[petalas]

Result

Claim type: scoped proof with exact finite corroboration. No new κ.

This works in Ali Romman's relaxed group-gate model from #5394 (research/group-subproduct-cover), unchanged. Let G be a group of prime exponent, possibly nonabelian, and K any field. Any diagonal target with d nonidentity entries, repetitions allowed, needs total projector rank at least d. Arbitrary finite clean or dirty scalar workspace and singular or rectangular free maps do not change this. Direct coordinate gates attain d, so the exact rank-one count is d.

This answers #5394's Open next question for these groups. The rank-one count exceeds the relaxed subproduct-cover cost exactly when the cover is shorter than d. The gap can be exponential: the 2^n − 1 nonidentity labels of (C2)^n have cover cost n and rank-one count 2^n − 1.

The proof (PROOF.md) is a representation rank cut: Σ_i rank(ρ(t_i) − I) ≤ Σ_j rank(N_j)·rank(ρ(g_j) − I) for any representation ρ over an extension field. The regular permutation representation over K has rank(ρ(g) − I) = |G|(1 − 1/ord g) in every characteristic. Prime exponent makes that value the same for every nonidentity label.

Evidence and reproduction

verify.py uses only the standard library, is read-only, and does not use the network or Git. It checks:

• 405 regular defect ranks over Q, F2, F3, F5 and F7, in (C2)^2, (C2)^3, (C2)^4, (C3)^2, C5, the order-27 Heisenberg group, C4 and Q8;
• 206 seeded random circuits with rectangular and singular free maps and projectors of every rank. Each one checks the augmentation identity, multiplicativity of the entrywise image, and the cut;
• exact cover costs 2, 3 and 4 for all nonidentity targets of (C2)^2, (C2)^3 and (C3)^2, with the attaining cover and direct words over Q, F2 and F3, plus the 15-label (C2)^4 basis cover;
• five controls.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -B -W error research/prime-exponent-rank-cut/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -B -W error research/prime-exponent-rank-cut/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/prime-exponent-rank-cut
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/prime-exponent-rank-cut --check-source-history

Any CPython 3.12 works. Local results on CPython 3.12.15, macOS arm64:

• both verifier runs print PASS_PRIME_EXPONENT_RANK_CUT regular_checks=405 random_circuits=206 separations=4 controls=5 no_new_kappa (0.51 s and 0.53 s);
• the package runner prints PASS_REGISTERED_RESEARCH_PACKAGE;
• the commit check prints PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

• This is an abstract count in an optimistic relaxed model. Physical group gates, scalar maps, source reads, precision, recursive children, hosting and cleanup are not priced. The selected frontier and κ are unchanged.
• Groups that do not have prime exponent are not settled. For three copies of a² in C4, the regular cut alone gives 2.
• The cyclic-group remark in PROOF.md section 6 (faithful character, characteristic not dividing m) follows from the same theorem but has no separate finite check.
• The finite checks corroborate the written proof; they do not replace it. No independent human or formal proof-kernel review is claimed. No receipt file is generated; the verifier's printed JSON is the run record.

Credit

• Ali Romman (Prove the all-group rank-one triple floor with free scalar workspace #5394): model, cover theorem, the question, and the all-group three-label floor. Cited, not re-verified.
• Wentao Li (research/elementary-abelian-diagonal-minimum): the rank-one minimum in F2[(C2)^k]. This result contains that statement and extends it to every field, every prime, nonabelian prime-exponent groups, rank-charged projectors and singular free maps.
• The rank-telescoping method was used before in Prove the exact order-21 normal-subgroup call tariff (no new κ) #5202 (research/order21-normal-tariff) and research/cut58-separable-query-floor.

Open next questions

1. For groups that do not have prime exponent, is the rank-one count always at least the number of nonidentity entries? The first case is three copies of a² in C4.
2. Which positive combination of permutation representations gives the best cut for a fixed group and target, for example D8 or Q8 over F2? These cuts are valid in every characteristic.
3. Does any physically realizable prime-exponent label family have a priced native implementation?

Related open work

• Prove an integer-shear target and its inverse need eight calls: no cross-role sharing (no new κ) #6011 (research/sl2z-inverse-pair-cover) works in the same Prove the all-group rank-one triple floor with free scalar workspace #5394 group-gate model. It proves the exact minimum of eight calls for diag([B],[B^-1]) over F2 in an integer-shear group. That group has elements of infinite order, so it is outside the prime-exponent hypothesis here. This package computes no number for that target, so there is no comparison.
• Searches for "subproduct", "group label", "regular representation" and "prime exponent" found no other open PR on this model or quantity.

CI notes

Head 9ba652f6e10ff296ca467fc69a24246c9a5b91af, run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40585468960.

• plan failed in the "Onboarding helper tests" step: 675 of 676 tests passed. The one failure is test_research_descendant_source_roles.DescendantSourceRoleTests.test_live_history_guard_is_not_a_historical_source_substitution. That test runs another package's research/isolated-descendant-bytecode/verify.py, whose nested test hit a tempdir cleanup race: OSError: [Errno 39] Directory not empty: '/tmp/isolated-descendant-proof-*/owned/*/.git/objects/pack'. This PR does not touch that package or that test. The same plan job passed on Accelerate exact mixed-point construction 1.51× (no new κ) #6007, Prove the char-2 jet obstruction on jf0..2 holds at every order (no new κ) #6009 and Prove an integer-shear target and its inverse need eight calls: no cross-role sharing (no new κ) #6011, which were opened in the same period.
• replay (research/prime-exponent-rank-cut) was skipped because plan failed. Hosted replay did not run, so this package is not CI-certified. Hosted certification needs a maintainer re-run.
• Local checks passed on CPython 3.12.15, macOS arm64: both verifier commands, research_package.py and check_research_commit.py --check-source-history.