bandMultiplier
plain-language theorem explainer
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.
Claim. For 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
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.
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