Research: prove even-subset R floor 82 among X=6 at seven queries (no new kappa)

[VictorLiwentao]

Change and scope

In the five-source common-B mutable-source alphabet of #1898, every restored straight-line word with exactly seven queries and even-subset connector-state readout needs at least 82 readouts. Among words with X = 6, that floor is sharp: the path-lowered champion (0,1,2,6,12,24,16,0) attains 7 Q / 82 R / 6 X (abstract tariff 442 at (b,a,x)=(38,2,2)).

Covering walk class	Distinct nonzero labels	Attainable even-subset R
Hamilton (distinct slots)	6	{82, 88}
One repeated vertex (GL basis census)	5 spanning	{96}
Rank < 5 label sets	≤4 independent	none (image ≤16)

This answers open question 1 of #2643: min R among restored seven-query words with X = 6 is exactly 82. In particular R ≤ 78 is impossible, so tariff 38·7 + 2R + 2·6 cannot undercut six-query endpoint 436 by lowering R at X = 6. Repeated prefixes cannot beat 82 (only add R = 96).

Scoped obstruction with sharp attainment; no new kappa. Leave-alone through #2677 (including #2643, #2593) was not reopened.

Result

Package: research/seven-query-r-floor/.
Tip: 0e5f24d033eba7a36e6fc990d7263dc4c2ee9183.

Exact: min_R_among_X=6 is 82 sharp; repeat layer R=96; tariff floor 442.

Validation

python3.12 -I -B research/seven-query-r-floor/verify.py
python3.12 -I -O -B research/seven-query-r-floor/verify.py
python3.12 -I -B research/seven-query-r-floor/independent.py
python3.12 -I -B research/seven-query-r-floor/test_package.py
python3.12 -B scripts/research_package.py --package research/seven-query-r-floor

Expected:

PASS_SEVEN_QUERY_R_FLOOR Q=7 min_R_among_X=6 is 82 sharp repeat_layer_R=96 answers_7qxor_oq1 tariff_floor=442 no_new_kappa
PASS_INDEPENDENT_SEVEN_QUERY_R_FLOOR min_R=82 ... tariff=442 answers_7qxor_oq1
PASS_SEVEN_QUERY_R_FLOOR_PACKAGE_TESTS groups=6
PASS_REGISTERED_RESEARCH_PACKAGE research/seven-query-r-floor outcome=completed

Default verify is read-only. Admission uses need(); -O also passes.

Overlap and credit

Closest priors: #2643 (X≥6 at Q=7; OQ1), #2677 (path-lowered distinct-slot spectrum), #1898. Freestanding; no competing open PR on this repeated-prefix R floor at refresh.

Credit the mutable-source / PR1898 / #2643 lineage. Prepared by Wentao Li (VictorLiwentao, @Is__Victor) with substantial AI assistance, under Apache-2.0.