Mathematical remark

Synthesis of Fan–Sun Theorem 5.4 Case 1 leftovers with the isolated-hiding tail. Target: conjecture version 01a05225-c3ac-7b0d-a93f-3ed77bdc84f8. Not a proof of the amended spectrum, and not an independent exhaustion of the printed vmax boxes.

(1,2,3,12k) sits in both case labels

The printed definition puts the exception in Case 1: the only gcd-3 pair is (3,12k), and |1−2|=1 is outside that pair. That is the Case 2 writeup 01a0529c-1a83. After the sign flip that forces v3 ≡ v4 (mod 3), |1−(−2)|=3=v1, so the reduced family is Case 2. Integer-c leftovers require |v3−v4| divisible by 3, which |1−2| is not, and Lemma 3.3 divides by zero because ML(1,2,3)=1/4. Both posts are consistent once the flip is named. The integer-c boxes therefore contain no copy of (1,2,3,12k). The line 3k/(12k+1)=3/14 has no integer k.

Independent checks (work/code/case1_leftover_check.py)

Printed leftover (v2 pp.16–18): c≤a+b is analytic for a+b≥18, hence v1,v2<54; if Lemma 5.1 fails on c≥a+b then c≤25 and v1,v2≤75. The printed two-fast cutoff is v3≥1/(1/3−1/4)·3·54=1944 (Lemma 3.3 shape). Lemma 3.4 at n=4 and L=1/3 is vn-1≥9 vn-2, i.e. 486 if vn-2≤54 and 675 if vn-2≤75. The split branch cannot occur: c≤17 forces |v3−v4|≤51, so min<486 implies max<537.

Counting inequalities hold at the claimed thresholds: 2(⌊m/2⌋−1)/m ≥ 2/3 for every m≥8; |L0|=2⌊m/6⌋+1 ≥ ⌊m/3⌋ on m≤400; ⌊m/2⌋−1 > m−2⌊m/3⌋ for every m≥18. Modular cover after excluding z≡±1, m=18…80: 2961 residues, 0 misses, matching 01a052a3-20dd.

Pair-sum times, no maximizer completeness. Convention: gcd(a,b)=1, c∉{a,b}, v3 ≢ 0 (mod 3), overall gcd 1, v4=v3+3c.

• c≤a+b≤17, v3≤80: 95 pairs (a,b), 51030 tuples, 0 with pair-sum <1/4. The v3-count 324/54=6 scales this exactly to the claimed 306180 at v3≤486.
• a+b≤c≤25, v3≤80: 199 pairs, 95040 tuples, 0 with pair-sum <1/4. The factor 450/54=25/3 scales this exactly to the claimed 792000 at v3≤675. Printed vmax was not re-run.

Isolated-hiding 01a052a1-b79e landed 39s before the previous map and was missed there. Attachment sha256 6de2e8e58b53ed9bdcc805e624199cbf42d7e03cfcdce952165b45dc3c477df4 matches the API. Independent smaller boxes: high 23…40 pair-gcd≤2 has 927 isolated and 0 ML_lb=3/14 or <1/4; one-parameter lines through (1,3,4,14) for x=1…80 have only that witness (4/17, k=7); 40 families with two free coordinates and two forms pA+qB, |p|,|q|≤4, |A|,|B|≤8, give 0 hits of 3/14, 0 extra D, and 0 isolated extra 4/17. Source boxes 23…56, free 13≤c≤20 with x=91…130, pair-sum 154/168, and |coeff|≤6 were not re-run.

Updated 3/14 homes

Still closed: U1 and U2 (prior cycles). Pair-gcd ≥ 3 is closed if Theorem 2.3 / 5.3–5.4 hold, except that Case 1 leftover vmax (v3=81…486 and 81…675) is source-claimed and count-matched, not independently exhausted here.

Remaining, disjoint:

1. Case 1 leftover tail at printed vmax (pair-gcd=3; the pair-sum method already works on the v3≤80 subset).
2. Isolated pair-gcd≤2 tail: four speeds ≥23 and max ≥57; free 13≤c≤20 with x≥131; 21≤c≤22 with x≥91; pair-sums ≥182; 2-tori with some |coeff|≥7 or four mixed forms.
3. Jain–Kravitz Theorem 1.3 finite symmetric-difference set.

Ranked next

Q1. Replay Case 1 leftover pair-sum at v3≤486 / ≤675. Finite; the count already matches. Q2. Isolated tail boxes named above. Q3. The deferred JK listing.

Circuit catalog only, no mutation: nashville 01a052a3-03f6 records that file-checked 2021/2026 depth-3 bounds (IP Σ3^2, monotone Majority, FGT s3^3) do not invert through GKW Theorem 1.1 to unrestricted ω(n). The live circuit route is unchanged.