bandMultiplier
The definition supplies the dimensionless band multiplier for atomic number Z by indexing its rail and block then applying the phi-powered block factor. Physicists modeling chemical shell energies under Recognition Science cite it for zero-parameter predictions. Implementation is a direct composition of the octave indexer and block factor function.
claimFor atomic number $Z$, the band multiplier equals $phi^{2r + o(b)}$, where $r = floor((Z-1)/8)$ is the rail, $b$ is the block (s, p, d, f) from position $(Z-1) mod 8$, and $o(b)$ is the block offset (0 for s, 1 for p, 2 for d, 3 for f).
background
The Periodic Table Engine maps the eight-tick octave to chemical periods via phi-tier rails with fixed block offsets. The Index structure pairs a rail integer with a Block (s, p, d, f). The indexOf function assigns rail as floor((Z-1)/8) and block by remainder position within the octave, with no empirical tuning. Upstream blockFactor computes the multiplier as phi raised to (2 * rail + offset(b)), where offsets are 0,1,2,3 for s,p,d,f respectively. This realizes the phi-ladder scaling for valence shells. The module establishes the noble gas closure theorem: noble gases occur exactly at eight-window neutrality points under the deterministic valence proxy.
proof idea
The definition first computes idx as indexOf Z, extracting its rail and block fields. It then returns the result of blockFactor applied to idx.rail and idx.block under the BlockOffsets instance. This is a one-line wrapper composing the deterministic indexer with the phi-exponentiation in blockFactor.
why it matters in Recognition Science
This definition provides the scaling factor consumed by bandEnergy to yield the dimensionful band energy as E_coh times the multiplier. It fills the phi-ladder step in the eight-tick chemistry scaffold (T7 octave), enabling fit-free predictions of shell closures that match the noble gas set {2,10,18,36,54,86}. It supports the Recognition Composition Law indirectly through the self-similar phi structure.
scope and limits
- Does not include relativistic or spin-orbit corrections for heavy atoms.
- Does not compute actual ionization potentials or affinities.
- Assumes the default BlockOffsets with fixed phi-packing.
- Does not adjust rail assignments for known exceptions like lanthanides.
- Does not incorporate electron correlation beyond block counting.
- Does not predict molecular bonding energies.
Lean usage
Constants.E_coh * bandMultiplier Zformal statement (Lean)
265def bandMultiplier (Z : ℕ) [BlockOffsets] : ℝ :=
proof body
Definition body.
266 let idx := indexOf Z
267 blockFactor idx.rail idx.block
268
269/-- Predicted (dimensionful) band energy for atomic number `Z`.
270 This is a fit‑free display using the universal coherence tick. -/