Research: quaternion alphabet central commutator pairs cost their direct bill (no new kappa)

[navinor]

PR: Quaternion alphabet: central commutator pairs still cost their direct bill (no new kappa)

Scope

Track bit-mechanisms, task card bit-new-register-model (scope key 7f5f4426ef796683c485,
chooser OS nonce 73aa3cb4cec915ba6c0a1125abe1e410, priority revision 3), scope refined
to the card's allowed variation "a specified nonabelian operation family", instantiated as
the quaternion group Q8. Exact base 0b1cccf86e2f5132ed286a26ef0a681a495740ea;
selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

The scoped three-register obstruction (research/bit-three-register-lower-bound) and the
dihedral predecessor (research/bit-dihedral-commutator,
#1443) both fix their address group and
state that other address groups are not excluded. This package changes exactly that one
hypothesis: address group Q8 = {+-1, +-i, +-j, +-k} instead of M_3(Q) / UT3(F2) = D8.
The commutator arithmetic differs: z = xyxy = x^2 = y^2 = -1 is a paid-label square, so
the direct bill is 4 rather than the dihedral 8.

Model: formal F2[Q8], paid labels x = i, y = j only, paid factors
D_N(g) = I + ([g]+1)N with N = u v^T, v^T u = 1, over r scalar registers (6 N / 12
generators at r = 2, 28 N / 56 at r = 3), free scalar GL_r(F2) mixing normalized by
augmentation (predecessor's argument; its finite GL-conjugation closure step is rechecked),
arbitrary dirty register inputs, every call charged, no free address maps, no COPY/ERASE, no
clean ancilla. Acceptance is equality in every coefficient of the full group algebra.

Result

endpoint	direct bill	exact minimum
r = 2: diag([z],[z]), z = xyxy = x^2 = y^2	4	4
r = 2: diag([xy],[yx]) (noncommuting pair, both orderings)	4	4
r = 3: diag([z],[z],[e]) with one restored dirty register	4	4

Scoped negative: no cross-register sharing compression exists in this alphabet. Complete BFS
layer expansion to depth 3 (layers 1,12,110,936 at r = 2 and 1,56,2324,89574 at r = 3,
ball SHA-256 digests in CERTIFICATE.json) excludes every word of length at most 3; the
direct four-call word attains 4. Unlike the dihedral generators the Q8 factors are not
involutions (D_N(x)^2 = D_N(x^2) = D_N(z) is checked for all generators), so no split
argument is needed.

Evidence and reproduction

Package: research/q8-central-commutator-pair/. Registered replay passes:

python3.12 scripts/research_package.py --package research/q8-central-commutator-pair
# PASS_REGISTERED_RESEARCH_PACKAGE research/q8-central-commutator-pair outcome=completed
python -I -O -B research/q8-central-commutator-pair/verify.py
# PASS_Q8_CENTRAL_COMMUTATOR diag_z_z_r2:4 diag_xy_yx_r2:4 diag_z_z_e_r3:4
#   r2_layers=[1, 12, 110, 936] r3_layers=[1, 56, 2324, 89574] no_new_kappa

The verifier recomputes everything from scratch (stdlib only) and checks: Q8 group laws
exhaustively plus agreement of two independent quaternion multiplications; rank-one
idempotent counts 6 / 28 and idempotence; D_N(g)^2 = D_N(g^2); GL_2(F2) (order 6) and
GL_3(F2) (order 168) conjugation closure of the generator sets; complete BFS layer sizes
and ball digests; engine A (bitmask convolution) and engine B (independent frozenset formal
matrices with Hamilton 4-tuple multiplication) agree on every witness and direct word;
omission controls (every single-call deletion of each direct word misses, 12 controls);
known-distance controls (I at 0, diag([x],[e]) at 1, diag([z],[e]) at 2); mutated-target
failure control; certificate equality. scripts/research_package.py --check-only reports
PASS_PACKAGE_METADATA_AND_PINS. The tests/test_research_*.py suite passes (42 tests OK,
2 skipped).

Limitations (stated, not checked)

No kappa, supplier, moment or frontier change is claimed; the bit side does not bind and no
physical invoice, native replay, hosting, precision or analytic obligation is invoked. The
result is exact only for this alphabet (labels x, y, these endpoints, r in {2,3}): a failed
search is a result about the searched domain, not an impossibility theorem. Not a
proof-assistant verification. The free-mixing augmentation normalization is inherited from
the predecessors; only its finite closure step is rechecked here.

Overlap

Refreshed scripts/find_overlap.py with the concrete object terms (2026-10-14T10:53Z,
complete pagination): no Q8 object in open PRs, discussions or local packages besides
this one. Deliberately distinct from the F2[GL2(F2)] coordinate-CNOT/SWAP line (order-six
group, isomorphic to S3), the UT3/D8 commutator line (closed at 4k for all registers;
third dirty register closed negatively), and the S4/S5/AGL2/GL3 target searches of
2026-10-14. Coordination claim:
#1317 (comment).

Credit

Prepared by Aaron Wang, with help from opencode (substantial AI assistance disclosed; no
independent human review). Apache-2.0. Model, normalization and presentation follow
woklerx's research/bit-dihedral-commutator
(#1443), which explicitly leaves other
address groups open. Scalar-idempotent register model: eumemic's PR 288 section 1.2; the
register-model lineage credits Ali Romman (#468),
Gareth Davidson (#1333) and the pinned
three-register note (research/bit-three-register-lower-bound). Frozen sources and
upstream/ are unchanged.

[Mister-Meeseeks]

Batch review at ee736b4b11bbd63b04d94c8edafb3790161ebdea.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Complete full-group-algebra BFS through depth3 and independent Hamilton-quaternion witness replay certify all three specified Q8 pair minima4. Free scalar mixing normalization and GL2/GL3 closure checked; inverse labels, more active roles and physical suppliers remain outside this result.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.