Anomaly Cancellation in the Metric Bundle Pati-Salam Theory: Four- and Fourteen-Dimensional Analysis

Paper 4 of 6 in the Metric Bundle Programme. We verify anomaly cancellation for the Pati-Salam gauge theory (4) × (2)_L × (2)_R that emerges from the metric bundle Y¹⁴ = (X^4). The fermion content derived in earlier work---the 16-plet of (10) ⊃ (6) × (4), decomposing as (4, 2, 1) ⊕ (4̄, 1, 2)---is shown to be free of all gauge anomalies. The purely cubic (4)^3 anomaly vanishes because 4 ⊕ 4̄ is a real representation; all (2)^3 anomalies vanish because the symmetric d-tensor of (2) is identically zero; and all mixed gauge-gravitational anomalies cancel because every gauge factor is non-abelian with traceless generators. The Witten (2) global anomaly is absent because each (2) factor sees four Weyl doublets (an even number), a consequence of the Pati-Salam unification of leptons with quarks as the fourth colour. After Pati-Salam breaking to (3)_c × (2)_L × (1)_Y, all six Standard Model anomaly conditions are verified, including the cubic (1)_Y^3 and the mixed (1)_Y-gravitational anomaly. Part of a six-paper series deriving the Pati-Salam gauge group, fermion content, gauge dynamics, anomaly cancellation, and the three-generation structure from the geometry of the metric bundle Y14 = Met(X4). v2 changes (March 2026): Fixed arithmetic error in U(1)_Y^3 anomaly computation (-28/36 corrected to -32/36, result 0 unchanged). Removed scratch-pad text. v3 changes (March 2026): v3: Added Paper5 bib entry for cross-referencing. v4 changes (March 2026): v4: Recompiled with updated cross-references. v5 changes (May 28, 2026): v5 (May 28, 2026): Section 6.3 now references Paper V's spin^c index theorem result N_G = (c_1^2 - sigma(K3))/8 = 3 directly, rather than treating the base-topology computation as future work. Open Questions item 2 reframed accordingly: the K3 base case is closed by Paper V; the full 14D bundle index on Y^14 remains open. v6 changes (May 28, 2026): v6 (May 28, 2026, post-referee-simulation revision): Section 6 (Pure Gravitational Anomaly and Generation Number) removed. The pure D=4 gravitational anomaly vanishes automatically and does not constrain the matter content; the mod-24 modular-invariance argument the section invoked is not a standard D=4 result and the previously cited reference did not support it. The substantive 14-dimensional anomaly analysis on Y^14 = Met(X^4) is deferred to a separate work in preparation. Proposition 3.1's proof clarified (cubic-only chirality-pairing). Historical citations added (Georgi-Glashow 1972; Frampton-Kephart 1983). Discussion section reorganised: relation-to-D=4-gravity paragraph added, Paper VII forward reference removed (not relevant to anomaly cancellation), open questions reduced to two. v7 changes (May 28, 2026): v7 (May 28, 2026): Errors flagged by an independent technical review addressed. (1) Section 4 Witten Remark corrected: the SU(3)-vs-SU(4) counterfactual was incorrect (the Standard Model already has 4 SU(2)_L doublets from 3 quark colours plus 1 lepton doublet, so the even count is not specific to SU(4) unification); replaced with an honest doublet-count remark. (2) Proposition 3.2 proof: dropped the misleading 'A(adj) = 0' justification (true for every simple group); replaced with the operative absence of a totally symmetric rank-3 invariant for SU(2). (3) Discussion: 'Spin(4k+2) anomaly-free for all subgroups' overstatement narrowed to SO(10) specifically, with the Spin(6) = SU(4) counterexample noted (the 4 of SU(4) has nonzero cubic anomaly on its own). (4) Gravitational-anomaly paragraph: garbled phrasing replaced with the correct Pontryagin-degree argument (classes live in degrees 4, 8, 12; no 6-form constructed from them, so A_hat_6 = 0 identically); Alvarez-Gaume-Witten 1984 now properly cited. (5) Cover letter: removed the now-incorrect 'most direct value-added' Witten claim. (6) References: pruned uncited entries (DeWitt 1967, Georgi-Glashow 1974, Kobayashi-Maskawa 1973, Paper 7). v8 changes (May 28, 2026): v8 (May 28, 2026): Major expansion. The paper now covers both the four-dimensional anomaly verification (Sections 1-6, as before) and a new fourteen-dimensional Atiyah-Singer index analysis on the full metric bundle Y^14 = Met(X^4) (Sections 7-10). Title and abstract extended. Main new results: (1) a bidegree argument shows that, because the fibre GL+(4,R)/SO(3,1) is contractible, the projection Y -> X is a homotopy equivalence and every characteristic class on Y is pulled back from the base, so the direct index of the ordinary gauge-twisted Dirac operator on Y^14 VANISHES -- re-deriving Paper V's fibre no-go from the total-space perspective; (2) the generation number is recovered solely through the base spin^c index on K3, N_G = (c_1(L)^2 - sigma)/8 = 3, with the fibre supplying the U(1)B-L spin^c line bundle; (3) explicit computation p_1(V_vert) = 6 p_1(X) for the vertical bundle Sym^2(T*X), shown invisible to the index for bidegree reasons; (4) explicit distinction between the field-space volume V_eff (which enters the gauge coupling) and the normalised fibre measure (which enters the index), clarifying the role of Paper VIII's FEP localisation. New references: Atiyah-Singer 1968/1971, Atiyah-Patodi-Singer 1975, Lawson-Michelsohn 1989, Paper VIII. Consistency with Paper V's K3 spin^c result verified throughout.