In gauge–Higgs unification the cancellation of gauge anomalies is necessary, but is it sufficient for a viable fermion sector? On an orbifold the answer involves genuinely local data — anomalies and tadpoles bound to the fixed points — that a bulk count misses. We make the question concrete for a colored quark block embedded in the dimension-60 representation (3,60) of SU(4) on T²/Z₂, built over the two-Higgs-doublet gauge sector of Akamatsu, Hirose, Maru and Nago (AHMN, arXiv:2603.05857); the fermion embedding is our own. We fix the unique unbroken hypercharge that reproduces the Standard-Model quark charges (Y = (-7q₈+4q₁₅)/18), show by an exhaustive scan that (3,60) is the minimal SU(4) representation able to host a full quark generation, and organize every consistency requirement into a reproducible hierarchy of exact conditions: the irreducible six- and four-dimensional anomalies, the localized non-Abelian cubic residues (certified axiom-free in the Lean 4 proof assistant), the localized tadpole, the global SU(2) (Witten) constraint, and the boundary-mass matching of exotic zero modes. We then exhibit — and independently verify in exact rational arithmetic — a 45-multiplet completion that cancels the eight anomaly coefficients and the localized tadpole together, with integral local cubic residues, so that here anomaly freedom and the tadpole condition are compatible. This is not accidental: we prove an Existence theorem — on the fermion library the tadpole is linearly independent of the anomalies (the combined map has rank 10 = 8+2) and freely adjustable (its image on the anomaly-neutral non-negative cone is all of R²), so every anomaly-free completion of the block admits a tadpole-compatible one; the witness realizes a structural guarantee. The certificate is machine-checked in Lean 4 (depending only on propext). The same exact rank-and-cone test is released as a reusable tool, ghu_preflight.py, that decides for any orbifold fermion library whether a localized functional is an independent obstruction or is always absorbable. The decisive remaining test — whether the exotics can be given mass without lifting the Standard-Model quarks — is cast as an exact bipartite-matching gate and left open pending the full brane spectrum. Every step is a finite exact computation; the results reported are certificates, not estimates. English and Spanish versions are included, together with two Lean 4 certificates, the reusable preflight tool, the exact SageMath verification scripts, and the full spectrum. The exact computations were carried out and cross-checked with Claude (Anthropic) as an AI research assistant against a common machine-verifiable ground truth.

This record belongs to a four-part series on six-dimensional SU(4) gauge-Higgs unification on the orbifold T²/Z₂ and the symmetric-function identity it exported. The parts are independent papers and are best read in order. Part I (this record) — Anomaly- and Tadpole-Compatible Fermion Completion of Six-Dimensional SU(4) Gauge-Higgs Unification — embeds a Standard-Model quark block in the (3,60) of SU(4) on T²/Z₂, and proves that on this fermion library the localized tadpole never obstructs an anomaly-free completion.DOI: 10.5281/zenodo.21432625 · code: github.com/karlesmarin/ghu-su4-completion Part II — Three Gates to a Quark Generation — the exact criterion for which SU(4) representations contain a Standard-Model quark cell among their T²/Z₂ chiral zero modes: (a+2b+3c) odd, b≥1, a+b+c≥3.DOI: 10.5281/zenodo.21432627 · code: github.com/karlesmarin/su4-sm-cell-criterion Part III — A Centre-Charge Selection Rule for the Wilson-Line Potential — a period of Wilson line is a central gauge transformation up to Weyl, so the centre charge decides which Fourier sector of the potential may exist; the fundamental domain of gauge-Higgs unification is representation-dependent.DOI: 10.5281/zenodo.21438226 · code: github.com/karlesmarin/centre-parity-selection Part IV — Schur Functions at (1,−1,t,t⁻¹) — on the non-identity component of O(4) every four-row Schur function is zero or ± a product of exactly three SU(2) characters read off the 2-quotient; Part III's vanishing class is its zero locus. Verified exhaustively, not proved.DOI: 10.5281/zenodo.21463000 · code: github.com/karlesmarin/schur-nonidentity-o4