Conditional Reconstruction Closure D; Conditional Selection of Constitutive Sectors- Gauge Group, Linearity, Dimension, and Common Causal Cone

This paper studies the large number of alternative theories contained in thepre-metric system, delineating which explicit additional conditions can select thefour-dimensional vacuum Maxwell sector and which selections are fundamentallyimpossible. Alternative fixed points are registered rather than pathological objects:SU(2), U(1)N, the non-compact one-dimensional group R, Born–Infeld theory, andtheories in other dimensions each pass the same set of foundational axioms withintheir respective sectors and have zero internal anomalies (the baseline caliber isgraded and registered according to the machine-check ledger in Chapter 2), so theunconditional uniqueness route is closed at the first hurdle (D-CM1). A conditional classification of the connection sector is given: when the four conditions ofsingle sector, connectedness, one-dimensionality, and compactness are conjoined,the gauge group is U(1); the objects excluded by each of the four conditions areregistered item by item, and none can be omitted (D-CT2). Born–Infeld theoryserves as an explicit witness refuting the inference from symmetry plus gauge closure to linearity: its constitutive relation H(E) = Eb/√b2 − E2 is nonlinear yetgauge invariant, and the failure of superposition is witnessed by machine check(gauge invariance is a structural-layer property, anchored to the original BornInfeld literature; the failure of superposition is a machine-check output, D-CM3,MC-D-01). The thirty-six components of the local linear constitutive relation arecounted according to three channels—principal, skew, and axion (20+15+1); evenassuming local linearity, one still needs metric-induced principal part, vanishingskew, and vanishing axion to obtain H = λ ⋆ gF (D-CT4). Component countingyields a precise criterion for dimension selection: in n-dimensional spacetime thereare n−1 electric-field components and C(n−1,2) magnetic-field components, andthe electromagnetic vector duality condition holds only for n = 4 (D-CT5, MC-D03). Signature is not determined by counting: the Lorentz-type principal symbolis hyperbolic, the Euclidean type is elliptic without Cauchy evolution, and thecommon causal cone is an independent premise; the product-cone model cannot beexcluded by anomalies (MC-D-04). The invariant speed c is the conversion constantbetween spatial and temporal units within the common causal cone; unit rescalingchanges its numerical value without touching the two axioms, so the axioms donot determine its value (D-CM6, MC-D-05). An eleven-row assumption-ablationmatrix rewrites the uniqueness theorem as a testable hypothesis-independence theorem (D-CM7, MC-D-06). All numerical assertions are machine-checked, and thestrength of machine checks is graded in three levels according to Appendix A.7.The honest boundary of this paper: no unconditional version of necessary U(1),necessary linearity, necessary four-dimensionality, or necessary Lorentz signature isrestored; that route is formally closed and will not be retried.