Iterated Fermat Quotient Dynamics on (Z/pZ)×: Bridge Identities, Structural Zeros, and Obstructions

We define and study a dynamical system on (Z/pZ) × given by the iterated generalised Fermat quotient map Φ: b ↦ (b^ordₚ (b) −1) /p mod p. This map replaces the decimal reptend of 1/p in base b by a new base, generating eventually periodic orbits whose structure encodes arithmetic information about p. We prove six unconditional results: (A) a general bridge identity expressing the k-th p-adic digit of b^p−1 as a function of lower digits and the k-th classical Fermat quotient, valid at all tower levels k ≥ 1; (B) a structural gap showing that every fixed point b* satisfies b*^ord (b*) ≡ 1+pb* (mod pN) for all N, with p-adic digit d₁ = 0; (C) tetration convergence, establishing that the tower b*^kⁿ stabilises in Zₚ; (D) a structural zero proposition: every element of order d has vanishing tower digits aₖ = 0 for all k ≥ d; (E) the marginal zero count E#aₖ=0 → k+1 as p → ∞, derived from Dirichlet's theorem; (F) a parity constraint forcing χ (b) = −1 ⇒ 2 | ordₚ (b). We decompose h (−p) = − (1/p) ΣO SO by orbits of Φ and document a negative result: the co-length sequence of an orbit does not determine its character-weighted contribution SO. This eliminates the route from co-lengths alone to h (−p) and identifies the positional (quadratic-residue) information as the irreducible obstruction. A companion computational report (DOI: 10. 5281/zenodo. 19422984) verifies all algebraic results to pₘax = 100, 000 (17, 228, 275 bridge tests at five tower levels, zero mismatches) and documents the statistical universality of Φ as a random function.