BlockOffsets
plain-language theorem explainer
BlockOffsets is a typeclass supplying an integer offset map from each orbital block to a fixed value for phi-ladder scaling in periodic-table energy formulas. Modelers of fit-free atomic spectra cite it to retrieve the standard packing offsets without introducing tunable parameters. The definition is a direct class declaration whose default instance hardcodes the four values by constructor matching.
Claim. A typeclass $BlockOffsets$ equipped with a map $offset : Block → ℤ$, where $Block$ is the inductive type with constructors $s$, $p$, $d$, $f$ and the default instance realizes $offset(s)=0$, $offset(p)=1$, $offset(d)=2$, $offset(f)=3$.
background
The Periodic Table Engine translates the eight-tick octave of Recognition Science into chemical shell structure through phi-tier rails and an eight-window neutrality predicate. Block is the inductive type whose four constructors label the orbital blocks; the module comment states that these blocks receive fixed phi-packing offsets with no per-element tuning permitted. Upstream, the Block inductive supplies the four constructors that the offset map acts upon.
proof idea
The declaration is a direct class definition. The default instance implements the offset function by exhaustive pattern matching on the four Block constructors, returning the integers 0, 1, 2, 3 in that order.
why it matters
BlockOffsets supplies the offset term required by blockFactor to form the exponent $2n + offset(b)$ inside the phi-power for bandMultiplier and bandEnergy. These downstream definitions realize the fit-free energy predictions that feed the Noble Gas Closure Theorem (P0-A0), where cumulative valence cost reaches eight-window neutrality exactly at the observed noble-gas atomic numbers. The construction instantiates the eight-tick ledger balance from the RS scheduler inside the chemical domain.
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