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.