Negative result: Corollary J fails on the whole #2111 121-line pool for every phi (rank 266, 30/30 immovable); exhaustive <=5-line immovable atoms all satisfy cost >= v

[thakrarsagar]

Status: Research note / scoped negative result (no κ claim). Track 40-49 Structural decoder research, sprint seed 43. Scope: #1166 OPEN ENDS item 5b. Claim: #1163 (comment)

What was learned

New package research/pool121-jet-rigidity/. It applies the #2162 B-independent first-order jet certificate (Corollary J) to the whole #2111 121-line T=3 unit-world pool for the first time.

1. Pool-level failure (complete finite certificate). In the unit world φ(±1)=1 is forced, so (J1) has the same F2 coefficients for every admissible φ and every characteristic-2 field. (J1) has 2646 rows, 510 unknowns and F2 rank 266; two code-disjoint eliminations agree. The 245 explicit gauge + torus motions span dimension 244 = 510 − 266, so the first-order solution space is exactly gauge + torus. All 30 of 30 harmonic lines are immovable, with 30 explicit Lemma F flows as dual witnesses (10–14 edges each, all even). Corollary J therefore cannot certify the pool for any φ.
   • Scope note (from review): the pool's incidence digraph has one 70-line strong component holding all 30 harmonic lines, plus 51 singletons stripped by Three-total root-unit decoders: reductions, a sharp example and the remaining lift question (no κ claim) #2111 Lemma SCC/Peel. The real test object is the 70-line core.
2. Higher orders. Proposition R (proved in the README by hand, not machine-checked; only its finite premise is checked): on a configuration whose solution space is gauge + torus, the uniform order-m version of Corollary J fails at every order. That applies to the pool and to 17 of the 42 stored atoms. 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 order-2 test fails on all 42 atoms.
3. Minimal immovable sub-configurations (exhaustive up to 5 lines). No such sub-configuration has 3 or fewer lines. The immovable 4-line ones form one orbit, with cost 5 = v+1. There are 353 orbits of minimal 5-line ones, and none is CR2 over GF(2^k) for k ≤ 4. Their exact min cost − v is {0:4, 1:103, 2:184, 3:55, 4:7}, with C and Python enumerators agreeing. So (A) holds on all of them.
   • Correction after review: the 4 tight orbits (cost = v) are not uncertified test cases. Each has a unit line and a non-strongly-connected digraph, so Three-total root-unit decoders: reductions, a sharp example and the remaining lift question (no κ claim) #2111 Lemma SCC/Peel plus Corollary J on components already gives cost ≥ v.
   • An independent reviewer script reports that this reduction covers 242/353 five-line orbits and 39/42 stored atoms. It also reports that the 111 strongly connected minimal 5-line orbits all have slack ≥ 1. That reduction count is reviewer evidence and is not replayed by verify.py.
4. Greedy atom search: 42 inclusion-minimal orbits (v = 4..10), each with exact cost v+1..v+5. The B-dependent jet passes on all 1305 minimisers; this is a consistency check, not a certificate.
5. Q4 (failed search): the largest jet-certifiable sub-pool found has 103 lines, giving cost ≥ 103 < Theorem U 106 ≤ MCC 119. This is not a bound on the method.

Jet failure is not evidence against (A). No counterexample to (A) was found and no κ is claimed.

Reproduce

• Base commit: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2 (main is unchanged as of submission).
• Pins (SOURCES.json, NOTICE.md): vendored cr2.py, enumall.c and tied_min.json 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 113a7182 (sha256 pinned); pool definition from Three-total root-unit decoders: reductions, a sharp example and the remaining lift question (no κ claim) #2111 head 98da76c2, re-implemented; tooling pattern from 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) #2274 67294e7d.

python3 -I -B research/pool121-jet-rigidity/verify.py

This uses only the stdlib plus cc, takes ~60 s, and should exit 0 with 30 PASS lines. The final line should be PASS pool121 jet rigidity: (J1) rank 266 / solution dim 244 (= gauge + torus) / 30 of 30 harmonic lines immovable with Lemma F flows; 42 minimal immovable atom orbits (v 4..10), all non-CR2 (GF(2..16)) with exact min cost >= v + 1, order-2 jets fail on all; minimum immovable size 4 (one orbit); 353 minimal 5-line orbits, all cost >= v, 4 tight (cost = v); Q4 search best certifiable S = 103 < 106 <= 119.

--full re-runs the exhaustive 5-line enumeration and the CR2 DFS: 32 PASS lines, 534 s measured. The default mode trusts the stored 5-line completeness and non-CR2 labels. All replays used Python 3.14.

Controls

• Positive control: tied_min is jet-movable (G8).
• Negative controls:
  • N1: a flow with one edge dropped is rejected.
  • N2: a non-incidence edge is rejected; a corrupted pool line changes the rank from 266 to 265.
  • N3: non-homomorphic ψ(−1)=ω breaks all 2646 order-0 incidences (an order-0 sanity control).
  • N4: every one-line deletion of every atom and orbit is jet-movable.
  • N5: the order-2 test without the X1·Y1 term is rejected.
• Reviewer mutations M1–M5 of the stored data each trigger exactly the targeted FAIL.

Scope, inherited assumptions, unresolved obligations

• Inherited and not audited here: ✈️ Characteristic-2 shadows of rank-3 decoders: the anti-Fano bound by first-order jets, with a Lean-checked core (no κ claim) #2162 Corollary J / VCR2 / Lemma F / CR2 / Theorem B2; Three-total root-unit decoders: reductions, a sharp example and the remaining lift question (no κ claim) #2111 Lemma SCC/Peel and the cost model. Independent audit of Lemma CR2 and Theorem B2 (item 2 of the 1166 independent-check request) #2293 lists Corollary J as unchecked.
• Proposition R and the order-2 reading of G11 are hand proofs; there is no formal proof.
• Search coverage: non-CR2 status covers GF(2^k), k ≤ 4 only. Atoms with more than 5 lines come from greedy search only (seed 43; flow-support starts with 40 orders per harmonic line, plus 8 whole-pool orders), so missed atoms are not excluded. Q4 used greedy insertion over 16 orders plus one scratch-only swap pass that was not replayed.
• Not run: the SCC/Peel classification is not yet in verify.py (open question 1).
• Not covered: orders ≥ 3 on the 25 non-rigid atoms, non-uniform VCR2, and other char-2 base points.
• Generated data hashes in SOURCES.json are informational. Correctness is checked semantically.

Credit and overlap

• Predecessors: DreamingOfClouds (Three-total root-unit decoders: reductions, a sharp example and the remaining lift question (no κ claim) #2111 pool/MCC/Theorem U/Lemma SCC-Peel; ✈️ Characteristic-2 shadows of rank-3 decoders: the anti-Fano bound by first-order jets, with a Lean-checked core (no κ claim) #2162 Corollary J, VCR2, Lemma F, CR2, tools; Where characteristic 2 starts to pay: a T² threshold for root-unit decoders (E6 ties, E7 first gain found; no κ claim) #2125; OPEN ENDS); navinor (Independent audit of Lemma CR2 and Theorem B2 (item 2 of the 1166 independent-check request) #2293, Research: dense-polar small-T gain screen - no T <= 6 gain in 1,056,861 exhaustive subsets (scoped negative) #2272); RossRichardson (Prove jf0 pairing cost exactly 19 with fixed minors #2176); petalas (Independently audit anti-Fano rank-three completions #2193, Certify exact pairing cost 21 for the 19-line configuration #2206); the Root-unit decoders: two lemmas, an erratum to #595, and open cases (partial; no κ claim) #1590/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 model authors.
• Concurrent related work (no overlap with the pool jet test): Research note: justify decoder split min-max via linking systems #2330 (theon199, ✈️ Characteristic-2 shadows of rank-3 decoders: the anti-Fano bound by first-order jets, with a Lean-checked core (no κ claim) #2162 §3 split min-max audit), Research: the MAS edge form of Theorem B2 (iii) is false (two exact counterexamples; no new kappa) #2340 (navinor, B2(iii) MAS edge form false), research: nine-line counterexample to the B2 acyclic-edge strengthening #2334 (ROTl24, B2 acyclic-edge strengthening counterexample).
• What this adds: ✈️ Characteristic-2 shadows of rank-3 decoders: the anti-Fano bound by first-order jets, with a Lean-checked core (no κ claim) #2162 ran Corollary J only on tied_min, random configurations and jf0..2, so this is the first whole-pool application, giving an exact φ-independent rigidity certificate and an exhaustive ≤5-line atom classification.

Contributor: @thakrarsagar, with substantial AI assistance from Claude (Anthropic, Claude Opus 5.5 via Claude Code, run in an automated sprint loop). Code, search, and drafting are AI-assisted. Three independent AI review passes were run; the SCC/Peel correction came from one of them.

🤖 Generated with Claude Code