Research (no kappa claim): classical Pappus/Desargues T=3 pairing-cost census — no cost < v falsifier; tight atoms through v = 6

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Result

Scoped negative, no kappa claim. A first answer to item 4 of the open-ends request
(discussion comment 18932893):
the classical rich configurations Pappus and Desargues, named there as untested
candidates for tight atoms or counterexamples of the T = 3 core (Q <= F2), were tested in
the exact PR 2111 model. No cost < v instance exists in the tested domain. New package
research/classical-t3-config-screen-seed3e11/ only; the selected record, both suppliers,
upstream/ and all frozen pins are unchanged.

Precise statement: over all admissible pairings B (0/1 over F2, unit diagonal, ones only
where p(t) dot q(s) = 0), cost(B) = sum_j rank_F2 B[R_j, A_j] with
R_j = {t : p_j(t) != 0}, A_j = {s : q_j(s) != 0}. Of 1,326,413 sub-configurations of
the Pappus (54 valid (p, q) pairs) and Desargues (70 pairs) pair sets, each priced exactly
over all 2^f admissible B (f = free incidences, f <= 20), none has minimum cost below
v. Every sub-configuration of size v <= 4 is exhaustive (1,319,225 instances); sizes 5..10
are fixed-seed samples (seed 20261015). The full pair sets survive a randomized falsifier
search (120 single-bit-descent restarts each).

Size v	Pappus tested / exact / tight (cost = v)	Desargues tested / exact / tight
1	54 / 54 / 20	70 / 70 / 25
2	1,431 / 1,431 / 198	2,415 / 2,415 / 308
3	24,804 / 24,804 / 1,310	54,740 / 54,740 / 2,510
4	316,251 / 316,251 / 6,579	916,895 / 916,895 / 15,318
5	3,000 / 3,000 / 20	3,000 / 3,000 / 15
6	1,500 / 1,500 / 3	1,500 / 1,500 / 2
7	300 / 300 / 0	300 / 300 / 0
8	60 / 56 / 0	60 / 60 / 0
9	20 / 13 / 0	20 / 17 / 0
10	10 / 1 / 0	10 / 6 / 0

"Exact" counts instances with f <= 20, fully enumerated; the rest are recorded as
skipped_f_cap and were not priced exactly. Two observations: tight atoms are common at
small sizes and disappear in the samples from size 7 on (sampled minima rise strictly above
v there), and the full pair sets are far from tight (Pappus v = 54: dense 109, identity
104, search best 108 with block ranks 36,36,36; Desargues v = 70: dense 145, identity
137, search best 146 with block ranks 49,48,49). So the rich pair sets do not approach the
cost >= v boundary at all.

Nearest overlapping work (surfaced by the pre-submission overlap refresh).
#2274 (thakrarsagar; claim
#1163 (comment))
already treats the same object (Desargues 10_3 and Pappus 9_3 as T = 3 decoder atoms,
answering the same open-ends item 4) by a different method: frame-universal char-2 (CR2)
certificates giving cost >= v for every configuration of non-incident pairs in a certified
frame, conditional on Lemma CR2 of the open PR 2162, plus exact pairing-cost minima only for
the 18 enumerable non-CR2 instances of one special Pappus frame (their min(cost - v) = 1).
This package is an independent, unconditional corroboration by direct exact enumeration of raw
pairing costs over all admissible B: it carries no CR2 or PR 2162 premise, prices 1,326,413
instances across both geometries including an exhaustive v <= 4 census and tight-atom counts
that PR 2274 does not compute, and adds the full pair-set landscape. PR 2274's frame-conditional
coverage is far broader in v; the two results are complementary and neither supersedes the
other. The overlap was missed at entry because the pre-computation overlap re-run used the
request's terminology (char2-jets, three-total-decoders) and not the geometry names
(Pappus, Desargues); this is recorded as a workflow failure in the session handoff.

Task card

Task ID / proposed scope key: coverage-law-change (drawn) -> changed; open request item 4 /
  classical-t3-config-screen. Claim: discussions/1166#discussioncomment-18944356.
Base commit; predecessor heads: main 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2;
  predecessor #2111 head 98da76c2839ba38538b510420dec54546ccabf0b (model + two fixtures);
  #2162 head 113a718265c62aec301c5695f4bfb9c421bef7ab (context only).
Concrete object and question: do the Pappus and Desargues valid (p, q) pair sets contain a
  tight atom (min cost = v) or a counterexample (min cost < v) to the T = 3 core (Q <= F2)?
Scope status: changed (recorded in TASK.json with reason). The drawn card coverage-law-change
  has its object answered and under active follow-up in discussion 1587; the other drawn
  cards collide with fresh work (PR 2391 centre sharing; the QQ16 observation sweep).
Existing restriction being changed / claim audited: item 4 of comment 18932893, the named
  untested classical configurations, in the unchanged #2111 cost model.
Method; parameter range: exact rational geometry construction; exhaustive enumeration of all
  2^f admissible B per instance for f <= 20; all subsets of size v <= 4 and fixed-seed samples
  of sizes 5..10 (random.Random seed 20261015); randomized single-bit-descent falsifier search
  on the full pair sets.
Nearest overlapping PRs and why this adds something: PR 2274 (thakrarsagar) covers the same
  Pappus/Desargues object by CR2 frame certificates conditional on #2162 and exact minima for
  18 non-CR2 instances of one special Pappus frame; this adds unconditional raw pairing-cost
  minima on 1.3M instances, the exhaustive v <= 4 census, tight-atom counts and the full
  pair-set landscape, by an independent implementation. PR 2111/2162 define the model; PR
  2176/2206 closed jf0 and cex_min_1; PR 2330/2351 work the 121-line pool. Overlap with PR
  2274 was found only at the pre-submission refresh (entry re-runs omitted the geometry names).
Optimistic full cost budget; required observable improvement: no kappa or cost-structure
  claim; the observable is a certified cost < v falsifier or a complete bounded negative.
Success evidence; falsification test; time/compute cap: engine must reproduce the two pinned
  #2111 control minima (18 and 12) before any instance is believed; falsifier = certified
  admissible B with cost < v; single process, about 8 minutes measured, 90-minute budget.
Predecessor authors; intended artifact: DreamingOfClouds (model, open question); package
  research/classical-t3-config-screen-seed3e11/ with registered verify.py.

Validation

CPython 3.12.14, Windows x64, standard library only. From the repository root:

python3 -B research/classical-t3-config-screen-seed3e11/verify.py
python3 scripts/research_package.py --package research/classical-t3-config-screen-seed3e11 --check-only

Measured 470 s for the verifier. Expected and observed marker:

PASS_CLASSICAL_T3_CONFIG_SCREEN controls=18/12 pappus_exact=347410 pappus_tight=8130 desargues_exact=979003 desargues_tight=18178 falsifiers=0 full_best=108/146 no_new_kappa
PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)

Positive fixtures, from pinned bytes of the PR 2111 head: data/cex_wholeTS_v9.json reproduces
its documented exact minimum 18 (block ranks 7,4,7, 16 minimisers) and data/cex_joint_v7.json
its documented minimum 12 (block ranks 4,4,4, 4 minimisers). Failure controls: two ill-formed
lines (p(t) dot q(t) = 0) are rejected; tampering with one fixture coordinate moves its
optimum to 19, so the positive control binds; disallowed B positions exist and are excluded.
The Pappus and Desargues constructions assert their collinearities and 3-uniformity
(3 points per line, 3 lines per point). RESULT.json records every per-size count and the
full-pair-set landscape.

Not checked: no lower bound is proved (cost >= v is verified instance by instance only); the
full pair sets carry upper bounds only; the Reye configuration and D4 line arrangements of the
same item are out of scope (no pinned point/covector encoding for them); the G1' lift witness
and the rational side (Lemma B) are not re-checked; no cold replay or supplier admission is
involved.

Open next questions

1. Reye and D4 line arrangements, the remaining two families of item 4: fix a
   point/covector encoding for the Reye 12-point/16-line structure, then rerun this census
   unchanged. Blocked by: no predecessor pins such an encoding.
2. A whole-set lower bound for the full pair sets (fixed minors as in
   Prove jf0 pairing cost exactly 19 with fixed minors #2176 or the PR 2111 multi-copy
   certificate) against v = 54. Blocked by: that certificate machinery lives in PR 2111/PR 2162
   package code not vendored here.
3. Tight atoms at sizes 7..12: enlarge the fixed-seed samples or bias toward incidences as
   the PR 2111 Hesse experiment did. Blocked by: exact enumeration stops at f <= 20.

Credit

The model, the T = 3 open core (Q <= F2), the lift route and the untested-classical-configurations
lead are DreamingOfClouds'
(#2111,
#2162,
comment 18932893
item 4); the PR 2111 Hesse-configuration experiment (G7) is the model precedent. thakrarsagar's
#2274
(research/three-total-classical-configs, claim
#1163 (comment))
is the nearest overlapping work on the same object and is not superseded here. The two control
fixtures are vendored unmodified from the PR 2111 head 98da76c2839ba38538b510420dec54546ccabf0b
(git blobs a843cce049a2aadd5352763edeedd911cc2ebd1e and
cf7a3db872787a5480d6d6d49b5e52218a1c3905; exact SHA-256 pins in the package README). The
exact-cost precedents #2176 (jf0) and
#2206 (cex_min_1) shaped the certification
discipline. Everything else in the package is new.

Contributor: Aaron Wang, with help from opencode (substantial AI assistance; no independent
human review). New work is Apache-2.0; predecessor notices and licenses are retained.

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