A385316: independent exact re-derivation of the five posted terms and a two-engine exhaustive floor extension to 10^14

Two independent implementations (Node.js: Eratosthenes sieve with sorted triple enumeration and running-sum early breaks; Python stdlib: incremental prime generation with a double two-pointer merge over sorted pair sums) re-derive the five posted terms of OEIS A385316 ("Smallest number that is the sum of 3 cubes of primes in exactly n different ways", Stijn Cambie, Sep 20 2025) exactly: 24, 185527, 8627527, 999979163, 10588881419, at bound B = 10,700,000,000. Each term carries a sufficiency certificate: every representation of any v ≤ B uses primes at most the cube root of B, all inside the sieve, so every count ≤ B is exact and "smallest with exactly n" is decided, not sampled. The two implementations agree on every census aggregate (4,631,825 triples; 4,606,359 distinct values ≤ B; maximum multiplicity 5), and the test battery enforces mutation controls in both directions. Both engines then close the ledger from 0 to 10^14, independently and exhaustively. Below 10^13: the complete count-5 census (exactly four values: 10,588,881,419 = a(5); 13,604,651,997; 4,602,075,977,773; 7,185,059,419,357 — each recounted to exactly 5 by independent cube-complement lookup), agreed by both engines, with an exact sorted-triple conservation identity. Over the fourteenth decade [10^13, 10^14): engine A swept 45,003 windows (12,134,632,181 representation emissions) and engine B independently re-swept the identical range on a provably boundary-disjoint window grid (the materialized boundary sets share 0 of 45,002 internal boundaries), the two agreeing on every census aggregate, on the decade's single count-5 value 22,771,734,649,091 (recounted twice), and on the exact conservation identity A-emissions = B-sorted-triples = 12,134,632,181 with no boundary term. The result, in the deposit's pre-agreed words: no value below 10^14 is the sum of 3 cubes of primes in 6 or more ways, verified by two independent exhaustive computations. This locates a(6) > 10^14 if a(6) exists, and says nothing about whether it does. It independently re-derives the floor a(6) > 499243435237 posted in the OEIS entry and extends the swept region about 200 times beyond it. A calibrated location heuristic (Cramér prime-density integral, exact congruence local factors, fitted to the measured band tallies; deposited with its full validation table including the check it fails) places the first count-≥6 value at conditional median 3.5e15 (data-extrapolated moments) to 1.6e18 (theory-ratio moments) — a heuristic, not a bound; the exhaustive floor above is the only statement this deposit certifies. Honest limits, stated as in the working notes: all exactness statements are statements about this code path — two independent implementations agreeing on disjoint decompositions, plus mutation controls in both directions, is strong evidence of correctness, not a proof. Floats appear nowhere in a counted quantity. No claim of novelty or priority over the OEIS entry's authors is made; this deposit records a computation and its programs so both can be re-run. A narrative report page (report.html) is generated from the deposited records and held to them by its own test battery, whose one conditional (the one-engine/two-engine decade sentence) flips on the deposited verdict record alone. The programs are MIT; this note and the generated page are CC-BY 4.0. The license split is declared file-by-file in priority.json inside the deposit, following the license-split pattern of the Erdős #290 deposit.