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 tower digit a_k as a recursive function of the sub-tower digits a_1⁽ᵈ⁾, …, a_k⁽ᵈ⁾ of b^d via the relation a_k ≡ n·a_k⁽ᵈ⁾ + T_k(a_1⁽ᵈ⁾, …, aₖ₋₁⁽ᵈ⁾; n) (mod p), where n = (p−1)/d and T_k collects binomial cross-terms, 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 periodicity, establishing that the tower b*k^n is eventually periodic modulo p^N with period ord_{pN−1}(k), and documenting a negative result: the sequence is not Cauchy in Z_p (ν_p = 1 constant); (D) a sub-tower truncation: for every element of order d, the base-p digits of b^d vanish at depth ≥ d, and a corollary showing that the tower digits a_k for k ≥ d are deterministic functions of a_1, …, ad−1; (E) the sub-tower marginal zero count E[#{a_k⁽ᵈ⁾=0}] → k+1 as p → ∞ (structural summand k unconditional via Dirichlet; equidistributional summand +1 heuristic for k ≥ 2); (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 (verified at p=251: two 2-cycles sharing co-lengths (1,1) yield S = −118 ≠ −272). 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.19430537) 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.