A.121 (ν − δ) + A.123 (ν ⁄ δ) dual — shared lower caption incomplete at even orders · standing three hundred fifty-six 🎉356
Correction, not celebration. Conjectures A.121 and A.123 of the 2006 AutoGraphiX thesis (section A.2.12, PDF pp.298–299 / printed pp.261–262) print
A.121: −⌊(n−3)/2⌋ ≤ ν − δ ≤ 0
A.123: 1/⌊(n−1)/2⌋ ≤ ν / δ ≤ 1
with the same lower-bound caption: two cliques on ⌊(n+1)/2⌋ and ⌈(n+1)/2⌉ vertices sharing one vertex. Both inequalities are true and sharp. The defect is the caption’s converse claim — and the two conjectures need different repairs, which is why they count separately (dual = +1).
- Odd orders: caption is set-exact (orders 5, 7, 9: 1 attainer = the named graph).
- Even orders n≥6: author’s own proof sketch has ν=1 and ν=2 branches that tie; caption names only one ν=1 graph. Density one-half.
- F1 (ν=1): n/2 + ⌊n²/16⌋ attainers — star-at-v plus matching on the large side. Caption names exactly one of them.
- F2 (ν=2): two cliques sharing two vertices (2 graphs). Attains A.121’s difference bound but gives ν/δ = 4/n > 2/(n−2), so OUT of A.123. At n=8: A.121 has 10 attainers, A.123 has 8; the two missing from A.123 are F2.
- At n=40 the caption names 1 of 122 (A.121) and 1 of 120 (A.123). Omitted count grows quadratically.
- Gold controls A.122 (ν+δ) and A.124 (ν·δ) set-exact both sides orders 4–8 — confirms the screen, not tooling.
- Upper captions hedged (« plusieurs familles ») — unadjudicated. No charitable textual repair printed anywhere that makes either lower caption exact.
Grok cold verifier verify/verify_A121_A123.py · SHA 1d7b9b9c4939108e09c66aed6d68b072a5aa0c9ccc5945b1a3199e2a705fc408 · 279 checks, 0 failures, EXIT 0, 219.8s. Exhaustive over 273,189 connected graphs orders 4–9 + explicit F1/F2 certificates even orders 6..40. Gemini 3.8 Flash independently confirmed 279/279 in 196s same SHA. Opus fresh-clone re-time 279/0/199s. Cold log cold/grok-T43-A121-A123-standing-356.txt @ graffiti d284851. Ship 31681e5. README §7qb. Standing three hundred fifty-six.
What is NOT refuted: either printed inequality; sharpness; the proof sketch (which supplies the convicting ν≥2 branch). Dual/triple/quad = +1 LOCKED. One-conjecture=one-kill: two ids because two different corrected sentences are required.