Detailed results for the remaining cases

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cubic_abc: Ordinary abc implies cubic finiteness. A single abc exponent 10/9 suffices; a specified constant K gives n < 10^263 K^69. No Hall–Lang assumption is needed.

cubic_abc_factorization: The endpoint conditions give coprime factors, the conic, and the paired identities used by the cubic abc proof.

cubic_after_cutoff_9: The actual-overlap bound saturates on odd columns of power-of-two rows. Higher Lucas digits give a simultaneous approximation problem in the cofactors, but no fixed finite set of odd primes can cover all rows.

cubic_conductor: Removing an odd quadratic twist bounds the conductor by Disc(P). Bounded cubic discriminant gives an effective finite reduction; global finiteness remains open.

cubic_curves: Siegel's theorem, polynomial abc, and a two-circle argument prove that every fixed algebraic curve contains only finitely many bad cubic pairs, with no restriction on the odd part of n. Global nondensity remains unproved.

cubic_elliptic: A hypothetical cubic counterexample forces an explicit point of infinite order on one of two twists. The correct quartic transformation and varying primes of bad reduction delimit the finiteness argument.

cubic_followup: An endpoint divisibility criterion, prime-power and column restrictions, and certified exclusions for n<=10^19 and v_2(n)<=48. Global cubic finiteness remains unproved.

cubic_irreducibility: Every bad triple gives an irreducible integer cubic with roots in (-1,0), and an associated monic integral depressed cubic. Global finiteness remains unproved.

cubic_residue_groups: A cubic counterexample needs three groups at (n-2)/2 and two large groups at n-1; bounded groups at (n-2)/2 and columns near the edge or centre are effectively finite.

cubic_uniform: Integer moments, an incomplete Vieta argument, and a two-prime full-Lucas criterion excluding rows with suitably sized prime-power factors of n-1 and n-2. Global finiteness remains unproved.

cubic_weighted: The degree-2m polynomial through the five remaining congruence points is unique. This explains why discounting powers of n alone cannot repair the cubic fat-point argument.

five_curve_obstruction: An exact nine-line covering proves that adding arbitrary fixed plane-curve factors cannot bring the residual quadratic cost below one at cofactor exponents .13 and .34 in positions zero and one.

index_5_cofactor_range: Beyond an effective threshold, both cofactors in a bad index-five row lie strictly between (n-4)^(1/100) and (n-4)^(57/100). The maximal positions are already confined to zero and one.

index_5_higher_digits: Exact higher Lucas conditions for the quintic cofactor system, saturated small-prime overlaps on a remaining candidate branch, and the local limitation of polynomial valuation gains. Compatibility across primes and bounds on growing slopes remain unresolved.

index_5_maximal_positions: Beyond an effective threshold, a bad index-five row must place the maximal powers of two and three at n and n-1. Four unequal-block Padé constraints further restrict their cofactors.

index_6_carry_constraints: An explicit degree-twelve product and an exact valuation identity exclude the low-residue Lucas branches at 2,3,5 in a cofactor box around the remaining hard profile. No interpolation or rank computation is used.

index_6_curve_obstruction: An exact nine-line covering proves a degree bound for every polynomial at the remaining .31, .36, .31 cofactor profile, even when its common curve zeros can be handled effectively.

index_8_carry_constraint: An exact prime-adic column identity and a small interpolation certificate exclude the low-residue Lucas branch in the difficult index-eight cofactor configuration. The whole index remains unresolved.

index_8_curve_obstruction: Thirteen lines prove that no polynomial can have a positive weighted multiplicity-versus-degree gain at the remaining uniform .3 cofactor profile. Controlling common curve zeros does not remove this obstruction.

indices_4_5_row_zero: Every bad triple at index 4 has 4|n or 9|n; at index 5 it has 8|n or 9|n. An elementary argument and 61 fixed rational-slope cases exclude the complementary classes completely.

prime_index_transfer: Prime-index exclusions impose explicit prime-square congruences on five remaining indices. Exact Lucas alternatives describe the forced common primes, but do not yet transfer complete exclusions across a threshold prime.

small_cofactors_effective: A uniform p-adic logarithm bound forces both selected cofactors to exceed exp(c log n/log log n). This excludes every fixed logarithmic cofactor range effectively, but does not make either whole index effective.

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The remaining-case guide and index-three guide explain which of these results matter for the unresolved problem. None of these notes completes a remaining index.