[petalas]

Summary

Proves exact two-call P112 COPY cost 128 for a new exceptional-hull family of directed cross forms, including a rank-(1,2) example that PR2395's sufficient test leaves undecided. Claim type: scoped written proof with exact finite corroboration.

beta01 = r_i v^T
rowspace(beta10) contained in span{s_0,...,s_7}
Schur feedback = 0
double price >= 16; incidence count => total price >= 128
sixteen coordinate terms attain 128

The new step extends the mixed-orientation transfer to the linear span of exceptional source functionals, allowing higher reverse rank. It does not settle the full rank-(1,2) class. The inherited support theorem, within-call floor and incidence count are cited premises from commdata2338's #2395 and Sagar Thakrar's #2294, following alephnan, Mihir Srivastava and Adam Gall. JoyboyBrian's #5141 is the adjacent rank-(1,1) predecessor. Exact source pins and the written proof are in research/p112-exceptional-hull-floor/.

This contribution was produced by Nick Petalas. Independent human and formal proof-kernel review are unclaimed.

Evidence

• Before: the explicit form pair has global ranks (1,2) and minimum cross prices (0,0), so Research: a stronger P112 two-call COPY floor for arbitrary cross forms #2395's test reads 0 >= 1 and abstains.
• After: the written proof gives the sharp floor 128 for its entire exceptional-hull family. Exact replay checks 8,192 transfer instances, all 6,561 support-span choices for that example, the 128 attainer and two actual price-15 hypothesis-removal controls.
• Local CPython 3.12.15 on macOS arm64: both registered warning-strict checks pass. Individual normal and optimized checks took 1.408 and 1.406 seconds in single validation runs. These are timeout checks, not performance claims.
• Metadata/pins, committed bytes and source history pass at 8f26fda2e817be8dc5ed3bf1edcde774b9f20f7c. The official eight-file export passes integrity checking; normal and optimized Git-free replays pass. The verifier emits generated hash-bound receipts to stdout. No predecessor verifier, Lean build or heavy replay was run.

Exact reproduction commands from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B scripts/research_package.py --package research/p112-exceptional-hull-floor
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/p112-exceptional-hull-floor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/p112-exceptional-hull-floor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B scripts/check_research_commit.py --package research/p112-exceptional-hull-floor --check-source-history

The portable interpreter is python3.12; no third-party dependencies are required. Expected marker: PASS_P112_EXCEPTIONAL_HULL transfer=8192 doubles=6561 floor=128 controls=2.

Limitations and unrun obligations

The complete support theorem is inherited, not re-audited. The universal argument is written, not kernel formalized. The finite enumeration concerns the new transfer and example; it does not enumerate arbitrary complex decoders. Physical chart realization, dirty restoration, hosting, precision, rank-power costs and recursive admission remain outside scope. No complete multiplication theorem or new kappa is claimed, and no selected-frontier file changes.

Open next questions

• Decide the remaining rank-(1,2) cases outside this exceptional-hull family by coupling nonzero Schur feedback to pure coefficient ranks.
• Identify and fully charge a physical realization of one surviving discounted cross form, including chart conversions and cleanup.

Related open work

• Prove the sharp rank-one two-call P112 COPY floor #5141 proves the full rank-(1,1) two-call floor 128. This extends its mixed-orientation mechanism to higher reverse rank; it leaves the other rank-(1,2) cases open.

• Copied-information floor of the selected centre path #3120 studies copied information at a literal centre consumer under its pinned coupling contract. This studies directed cross-form prices in a different algebraic interface.

• Decoder floors: universality, projection transfer and batching reductions; C(P112) ≥ 64 reduced to named conjectures, C(P120) ≥ 72 restated on N₀ (no κ claim) #2832 gives broader decoder and batching reductions, including factors outside the root code. This keeps both factor families in the root code and changes the two directed cross forms.

• C(P112) >= 64: complete reduction to stated conjectures #3154 reduces the broader single-call P112 floor to conjectures. This result addresses a restricted two-call root-code family.

• Non-dense fitting B: every E8 term with x·y ≠ 0 has rank ≥ 9; square root-code floors E8 ≥ 72, P112 ≥ 64, P104 ≥ 56, D4 ≥ 12 for any fitting B (no κ claim) #3045 treats square root-code decoders with fitting pairings. This permits any finite number of factors and two independent directed cross forms.

• One-sided padding lemma: P112 to P120 at cost exactly +8 (L120 suffices for Delta >= n_nf) #3074 gives one-sided P112-to-P120 padding. This changes cross-call pairings without introducing outside-code factors or padding.

• Checkpoint symmetric two-star P112 supports and inconclusive COPY63 search #5088 explores a masked two-star P112 support dictionary and an inconclusive COPY63 search. This proves a floor under a separate all-pairs two-call root-code contract.

• Prove binary E8 factor-rank-sum barrier 21 at every term count #5155, opened on this account after the CLAIM, proves a factor-rank-sum barrier for the fixed binary E8 matrix with rational pairing charges. Its scalar field, matrix and charge differ from this complex P112 two-call theorem; no numerical comparison or implication is asserted.

CI notes

Both hosted checks passed at 8f26fda2e817be8dc5ed3bf1edcde774b9f20f7c in run 39943764557: plan and replay (research/p112-exceptional-hull-floor) / replay (research/p112-exceptional-hull-floor). The completed run has conclusion success and its head SHA matches this contribution. The replay runs both registered normal and optimized checks. All local checks listed above also passed; no local obligation was deferred to CI. Passing automated checks does not constitute independent acceptance of the written proof or inherited premises.

Merge Danger

Door: two-way. Blast radius: one isolated research package. Shared code, predecessor packages and the selected frontier are unchanged.