Endpoint structure of graceful permutations of paths: alpha-labelings, a halving congruence, and reductions toward 8 | G(n)

Technical note on the endpoint structure of graceful permutations of thepath P_n, aimed at the conjecture 8 | G(n) (OEIS A006967). Each claim islabelled Proved, Verified (finite computation, range stated) or Observed. Proved: (i) 8 | G(n) is equivalent to a parity statement about two finitefamilies of endpoint cells; (ii) the extremal endpoint cells consistexactly of alpha-labelings, with bijections between n and n+1 obtained bydeleting the middle value; (iii) a halving congruenceG(2r+2; a, a+r+1) = R(r+1, r-a) (mod 2) via an involution whose fixedpoints are graceful permutations of half length; (iv) the extremal partof the mod-4 obstruction reduces to the parity of R(2s+1, s)/2, whichchecks the predicted behaviour up to n = 129, 130. Verified: endpoint data for n = 5..71; alpha-labeling sums agree withOEIS A395104 for N = 5..55, with further terms N = 56..71; predictionsrecorded before computation were confirmed for n = 51..64 and 67..71. Observed (open): the row sums satisfy a Lucas-type lawR(n, c) = C(n-1, c) (mod 2); an explicit odd-class law; thecomplementary cells modulo 4. A companion file states these as openproblems for specialists. Files: English note (PDF), Japanese version (PDF), open-problems list(PDF), Markdown sources. Prepared with AI assistance.