Research note: Desargues 10_3 / Pappus 9_3 as T=3 decoder atoms — frame-universal char-2 (CR2) certificates and the one uncertified Pappus frame (conditional on #2162, no κ claim)

[thakrarsagar]

Status: Research note (scoped result, conditional on open PR #2162's Lemma CR2). No κ claim; certificates/selected-result.json is untouched.

This answers OPEN ENDS item 4 of the #2111 handoff (#1166 comment) for Desargues 10_3 and Pappus 9_3 only: are classical rich configurations tight atoms or counterexamples for (A), cost ≥ v at T = 3? Reye 12_4 and D4 are not covered.

What was learned

1. Frame-universal char-2 certificates. A frame uses three configuration points (or three configuration lines) as the coordinate triangle. For every frame, data/universal.json stores one char-2 assignment X(point), Y(line). Each assignment respects the frame's supports and has X·Y = 0 exactly on incidences. Coverage: all 221 Desargues frames (GF(16)) and all 151 Pappus frames (GF(4)), plus 1793 certificates for every structured frame of 12 special (extra-triple) Pappus realisations. The verifier checks each one exactly, with no search.
   • Lemma U (hand proof in the README): one certificate gives a valid CR2 assignment for every decoder configuration built from non-incident (configuration point, configuration line) pairs in that frame, for any v and with repeats allowed. So (A) holds for that whole class, conditional on CR2.
   • Lemma Z: a frame's zero pattern depends only on the incidences plus any extra collinear or concurrent triples. Desargues has no extra triples (verified: the non-collinearity graph is triangle-free), so the Desargues result covers every exact realisation over Q. The Pappus result covers the generic signature and the 12 special signatures that the grid search found (a₂, b₁, b₂ ∈ [−15, 15]).
2. Side fact (likely classical; not claimed as new). No exact Desargues 10_3 exists in PG(2,4) or PG(2,8). One parameter fixed, there are 1344 in PG(2,16). Desargues frame certificates therefore need GF(16). On the 740 sampled instances, the per-instance search never needed more than GF(4).
3. Where the certificate search fails. I examined 2,166 coordinate systems: every structured frame, plus 13 identity frames that are all of one mixed type (e₁ = l∩m, e₃ = A₀, x = 0 the join A₀B₀). Exactly one has no frame-universal certificate over GF(2..16). It is the special Pappus realisation a=(0,1,-1), b=(0,-15,15). In that frame I sampled 2000 decoder configurations (seed 22, v = 6..10):
   • 1912 are CR2-certified.
   • 88 have no CR2 realisation over GF(2), GF(4) or GF(8). This is a bounded search; CR2 allows any char-2 field.
   • 18 of the 88 have f ≤ 26. Their exact minima give min(cost − v) = 1, at the instance (v, f, min) = (8, 24, 9).
   • The other 70 (f = 27..45) were skipped and counted.
   • Within this search there is no counterexample and no tight atom without a CR2 realisation. This is a bounded failed search, not a proof of (A) for this frame.

Reproduce

Base commit 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2 (origin/main at submission). From the repository root, using the standard library only:

python3 -I -B research/three-total-classical-configs/verify.py

Expected: every check line (checks 0–8) prints PASS, then
counts {'CR2_certified': 2652, 'noCR2_GF2to8_exact': 18, 'skipped': 70, 'universal_frames': {'desargues': 221, 'pappus': 151}}; min(cost - v) ...: 1,
then PASS_THREE_TOTAL_CLASSICAL_CONFIGS, exit 0. It took about 52 s on CPython 3.14.4. I tested only that version; the syntax is 3.10-compatible.

Lower-bound scope of the default run. Default mode re-derives exact minima by pure-Python exhaustion only for the 3 instances with f ≤ 18. For f = 19..26 it checks witness upper bounds, and the lower bounds are recorded enumall output. Check 8 asserts the counts and min(cost − v) = 1. To replay all 18 minima natively, run gcc -O2 -o /tmp/enumall research/three-total-classical-configs/c/enumall.c and add --enumall /tmp/enumall (PASS; 3–6 min depending on load). That replay re-runs the same vendored C code, so it is not an independent implementation. A local-only pre-submission review re-enumerated all 18 minima with its own C enumerator and found no B with cost < v. Those scripts are not included.

Controls.

• tied_min and the ✈️ Characteristic-2 shadows of rank-3 decoders: the anti-Fano bound by first-order jets, with a Lean-checked core (no κ claim) #2162 six-line atom are classified as having no CR2 realisation over GF(2/4/8). The six-line atom has pure-Python exhaustive minimum 6 = v.
• The stored ✈️ Characteristic-2 shadows of rank-3 decoders: the anti-Fano bound by first-order jets, with a Lean-checked core (no κ claim) #2162 GF(4) realisation of cex_min_1 passes the checker.
• Seven negative controls must be rejected: a corrupted certificate entry, a certificate checked against another frame, a GF(4) certificate read over GF(8), a corrupted realisation, a Pappus certificate with two line vectors swapped (incidence-only corruption), a zeroed point, and a constant assignment. The last two are rejected for trivial reasons; the README says so.
• Mutation tests on scratch copies with MANIFEST re-pinned:
  • One flipped certificate bit makes the verifier FAIL.
  • Raising the headline instance's minimum from 9 to 10, with a consistent cost-10 witness, now FAILs check 8 in default mode. Before this revision, only --enumall caught it.

Other checks run: python3 scripts/research_package.py --package research/three-total-classical-configs --check-only → PASS_PACKAGE_METADATA_AND_PINS; python3 -B -m unittest discover -s tests -p 'test_research_*.py' → 65 tests OK (2 skipped). git diff 0fdd7d82 -- upstream certificates is empty.

Pins, scope, obligations

• Vendored, byte-identical, with Apache-2.0 notice in NOTICE.md: lib/cr2.py, c/enumall.c and four data files from ✈️ Characteristic-2 shadows of rank-3 decoders: the anti-Fano bound by first-order jets, with a Lean-checked core (no κ claim) #2162 head 113a718265c62aec301c5695f4bfb9c421bef7ab, and enumall.c from Three-total root-unit decoders: reductions, a sharp example and the remaining lift question (no κ claim) #2111 head 98da76c2839ba38538b510420dec54546ccabf0b. All are pinned in MANIFEST.sha256 and SOURCES.json.
• Inherited: ✈️ Characteristic-2 shadows of rank-3 decoders: the anti-Fano bound by first-order jets, with a Lean-checked core (no κ claim) #2162 Lemma CR2 and the a = 0 2-adic corollary (open PR, not re-reviewed here), and the three-total model and cost formula (Three-total root-unit decoders: reductions, a sharp example and the remaining lift question (no κ claim) #2111 §1, from Format obstructions (no new κ): exact bounds on window width, copy cost and label dimension #446/Format limits (no κ claim): certified ceiling κ ≤ 5.1595e-3, spec census, structural lemmas, helper bounds and open problems #595).
• Not formalised: Lemmas U and Z are short hand proofs. The verifier checks only their finite inputs.
• Unresolved:
  • The 70 skipped instances (f = 27..45) in the special frame.
  • Mixed frames of other types, and intersection points of non-configuration lines used as p.
  • Pappus signatures outside the grid, including whether two extra point triples can coexist over Q.
  • Reye 12_4 and D4.
• tied_min control. The card asked for tied_min's exact minimum 12. Its f = 72 cannot be enumerated, so that value is inherited from Three-total root-unit decoders: reductions, a sharp example and the remaining lift question (no κ claim) #2111/✈️ Characteristic-2 shadows of rank-3 decoders: the anti-Fano bound by first-order jets, with a Lean-checked core (no κ claim) #2162, and the six-line atom serves as the enumerable tight control.
• Search record:
  • Frame-universal DFS over GF(2^k), k ≤ 4, with a 2e5 node limit; no timeouts.
  • Special realisations: a Desargues grid of 10584 points, which found 0 signatures (also proved impossible), and a Pappus grid of 27869 points, which found 12 signatures.
  • Samples: 740 structured instances and 2000 special-frame instances, all with seed 22.
  • Stopping rule: f > 26 skipped and counted, stop by 2026-10-15T00:34Z. All requested coverage was completed. "No cost < v" is not an impossibility theorem.
• Open next questions are in the README and TASK.json.

Credit and overlap

• Predecessors: DreamingOfClouds Three-total root-unit decoders: reductions, a sharp example and the remaining lift question (no κ claim) #2111 (model, enumall.c, tied_min, Hesse X2, the OPEN ENDS handoff), ✈️ Characteristic-2 shadows of rank-3 decoders: the anti-Fano bound by first-order jets, with a Lean-checked core (no κ claim) #2162 (Lemma CR2, cr2.py, the atoms without a CR2 realisation) and Where characteristic 2 starts to pay: a T² threshold for root-unit decoders (E6 ties, E7 first gain found; no κ claim) #2125; Root-unit decoders: two lemmas, an erratum to #595, and open cases (partial; no κ claim) #1590 (Lemma B); Format obstructions (no new κ): exact bounds on window width, copy cost and label dimension #446/Format limits (no κ claim): certified ceiling κ ≤ 5.1595e-3, spec census, structural lemmas, helper bounds and open problems #595.
• Adjacent: petalas Independently audit anti-Fano rank-three completions #2193 and RossRichardson Prove jf0 pairing cost exactly 19 with fixed minors #2176.
• Concurrent: petalas Certify exact pairing cost 21 for the 19-line configuration #2206 certifies cex_min_1's exact cost 21. That is a different question, and this package does not use it.
• What this adds: in full-support coordinates the result already follows from CR2 (compare Three-total root-unit decoders: reductions, a sharp example and the remaining lift question (no κ claim) #2111 X2). The new content is the structured, support-restricted frames, the special Pappus realisations, and the one located coordinate system where the certificate search fails.
• Overlap refresh at submission: origin/main unchanged since base. No PR or discussion mentions Desargues or Pappus apart from the Handoffs and requests for independent checks #1166 handoff and the claim. The Three-total root-unit decoders: reductions, a sharp example and the remaining lift question (no κ claim) #2111/✈️ Characteristic-2 shadows of rank-3 decoders: the anti-Fano bound by first-order jets, with a Lean-checked core (no κ claim) #2162 heads are unchanged.

Sprint: loop 9d24, iteration 2, track 20-39 (complex supplier, decoder sub-line), seed 22. Claim: #1163 (comment)

Contributor: @thakrarsagar, with substantial AI assistance from Claude (Anthropic, Claude Opus 5.5 via Claude Code, run in an automated sprint loop). The package is Apache-2.0.

🤖 Generated with Claude Code