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feat(research): price a carrier for a third frame-593 rung removal
In the raw moment crossing of the #6030 profile, a free repeat of the twin removal gains at least 5.2263e-7, but one added paid call of any rank 1..145 lowers the crossing below #6030's, in total or per LOCAL.
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‎research/p104-rung-carrier-break-even/README.md‎

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# Break-even price of a carrier for a third frame-593 rung removal
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**Scoped negative result in the raw-moment relaxation; no new κ.**
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DreamingOfClouds's open [PR #6030](https://github.com/CrocSwap/integer-mult-bounds/pull/6030)
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removes helper 470's frame-593 rung, the twin of #5907's helper-468 removal, and
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notes that removing helper 469's or 471's rung is exclusive with both "unless a
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new carrier of that content is added". This package prices that escape route.
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Grant a third removal for free, with exactly #6030's per-LOCAL profile delta
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again (paid ranks 1 −2, 2 +1, 3 −2, 4 +1, 6 +1, 7 −1, 8 +1; credited helper ranks
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1 −1, 2 +1, 3 −1, 6 +1, 7 −1, 8 +1, each times the 18 LOCAL invocations). Then:
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- With a free carrier, the raw crossing would rise from
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(.00312581155791, .00312581155792) to (.00312633419236, .00312633419237),
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a gain of at least 5.2263e-7.
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- **One added paid call of any rank 1..145, anywhere in the complete invoice,
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defeats it.** For each of those 145 ranks the edited profile's crossing is
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certified below #6030's lower endpoint. One added rank-1 call alone moves
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#6030's crossing to (.00312167101310, .00312167101311). That loss is at least
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7.92 times the free repeat's gain.
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- A carrier inside the local word is paid once per LOCAL. One added paid call
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per LOCAL, of any rank 1..145, also defeats the repeat. One rank-1 call per
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LOCAL moves #6030's crossing to (.00305129298944, .00305129298945).
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So a carrier is useful only if its paid cost is zero, or if it also earns
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productive target credit or enables further savings worth more than one paid call.
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A twin-sized rung removal cannot fund a single extra paid Gaussian call.
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## Claim type and scope
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This is a **finite certificate** about hypothetical profile edits in the raw
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moment model `Σ_r T_r r^(1-k) − Σ_r P_r r^(1-k)` that #5827/#5907/#6030 bisect. It
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is not a word, not a construction, and not a complete multiplication theorem.
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The free third removal is a hypothesis: no rung removal for helpers 469 or 471
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is constructed, and its true delta may differ. The tail ratios, compiler and outer
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checks that a complete bill also needs are not evaluated. They only add
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conditions, so they cannot rescue an edit that already fails the raw crossing.
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The analysis does not cover a carrier that also adds productive targets.
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The profile is copied as a literal from #6030's `RESULT.json` at head
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`772dd10a5859a1d15f4851feb371519789213ad8`
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(`research/p104-helper470-twin-rung-removal/RESULT.json`, SHA256
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`84fcbe2a0b64307b5703c363408042bdd91555b0419509aa50daa43b3b1924ad`), whose
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`complete_cost.targets` and `complete_cost.paid` it reproduces. That PR head is
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not on main, so it cannot be a Git pin here. As positive controls, the
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verifier reproduces #6030's published crossing enclosure and, after removing
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the stated delta once, #5907's enclosure (.00312528909884, .00312528909885). It
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also checks the published counts: 13,444 targets, 66,834 paid calls, rank surplus 760.
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These are consistency checks of this package's pricing, not a review of either PR.
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Signs are certified with Python's correctly rounded `Decimal` `ln`/`exp` at 60
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digits and an explicit relative error allowance of 10^-50 per moment. Every
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checked sign must clear that allowance. Three damaged profiles must be rejected.
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## Reproduce
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From the repository root, with CPython 3.12 and the standard library:
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```sh
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python3.12 -B research/p104-rung-carrier-break-even/verify.py
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python3.12 -B scripts/research_package.py --package research/p104-rung-carrier-break-even
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```
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Expected last line: `PASS_P104_RUNG_CARRIER_BREAK_EVEN`. It runs in about one
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second. `--write` regenerates `RESULT.json`.
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## Open next questions
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- Can a carrier for the content of sources 703/704 be a productive helper itself,
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so its target credit offsets its paid call? This matters because that is the
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only escape this package leaves. First step: name the helper rank and the
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frame at which helper 469 or 471 would read it. Then price target plus
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paid deltas with this verifier's `edit`/`crossing` before building a word.
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- Is there an existing paid call in #6030's word whose content could carry 703
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or 704 to frame 593 without adding a call? This would be a zero-cost carrier.
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First step: list the helpers live at frame 593 that read 703 or 704. Blocker:
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this package does not parse the word.
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## Credit
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Produced by Nick Petalas. The rung removals and the profile are by
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DreamingOfClouds (#6030, #5907). They edit Ali Romman's #5827 word and its lineage
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back to #5565, whose credit is retained in those packages. The raw moment
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crossing follows the pricing of that line. Apache-2.0.

‎research/p104-rung-carrier-break-even/RESULT.json‎

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{
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"claim": "Raw-moment relaxation only; hypothetical edits of the PR #6030 published profile.",
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"controls": {
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"pr5907_crossing": [
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"0.00312528909884",
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"0.00312528909885"
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],
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"pr6030_crossing": [
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"0.00312581155791",
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"0.00312581155792"
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]
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},
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"free_twin_repeat_crossing": [
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"0.00312633419236",
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"0.00312633419237"
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],
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"free_twin_repeat_gain_lower": "5.2263444E-7",
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"one_extra_rank1_call_crossing": [
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"0.00312167101310",
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"0.00312167101311"
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],
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"one_extra_rank1_call_loss_over_twin_gain_lower": "7.92",
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"one_extra_rank1_call_per_local_crossing": [
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"0.00305129298944",
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"0.00305129298945"
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],
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"ranks_checked_for_added_call": [
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1,
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145
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],
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"rejected_controls": [
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"paid_count",
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"target_rank_moved",
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"paid_rank_swapped"
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]
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}

‎research/p104-rung-carrier-break-even/SOURCES.json‎

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{
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"schema_version": 1,
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"base_commit": "19fdb086ea96af9a9f87936a2b888e10af4ed810",
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"files": [
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{
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"path": "certificates/selected-result.json",
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"usage": "provenance",
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"git_blob": "581f8d87008354eb5b869f7ed9de1317ad837ce8",
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"sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
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},
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{
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"path": "docs/research/agent-task-catalog.json",
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"usage": "provenance",
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"git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
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"sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
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}
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]
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}

‎research/p104-rung-carrier-break-even/TASK.json‎

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{
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"schema_version": 1,
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"task_id": "p104-side-correction",
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"priority_revision": 8,
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"base_commit": "19fdb086ea96af9a9f87936a2b888e10af4ed810",
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"original_card": {
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"id": "p104-side-correction",
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"question": "Can joint helper bases and a legal causal split improve the complete P104 productive/paid invoice?",
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"starting_points": [
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"docs/research/current-priorities.md",
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"docs/research/research-family-status.md",
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"certificates/selected-result.json",
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"docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
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"research/p104-helper-subspace-splits/README.md",
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"research/dyadic-helper-channel-capacity/README.md"
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],
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"methods": [
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"Construct and independently check one source-bound instance of the stated first experiment.",
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"Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change."
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],
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"acceptance_gate": "An exact arbitrary-input identity, source/target frame availability and restoration at the chosen cut, plus the complete channel+correction+idle/hosting bill or an explicit favorable bound. The native channel alone is not a supplier. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
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"first_step": {
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"command": [
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"python3",
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"-B",
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"scripts/research_selected_baseline.py"
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],
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"expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
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"inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.",
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"scope": "Source-bound orientation only, not a supplier replay or theorem proof."
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},
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"first_experiment": "Start with the selected #5599 emitted word and review pending #5718 distinct input/output bases plus #5717/#5719 coverage before claiming work. Pick one uncovered source word, helper group and cut region. Prove the arbitrary-input relation and legal entry/exit corrections, then compare full target and paid profiles. Reject if completion, owned lifetimes or consumer costs repay the gain. A static pairing rank alone is not a physical split.",
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"bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.",
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"allowed_variations": [
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"One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes."
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],
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"scope_change_triggers": [
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"Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.",
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"Treating an archived proposal or local numerical pass as complete supplier admission."
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],
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"overlap_terms": [
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"P104",
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"helper basis",
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"helper split",
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"5718",
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"dyadic capacity"
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],
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"existing_checks": [
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{
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"scope": "PR #4133: literal-slot defect",
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"evidence": "research/p104-literal-slot-defect/README.md",
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"remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
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"status": "author-checked",
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"source_commit": "88666f1dee93fbd85b31fd5d6f91a219e111b5f4",
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"selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
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},
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{
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"scope": "PR #4136: uniform commutation gap",
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"evidence": "research/p104-uniform-commutation-gap/README.md",
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"remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
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"status": "author-checked",
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"source_commit": "281ce601a8be9db470de630759179c23e1dfaa88",
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"selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
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},
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{
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"scope": "PR #4140: uniform encoding gap",
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"evidence": "research/p104-uniform-encoding-gap/README.md",
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"remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
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"status": "author-checked",
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"source_commit": "de25c6525aa866962309df5d107b32d55e6e5628",
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"selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
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},
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{
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"scope": "PR #4153: source-chart shortcut exhausts its optimistic allowance",
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"evidence": "research/p104-source-chart-shortcut/README.md",
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"remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
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"status": "author-checked",
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"source_commit": "9123241fea53df77e51ffe30d9aca1625d4bf3a8",
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"selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
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},
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{
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"scope": "PR #4154: fiber-tag seam mismatch",
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"evidence": "research/p104-fiber-tag-seam/README.md",
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"remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
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"status": "author-checked",
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"source_commit": "4ffcdaee5d6255a9cea1fe53a5424435068089d0",
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"selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
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},
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{
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"scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.",
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"evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
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"remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.",
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"status": "reviewed",
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"source_commit": "09dd0965711ca7eeb169057a420abab1c9052137",
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"selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
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}
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]
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},
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"scope_status": "changed",
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"actual_scope": "Break-even paid cost of a new carrier that would let a third frame-593 rung removal (helper 469 or 471) repeat PR #6030's per-LOCAL profile delta, in the raw moment relaxation of PR #6030's published complete profile.",
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"scope_change_reason": "PR #6030 leaves the carrier route open; pricing it before anyone builds a carrier word tells the next contributor which carriers can pay.",
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"bottleneck_link": "Complex P104 supplier binds the selected κ; rung removals are the current hill-climbing route on that supplier, and this prices their only noted continuation.",
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"budget_minutes": 60,
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"stop_rule": "Stop after certifying the free-repeat gain and the cost of one added paid call at every rank 1..145; do not construct a word.",
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"outcome": "negative-result",
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"evidence": "verify.py certifies all signs with correctly rounded Decimal arithmetic and an explicit error allowance; RESULT.json is regenerated by verify.py --write and compared on every run.",
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"author": "Nick Petalas",
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"ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
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"open_questions": [
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{
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"question": "Can a carrier for sources 703/704 itself earn productive target credit that offsets its paid call?",
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"why_it_matters": "It is the only escape this raw-crossing certificate leaves for a third rung removal.",
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"first_step": "Name the carrier helper rank and frame, then price target and paid deltas with this verifier before building a word.",
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"blocked_by": "No candidate carrier word exists."
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},
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{
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"question": "Does an existing paid call in the #6030 word already carry 703 or 704 content near frame 593?",
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"why_it_matters": "A zero-cost carrier would make the third removal additive.",
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"first_step": "List helpers live at frame 593 that read 703 or 704.",
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"blocked_by": "This package does not parse the word."
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}
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]
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}

‎research/p104-rung-carrier-break-even/VERIFY.json‎

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{
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"schema_version": 1,
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"status": "ready",
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"python": "3.12",
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"requirements": [],
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"inputs": [],
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"commands": [
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{
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"argv": [
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"{python}",
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"-B",
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"{package}/verify.py"
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],
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"timeout_seconds": 120,
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"checks": "Positive controls reproduce the #6030 and #5907 crossings; certified free-repeat gain; one added paid call of every rank 1..145, in total or per LOCAL, defeats it; RESULT.json equality; three damaged profiles rejected."
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}
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]
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}

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