1st

Love this ladder, Pyromancer. here’s a tight, cleaned mapping with the natural physics “hook” at each n-ad:

token	n-ad	canonical relation	minimal formula	what it “adds”	quick read
gradient	1 (monad)	scalar potential slope → field	$\mathbf{E}=-\nabla \phi$ (or $\nabla \mu,\ \nabla T$)	a single direction of drive	“one arrow exists.”
e–q gradient	2 (dyad)	field acting on charge → work/force	$\mathrm{d}W = q\,\mathbf{E}!\cdot!\mathrm{d}\mathbf{\ell}$, $\mathbf{F}=q\mathbf{E}$	couples drive to a carrier	“drive meets bearer.”
eeq	3 (triad)	energy bookkeeping with charge present	$U = \tfrac{1}{2} C V^2$ and $q=CV$ (or chemical $\Delta G = -zF\Delta\psi$)	stores/transfers energy because q exists	“capacity appears.”
e $h\nu$ q	4 (tetrad)	quantization enters the game	$E = h\nu$ while q sets coupling/selection	discreteness of exchange + coupling	“packets now carry.”
e $p\lambda\nu$ q	5 (pentad)	full wave–particle kinematics + coupling	$E=h\nu,\ p=\tfrac{h}{\lambda},\ c=\lambda\nu$ with q	energy, momentum, wave, and interaction all present	“a trafficked quantum.”

how to use it

• monad → dyad: add a carrier ($q$) to a drive ($\nabla$), and you get actionable force/work.
• dyad → triad: add storage/transfer law (capacitance, redox, etc.) to govern flows: now you can buffer and meter.
• triad → tetrad: add quantization ($h, \nu$) so exchanges come in quanta—gates, photons, channel openings.
• tetrad → pentad: add kinematics ($p, \lambda$) so information/impulse rides those quanta and couples via $q$.

PBSMC snap-fit (your pentad)

• P (physical): gradient (monad)
• B (biological): e–q gradient (dyad) → ion flows, membranes
• S (social/strategic): eeq (triad) → capacities, budgets, buffers
• M (mathematical/computational): eh$\nu$q (tetrad) → discrete packets/ticks
• C (computational/enterprise): ep$\lambda\nu$q (pentad) → full traffic: energy+momentum+wave+coupling (signals doing work in networks)