Supporting proofs for the completed indices

../

cutoff_9: Complete numerical source covers and finite certificates settle nine and seven. Every i>=9 and i=1,2,7 is excluded; only four fixed indices at least four remain.

cutoff_and_cubic: A certified interval argument excludes i=29 and every i>=35. The cubic case is reduced to n=2^h u with u odd and 3u^3<2^h, verified through 10^12, and its uniform fat-point obstruction is determined exactly.

effective_interval_certificates: Complete interval proofs exclude every i>=26 and the isolated indices 17 and 23, using numerical prime-pair bounds both in the tail and inside the interval proof.

further_interval_certificates: Exact prime-power certificates, interval proofs, and small residual-row checks complete the original claim for every i>=15, and also for i=10,13.

index_10_effective: Position-dependent interpolation and constant-gap product identities prove that every bad row at index 10 is below 10^15000000.

index_11_complete: An asymmetric Padé bound, exact finite prime-power certificates and interval inequalities exclude every bad triple with i=11; eleven small remaining rows are checked directly.

index_11_effective: A weighted Kummer polynomial and the maximum constraints in Bennett–Filaseta–Trifonov give a positive exponent 7/100 and an effective last row for i=11.

index_12_complete: Numerical unequal-block Padé bounds, two finite prime-pair cliques, and a completed interval certificate exclude every bad triple with i=12.

index_12_ramp: Twenty off-diagonal pair estimates and one explicit degree 72 product replace the large interpolation certificates; the algebraic part of the bound becomes 10^845.

index_13_effective: A surjective finite interpolation system and asymmetric prime-pair estimates prove n<10^56000 for every bad triple with i=13.

index_13_interpolation: A fixed polynomial space of dimension at least 108 gives a positive exponent, leaving an explicit common-divisor computation as the nonvanishing gap.

index_13_interpolation_gcd: An exact modular rank and polynomial gcd certificate proves that the complete degree-106 fat-point system has a finite, effectively bounded common zero set.

index_14_effective: Off-diagonal prime-pair alternatives and one universal polynomial interpolation certificate make the remaining index fourteen effective.

index_14_finite: A degree-90 ramp and 400,818 certified intervals exclude every index-fourteen bad triple below 10^15170264, ready to combine with a numerical infinite tail.

index_14_numerical_tail: Sixty-four numerical pair sources, finite coefficient-gcd bootstraps, and the existing interpolation family give n<10^8516269 for every bad triple at index fourteen.

index_15_numerical_tail: Seven explicitly bounded asymmetric prime-pair estimates strengthen the existing interpolation gain to 7993/10000 and prove n<10^15170264.

index_16_complete: An asymmetric cofactor estimate reduces the stopping bound to 10^914; finite prime-power certificates and 100204 certified intervals exclude every remaining row.

index_18_complete: A numerical interpolation bound and a completed finite prime-pair and interval certificate exclude every admissible bad triple with i=18.

index_18_effective: An exact check of 65 cofactor vectors and 5707 collision capacity lists gives gain 186/625. The full stopping bound includes every symmetric and asymmetric source threshold.

index_20_numerical_tail: Twelve explicit cofactor estimates and a surjective degree-118 interpolation map prove n < 10^27027659 for every bad triple with i=20.

index_20_small_height: An explicit basis of 51 integer interpolation polynomials reduces the algebraic index-20 tail from 10^4000000 to 10^6396, after the cofactor source thresholds.

index_7_complete: Adjusted Padé blocks and complete numerical source covers meet the finite exclusion, proving the original common-prime assertion for every admissible triple at index seven.

index_7_effective: Joint cofactor inequalities from unequal Padé blocks, integer polynomial certificates, and exact rational separation give an effective bound for every bad triple at index seven.

index_7_finite: Pair and three-prime CRT certificates reduce the finite range to 18,000 intervals and 28 explicitly checked rows. The numerical infinite tail remains separate.

index_7_initial_height_cover: Two exact boxed Padé covers lower the remaining initial pair thresholds to h=10^100000000, with all 122868 witnesses retained and independently replayed.

index_7_nonvanishing: Three modular kernel sections and two local p-adic calculations prove uniform arithmetic nonvanishing in degree 196.

index_7_polynomial_bounds: A monic Newton basis, exact Farkas margins, and uniform arithmetic nonvanishing reduce the polynomial height exponent from 19813540001 to 44245906.

index_7_relaxed_sources: A slightly weaker set of cofactor inequalities still gives exact complete coverage at index seven, with polynomial height exponent 99990923.

index_9_baseline: Seventy modular interpolation certificates cover 104059 of 104976 labelled position/witness cases; the retained 917 residual cases include every collision not yet covered.

index_9_complete: Complete Padé source covers meet the finite exclusion, proving the original common-prime assertion for every admissible triple at index nine.

index_9_effective: Integer polynomial certificates and exact rational branch trees give an effectively computable bound for every bad triple at index nine, including all prime-position collisions.

index_9_finite: Two finite cofactor cliques and degree-42 weighted ramps reduce index nine to 607,975 intervals and thirteen exact rows. The numerical infinite tail remains separate.

index_9_initial_height_cover: Complete boxed Padé interval chains establish all eleven initial pair sources and four ordinary AP sources from h=10^241547627.

index_9_nonvanishing: Local certificates modulo 251 remove the common-zero thresholds for round3_000 and round3_001 without increasing their polynomial degrees.

---

Use the proof guide for a direct reading route and a table covering every completed index. The pages above retain the detailed dependencies, including earlier results still used in those proofs.