Erdős 699: all initial index-nine sources at the polynomial height

P699 research · Choosing the Padé index · The earlier numerical stopping bound

All eleven pair sources and four ordinary arithmetic-progression sources in the relaxed index-nine proof hold for

The checker and complete interval chains establish these fifteen target-box exclusions. The pair sources imply the nineteen weakened baseline alternatives. This addresses the initial sources; the 790 further joint clauses and the whole-index coverage are separate ingredients.

Complete result. The combined proof excludes every admissible bad triple at nine. The estimates below are retained as components of that proof.

Fixed-index inequalities

The argument uses the boxed sparse-witness lemma. Under upper bounds $u_p\le h^{e_p}$, the cost of clearing residual denominators at a fixed integer Padé index $m$ is at most

Each logarithmic coefficient is rounded upwards. The first term of the integer estimate decreases with $x$. The second term is the maximum of affine functions indexed by all subsets of the prime components. Exact rational endpoint inequalities therefore prove the estimate throughout an interval of real logarithmic heights.

Every source starts at the integer 556183965, strictly below $241547627\log10$. Accepted intervals have adjoining endpoints and continue beyond the source's previously established numerical tail. The symmetric $(3,7)$ pair already has a smaller tail threshold and needs no additional interval.

The arithmetic-progression families

For the ordinary AP sources, the sparse argument uses newly specified off-diagonal families. If the old seed record is $(P,Q,a,b,c,d)$, the directions are $(d,c-d,d)$, with adjacent rectangles

The source preparation verifies their polynomial constants independently. The old AP coefficient-gcd bounds are not transferred to these new rectangles: each sparse witness receives its own pointwise bound. Both methods exclude the same target box, so the existing AP tail remains applicable.

The pair gap is at most eight. Each AP has position step of absolute value at most four, so its product gap is at most sixteen. All factors lie in $[h,h+8]$.

Retained certificate

The fifteen sources use twelve distinct direction triples. The complete certificate contains 299324 interval witnesses. Each records the Padé index, right logarithmic endpoint, coefficient-gcd lower bound on the $2^{-32}$ grid, and number of BFT intervals used. The source identities, targets, arithmetic hypotheses, and numerical tail bounds are retained with them.

The calculation reuses the read-only prime-log prefix through $10^9$ and proper prime powers through $10^{14}$. Every accepted witness was checked during construction. The serialized-chain verifier also checks every interval inequality and endpoint, all prepared constants, and the presence of every named input. No interpolation rank or binomial row was recomputed.

Run the serialized-chain checks with:

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_9_initial_height_cover.py

The optional --replay-gcd flag independently recomputes every pointwise coefficient-gcd bound using the shared prime cache. A combined proof wrapper can call check(data, replay_gcd=False); the default performs all source-binding and rational interval checks.