Detailed mathematical tools

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ap_cofactors: Composite-base Padé estimates give effective three-prime restrictions at AP positions and four-prime restrictions when two pairs have equal position sums.

asymmetric_cofactors: Keeping the two seed prime-power logarithms separate strengthens the BFT cofactor inequalities and gives a direct numerical threshold.

bft_numerical_thresholds: Explicit bounds for the Padé coefficient gcd turn stronger prime-pair estimates into numerical stopping bounds, including index 11.

bft_prime_bootstrap: An exact fixed-point certificate lowers six BFT coefficient-gcd thresholds from millions of Padé parameters to between 1046 and 28602, preserving all cofactor exponents.

bft_rational_blocks: Clearing the denominators of unmatched Padé blocks gives a piecewise linear cofactor restriction and improves the effective 2,3 maximum exponent to .2926. No additional index is settled.

coprime_kernels: Two full-degree coprime modular kernel polynomials remove the extra degree in interpolation, at the cost of an effectively bounded finite set of common zeros.

dusart_gcd_bounds: Two retained Dusart bounds improve the uniform prime-interval calculation used to instantiate BFT cofactor estimates.

finite_prime_pairs: Exact exponent and CRT certificates give one-third cofactor bounds through height 10^1732 for the prime pairs needed by the complete interval proofs.

gcd_localization: An exact Kummer localization lemma replaces each selected small-prime power by its overlap with the column binomial coefficient, without an extra prime-power loss.

jet_nonvanishing: A finite-dimensional multiplication operator turns a surjective symmetric jet map into degree-D+1 polynomials that cannot vanish simultaneously when X+Y exceeds the grid.

joint_numerical_thresholds: Finite integral constants and uniform coefficient-gcd bounds give numerical starting heights for the positive-part Padé inequalities, including zero actual product gap.

off_diagonal_numerical_thresholds: Explicit factorial, integral, and coefficient-gcd bounds turn the twenty index-twelve pair alternatives into numerical stopping bounds.

off_diagonal_pade: General Padé rectangles give effective joint cofactor estimates with different block lengths on the two sides, improving several difficult small-index configurations.

pade_block_adjustment: A smaller Padé block reduces the residual denominator cost. A positive-part inequality quantifies that saving for both finite witnesses and uniform numerical source bounds.

prime_power_gcd: The coefficient-divisibility argument holds for each Legendre summand, so the numerical gcd estimate can use Chebyshev psi instead of theta. The common interval construction does not require floor stability.

prime_power_strips: A modular coefficient bound replaces the quadratic exponent search by a short, exactly checked covering of exponent quadrants; height caps with millions of digits become practical.

product_thresholds: A direct numerical threshold for the composite-base Padé argument, with explicit AP and equal-sum product certificates.

rational_product_bounds: A numerical product lemma retains denominator costs and optimizes the Padé block; it gives stronger joint cofactor restrictions than separate endpoint alternatives.

row_zero_effective: For each fixed i>=4, bad triples whose row has bounded small-prime part form an effectively bounded set. Beyond that bound a selected small-prime maximum must occur at position zero.

runge_curve_criterion: Bad pairs on a fixed curve are effectively bounded when sufficiently many numerator fibers contribute poles to explicit S-integral functions; at indices four and five, three finite pole fibers suffice.

sharp_theta_gcd: Broadbent's square-root bound through 10^19 and global theta tables remove the old Dusart error floor; verified finite-RH estimates are patched to unconditional tails.

small_index_discriminant: The exact orbit discriminant gives a weighted S-part inequality, and an effective cutoff for additional patterns of small-prime maximal positions. It does not give a cutoff for every remaining index.

terminal_pair_corners: An independently numerical cofactor estimate can eliminate the last exponent orthant without another large modular inverse.

triple_prime_power_strips: Exact three-dimensional exponent covers give .61 through 10^1000 and .66 through 10^3000; direct product checks remove every large exception to the latter.

weighted_ramps: Position-sensitive polynomial bounds, with all coincidences of small-prime maxima covered, make index 21 effective and sharply reduce several earlier stopping bounds.

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Use the methods guide to select the relevant statements. These notes retain the full arguments and constants.