[petalas]

Result

Scoped written theorem in a formal group-algebra model, with a finite corroborating certificate. No new κ.

For every group G, every field F, every pair x, y ∈ G with xy ≠ yx and every width r ≥ 2, the target diag([xy],[yx],[e],…,[e]) in M_r(F[G]) has exact paid-call minimum four. Paid factors are I + ([g]-[e])N with g ∈ {x,y} and N a rank-one idempotent, as in #4089.

This answers Open next question 2 of Wentao Li's #4089 (research/a5-conjugate-any-registers) for all conjugate-pair alphabets together. #4089 (A5) and #4093 (PSL(2,7)) rely on a support filter that is specific to each group. In a general group, coincidences such as x²y = yx defeat that filter, so the patterns xxy, yyx, xyy and yxx need a new argument.

New step (PROOF.md): each factor is written as [g]N + (I-N). Sandwiching a coefficient between N_1 or I-N_1 and N_3 or I-N_3 isolates one corner of call subsets. Three uniform lemmas follow:

1. the single-call products vanish;
2. the complementary corner of the xy/yx coefficients vanishes;
3. P_{12} and P_{23} cannot both be nonzero.

Every three-call pattern then fails, either by Lemma 3 or by a rank-one row/column mismatch. The proof uses no generation hypothesis, element order or field characteristic. For y = x⁻¹ the target is I, so the noncommuting hypothesis is necessary.

Evidence and reproduction

verify.py checks the following:

• The expansion and corner identities, on 200 seeded rational triples.
• 82 distinct length-4 coincidence signatures of noncommuting pairs in S3, S4, S5, Q8 and the free group. The direct four-call word and all four one-call deletions are checked on each.
• Exhaustive exclusion of every word of length 1..3. These words fall into 26 subset-class keys. All are checked over F2 at widths 2 and 3 and over F3 at width 2.
• The inverse-pair control.

The finite check corroborates the proof. It does not cover all groups or widths.

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error research/conjugate-pair-every-group/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -B -W error research/conjugate-pair-every-group/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/conjugate-pair-every-group
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/conjugate-pair-every-group --check-source-history

All four commands passed locally on head 0cee0e104b68ce6f06bf84af03f46ac30680e64c, base 19fdb086ea96af9a9f87936a2b888e10af4ed810. Markers were PASS_CONJUGATE_PAIR_EVERY_GROUP, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Each verifier run takes about 5 s.

Limitations

• The theorem holds only in the formal group-algebra model. No native address operation, conversion, restoration or recursive caller is priced, and this is not a multiplication theorem.
• The selected κ is unchanged.
• The written proof has had no independent human review and no proof-kernel check.
• The results of Research: any-register A5 conjugate 5-cycle four-call floor (no new kappa) #4089, Research: PSL(2,7) conjugate any-register four-call floor (no new kappa) #4093 and Research: exact four-call floor for A5 conjugate 5-cycle pair (no new kappa) #2246 are cited, not re-checked.

Open next questions

1. Which pair targets diag([g],[h]) in an arbitrary group allow cross-register sharing below the sum of their positive word lengths? Corner sandwiches might give uniform floors in place of per-group censuses. A first step is diag([xyx],[yxy]) at length five.
2. Does an admitted BIT supplier have a priced native nonabelian address alphabet? This theorem rules out any saving on this target in every group, so such a proposal has to find its gain elsewhere.

Related open work

No open PR studies this target or the general-group conjugate-pair floor.

• Classify mixed Q8 call-space shear identities; no pair of shear images separates (no new κ) #6039 (opened after this PR's claim) works in the same rank-one-idempotent group-algebra model over F2[Q8], but on a different target: the mixed shape of four [zi] and one [zj] targets. Its target is not computed here, so there are no numbers to compare.

CI notes

Both hosted checks passed at head 0cee0e104b68ce6f06bf84af03f46ac30680e64c, plan and replay (research/conjugate-pair-every-group), in run 40608973638.