Other approaches and their limitations

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adapted_certificates: An explicit weighted polynomial confines the remaining position patterns at indices 27, 30, and 33 to 67, 97, and 139 possibilities, with effective bounds for all other patterns.

additional_product_constraints: Eleven AP, five equal-sum, and one cubic product estimate give certified cofactor restrictions. They do not by themselves settle index 12.

cubic_descent: An explicit Kummer class and a uniform local obstruction to halving the marked point. This does not exclude bad triples or prove global finiteness.

cubic_factorization: A uniform factorization, including the canceled factor of three, identifies the odd conductor exactly away from 6. The resulting equations have no positive integral descent by the displayed coordinate Vieta mutations.

cubic_label_families: At index three, the Lucas conditions coming from n−1 and n−2 hold along an explicit cubic curve and along Pell conics. Congruences modulo 27 and 73 show that the cubic curve contains no bad row, but the families show that a proof must use the power-of-two form of n or higher Lucas digits.

cubic_multiplication: A rational reconstruction of the row from an elliptic curve and marked point. Doubling always produces a nonintegral row; tripling preserves the two-adic row and column valuations but has uncontrolled odd denominators.

cubic_nested_heights: A Bezout argument rules out the required strict inequality for every big divisor with arbitrary endpoint weights when log j / log n tends to one; earlier explicit symmetric calculations are retained.

cubic_sections: For every possible bad row, the marked elliptic family has Mordell–Weil group generated by the known point over the rational function field in the column. This does not bound ranks after specialization or settle the bad triples.

cubic_surface: A factorization and birational involution of the integer-moment cubic. Every even column obstructs dyadic normalization of the transformed moments; odd columns and global finiteness remain unresolved.

factorial_divisor_obstruction: Every fixed product of shifted factorials that divides a Padé coefficient gcd for all sufficiently large parameters has only polynomial growth. A reflection identity for factorial valuations proves the obstruction.

fourteen_ap_source: The seed 5·13−2^6=1 gives a certified cofactor disjunction when the maximal positions for primes 5, 2, and 13 form a nonconstant arithmetic progression.

index_14_boundary: A shifted application of the complete index-13 result forces large prime divisors of both columns; its numerical certificate gives explicit power growth of these boundary gcds.

index_14_product_library: Nine numerical constant-gap product bounds strengthen the cofactor restrictions involving 11 and 13. Whole-index effectivity remains unresolved.

index_6_overlap_obstruction: In the pattern r2=2, r3=1, r5=0, four column residue classes realize the entire selected prime powers as common-prime overlap. The forced valuation gap at row zero does not itself improve the polynomial capacities.

index_8_next_obstruction: An exact compatibility check isolates a profile that survives the retained sources; a necessary condition explains why a natural quartic identity does not readily improve them. No index-eight effectivity claim.

low_crt_row_zero: Every bad triple at index six or seven has a gap of at least three at 2, or two at 3 or 5, between the row and column valuations. Exact slope divisibilities exclude all complementary cases.

low_off_diagonal_ap: Unequal Padé blocks at the seed 49−48 give an effective alternative for AP positions carrying the primes 2,3,7.

low_rational_four: Rational Padé blocks give a numerical cofactor restriction for the position relation r2+r7=r3+r5, including low-index configurations missed by the separate endpoint alternatives.

more_prime_pairs: Twelve certified Padé parameter choices strengthen two published pair estimates, add bounds involving 17,19,23, and improve the cofactor exponents available to the small-index argument.

multipoint_minor_divisibility: A Frobenius calculation makes normalized integral rows proportional modulo primes in explicit intervals, supplying an additional exponential divisor of all two-by-two minors.

multipoint_moments: Polynomial moments provide three integral approximation rows with separate prime-power blocks and an explicit nonzero determinant.

multipoint_two_adic: For the seeds 10,9,8, an integer beta expansion proves a common 2-power in the cross product and a further 2-power in its third coefficient, without estimating a maximal gcd.

polynomial_factor_limits: Exact rational certificates show that adding factors of arbitrary degree cannot repair the usual residual dimension criterion for three remaining cofactor profiles.

symmetric_moment_reduction: An exact arithmetic-progression moment construction has only three singular points after a quadratic substitution; the calculation identifies its stronger prime-power blocks and the normalization issue.

twelve_rational_pairs: The seeds 49−3·16=1 and 13·512−5·1331=1 give effective rational-block restrictions excluding three remaining exploratory cofactor boxes.

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These approaches did not resolve their targets. The current guides summarize the limitations that matter now; the notes here preserve the calculations and precise hypotheses.