From the String Landscape to the Fine-Structure Constant: A Pascal Anti-Diagonal of M-theory Sector Dimensions

A systematic survey of characteristic integers across mainstream and adjacent string-theoretic frameworks finds twenty-seven distinct Pascal-triangle entries in rows 0–14 each carrying at least one such anchor. Within this catalog, a single anti-diagonal traversing rows 8–13 identifies seven integers Vk = 28, 126, 210, 330, 165, 66, 13, each independently identified with a dimension or root count of a canonical M-theory / F-theory bosonic sector before α−1 is computed. Their nested-reciprocal closed form α−1 (0) = (V5 − V1) + Σk=17 1/ (V1 V2 ··· Vk) = 137. 0359990862878849… matches the low-energy (Thomson-limit) inverse fine-structure constant CODATA 2018 α−1 = 137. 035999084 (21) at 0. 11σ (Δα−1 = 2. 29 × 10−9 vs. CODATA σ = 2. 1 × 10−8) with seven integers and zero free parameters. The result is consistent with α0−1 being a topological invariant of the M-theory IR vacuum. The methodology distinguishes the proposal from numerical-coincidence claims on five independent grounds: Strong Law of Small Numbers defense (Guy 1988): each Vk is a mandatory exterior-calculus dimension binom (n, p) = dim Λpℝn, anchored before α−1 is computed. Pascal-uniqueness audit: of all 105 Pascal entries on rows ≤13, no alternative anti-diagonal satisfies the anchoring criteria; a five-alternative-sequence null check at 15-digit precision returned zero candidates. Engel-theorem forcing: the first three coefficients (28, 126, 210) are mathematically forced by the value of ε = α−1 − 137 alone via the greedy nested-reciprocal algorithm ak = ⌈1/xk⌉; the cascade departs from Engel-canonical at V4 = 330 exactly where the M-theory F4 = dC3 bridge inserts. Wyler differentiator: Wyler's 1969 continuous bounded-symmetric-domain volumes could drift with measurement refinement; the Vk are integer-quantized charges that cannot deform. Quantum Hall Effect precedent: integer-quantized Hall conductance is topologically protected at 10−9 via Chern-number invariance (Thouless–Kohmoto–Nightingale–den Nijs 1982), fixing the von Klitzing constant RK = Z0/ (2α) exactly. Laboratory precedent for “integer topology fixes a continuous coupling at 10−9 precision. ” Falsifiable by confirmed |α̇0/α0| > 0 at ≥5σ, or by any canonically-anchored higher-dimensional object on Pascal rows ≤13 missed by the anti-diagonal. This Letter is the abbreviated counterpart to a longer companion archived at Zenodo DOI 10. 5281/zenodo. 20013547, which contains the full derivation chain, the per-Vk string-theory identifications, the matrix-theory algebra supplying the boundary 112 + 42 = 137 reading, the Pascal-uniqueness audit, the Python verification code with full reproducibility log, and the string-theory heat-map source dataset and generation script. Submitted to Physical Review Letters on 6 May 2026.