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 ↦ (bord_p(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 bᵖ⁻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 p^N) for all N, with p-adic digit d₁ = 0; (C) tetration convergence, establishing that the tower b*k^n stabilises in Z_p; (D) a structural zero proposition: every element of order d has vanishing tower digits a_k = 0 for all k ≥ d; (E) the marginal zero count E[#{a_k=0}] → k+1 as p → ∞, derived from Dirichlet's theorem; (F) a parity constraint forcing χ(b) = −1 ⇒ 2 | ord_p(b). We decompose h(−p) = −(1/p) Σ_O S_O by orbits of Φ and document a negative result: the co-length sequence of an orbit does not determine its character-weighted contribution S_O. 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 p_max = 100,000 (17,228,275 bridge tests at five tower levels, zero mismatches) and documents the statistical universality of Φ as a random function.