Information-Theoretic Constraints on Black Hole Interiors: Layer Stripping, Monopole Convergence, and the No-Singularity Theorem

We derive six rigorous constraints on the fate of quantum information inside a Schwarzschild black hole using three premises from established physics: the Bekenstein entropy bound, the data processing inequality for completely positive trace-preserving (CPTP) maps, and the spectral theorem for iterated quantum channels. The derivation assumes no specific quantum gravity theory; the six statements follow from quantum information theory and the semiclassical Bekenstein bound alone.

The Six Statements.

(S1) Layer Stripping Theorem. As a quark falls toward the horizon and then inward, the shrinking Bekenstein capacity forces its internal degrees of freedom to be sequentially traced out in a specific order determined by their energy scales. Momentum is traced out first (its continuous spectrum hits the Bekenstein capacity immediately), followed by spin, then flavor (at r ~ 10⁻¹⁸ m), then color (at r ~ 10⁻¹⁵ m). By the time the matter reaches the critical radius, only the monopole (maximally mixed state) remains.

(S2) Critical Radius Theorem. The critical radius at which the total information content of a black hole saturates the Bekenstein capacity is r_c = ℓ_P √(S/π), where S is the total entropy of the infalling matter. For a stellar-mass black hole with S ≈ 3.6 × 10⁵⁷ nats, this gives r_c ≈ 0.55 μm, well above the Planck scale, in the deeply semiclassical regime where the Kretschmann scalar is ~ 10⁻⁹³ of the Planck curvature.

(S3) Monopole Convergence Theorem. Iterated partial traces of any initial quark state converge to the maximally mixed state on the residual Hilbert space. The convergence is exponential in the number of stripping steps, with rate set by the spectral gap of the CPTP map. The fixed point is unique and is reached at r = r_c.

(S4) No-Singularity Theorem. The classical singularity at r = 0 is unreachable by any infalling trajectory, because the effective information-theoretic dimension reaches zero at r = r_c > 0. Between r_c and r = 0, there are no physical degrees of freedom for a geodesic to carry. The singularity is excised by information theory alone, without invoking quantum gravity corrections to the metric.

(S5) Entropy Conservation. The total von Neumann entropy of the quark-plus-environment system is conserved throughout the interior trajectory. Each stripping step is a CPTP map that moves information from the quark into the environment without destroying it. The Bekenstein bound constrains where information can live, not whether it is preserved.

(S6) Microscopic Entropy Accounting. The Bekenstein-Hawking entropy S_BH = A/(4G) decomposes microscopically into two contributions: (i) the entropy of the particles making up the black hole, summed over their internal degrees of freedom and bounded by the Bekenstein capacity at r_s; and (ii) the correlation entropy between the particles and the horizon encoding, arising from the CPTP boundary map. The sum matches A/(4G) to the accuracy of the semiclassical approximation.

Cosmological and Particle Physics Consequences.

The framework has several consequences that follow from combining the six statements with cosmology and the Standard Model.

Timescale argument for the bounce mechanism. The expansion from the critical radius proceeds on a causal timescale t_bounce ~ r_c/c ~ 10⁻¹⁵ s for a stellar-mass black hole. Hawking evaporation, by contrast, takes t_evap ~ 10⁶⁷ years. The ratio t_bounce/t_evap ~ 10⁻⁸⁹ implies that less than one bit of Hawking radiation is emitted before the bounce occurs. Information is preserved not by subtle correlations in Hawking radiation but by the bounce mechanism.

De Sitter temperature problem for heat death. Hawking evaporation is thermodynamically forbidden for the final supermassive black hole in an accelerating universe because its Hawking temperature falls below the de Sitter temperature of the surrounding cosmological horizon. For M ~ 10¹¹ M_sun, T_Hawking ≈ 1.2 × 10⁻³⁰ K while T_de Sitter ≈ 2.7 × 10⁻³⁰ K. The standard heat death scenario requires black hole evaporation to complete; this calculation shows it cannot.

CMB fingerprint analysis. Four scenarios are identified under which a bounce origin of the current cosmological epoch would leave no detectable signature in the CMB: complete thermalization at the bounce, inflationary erasure of pre-bounce correlations, nonlinear oscillatory encoding, and reionization-coupled imprint. Each scenario is assessed for compatibility with Planck and SPT data.

K-sector fermion mass framework. The layer stripping order, combined with the three-sector structure of the K-functional (Paper 1), produces a Yukawa texture with adjacent-sector mixing singly suppressed and non-adjacent mixing doubly suppressed. The resulting CKM matrix reproduces |V_us| to 2.5%, |V_cb| to 3.9%, |V_ub| to 7.8%, |V_td| to 7.0%, |V_ts| to 7.4%, and the Jarlskog invariant J to 8%. All three PMNS neutrino mixing angles are reproduced because neutrinos couple only to K_rec, producing a democratic mass matrix whose large mixing is natural rather than tuned.

Zero-parameter Higgs VEV. With the dark matter coupling derived as y_DM = N_c √α_K = 3√0.082 = 0.860 (not fitted), the Higgs VEV computation yields v = 250.5 GeV against the observed 246.2 GeV (1.75% error). No free parameters enter.

Uniqueness of the gauge algebra. An exhaustive classification of rank-4 compact Lie algebra products shows that 𝔰𝔲(3) ⊕ 𝔰𝔲(2) ⊕ 𝔲(1) is the unique choice consistent with three structural constraints: three simple/abelian factors (one from each K-sector), one factor with a 3-dimensional complex fundamental, and total rank 4. This result is conditional on the three-sector structure of the K-functional, which is itself built into axiom (U5) of Paper 1 rather than derived from more primitive principles. The paper (Remark 9.X, new in v3) is explicit about this conditionality: given three K-sectors, the gauge algebra is uniquely determined; the three-sector count itself remains an axiom.

Laboratory Test.

The underlying K-functional framework predicts a 72% enhancement of the spectral linewidth decoherence rate for superconducting transmon qubits at 10 mK near-thermal populations, relative to the standard Lindblad value. This is a near-term laboratory test of the K-functional's off-equilibrium sector gradients, described in detail in Paper 4.

Scope and Assumptions.

The six statements rest on quantum information theory plus the Bekenstein bound. They are agnostic about the specific dynamics beyond r_c. The fermion mass framework, gauge algebra uniqueness, and Higgs VEV derivation additionally assume the three-sector K-functional structure of Paper 1. The timescale and CMB arguments additionally assume the bounce interpretation of the post-r_c evolution, which is motivated by the causal structure of the Schwarzschild interior (r is timelike, r < r_c is forbidden, the only timelike direction available is r > r_c).