A Base-Invariant Numerical Alphabet via Carry Propagation and Valuation Duality

We construct a base-invariant numerical alphabet by treating digit strings as finite generativedescriptions ordered by carry propagation rather than as primitive infinite objects. Numericalsize is shown to be determined by information density under valuation, while cardinality remainsbounded by infinity. Archimedean and p-adic representations are unified via mirror reindexing,and recurring digit tails are shown to be valuation-dependent normal forms rather than literalequalities. A M¨obius-boundary interpretation is presented as a geometric model of valuationreversal.