bel

Law 7 which-came-first, from independent historical reasoning, not the graph epoch. The logarithm was invented by John Napier in 1614 (Canon Mirificus) as the computation-reducing function that turns products into sums; the bel was coined at Bell Telephone Laboratories in the 1920s as a named unit for logarithmic ratios in telephone transmission. The bel's entire operative content is the base-10 logarithm of a power ratio — L = log10(P1/P0) bels: the unit is the log10 value of the ratio given a name, and without the log10 function there is nothing to name. Pinned sense (Law 11d): the bel per its accepted definition as the whole-logarithm ratio unit, the logarithm per its accepted definition as the function log_b(x) = y iff b^y = x. Reverse fails: the logarithm predates and does not require the bel. The board already accepts this family's parallel lineage — exponential-function DERIVED_FROM logarithm, ruled on the same 'Napier's 1614 canon predates' reasoning — so direction and type are consistent with the accepted record.

Law 9, nearest kind, pinned sense per both accepted carves (Law 11d). The bel's carve: 'the human-made whole-logarithm ratio unit — the level corresponding to one power of ten of a power ratio, L = log10(P1/P0) bels.' The unit's accepted carve: a standardized reference magnitude against which quantities are compared, with parameters of quantity kind, reference, and authority. The bel fills all three: (1) quantity kind — ratio (power ratio or field-amplitude ratio, the bel's first parameter); (2) reference — the fixed base-10 logarithmic scale (one bel = the power ratio 10, 1 B = 10 dB), pure convention — nothing in nature is a bel; (3) authority — the ITU/IEC standardization of the bel-decibel relation (1 B = 10 dB) named in the bel's persistence mechanism. No intermediate 'logarithmic unit' kind exists on the board, and the sibling decibel-to-unit edge is already ACCEPTED with the same three-parameter reading, so unit is the nearest accepted kind — no leap (Law 11e).

Law 7 which-came-first, from independent historical reasoning, not the graph epoch. The bel was the original whole-logarithm ratio unit at Bell Telephone Laboratories in the 1920s, named for Alexander Graham Bell; the decibel was introduced shortly after as its practical one-tenth subdivision because a 10:1 power step is too coarse for telephone line gain/loss work. The board's own accepted decibel definition pins this lineage: 'one decibel is one-tenth of the bel, the whole-logarithm ratio named for Alexander Graham Bell' — the decibel's scale is the bel's scale at one-tenth size, so the subdivision derives from the parent unit, not vice versa. Pinned sense (Law 11d): the bel as the parent scale of the decibel per both accepted definitions. Reverse fails: the bel's definition and standardization predate and do not require the decibel; the decibel is what was derived.