Structural Universality and Exact Terminal Bifurcation

One-Higgs Naturality, Projection Firewalls, and the Branch-Complete Fermionic Endpoint This paper is the capstone of the Fermion Carrier–Generation Structure programme. It integrates the accepted FCGS-I exact-three structural result and the FCGS-II terminal factorization theorem without modifying either predecessor chain. The central conclusion is a precise bifurcation: the fermionic architecture is universal at the structural root-and-response level; terminal physical standing is target-relative and remains branch-dependent. Across the neutral, charged-lepton, up-coloured, and down-coloured matter kinds, the paper establishes one common three-root structural geometry \[ I_3,\qquad |I_3|=3, \] with four kind-preserving fibres \[ E_f=I_3\times{r_f}, \qquad f\in{N,L,U,D}. \] On every maximal lawful same-target domain, the accepted rooted correspondences \[ \operatorname{Disc}(E_f) \longrightarrow S_f(C_f) \longrightarrow Q_{Y_f}=\operatorname{Disc}(A_f) \] are natural equivalences. Each kind therefore carries the same exact-three structural architecture and three rooted response components, while retaining its own charge, chirality, color, Higgs, boundary, continuation, provenance, failure, and physical-authority data. The paper carefully distinguishes full-target comparison from structural comparison. Full cross-kind arrows are absent unless independently licensed. After explicit erasure to the common structural reduct, effective codescent becomes available, but this does not identify the complete matter-kind targets or collapse their distinct physical roles. A single Higgs bridge/fixation carrier is shown to support every accepted rooted fermionic response: \[ C_H=H_{\mathrm{br/fix}}. \] This one-carrier result preserves the distinct matter-kind incidences and does not turn a Yukawa display, response slot, or matrix representation into a primitive cardinality argument. The integrated structural theorem establishes: maximal-domain same-target naturality; explicit structural erasure and effective codescent; one common exact-three root geometry; three rooted response components per matter kind; one shared Higgs bridge/fixation carrier; typed projection and process domains; preservation of kind, root, response, carrier, authority, degeneracy, and process-role fields; a fixed-authority global no-backflow theorem; an exact numerical and projection firewall. Exact terminal bifurcation Terminal physics is represented by a tagged disposition family rather than by analogy across matter kinds. The neutral fibre contains two certified child scopes with one terminal bearer each and a distinct (3+1) spectral core. The neutral parent complement remains a separate typed branch. The charged-lepton fibre retains the exact evidence-relative disposition \[ \mathrm{NotReady}\big[(C_L,M^0_L)\big]. \] The up- and down-coloured fibres retain \[ \mathrm{NotReady}\big[(\mathrm{Confinement},M^0_U)\big], \qquad \mathrm{NotReady}\big[(\mathrm{Confinement},M^0_D)\big]. \] Their terminal Role-Occupancy Closure objects remain unformed and uncounted. They are not completed by cross-kind analogy, particle naming, mass data, matrix structure, or detector information. A common physical apex requires all four positive terminal factors, one complete joint certificate, and one common target–parameter state. At the frozen evidence package, the certified common-apex witness set is empty. This is an exact activation result, not a theorem that future physical realization is impossible. The final status therefore has the structured form \[ \mathrm{StructuralUniversal} ;\oplus_{\mathrm{tag}}; \mathrm{TerminalPartial} ;\oplus_{\mathrm{tag}}; \mathrm{NotReady}[\mathrm{CommonApex};E^0]. \] This status preserves every positive child-scope result, every exact readiness obstruction, and every unresolved physical branch. Projection and no-backflow boundaries Carrier-preserving projection, the particle dictionary, and the process fibration are explicitly partial. Masses, matrices, singular values, poles, widths, fits, detector records, and HNL scouting coordinates remain downstream data. None of these can: reverse-authorize a carrier; alter the exact-three root count; create a second Higgs carrier; complete a terminal ROC; supply confinement authority; establish an unstable-pole theorem; create an interacting quantum-field completion; populate a common physical apex. A later strengthening must identify at least one new source theorem, certificate, target field, or inference rule. It must then be proved as a successor result rather than presented as a restatement of FCGS-III. Scope and realization boundary FCGS-III is greatest only relative to its frozen target, evidence package, claim language, and proof constructors. It is not a completeness theorem for all fermionic physics, a universal terminality theorem, a numerical prediction theorem, or an unrestricted common-apex theorem. The paper does not assert the physical nonexistence of charged leptons, quarks, heavy neutral states, confined sectors, or a future common realization. It records exactly where the required terminal or common-apex certificates are absent at the declared evidence basis. The exact-three result remains structural. Terminal status remains target-relative. Numerical, spectral, confinement, interacting-QFT, and dynamical-gravity conclusions retain their own authority requirements. Role in the FCGS programme FCGS-III is the branch-complete fermionic endpoint. It closes structural universality, preserves the exact terminal bifurcation, establishes the projection and numerical firewalls, and exports an immutable claim-bounded handoff to the BCRS and CSP programmes. Its principal contribution is to state precisely where universality holds and where it must stop: universal three-root structure and one-Higgs response at the structural level, but bifurcated and readiness-valued terminal physics at the physical level. Keywords structural universality; exact terminal bifurcation; FCGS; three-root structure; one-Higgs bridge; cross-kind naturality; structural codescent; terminal Role-Occupancy Closure; physical projection firewall; common-apex readiness; NotReady disposition; global no-backflow; HNL nonpromotion; fermion carrier structure; AASC