Maxwell's Equations Are the Monadic Address-Transport Theorem in Quantum Traction Theory

Bold thesis. Within Quantum Traction Theory (QTT), Maxwell's four equations are not postulated. They are theorems of substrate dial geometry. The electromagnetic U (1) gauge symmetry, integer charge quantization, the inverse-square Coulomb law with its forced 4π, and the structural form of the fine-structure constant all follow from one substrate primitive: a real two-component address dial (ℝ2, J) with quarter-turn J2 = −𝟙, A7 2π bundle closure, A5 visible/hidden factorisation, and A6 finite capacity. Central result. The unique local, gauge-invariant, Lorentz-scalar, parity-even, second-order capacity-stiffness action is Sfield = − (κ / 4c) ∫ fμν fμν d4x under the convention x0 = ct. Variation gives the inhomogeneous Maxwell pair (κ / c) ∂μ fμν = jν. The Bianchi identity gives the homogeneous Maxwell pair ∇·B = 0 and ∇×E = −∂tB from connection geometry alone. Continuous circle characters Homcont (U (1), U (1) ) = ℤ force q = N e0, N ∈ ℤ. Static spherical closure gives Coulomb's law with the 4π fixed by the S2 direction-bundle measure. The fine-structure coefficient takes the structural form α = e02 / (4π ε0 ℏ c) = ℏc / (4π κ) so the elementary charge unit cancels from α; the dimensionless coupling is the inverse stiffness reading of the dial-medium κ in units of ℏc. Visible-share firewall. A7 says every physical address is completed by visible plus hidden budget; A5 says laboratory matter couples through the visible factor. The two readings live on different ledgers. An electron can therefore have integer dial winding q = −e0 and a tiny visible inertial share me / m* ≃ 4. 19×10−23 simultaneously, with the hidden complement closing Qbundle = 2π without contributing an extra ordinary Planck-mass or extra hidden electric source. This share–winding separation is what makes sub-Planck charged particles (electrons, leptons, quarks) consistent with A7 bundle completion. Matter coupling. The same dial holonomy that gives Maxwell's equations gives the Lorentz force F = q (E + v × B) in the large-action many-address limit and the real-J charged wave equation Jℏ ∂tΨ = ( (1 / 2M) (−Jℏ∇ − qA) 2 + qΦ) Ψ in the single-address small-share limit. The classical–quantum boundary is the same Nb ≫ 1 vs Nb ≲ 1 transition that controls inertia in the QTT inertia paper. What v2. 1 hardens (relative to v2. 0). Compactness honesty. The paper does not pretend to derive circle compactness from no compact premise. A7 is the compact 2π bundle closure, exposed in a formal premise–forbidden alphabet audit, not hidden as a U (1) gauge postulate. Scalar-cutoff Lorentz realization. The Artian ruler ℓ̃ is a scalar capacity scale, not a cubic ether lattice vector. Preferred-frame photon operators (nμ fμν) (nα fαν) are exposed as deformations, not quietly tolerated regulator artifacts. This is the QTT answer to the lattice-gauge-theory objection. A6 Hooke limit. The linear stiffness response is now derived rather than postulated. From neutrality at zero curvature, scalar-cutoff Lorentz, parity-even closure, and convexity at the neutral state, the small-curvature expansion of the local capacity cost density begins as κ f2 / 4 plus O (ℓ̃2) corrections. Stiffness flow with no-retune audit. α (μ) = ℏc / (4π κ (μ) ) is now a theorem with a one-loop flow equation and an explicit Xκ2 single-stiffness audit statistic for precision-QED windows. A persistent structured cross-window stiffness drift after standard hadronic, weak, recoil, and structure corrections is a falsifier. Source firewall central. Integer dial winding q = N e0 and visible inertial share Mvis = λ m* are structurally separated. Hidden completion is not an extra ordinary electric current and not an extra ordinary Planck-mass source. Deformation/no-go theorem. A six-slot deformation theorem exposes every place a rival theory could hide a dimensionless coefficient: stiffness rescaling χstiff, parity-odd density β, higher-derivative cn, internal-symmetry rank n, charge–share contamination ζ, and hidden-source contamination σhid. Each is closed by a named QTT axiom (dial topology, linear-stiffness, A6 capacity, share–winding separation, A5 visible factor). The closing claim is sharp: there is no unspoken coefficient left in the electromagnetic sector. Falsifiers. Ten dated falsifying conditions are listed: (F1) a non-abelian photon signature at low energies; (F2) photon mass above current bounds; (F3) free fractional electric charge; (F4) charge–mass share mixing q ∝ N λζ with ζ ≠ 0; (F5) hidden-complement ordinary sourcing; (F6) Lorentz-violating photon signal; (F7) parity-odd photon vacuum term; (F8) coherent one-cell visible excitation above m*; (F9) precision-QED single-kernel failure with Xκ2 excess after standard corrections; (F10) cross-paper coherence failure within the QTT programme. Literature placement. The paper is placed explicitly against four bodies of work: standard fibre-bundle electromagnetism (Dirac 1931, Wu–Yang 1975) ; Wilson lattice gauge theory (Wilson 1974, Kogut 1979) and the Nielsen–Ninomiya obstruction; condensed-matter string-net and spin-liquid emergent-gauge programmes (Wen 2004, Levin–Wen 2005) ; and quantum-gravity arguments for compact gauge groups and forbidden global symmetries (Banks–Seiberg 2011, Harlow–Ooguri 2019). QTT's A7 closure is structurally aligned with the swampland-class compactness pressure, with a substrate-level antecedent rather than a top-down conjecture. Cross-paper coherence within QTT. The same A1+A4+A5+A6+A7 axiomatic core supplies, in parallel papers and without re-fitting: the parameter-free Newton constant G = ℓ̃2c3 / ℏ via the angular-capacity theorem (10. 5281/zenodo. 20057431) ; the centre-dictionary partition n ↦ U (1), SU (2), SU (3) of the Standard Model gauge group (10. 5281/zenodo. 20045141) ; the three-way capacity-counting identity for inertia, gravitational source, and Reality-Work power (10. 5281/zenodo. 20059780) ; the homogeneous-shift invariance theorem of the cosmological-constant source (10. 5281/zenodo. 20043007) ; the V−A parity-violation and active-neutrino mass-spectrum derivations (10. 5281/zenodo. 20051962) ; and the modular-charge entropy ledger (10. 5281/zenodo. 20045307). A falsification of any dated parallel result propagates back through the shared substrate. Status legend. S = standard mathematics or established experimental physics. Q1 = explicit QTT axiomatic input. Q2 = QTT theorem conditional on those axioms, with the coefficient algebra exposed. Q3 = paradigm-shifting consequence if the same one-ruler, one-dial closures survive independent cross-sector tests. Forbidden. A pure-integer prediction of α−1 = 137. 036 from no dimensional input. One substrate stiffness κ must be postulated or measured. What QTT removes is the independence of α as a coupling, not the need for a substrate stiffness scale. Sole author: Ali Attar, Independent Researcher, Colombes, France. ORCID 0009-0008-9931-2691. Framework manuscript: 10. 5281/zenodo. 17527179.