsets its paid call? Name the rank and frame, then price both deltas with this verifier before building a word.
• Does an existing paid call in Conditional κ .003125804 (+5.22e-7 over #5907): remove helper 470's frame-593 rung, the twin of #5907's removal #6030's word already carry 703/704 content near frame 593? That would be a zero-cost carrier. This package does not parse the word.

Related open work

• Conditional κ .003125804 (+5.22e-7 over #5907): remove helper 470's frame-593 rung, the twin of #5907's removal #6030 (DreamingOfClouds): source of the profile and of the question. This package reproduces its stated crossing enclosure as a control and does not otherwise check it.
• Conditional κ .003125282 (+5.22e-7 over #5827): remove helper 468's frame-593 rung with rerouted reads #5907 (DreamingOfClouds): the twin removal. Its crossing enclosure is reproduced from Conditional κ .003125804 (+5.22e-7 over #5907): remove helper 470's frame-593 rung, the twin of #5907's removal #6030's profile minus the stated delta.
• Exhaustion census: no PR5599-type helper split remains on the PR5907 word (no new kappa) #5994 (Matarener): exhaustion census of PR5599-type helper splits on the Conditional κ .003125282 (+5.22e-7 over #5827): remove helper 468's frame-593 rung with rerouted reads #5907 word. It studies a different edit class on the same word, and this package computes no number it reports.

CI notes

At head c7e171a9e9ee25e69b2533378abc2ada1a4fafa2, both hosted checks passed, plan and replay (research/p104-rung-carrier-break-even), in run 40608040511.

This account's open PRs created since the claim (#6037, #6038, #6039) study different objects and share no quantity with this package.

Credit

Produced by Nick Petalas. The rung removals and the profile are DreamingOfClouds's (#6030, #5907). Their base word is Ali Romman's #5827, with lineage back to #5565.