Research: exact Q8 pair minima under unit-priced inverse labels (complete r=2 census, no compression)

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Summary

Complete finite group-algebra census in the formal F2[Q8] register model at
r = 2: every one of the 64 ordered diagonal pair targets diag([g1],[g2])
costs exactly its direct bill, both with unit-priced inverse labels
x = i, x^-1 = -i, y = j, y^-1 = -j (24 paid factors) and under the
positive-label baseline {x, y} (12 paid factors). Unit-priced inverse
labels therefore enable no cross-register sharing compression; they change
only the re-pricing of the elements -i, -j from three calls to one. No
kappa, supplier, frontier or multiplication-theorem claim.

Package: research/q8-inverse-label-pair-census/. Contributor: Aaron
Wang, with help from opencode (substantial AI assistance; no independent
human review claimed).

Scope and changed premise

This answers open question 1 of the merged predecessor
research/q8-central-commutator-pair
(#1822): "does a label set
including the inverses compress any pair target?", extended from its two named
endpoints to the complete pair family, with the positive-label pair table as
the comparison baseline.

• Roles: exactly r = 2, both registers arbitrary dirty inputs, endpoint
  restored exactly. Every factor D_N(g) = I + ([g]+1)N is charged
  (N = u v^T binary rank-one idempotent, v^T u = 1 mod 2; six N).
  No free address maps, no COPY/ERASE, no clean ancilla.
• Free scalar GL_2(F2) mixing is normalized by augmentation; conjugation
  closure of the idempotent list is rechecked for all six S in GL_2(F2).
• Acceptance is equality in every coefficient of the eight-element group
  algebra F2[Q8] (a uniform identity on arbitrary dirty inputs), never a
  finite-column test.
• The factor set is closed under inversion: D_N(g) D_N(g^-1) = I, checked
  for every factor of both alphabets.

Exact result

Rows index the first target entry, columns the second, order
1, -1, i, -i, j, -j, k, -k. In both alphabets minimum = direct bill for
all 64 pairs (compressed_inv=0, compressed_pos=0):

inverse labels (min = bill)          positive labels (min = bill)
     1  -1   i  -i   j  -j   k  -k       1  -1   i  -i   j  -j   k  -k
 1   0   2   1   1   1   1   2   2    1   0   2   1   3   1   3   2   2
-1   2   4   3   3   3   3   4   4   -1   2   4   3   5   3   5   4   4
 i   1   3   2   2   2   2   3   3    i   1   3   2   4   2   4   3   3
-i   1   3   2   2   2   2   3   3   -i   3   5   4   6   4   6   5   5
 j   1   3   2   2   2   2   3   3    j   1   3   2   4   2   4   3   3
-j   1   3   2   2   2   2   3   3   -j   3   5   4   6   4   6   5   5
 k   2   4   3   3   3   3   4   4    k   2   4   3   5   3   5   4   4
-k   2   4   3   3   3   3   4   4   -k   2   4   3   5   3   5   4   4

Fact	Value
Pairs checked per alphabet	64 (all ordered pairs in Q8)
Pairs compressed below their bill	0 (inverse), 0 (positive)
Pairs whose minimum changes between alphabets	28 (exactly those with -i or -j as an entry)
Maximum minimum	4 (inverse), 6 (positive)
Predecessor endpoints reproduced	diag([z],[z]) = 4, diag([xy],[yx]) = 4 (positive alphabet)

Every difference between the two tables is the re-pricing of -i, -j
(length 3 to 1 per entry), not sharing.

Evidence

python -I -B research/q8-inverse-label-pair-census/verify.py (stdlib only,
about 40 s) recomputes everything and compares with CERTIFICATE.json:

• Q8 group laws exhaustively (512 associativity triples, x^2 = y^2 = z,
  xy = k, yx = -k, z central, [x,y] = z) and agreement of two
  independently built group tables.
• Idempotent completeness (6 of 9 binary pairs), GL_2(F2) conjugation
  closure, D_N(g)^2 = D_N(g^2) and the inverse identity for all factors.
• Complete BFS layers and ball SHA-256 digests from two independent engines
  (bitmask convolution from a (sign, axis) quaternion table vs frozenset
  convolution from explicit Hamilton 4-tuple integer products), agreeing on
  every layer and ball: inverse [1, 24, 358, 4104] (depth at most 3),
  positive [1, 12, 110, 936, 6945, 48080] (depth at most 5).
• Calibration with the merged predecessor: its recorded positive-alphabet
  layers [1, 12, 110, 936] and its two endpoint minima are reproduced
  exactly (source pinned in SOURCES.json).
• All 128 attaining words replay coefficientwise in both engines; 372
  single-call omission controls miss their targets; one mutated-target
  control fails; known-distance controls (1 at 0, diag([i],[1]) at 1,
  diag([z],[1]) at 2); alphabet monotonicity holds for all 64 pairs.

Complete shorter-word exclusion is the BFS itself: every word of length at
most 3 (inverse) or at most 5 (positive) lies in the complete ball, and no
ball contains a target below the recorded minimum.

Reproduce

python3.12 -I -B research/q8-inverse-label-pair-census/verify.py
python3.12 -I -B scripts/research_package.py --package research/q8-inverse-label-pair-census --check-only
python3.12 -B -m unittest discover -s tests -p 'test_research_*.py'

Observed on this branch (CPython 3.12 and 3.13, Windows):

PASS_Q8_INVERSE_LABEL_CENSUS pairs=64 inv_layers=[1, 24, 358, 4104] pos_layers=[1, 12, 110, 936, 6945, 48080] compressed_inv=0 compressed_pos=0 omission_controls=372 no_new_kappa
PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
Ran 65 tests in 25.030s  OK (skipped=3)

Limitations and unrun obligations

• Scoped to these two alphabets, r = 2, and these 64 targets. It is not an
  impossibility theorem for other alphabets, r >= 3, or the rational-frame
  model. No written structural lower bound is claimed; the minima rest on the
  complete search.
• No supplier mapping, paid reserve, moment or kappa change; the a < D/E
  diagnostic does not apply (no candidate improvement is priced).
• Not a proof-assistant verification: two independent Python engines and
  finite exhaustions; no Lean or native C++ replay. The r >= 3 extension
  and a second-order lower bound are recorded as open questions in the
  package README.
• Hosted CI checks were pending at submission time.

Overlap and predecessor credit

• Model and free-mixing normalization: eumemic's PR 288 section 1.2; woklerx's
  research/bit-dihedral-commutator
  (Research: prove the eight-call floor for two dihedral commutator targets #1443); the selected
  Q8 pair result research/q8-central-commutator-pair
  (Research: quaternion alphabet central commutator pairs cost their direct bill (no new kappa) #1822), whose open
  question 1 this closes. Register-model lineage: Ali Romman
  (Prove nine calls are necessary for three abelian bit registers #468), Gareth Davidson
  (Research theorem: exact paid cost for binary affine register words #1333),
  research/bit-three-register-lower-bound.
• Concurrent work this is deliberately distinct from: the r = 3 three-role
  positive-label minima (Research exact Q8 three-role target minima #2248,
  HANDOFF: exact Q8 three-role target minima (Cycle AF) #2243) and the
  one- and two-factor words for diag(i,j,k) with all-nonidentity labels
  (Research exact Q8 mixed-unit register minimum #2303,
  CLAIM: two-call Q8 register word for diag(i,j,k) (seed 189ad74f) #2300). The
  predecessor's open question 2 (diag([z],[z],[z]) at r = 3) is not taken,
  to avoid that concurrent object.
• scripts/find_overlap.py refreshes at 2026-10-15T04:34Z and before
  submission (complete pagination) found no inverse-label Q8 pair object.

Claim thread: #1317 (comment)