Erdős 885 (k=5) — Secondary Verification

Profile: results-envelope/1, results-claims/1, results-pairs/1, results-figures/1

Status

Exact reductions and certificates plus bounded exhaustive searches; no global k=5 witness or impossibility theorem.

Verified claims

ID	Claim
V1	an ERDOS-885 K_(5,5) witness transposes to a K_(6,4) factor-difference packet
V2	the displayed integer packets are exact: a K_(6,3) precursor, a transposed K_(4,5), and a K_(4,4)
V3	no K_(6,4) and no K_(5,4) exists over every canonical anchor 0 <= p < q <= 6000, every compatible row, every four-row subset — 0 and 0 hits across 18003000 canonical anchors
V4	that finite search implies the bounded necessary condition sqrt(d0^2 + 4N_min) > 3000, equivalently d0^2 + 4N_min > 9,000,000
V5	across the 71 archived public 4x6 packets, root-to-positive-shift and positive-shift-to-root are exact on all 71, and 426 five-root subsets produced 0 fifth-difference hits

Explicit non-claims

What this line does not establish, stated by the line itself — rendered on the same screen as the claims above, not behind a toggle.

ERDOS-885 k=5 is not solved here
search exclusions are exhaustive only inside their stated bounds and relative to the checked implementations; they are not global impossibility theorems
the direct search yields a bounded necessary condition, not a global obstruction
literature novelty is not claimed; in particular the main K_(6,3) packet independently rediscovered here is already present on Thomas Egense's public 2019 4x6 square-sumset page
a no-hit result at one level does not assert uniqueness at that level

Verification scale

Figure / against	What it measures
0 / 18003000	K_(6,4) hits, against canonical anchors searched
0 / 18003000	K_(5,4) hits, against canonical anchors searched
71 / 71	packets exact under root-to-positive-shift
71 / 71	packets exact under positive-shift-to-root
71 / 71	packets exact under signed factor difference
0 / 426	fifth-difference hits, against subsets checked
284000 / 284000	signed closure entries, against entries expected by inheritance
0 / 71000	new fifth differences, against packet-multiplier pairs scanned
0 / 40000	new fifth differences in the extended scan, against its entries
0 / 279936	candidate hits, against parameter tuples enumerated

14040082eligible pair fibers

209917507pair-fiber rows

465004retained differences

767726four-row support updates

4largest signed-shift closure

847896082divisors enumerated

116720nondegenerate packets

10777unique exact packets

2maximum positive-shift closure

Provenance

Rendered 2026-10-25 from agent/archive-chengxu-erdos-885-k5 at build time — not vendored into this site. Full reports and gate logs: the private repository amral-research-trees, folder erdos-885-k5-chengxu.