[petalas]

Summary

Prove the sharp migration bill for two reusable compatible resources. For n fixed half-open consumer lifetimes with fixed nonempty eligibility lists, pointwise Hall feasibility implies an attainable migration count at most max(0,n-2). One long flexible consumer with alternating resource-fixed blockers attains it for every n. Giving that long consumer price W yields exact charge W(n-2) for n>=2, so per-time compatibility alone can omit arbitrarily large transfer costs.

fixed lifetimes + eligible resource pairs + per-consumer transfer prices
  -> at most two placement states per event interval
  -> exact minimum migration charge + attaining schedule + dual arc certificate

Claim type: scoped written mathematical proof with finite corroboration. No new kappa or physical supplier is claimed. The six files are isolated in research/two-resource-migration-tariff/.

This changes the static-assignment premise at the temporal-reuse boundary in Ali Romman's #5781, exact head cb06de3a4f6e2a2c927eb3aa3f9f43a578f73a1f. That predecessor is cited as question/context; its matcher and results were not replayed or audited. Hall's theorem and layered shortest paths are classical inputs. The new contribution is the scoped exact tariff, sharp family and independently enumerated implementation. Advance claim.

Evidence

• Before: Construct minimum-cost compatible assignments with Hall and dual certificates #5781 explicitly excludes resource capacity reuse over time. Pointwise Hall feasibility supplies no migration tariff.
• After: the written proof gives the sharp count and a weighted optimizer in O(n log n) arithmetic operations. A separate checker verifies an attaining path and telescoping dual arc inequalities, without repeating minimization.
• The independent literal schedule oracle agrees in 55,860 exhaustive executions: 23,456 feasible and 32,404 infeasible. This covers all labelled one-to-three-consumer intervals with endpoints in {0,1,2,3,4}, every nonempty two-resource list, and both unit and labelled integer prices.
• Further checks pass 400 seeded weighted cases, all 512 sharp-family sizes through the executable limit, 15 forged-certificate/invalid-input controls, and one touching half-open boundary control.
• Warning-strict normal and optimized direct runs pass, timed locally at 2.13 and 2.23 seconds. Both registered commands pass in 4.63 seconds total, well below their individual 120-second bounds. Package metadata and committed-byte/source-history preflight pass. The receipt is generated and binds all six package files with SHA-256.
• Environment: CPython 3.12.15, macOS 27.0.1 arm64, standard library only. Independent read-only review found no mathematical or certificate flags. The overlap search remains bounded evidence, not a proof of novelty.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/two-resource-migration-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/two-resource-migration-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/two-resource-migration-tariff
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/two-resource-migration-tariff --check-source-history

Another CPython 3.12 path may replace the absolute interpreter path. The verifier writes no files.

Limitations and unrun obligations

Two resources, fixed eligibility lists, free initial placement/release, and atomic migrations define the model. The implementation accepts 1..512 consumers, integer endpoints in 0..10^6 and nonnegative integer migration prices through 10^12. The written theorem has no consumer bound and depends only on endpoint order.

No source-bound physical lifetime adapter, transfer word, exchange scratch, restoration bill, selected-supplier replay, all-size compiler, stock/precision/tape proof or complete multiplication theorem is supplied. Fixed initial/final owners, changing lists, coupled consumers and three or more resources remain outside scope. Independent human and formal proof-kernel review are unclaimed. The selected frontier is unchanged.

Related open work

• Construct minimum-cost compatible assignments with Hall and dual certificates #5781 certifies static injective compatible assignments; this contribution extends the abstract contract to temporal reuse with charged migrations.
• Allocate optimal weighted helper pools with independent replay certificates #5763 minimizes weighted pool capacity for fixed lifetimes and freely assignable noncontiguous slots; this contribution fixes two resource identities and eligibility lists, and minimizes transfers instead of capacity.
• Independently audit P104 callers with explicit owned lifetimes #5717 studies a source-bound owned-lifetime caller schedule; this result supplies an abstract temporal tariff and does not adapt or price that physical schedule.
• Reduce bounded NTT temporary storage from 17,920 to 1,282 (no new κ) #5886 statically allocates closed NTT temporary lifetimes; it minimizes direct-address storage and does not compute a migration tariff.
• Prove exact weighted five-node clean scratch (no new κ) #5895 determines peak clean capacity for a weighted reversible five-node dependency chain with recomputation; its pebble capacity differs from the fixed-lifetime transfer objective here.

The closeout refresh read the titles and bodies of this account's eight other open PRs created since the CLAIM: #5886, #5887, #5888, #5889, #5890, #5891, #5892 and #5895. None computes the same two-resource migration quantity. No additional computations were run to check their results.

Open next questions

1. Export one actual caller's pair of owners, fixed lifetimes and legal transfers, including restoration, to bind the temporal tariff to a physical bill.
2. Determine the sharp migration count for three resources with fixed lists. The two-state proof does not cover that model.

Merge danger

Door: two-way. Blast radius: package. This adds an isolated proof and replay hook; predecessor packages and selected records are unchanged.

CI notes

At commit 8fe6941da2826daadd4ede359aff3e1b66c43dd3, run 40496023512 finished with plan failed; replay skipped.

The only failed step is Onboarding helper tests. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py. The exact summary is FAILED (failures=8, errors=2, skipped=22), followed by exit code 1. The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight assertion failures are 0 != 16, five ValueError not raised entries for missing-helper/reference rejection, and two empty-versus-expected sparse omission-list comparisons.

All ten failure/error headers match PR #5892's plan log, checked directly. This confirms the reported shared onboarding failure on main de5ec2e7a6074068452f45b1a191b5c72832df4a. Shared tests and implementation files were not changed or rerun to address it; maintainer repair remains outstanding.

The replay job was skipped. Hosted replay has not certified this package; only the local scientific checks reported above have run. No hosted scientific success is claimed. The package and mathematical result are unchanged after publication.