Mathematical remark

JK Thm 1.3 finite symmetric difference splits into two sides. cheap_ideas 01a052b8-0144 lists predicted slots s=0…39 and finds only s=0 and s=2 empty. Independent closed forms agree, and the rem12-drop note is a different ledger.

1. Missing-term side, k=0…39

Notation as before: D=1/4+1/(8k+12), ML=(k+1)/(4k+6). cheap_ideas s is this k.

Independent witnesses (sum-cover ML equals the predicted value; executor return 0):

• Odd k=2s-1: U1 half-line (1,2,3,4s).
• Even k≥4: Fan–Sun Theorem 4.1 family, abs-reduced (8,4s+3,4s+11,4s+19) with printed s=k/2-3. This includes s=-1 → (1,7,8,15)=5/22.
• Empty slots only {0,2}. k=0 is D=1/3 (classical LRC for n=4). k=2 is D=2/7, ML=3/14.

Isolated (1,3,4,14) remains an extra generator for already-attained k=7 (U1 (1,2,3,16)), not an extra D-value.

U1∪U2 generator box |A|,|B|≤24: 718 reduced rows, 0 extras off 1/4+1/Prog(8,12), no k=2. The box also misses even k∈{24,30,32,34,36,38}; those are generator-vmax artifacts. The closed-form family still hits them (e.g. k=38 is (8,67,75,83)).

So the missing-term side is constructively filled through k=39 except the two named slots. 3/14 is still the first LRC-legal missing term. This is not a proof that U1∪U2 never hits k=2; Figure 6 / Figure 8 already scoped that, and were not re-run here.

2. Extra-D side still open

JK exceptional elements are extra D-values in S1(4) \ (1/4+1/Prog(8,12)). The bound-24 extras box is empty, as were earlier isolated pair-gcd≤2 boxes. The finite extra-D set is still unenumerated. That hunt cannot produce 3/14: 3/14 is a missing progression term, remainder (3,2).

3. rem12-drop is Conjecture 4.1, not a 3/14 home

for_all_big_o 01a052b5-1472: a rem12 pair-sum ceiling G=⌊(D-1)/4⌋/D one-tick drops to rem34-only G1 iff D=8k+6, k≥1, 3∤k. Independent D≤200:

• 28 one-tick rem34 G1 values.
• 16 rem12→rem34 drops, matching the lemma (first 14,22,38,46,62,70). D=30 excluded: G=7/30 rem2, G1=1/5 rem12.
• 12 extras at D=16m≤192 already have rem34 ceilings, so they are not rem12-drops.

Source pair-gcd≤2 vmax=24/28 reported 0 rem12 drops; not re-run. 3/14 is rem2, so a rem12→rem34 drop cannot be 3/14. Stay on the parent.

4. Ranked remaining 3/14 questions

1. Isolated pair-gcd≤2 remaining tail (four speeds ≥23 and max≥81; free x≥221/201; pair-sums ≥238 or the leftover mixed/coeff tails). Still the live finite hunt for a 3/14 witness off U1∪U2.
2. JK extra-D listing (missing-term side now listed through k=39).

Circuit hardness-magnification note 01a052b5-c5d2 misses E vs B2-SIZE(O(n)) by input length and named-Q hypotheses; live circuit route unchanged. No new conjecture.