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The numerical approximation α−1≈4π3+π2+π≈137.036 was first noted symbolically by Wolfgang Pauli through his "World Clock" dream (1932-34), documented in the Collected Works of C. G. Jung [1], and has since been listed among known polynomial approximations to α−1 [5]. Despite 90 years of recognition, no physical derivation of this formula has been achieved. We present the first topological identification of each coefficient within Discrete Topological Torsion Theory (DTTT), a framework that models spacetime as a nonlinear Cosserat micropolar elastic continuum [6, 7] with topological knot solitons. We do not claim the formula; we claim its derivation from the topology of a specific 3-manifold—the trefoil knot complement S3\N(T(2,3))1—with each coefficient identified as an independently verifiable topological invariant. The integer coefficients (4, 1, 1) are proven invariants of the trefoil: the bulk coefficient c3=p+q−1=4 is established by Chen-Ruan orbifold cohomology, with six supporting computations from independent branches of mathematics (Chen-Ruan orbifold Betti number, Seifert monodromy count, reduced deck module dimension, non-cohomological character count, Milnor fibre twisted cohomology, and additional routes requiring independent verification) ; the boundary and fibre coefficients follow from the Kronheimer-Mrowka theorem. The powers πd arise from the projective identification ψ≡−ψ of the Cosserat vacuum (the projective identification (SO(3)=SU(2)/Z2)), which halves the angular integration domain from [0,2π) to [0, π) per degree of freedom. We further derive: a correction formula reducing the 2.2 ppm residual to 0.3 ppb with zero new parameters ; compatibility with the standard QED running of a (βCS=0, Witten 1989) ; and most significantly the demonstration that the same one axiom (the Cosserat vacuum) derive 42 Standard Model constants [18] (28 DERIVED given Axiom A + identification, 8 STRONG CONJECTURE) including sin2θW=3/13, αs=3/(2π), the PMNS CP phase δCP=32π/27 [26], and the complete charged lepton spectrum. The assembly rule α−1=Σcdπd follows from Axiom A (mode counting + projective volumes) plus the physical identification (boundary U(1)= electromagnetism). It was previously classified as an irreducible postulate (11 topological derivation routes closed). A five-step proof now establishes it as DERIVED: (1) α−1=log Z [THEOREM, statistical mechanics]; (2) log Z=Σ log Zd [THEOREM, Beasley-Witten 2005]; (3) log Zd∝Vold [THEOREM, Cheeger-Müller 1978/79]; (4) Vold=πd [DERIVED, SO(3) projective measure]; (5) cd=(4,1,1) [THEOREM, Chen-Ruan 2004]. Four of five steps are proven mathematics; one requires the physical identification (the projective volume πd from ψ≡−ψ). This is the strongest status a physical law can achieve. The physical interpretation: α−1 is the total elastic impedance of the Cosserat vacuum around a topological soliton, decomposed by the Seifert strata of the knot complement. The vacuum is stiff (α−1≫1) which is why electromagnetic coupling is weak and stable atoms exist.