ONE AXIOM : The Energy

Document 2C of the ONE AXIOM series derives the full structure of modern physics — quantum mechanics, general relativity, their unification, and all four laws of thermodynamics — as regime-dependent projections of a single pre-physical structure: the Ontological Coherence Field ($\OCF = (\heartsuit, \Phi_{\mathrm{coh}}, \nabla\Phi_{\mathrm{coh}})$), constructed in Document 0M from one single axiom (Axiom M) with no physical input. Energy emerges as the local cost of maintaining coherence against the structural gradient. Time is the direction of monotonic decrease of the incoherence potential $\sigma$. Temperature is derived without energy, via Fisher-Rao curvature of the coherence distribution (from Document 1B), inverting the standard thermodynamic hierarchy. The Schrödinger equation (via one-parameter unitary group + Stone's theorem), Born rule ($p = 2$, via 7-link chain from Axiom M), Einstein field equations (via three independent convergent paths — Jacobson thermodynamic, Ollivier-Ricci discrete, and variational with explicit verification of all four Lovelock conditions), and the Hamiltonian/Lagrangian formalism (with Ostrogradsky exclusion enforced by the Predictive Closure Invariant $\mathbb{T} = 0$) are all derived as $\pi_6$-projections. The complex structure of Hilbert space is derived non-circularly from the spectral decomposition of the Schreier graph Laplacian $\Delta_\Gamma$, and Noether's theorem is applied via $\Gamma$-convergence continuum limit of the discrete group $G = S_4 \times \mathbb{Z}_2^3$. The dynamics is exact and finite-dimensional: $64$ states on the coset space $G/H$, with eigenvalues $\lambda_k = 2k$ and multiplicities $\binom{6}{k}$, providing a closed-form spectral solution at all scales including the intermediate (Planck) regime. The built-in UV cutoff $E_{\max} = 1536 = |G| \times |\mathbb{Z}_2^3|$ eliminates divergences structurally. A QM–GR regime comparison table (9 rows) exhibits how the same OCF objects project differently at $\sigma \approx 0$ (quantum) and $\sigma \gg 0$ (gravitational). The document resolves three long-standing open problems: (i) black-hole information preservation (unitarity of $\Delta_\Gamma$ + Page's theorem forces the Page curve), (ii) the cosmological singularity (Fisher–Rao metric degeneration near the Source — an artifact of the manifold representation, not a physical singularity), and (iii) intermediate-regime (Planck-scale) dynamics (exact closed-form 64-mode spectral solution, no terra incognita). The dark sector emerges as $\pi_5$-structure inaccessible to $\pi_6$-observers, with the FORBIDDEN graph decomposing as $13 = 7 + 2 + 4$ (cold/warm/transitional dark matter analogs); the cosmological constant is derived from the structural resolution limit $\delta_{\mathrm{rel}} = 1/64$, resolving the 120-order QFT discrepancy without regularization. A fully hand-computable 8-state toy model ($Q_3$ hypercube) makes the projection operator, Laplacian spectrum, Born rule mechanism, and Schrödinger dynamics explicit and independently verifiable. All 53 formal results (28 theorems, 17 propositions, 6 definitions, 1 lemma, 1 corollary) follow a strict dual-track methodology — ontological (top-down from axiom) $\cap$ epistemological (bottom-up recovery of standard physics) — with mandatory convergence across 137 track tags. The framework is metrologically decoupled: all OCF equations use dimensionless ratios, invariant under redefinition of SI base units. Natural SI anchoring is provided for $\hbar$, $c$, $G_N$, $k_B$, $\Lambda$, and $E_{\max}$. No physical law is assumed; every equation is derived. No new axiom is introduced beyond Axiom M. This is a standalone publication readable after Document 0M alone.