Entropy–Order Balance V: The Strong-Field Regime, Singularity Resolution, and Black Hole Information

Any finite-resolution description of a physical system incurs two costs: an information cost, quantifying microscopic complexity via relative entropy, and an ordering cost, quantifying geometric or structural organization. The Entropy-Order Balance (EOB) framework postulates that realized physical laws correspond to Lyapunov-stable equilibria of the total cost functional. Parts I-II derived gravity and quantum mechanics; Part III derived local quantum field theory and theory-space stability; Part IV derived the Standard Model. All previous parts assumed the sub-curvature regime (l_s << L_c). This paper removes that restriction, extending the framework into the strong-field regime (l_s ~ L_c) to resolve the deepest paradoxes of quantum gravity.

Starting from the seven inputs established in Parts I-III — five operational axioms, Lyapunov stability, and dilation covariance, with no new axioms introduced — the paper derives:

• A universal F_s divergence mechanism that assigns infinite information cost to curvature singularities, forcing a resolution-dependent effective geometry with strictly bounded curvature.

• The absolute exclusion of singularities and the existence of an effective bounce inside black holes at r_min ~ (GM l_P^2)^(1/3) > 0, rigorously extended to Schwarzschild, Kerr, and Reissner-Nordstrom spacetimes.

• The exact Bekenstein-Hawking entropy S_BH = A/(4 l_P^2) as the total information cost of Planck-scale horizon windows, and the Generalized Second Law (dS_gen/dt >= 0) derived directly from the Data Processing Inequality (DPI).

• The Page curve for evaporating black holes S_rad(t) = min{S_thermal(t), S_BH(t)} + O(1), and the dissolution of the AMPS firewall paradox via the algebraic exclusion of trans-Planckian modes.

• A cosmological bounce replacing the Big Bang singularity at a_min > 0, yielding testable low-l power suppression in the CMB, alongside the derivation of the Past Hypothesis as a conditional consequence of theory-space stability.

An ancillary Python script provides independently reproducible numerical predictions for bounce radii, quasi-periodic ringdown echo time delays (e.g., Δt ≈ 36 ms for a 30 solar-mass black hole), Page times, and Kerr/RN scaling verification. The total input count remains seven; the resolution of singularities and the Page curve are outputs.

Parts I-IV (companion papers) derive the axiomatic framework, quantum mechanics, general relativity, local quantum field theory, and the Standard Model architecture from which these strong-field results follow.