PUH Theorem 212 — The Weak Force from the E8 → E7 × SU(2) Decomposition: W and Z Bosons as the SU(2) Adjoint, Matter Doublets from the E7 56-plet, Parity Violation from the Rebound Asymmetry, and the Completion of the Fundamental Forces Programme in PUH
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2026
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| _version_ | 1866901151768641536 |
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| author | Martell, Brian |
| author_facet | Martell, Brian |
| contents | <p>The PUH archive has developed structural accounts of three of the four fundamental forces: electromagnetism via Theorem 169 (photon as U(1) Cartan from E8 → E7 × U(1)), gravitation via Theorem 167 (rank-4 of E8 Cartan), and the strong force via Theorem 211 (quantum chromodynamics from E8 → E6 × SU(3)). </p> <p>The weak force remained the one major structural gap.</p> <p>This paper closes it.</p> <p>The weak force emerges from the alternative maximal decomposition E8 → E7 × SU(2), in which the 248-dimensional adjoint branches as 248 = (133, 1) + (1, 3) + (56, 2). The (1, 3) factor is the SU(2) adjoint — the three weak gauge bosons (W⁺, W⁻, Z in broken phase, or W¹, W², W³ in unbroken phase).</p> <p>The (56, 2) factor is the E7 fundamental 56-plet paired with the SU(2) doublet — the standard matter doublets of weak interactions, each left-handed fermion in an SU(2)_L doublet.</p> <p>The (133, 1) factor is the E7 adjoint carrying weak-singlet physics (including electromagnetism and the strong force through further branching).</p> <p>Four structural results follow.</p> <p>First, this decomposition is complementary to Theorem 169. The U(1) of T169’s E8 → E7 × U(1) is the Cartan T₃ of the SU(2) in this paper — i.e., SU(2) broken to its U(1) Cartan generates the electromagnetic photon, while the full SU(2) describes the weak gauge structure before electroweak symmetry breaking.</p> <p>Second, the (56, 2) decomposes under SU(2) → U(1)_T₃ as (56, +1) + (56, -1), directly producing the ‘(56, ±1) matter fields’ flagged as Compendium Open Problem F. T212 therefore structurally closes OP-F: these are the weak matter doublets under hypercharge projection, and their QED running (OP-F) is computable in principle from the standard model running that includes all charged matter in the (56, ±1) content.</p> <p>Third, parity violation — the observation that weak interactions couple only to left-handed fermions — is proposed to have its structural origin in the rebound asymmetry δ_U = 2/27 established in Theorem 155. The cosmic rebound selects a preferred chirality for our universe; SU(2)_R is broken at a higher scale (the rebound scale itself) while SU(2)_L remains unbroken down to the electroweak scale. The throat / anti-universe branch would carry the complementary chirality. This is a structural hypothesis within PUH, not a derivation from first principles, but it is natural within the existing rebound geometry.</p> <p> </p> <p>Fourth, the Higgs mechanism is interpreted as a collective mode of the E8 condensate — the direction along which the condensate tension points to break SU(2)_L × U(1)_Y down to U(1)_EM at approximately 246 GeV. The Higgs boson itself is the fluctuation of the condensate around this symmetry-breaking VEV direction.</p> <p>This paper completes the PUH fundamental forces programme.</p> <p>All four forces now have structural treatments within the E8 substrate: electromagnetism (T169), gravitation (T167), strong (T211), weak (T212).</p> <p>What remains open are specific quantitative derivations — the Weinberg angle, the Higgs mass, the Higgs VEV value, the CKM and PMNS mixing matrices — all of which are noted as open questions. Five formal results and three falsifiable predictions follow.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19745004 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | PUH Theorem 212 — The Weak Force from the E8 → E7 × SU(2) Decomposition: W and Z Bosons as the SU(2) Adjoint, Matter Doublets from the E7 56-plet, Parity Violation from the Rebound Asymmetry, and the Completion of the Fundamental Forces Programme in PUH Martell, Brian weak force, SU(2)_L, W and Z bosons, parity violation, electroweak unification, Weinberg angle, Higgs mechanism, chirality, rebound asymmetry, E7 × SU(2) decomposition, matter doublets, 56-plet, Photonic Universe Hypothesis PUH <p>The PUH archive has developed structural accounts of three of the four fundamental forces: electromagnetism via Theorem 169 (photon as U(1) Cartan from E8 → E7 × U(1)), gravitation via Theorem 167 (rank-4 of E8 Cartan), and the strong force via Theorem 211 (quantum chromodynamics from E8 → E6 × SU(3)). </p> <p>The weak force remained the one major structural gap.</p> <p>This paper closes it.</p> <p>The weak force emerges from the alternative maximal decomposition E8 → E7 × SU(2), in which the 248-dimensional adjoint branches as 248 = (133, 1) + (1, 3) + (56, 2). The (1, 3) factor is the SU(2) adjoint — the three weak gauge bosons (W⁺, W⁻, Z in broken phase, or W¹, W², W³ in unbroken phase).</p> <p>The (56, 2) factor is the E7 fundamental 56-plet paired with the SU(2) doublet — the standard matter doublets of weak interactions, each left-handed fermion in an SU(2)_L doublet.</p> <p>The (133, 1) factor is the E7 adjoint carrying weak-singlet physics (including electromagnetism and the strong force through further branching).</p> <p>Four structural results follow.</p> <p>First, this decomposition is complementary to Theorem 169. The U(1) of T169’s E8 → E7 × U(1) is the Cartan T₃ of the SU(2) in this paper — i.e., SU(2) broken to its U(1) Cartan generates the electromagnetic photon, while the full SU(2) describes the weak gauge structure before electroweak symmetry breaking.</p> <p>Second, the (56, 2) decomposes under SU(2) → U(1)_T₃ as (56, +1) + (56, -1), directly producing the ‘(56, ±1) matter fields’ flagged as Compendium Open Problem F. T212 therefore structurally closes OP-F: these are the weak matter doublets under hypercharge projection, and their QED running (OP-F) is computable in principle from the standard model running that includes all charged matter in the (56, ±1) content.</p> <p>Third, parity violation — the observation that weak interactions couple only to left-handed fermions — is proposed to have its structural origin in the rebound asymmetry δ_U = 2/27 established in Theorem 155. The cosmic rebound selects a preferred chirality for our universe; SU(2)_R is broken at a higher scale (the rebound scale itself) while SU(2)_L remains unbroken down to the electroweak scale. The throat / anti-universe branch would carry the complementary chirality. This is a structural hypothesis within PUH, not a derivation from first principles, but it is natural within the existing rebound geometry.</p> <p> </p> <p>Fourth, the Higgs mechanism is interpreted as a collective mode of the E8 condensate — the direction along which the condensate tension points to break SU(2)_L × U(1)_Y down to U(1)_EM at approximately 246 GeV. The Higgs boson itself is the fluctuation of the condensate around this symmetry-breaking VEV direction.</p> <p>This paper completes the PUH fundamental forces programme.</p> <p>All four forces now have structural treatments within the E8 substrate: electromagnetism (T169), gravitation (T167), strong (T211), weak (T212).</p> <p>What remains open are specific quantitative derivations — the Weinberg angle, the Higgs mass, the Higgs VEV value, the CKM and PMNS mixing matrices — all of which are noted as open questions. Five formal results and three falsifiable predictions follow.</p> |
| title | PUH Theorem 212 — The Weak Force from the E8 → E7 × SU(2) Decomposition: W and Z Bosons as the SU(2) Adjoint, Matter Doublets from the E7 56-plet, Parity Violation from the Rebound Asymmetry, and the Completion of the Fundamental Forces Programme in PUH |
| topic | weak force, SU(2)_L, W and Z bosons, parity violation, electroweak unification, Weinberg angle, Higgs mechanism, chirality, rebound asymmetry, E7 × SU(2) decomposition, matter doublets, 56-plet, Photonic Universe Hypothesis PUH |
| url | https://doi.org/10.5281/zenodo.19745004 |