the rosetta addresses any position — π hex digit and n-th prime
Theorem.the rosetta addresses any position — π hex digit and n-th prime — π = Σ_{k≥0} 16^{−k} (4/(8k+1) − 2/(8k+4) − 1/(8k+5) − 1/(8k+6)) — BBP: the n-th hex digit without the neighbours.
Proof.a fixed decoder maps an index straight to its value, none of the neighbours needed: BBP's four-ray table {(1,+4),(4,−2),(5,−1),(6,−1)} reproduces π's hex expansion position-by-position (base 16); the sieve addresses the n-th prime to the Rosser bound pₙ < n(ln n + ln ln n) and π(x) inverts it exactly (π(pₙ) = n); hue = d·360/9 is the faithful dimensional coordinate the vortex ×2 six-cycle 1·2·4·8·7·5 moves through (multidimensional movement by colour); the primes thin as π(x)ln x/x → 1 (PNT density) — DRY on the sealed src/7/3 addressers; BBP (1997) and the PNT asymptotic cited.
Checked by exact arithmetic over the stated finite range — a verified witness, evidence toward the claim, not a ∀-proof.
src/water/digit/index.ts#theRosettaAddressesAnyPosition
1 · Classification
bounded-witness — computed witness within a cited frame (the unbounded form leans on the cited literature)
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"the rosetta addresses any position — π hex digit and n-th prime" is a re-derivation, acknowledged to documented mathematics — the original proof is the prior art this re-derivation acknowledges; not new to humanity — the contribution is the reproducible computation theRosettaAddressesAnyPosition.
- Prior art
- documented mathematics — the original proof is the prior art this re-derivation acknowledges
- Novelty
- not new to humanity — a re-derivation (humanityNovel = false)
- Contribution
- a reproducible computation (theRosettaAddressesAnyPosition @ src/water/digit) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
Recompute from source: npm run theorems:verify recomputes theRosettaAddressesAnyPosition (src/water/digit/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.