Arc 7: The Unified Invariant

The structural companion to Arc 6, completing the program's second arc-pair. Arc 7 derives the unified developmental invariant — a scalar functional structure on the developmental state space that articulates a single conserved quantity across both the tame and wild regimes. The invariant takes the form E(x)=kArc 2⋅SD(x)+b(SD(x))⋅MD(x)\mathcal{E}(x) = k_{\text{Arc 2}} \cdot \mathcal{S}_\mathcal{D}(x) + b(\mathcal{S}_\mathcal{D}(x)) \cdot M_\mathcal{D}(x) E(x)=kArc 2⋅SD(x)+b(SD(x))⋅MD(x), where SD\mathcal{S}_\mathcal{D} SD is the cone speed (timing-scale coupling from Arc 2), MDM_\mathcal{D} MD is the developmental mass (heat plus tangling, from Arc 6), and kArc 2k_{\text{Arc 2}} kArc 2 is the Arc 2 timing-curvature coupling constant. The volume is structured around six structural axioms (regime-compatibility, three conservation axioms under collapse, redistribution, and inversions, additivity in mass, and boundary-value identification at the heat-extremal reference state). The central derivation establishes that any scalar functional satisfying these axioms takes the form above with a(SD)=kArc 2⋅SDa(\mathcal{S}_\mathcal{D}) = k_{\text{Arc 2}} \cdot \mathcal{S}_\mathcal{D} a(SD)=kArc 2⋅SD pinned and b:R≥0→R≥0b: \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0} b:R≥0→R≥0 a continuous positive function not pinned by the present axiom system. The program's structural commitment is b(S)=kArc 2⋅Sb(S) = k_{\text{Arc 2}} \cdot S b(S)=kArc 2⋅S, giving the closed form E=SD⋅kArc 2⋅(1+MD)\mathcal{E} = \mathcal{S}_\mathcal{D} \cdot k_{\text{Arc 2}} \cdot (1 + M_\mathcal{D}) E=SD⋅kArc 2⋅(1+MD) as the simplest extension of the tame-regime coupling structure to the wild regime. Pinning bb b's functional form from an additional structural axiom is registered as the volume's central open mechanism (severity: major), deferred to a future treatment. The volume develops the inversion atlas (three involutive symmetries forced by Arc 6's binary cone duality), the regime-stable scalar classification (Mass and Cone Speed as the unique regime-stable scalars in the program's primitive-quantity budget), the central derivation, a standalone uniqueness theorem, the placement of the unified invariant in the DG invariant catalog, and a closing-insight chapter on collapse-and-expansion as a dual structural pair. The volume completes the structural foundation for Arc 8's interpretive content. Five additional minor open mechanisms are registered honestly (axiom independence verification, analytic light-extremal face, formalization of disjoint composition, ϵ\epsilon ϵ-dependence inherited from Arc 6, and a possible program-level rename of "mass" to a more neutral term). Arc 7 was developed through the program's dual-CI referee discipline: three rounds of independent-referee passes, with foundational reformulations between rounds. The volume forms a structural pair with Arc 6 (DOI: 10.5281/zenodo.19899286) and sits within the broader Developmental Geometry program (ORCID: 0009-0007-5101-6044).