Research: D10 inverse rotations retain four-call minimum

[1337isnot1337]

Result

This package gives an exact minimum of four paid reflection calls for
diag([st],[ts],[e]) in the formal group algebra of the order-10 dihedral
group D10=<s,t | s^2=t^2=e,(st)^5=e>, on three arbitrary dirty scalar
registers and with paid labels restricted to {s,t}.

The proposed shared s,t,s schedule does not work. The verifier exhausts all
rank-one idempotent choices in M3(F2) and all ordered label words through
length three: 1, 56, 3,136 and 175,616 cases by length, with zero exact
endpoints. A direct four-call word attains the target. A second engine builds
the group by permutations of the pentagon and uses row-mask outer products;
it independently covers the same domain with distinct per-level hashes.

This tests the finite torsion boundary left by PR #1911's D_infinity
reflection theorem. It does not extend that theorem to other quotients, groups,
register counts or alphabets.

Reproduction

python3 -I -B research/d10-reflection-sharing-seeda3def8/verify.py
python3 -I -B research/d10-reflection-sharing-seeda3def8/independent.py
python3 -I -O -B research/d10-reflection-sharing-seeda3def8/test_package.py
python3 scripts/research_package.py --package research/d10-reflection-sharing-seeda3def8 --check-only
make selected-record-check

Both mathematical replays and the optimized controls passed locally under
CPython 3.14.7; metadata and source pins passed; make selected-record-check
passed. This machine has no Python 3.12 interpreter, so the registered Python
3.12 replay is left to the repository workflow. The research baseline also
passed as PASS_RESEARCH_BASELINE_BINDINGS_ONLY earlier in this cycle.

Pins, limits and attribution

Base: 0b1cccf86e2f5132ed286a26ef0a681a495740ea. Selected-result SHA-256:
10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.
The task seed is a3def8e7b944810f5d933bfa65cba990; the exact search certificate,
source manifest, task card and all source hashes are in the package.

The result is a scoped finite-group obstruction only. No supplier/compiler,
whole-interface cost, physical hosting, precision, recursive moment, new
kappa or selected-frontier change is claimed. The selected conditional
kappa remains 836330098538459/500000000000000000.

PR #1911 (infinite dihedral reflection bills), PR #1821 (D16 central pair),
PR #1573 (D8 central target), and PR #1870 (S3 target pairs) are credited and
pinned in SOURCES.json. New open PR #1941 (head
5938ce3e91bbb8b5c2227af7976d444b2fbac7d4), claimed in Discussion #1929
comment 18925900, proves tariffs for every even rotation order and expressly
leaves odd periods out. This package treats odd rotation order five and the
inverse-rotation pair. The final overlap refresh at 2026-10-14 14:56:57 UTC
covered 601 open PRs and 575 Discussions with complete pagination and no
warnings; no D10 target claim was found. Prepared with
substantial OpenAI Codex assistance. No independent human review is claimed.

[Mister-Meeseeks]

Batch review at 0f26ac2016777224e3c66c8ead77defbf3e526d2.

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

Reviewed scope: Exact full group-algebra search of all 178809 words of length at most three on three roles in the stated D10 rank-one projector alphabet, with independent traversal and four-call attaining endpoint. This excludes sharing only in this alphabet and role budget; no physical supplier.

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.