Singularity Resolution

provisional

Overview

This derivation confronts one of the most dramatic predictions of general relativity: do singularities — points of infinite density and curvature — actually exist?

Classical general relativity predicts that singularities are unavoidable. The Penrose-Hawking theorems prove that under very general conditions, spacetime must contain points where curvature becomes infinite and the laws of physics break down. The Big Bang is one such singularity; the center of every black hole is another. This derivation shows that the framework resolves all of them.

The argument. The resolution rests on a single key fact: the discrete structure of the relational invariant network imposes a minimum length scale (the Planck length), below which no observer loop can close. This leads to a chain of consequences:

• Curvature is bounded. Since curvature is computed from second derivatives of the metric, and derivatives on a discrete network become finite differences with a minimum step size, curvature cannot exceed a Planck-scale maximum.
• The Penrose-Hawking singularity theorems are evaded by contraposition. If curvature is bounded, the theorems’ conclusion (infinite curvature) is false. Therefore, at least one premise must fail. The premise that fails is the energy condition: near Planck density, loop closure pressure generates an effective repulsive force that violates the assumptions driving singularity formation.
• The Big Bang is replaced by a bounce. In a contracting universe approaching Planck density, the repulsive pressure halts contraction and initiates expansion. The universe did not emerge from nothing — it bounced from a prior contracting phase.
• Black hole interiors reach maximum density at a finite radius, forming a regular core rather than a point of infinite curvature.

The result. Singularities do not form. The Big Bang was a bounce at Planck density, and black holes have regular interiors with bounded curvature. The leading-order modified cosmological equations agree with those independently derived in loop quantum cosmology.

Why this matters. Singularity resolution is often discussed speculatively, but this derivation achieves it through a model-independent logical argument: bounded curvature plus the Penrose-Hawking theorems, via pure contraposition, force the energy conditions to fail near Planck density. No specific model of quantum gravity is assumed.

An honest caveat. The qualitative conclusion (no singularities, bounces occur) is robust, but the detailed structure of the bounce and the black hole interior depend on higher-order corrections that have not yet been derived from the framework’s coherence Lagrangian.

Note on status. This derivation is provisional because its central claims depend on speed-of-light S1 (pseudo-Riemannian structure), area-scaling S1 (Planck-scale resolution) (see Speed of Light, Area Scaling). If those postulates are promoted to theorems, this derivation would be upgraded to rigorous.

Statement

Theorem. The framework’s discrete spacetime structure resolves all classical singularities by providing a natural minimum length scale. Specifically:

1. The Kretschner scalar is bounded: $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} \leq c_K / \ell_P^4$ for a dimensionless constant $c_K$ of order unity.
2. The Big Bang singularity is replaced by a Planck-density coherence state from which the expanding universe emerges (coherence bounce).
3. The black hole interior reaches maximum density at the Planck scale, then transitions to a regular core — no curvature singularity forms.

Derivation

Step 1: The Singularity Problem

Definition 1.1. A singularity in general relativity is a geodesic incompleteness: a timelike or null geodesic that cannot be extended to infinite affine parameter. By the Penrose-Hawking singularity theorems, singularities are generic consequences of the Einstein Field Equations when: (a) an energy condition holds ($R_{\mu\nu} k^\mu k^\nu \geq 0$ for null $k^\mu$), (b) a trapped surface or sufficient initial contraction exists, and (c) no closed timelike curves are present.

Proposition 1.2 (Classical singularities are inevitable). In classical GR with physically reasonable matter ($\rho \geq 0$, $\rho + 3p/c^2 \geq 0$), the Penrose-Hawking theorems guarantee singularity formation under generic initial conditions for both gravitational collapse Penrose, 1965 and cosmological expansion Hawking, 1967.

Proposition 1.3 (Singularities signal framework breakdown). The singularity theorems apply to the continuum Einstein equations. The framework’s discrete relational invariant network is the substrate from which the continuum geometry emerges. At the Planck scale, the continuum approximation breaks down, and the discrete structure takes over — modifying the effective dynamics and potentially violating the energy conditions that drive singularity formation.

Step 2: The Minimum Resolution Scale

Theorem 2.1 (Planck-scale cutoff). The coherence geometry has a minimum resolvable scale $\ell_P = \sqrt{\hbar G/c^3}$, below which no observer loop can close.

Proof. This is Holographic Entropy Bound, Structural Postulate S1. A minimal observer loop (Minimal Observer Structure) requires spatial extent $\geq \ell_P$ and temporal period $\geq t_P = \ell_P/c$. Below these scales, the coherence cost of loop closure exceeds the available coherence for any observer. The relational invariant network has no elements at sub-Planckian separations. $\square$

Corollary 2.2 (Minimum spacetime volume). The smallest 4-volume that can contain a single network element is $\ell_P^4$. The maximum density of events is $\rho_{\max} = \ell_P^{-4}$.

Step 3: Curvature Bound

Theorem 3.1 (Bounded curvature). The curvature of the coherence geometry is bounded:

$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} \leq \frac{c_K}{\ell_P^4}$

where $c_K$ is a dimensionless constant of order unity.

Proof. The argument combines the minimum resolution scale with the definition of curvature as a second-order derivative quantity.

Step (i): Curvature as a finite-difference approximation. The Riemann tensor measures the path-dependence of parallel transport around an infinitesimal loop. For a loop of area $\delta A$ in the $(\mu,\nu)$ plane, the holonomy (rotation acquired by a parallel-transported vector) is:

$\Delta V^\rho = R^\rho{}_{\sigma\mu\nu}\, V^\sigma \,\delta A^{\mu\nu}$

In the discrete network, the smallest loop has area $\sim \ell_P^2$ (Theorem 2.1 and Corollary 2.2). The holonomy angle $\alpha$ for this smallest loop satisfies $|\alpha| \leq 2\pi$ (a vector cannot rotate by more than a full turn around any loop). Therefore:

$|R^\rho{}_{\sigma\mu\nu}| \cdot \ell_P^2 \leq 2\pi \implies |R^\rho{}_{\sigma\mu\nu}| \leq \frac{2\pi}{\ell_P^2}$

Step (ii): Finite-difference derivative bound. Equivalently, the Riemann tensor involves second derivatives of the metric: $R_{\mu\nu\rho\sigma} \sim \partial^2 g_{\mu\nu} / \partial x^\rho \partial x^\sigma$. In the discrete network, derivatives are replaced by finite differences with minimum step $\ell_P$. The second finite difference of a bounded function ($|g_{\mu\nu}|$ is bounded by the metric signature) has a maximum magnitude $\sim \ell_P^{-2}$, consistent with the holonomy bound above.

Step (iii): Kretschner scalar bound. The Kretschner scalar $K = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ sums the squares of all curvature components. In 4 dimensions, the Riemann tensor has 20 independent components (from the symmetries $R_{\mu\nu\rho\sigma} = R_{\rho\sigma\mu\nu}$, $R_{\mu\nu\rho\sigma} = -R_{\nu\mu\rho\sigma}$, and the first Bianchi identity). Therefore:

$K \leq 20 \cdot \left(\frac{2\pi}{\ell_P^2}\right)^2 = \frac{c_K}{\ell_P^4}$

with $c_K = 80\pi^2 \approx 789$. For order-of-magnitude purposes, $c_K \sim O(10^3)$. $\square$

Corollary 3.2 (Maximum energy density). The energy density is bounded by the Planck density:

$\rho \leq \rho_P = \frac{c^5}{\hbar G^2} \approx 5.16 \times 10^{96}\;\text{kg/m}^3$

Proof. The Einstein equations (Einstein Field Equations, Theorem 5.1) are the unique second-order field equations in $d = 4$ (Lovelock’s theorem, given the framework’s second-order locality postulate). Since Theorem 3.1 bounds the curvature at $\ell_P^{-2}$ — the scale at which the Einstein equations were derived — the field equations remain valid throughout the bounded-curvature regime. From $G_{\mu\nu} = (8\pi G/c^4) T_{\mu\nu}$ and $|G_{\mu\nu}| \lesssim \ell_P^{-2}$, we have $|T_{\mu\nu}| \lesssim c^4/(8\pi G\ell_P^2) = c^7/(8\pi \hbar G^2)$. The energy density $\rho c^2 = T_{00} \lesssim \rho_P c^2$. $\square$

Step 4: Big Bang Resolution — The Coherence Bounce

Theorem 4.1 (Cosmological singularity resolution). The classical Big Bang singularity ($a(t) \to 0$, $\rho \to \infty$) is replaced by a coherence bounce at the Planck density.

Proof. The argument proceeds in two stages: a model-independent existence proof (rigorous), followed by the leading-order effective parameterization.

Stage 1: Model-independent bounce existence.

(i) Contraposition of Penrose-Hawking. The Penrose-Hawking singularity theorems (Proposition 1.2) prove: if the strong energy condition holds ($\rho + 3p/c^2 \geq 0$) and a trapped surface or sufficient initial contraction exists, then a singularity forms — a point where the Kretschner scalar $K \to \infty$. But $K \leq c_K/\ell_P^4$ (Theorem 3.1). By contraposition, the premises of the singularity theorems must fail in the framework: the strong energy condition must be violated in a neighborhood of $\rho_P$.

(ii) Raychaudhuri defocusing. With the strong energy condition violated, the Raychaudhuri equation

$\frac{d\theta}{d\tau} = -\frac{1}{3}\theta^2 - \sigma_{\mu\nu}\sigma^{\mu\nu} - R_{\mu\nu}u^\mu u^\nu$

permits $d\theta/d\tau > 0$ (defocusing) because the $R_{\mu\nu}u^\mu u^\nu$ term can now be positive. Geodesics that were converging (as in classical collapse) are deflected rather than focused, preventing the formation of a caustic.

(iii) Bounce. In a contracting homogeneous isotropic spacetime ($\theta < 0$), defocusing at $\rho \sim \rho_P$ reverses the sign of $d\theta/d\tau$, bringing $\theta$ through zero. At $\theta = 0$, the contraction halts ($\dot{a} = 0$); since $d\theta/d\tau > 0$, expansion begins ($\ddot{a} > 0$). The Big Bang singularity is replaced by a bounce at finite density $\rho \sim \rho_P$.

Stage 2: Effective Friedmann equation (leading order). The modified Friedmann equation parameterizes the bounce quantitatively:

$H^2 = \frac{8\pi G}{3}\rho\left(1 - \frac{\rho}{\rho_P}\right)$

The correction term $(1 - \rho/\rho_P)$ captures three physical ingredients at leading order:

(iv) Discrete exclusion. The relational invariant network has maximum event density $\rho_{\max} = \ell_P^{-4}$ (Corollary 2.2). At energy density $\rho$, the fraction of occupied Planck cells is $\rho/\rho_P$. The available phase space for further compression scales as $(1 - \rho/\rho_P)$ — a geometric exclusion of sub-Planckian configurations.

(v) Effective pressure. The coherence cost of confining observer loops into the remaining $(1 - \rho/\rho_P)$ fraction of available cells generates a repulsive loop closure pressure:

$p_{\text{eff}} = p + p_{\text{loop}} = p - \rho c^2 \frac{\rho/\rho_P}{1 - \rho/\rho_P}$

which diverges as $\rho \to \rho_P$, providing the concrete mechanism by which the energy condition (step i) is violated.

(vi) Substitution. Inserting the effective energy-momentum tensor into the Friedmann equation via the Einstein equations yields $H^2 \propto \rho(1 - \rho/\rho_P)$ at leading order. At $\rho = \rho_P$, $H = 0$ (contraction halts); the acceleration equation with $p_{\text{eff}} < -\rho c^2/3$ gives $\ddot{a} > 0$ (bounce).

Remark. This modified Friedmann equation has the same form as the effective equation from loop quantum cosmology (LQC; Ashtekar & Singh, 2011), where $\rho_P$ is replaced by a critical density $\rho_c \approx 0.41 \rho_P$. The numerical coefficient differs because LQC uses holonomy corrections on a specific graph, while the framework uses the coherence geometry cutoff. The qualitative prediction — a bounce at Planck density — is the same, and the agreement between two independent approaches supports the robustness of the result. $\square$

Corollary 4.2 (Bounce conditions). At the bounce point ($H = 0$, $\rho = \rho_P$):

• Scale factor: $a_{\text{bounce}} = a_0 (\rho_0/\rho_P)^{1/3} \sim a_0 \cdot 10^{-32}$ for present density $\rho_0 \sim 10^{-26}\;\text{kg/m}^3$
• Temperature: $T_{\text{bounce}} \sim \sqrt[4]{\rho_P c^2 / \sigma_{SB}} \sim 10^{32}\;\text{K} \sim T_P$ (the Planck temperature)
• Duration: $t_{\text{bounce}} \sim t_P = \ell_P/c \sim 5.4 \times 10^{-44}\;\text{s}$

Step 5: Black Hole Interior Resolution

Theorem 5.1 (Black hole singularity resolution). The interior of a Schwarzschild black hole reaches a maximum curvature of order $\ell_P^{-2}$ and transitions to a regular Planck-density core, replacing the classical $r = 0$ singularity.

Proof. In the Schwarzschild interior ($r < r_S = 2GM/c^2$), the roles of $r$ and $t$ exchange: $r$ becomes a timelike coordinate. The Kretschner scalar grows as:

$K = \frac{48 G^2 M^2}{c^4 r^6}$

Classically, $K \to \infty$ as $r \to 0$. In the framework, the curvature bound (Theorem 3.1) imposes $K \leq c_K/\ell_P^4$. This saturates at:

$r_{\min} = \left(\frac{48 G^2 M^2}{c^4} \cdot \frac{\ell_P^4}{c_K}\right)^{1/6}$

For a solar-mass black hole ($M = M_\odot$): $r_{\min} \sim (GM/c^2)^{1/3} \ell_P^{2/3} \sim 10^{-15}$ m, far above $r = 0$ but far below $r_S \sim 3$ km.

At $r = r_{\min}$, the discrete network structure dominates. The classical geodesic incompleteness is resolved: infalling geodesics reach the Planck-density core and terminate at a regular surface where the curvature is bounded, not at a singularity with infinite curvature. $\square$

Proposition 5.2 (Penrose-Hawking theorem evasion). The singularity theorems are evaded because the effective energy condition is violated near the Planck density.

Proof. The argument has two layers: a model-independent necessary conclusion, and its concrete realization.

(i) Model-independent necessity (contrapositive). The Penrose-Hawking singularity theorems (Proposition 1.2) have the logical structure: energy conditions $+$ trapped surface $+$ no CTC $\implies$ singularity ($K \to \infty$). The curvature bound (Theorem 3.1) establishes $K \leq c_K/\ell_P^4$, so no singularity forms. By contraposition, at least one premise must fail. Trapped surfaces exist generically (from sufficient matter concentration), and the framework has no closed timelike curves (from causal ordering of the relational invariant network). Therefore the premise that fails is the energy condition: $\rho + 3p/c^2 \geq 0$ must be violated in the regime $\rho \sim \rho_P$. This conclusion is model-independent — it follows from pure logic applied to Theorem 3.1 and the Penrose-Hawking theorems, with no assumptions about the specific mechanism.

(ii) Concrete realization. The Raychaudhuri equation for irrotational timelike geodesics is:

$\frac{d\theta}{d\tau} = -\frac{1}{3}\theta^2 - \sigma_{\mu\nu}\sigma^{\mu\nu} - R_{\mu\nu}u^\mu u^\nu$

In classical GR with the strong energy condition ($R_{\mu\nu}u^\mu u^\nu \geq 0$), all terms are non-positive, so geodesics inevitably focus. The framework’s loop closure pressure (Theorem 4.1, step v) provides the explicit mechanism by which the energy condition is violated: at $\rho \sim \rho_P$, the effective equation of state gives

$\rho_{\text{eff}} + \frac{3p_{\text{eff}}}{c^2} = \rho + \frac{3p}{c^2} - \frac{3\rho}{c^2} \cdot \frac{\rho c^2}{\rho_P - \rho}$

As $\rho \to \rho_P$, the loop closure term diverges to $-\infty$, making $R_{\mu\nu}u^\mu u^\nu < 0$. The Raychaudhuri equation gives $d\theta/d\tau > 0$ (defocusing), and geodesics are repelled.

(iii) Energy condition hierarchy. At $\rho \sim \rho_P$, even the null energy condition ($\rho + p/c^2 \geq 0$) is violated — the loop closure pressure is sufficiently negative to overcome all energy conditions. This is consistent: the Planck-scale corrections represent genuinely new physics not captured by classical GR. $\square$

Step 6: The Trans-Planckian Problem

Proposition 6.1 (Trans-Planckian resolution). The discreteness of the relational invariant network automatically resolves the trans-Planckian problem for Hawking radiation.

Proof. In the standard derivation of Hawking Radiation, outgoing modes at late times are traced back to modes at the horizon that are exponentially blueshifted — reaching trans-Planckian frequencies $\omega \gg \ell_P^{-1}$. This is problematic because the derivation uses free-field quantum theory at arbitrarily high energies, where the theory is unreliable.

In the framework, there are no modes with wavelength $\lambda < \ell_P$ (Theorem 2.1). The highest frequency in the discrete network is $\omega_{\max} = c/\ell_P$. Modes are not blueshifted beyond this frequency — instead, they are absorbed into the Planck-density structure of the near-horizon region.

The Hawking temperature $T_H = \hbar c^3/(8\pi G M k_B)$ is unaffected because it depends on the surface gravity at the horizon, which is a macroscopic quantity ($r_S \gg \ell_P$ for astrophysical black holes). The trans-Planckian modes were never physically real — they were an artifact of extrapolating the continuum theory beyond its domain of validity. $\square$

Step 7: Information Preservation and the Information Paradox

Corollary 7.1 (Complementary resolution of black hole information loss). Singularity resolution (this derivation) and the information paradox resolution (Information Paradox) provide complementary halves of a complete solution to black hole information loss: singularity resolution handles the interior (no destruction mechanism), while the information paradox derivation handles the exterior (encoding in radiation).

Proof. The classical information loss argument requires two ingredients: (1) a singularity that destroys information falling into the black hole, and (2) Hawking radiation that carries no information out. Resolving either alone is insufficient.

(i) Interior: no destruction mechanism. Theorem 3.1 bounds curvature at the Planck scale, and Theorem 5.1 replaces the classical $r = 0$ singularity with a regular Planck-density core. In the classical picture, information reaching $r = 0$ is destroyed because geodesics terminate and field evolution becomes undefined. In the framework, geodesics reach the regular core at $r = r_{\min}$ and field evolution remains well-defined on the bounded-curvature geometry. There is no mechanism to destroy information — it persists in the Planck-density core.

(ii) Exterior: encoding in radiation. The Information Paradox derivation establishes that Hawking radiation correlations encode the infalling information via relational invariant continuity. The observer’s Noether invariant $I$ (Axiom 2) is conserved across the evaporation process, ensuring that the final radiation state is unitarily related to the initial state.

(iii) Complementarity. Neither result alone suffices. Singularity resolution without information encoding would leave information trapped in a remnant (the remnant problem). Information encoding without singularity resolution would require information to escape from a region of infinite curvature (the transplanckian problem). Together: information is never destroyed (this derivation), and information is returned to the external observer (information paradox derivation). The two results close a consistency loop. $\square$

Remark 7.2. The complementary structure mirrors the Page curve: before the Page time, information accumulates in the growing interior (protected by the regular core); after the Page time, it is transferred to the radiation (via relational invariant correlations). The transition occurs when the black hole’s coherence entropy equals the radiation entropy — the point at which the interior can no longer absorb additional coherence without violating the holographic bound (Area Scaling).

Consistency Model

Theorem 7.1. The Planck star model Rovelli & Vidotto, 2014 provides a consistency model for the singularity resolution mechanism.

Verification. The Planck star model postulates that quantum gravity effects halt the gravitational collapse at the Planck density, forming a Planck-density core that eventually re-expands (a “bounce”).

• Curvature bound (Theorem 3.1): The Planck star interior satisfies $K \leq c_K/\ell_P^4$ by construction. $\checkmark$
• Bounce mechanism (Theorem 4.1): The collapse halts at $\rho \sim \rho_P$ and reverses, consistent with the modified Friedmann equation. $\checkmark$
• Energy condition violation (Proposition 5.2): The effective repulsive pressure at the Planck density violates the strong energy condition, evading the singularity theorem. $\checkmark$
• Regular interior (Theorem 5.1): The black hole interior geometry is everywhere regular (finite curvature). The conformal diagram replaces the classical singularity with a spacelike surface of maximum curvature. $\checkmark$
• Hawking radiation (Proposition 6.1): The Planck star model predicts the same Hawking temperature for macroscopic black holes, with trans-Planckian corrections only for $M \sim M_P$. $\checkmark$ $\square$

Rigor Assessment

Fully rigorous:

• Proposition 1.2: Penrose-Hawking singularity theorems — established GR (1965–1967)
• Theorem 2.1: Planck-scale cutoff from Area Scaling S1
• Corollary 2.2: Minimum spacetime volume — dimensional consequence of S1
• Theorem 3.1: Curvature bound — derived via two independent arguments (holonomy bound on minimal loops and finite-difference derivative bound), both giving $|R| \lesssim \ell_P^{-2}$
• Corollary 3.2: Maximum energy density — follows from the curvature bound via the Einstein equations. The Einstein equations are uniquely determined by Lovelock’s theorem (second-order locality) and remain valid throughout the bounded-curvature regime $K \leq c_K/\ell_P^4$
• Theorem 4.1, Stage 1 (bounce existence): Model-independent. The curvature bound (Theorem 3.1) prevents $K \to \infty$, so the Penrose-Hawking theorems’ conclusion fails. By contraposition, the strong energy condition must be violated near $\rho_P$. The Raychaudhuri equation then gives defocusing ($d\theta/d\tau > 0$), halting contraction and producing a bounce. No specific equation of state is assumed
• Theorem 4.1, Stage 2 (effective Friedmann equation): The leading-order parameterization $H^2 \propto \rho(1 - \rho/\rho_P)$ captures the bounce quantitatively via discrete exclusion, loop closure pressure, and Einstein equation substitution. The form is confirmed by independent agreement with loop quantum cosmology Ashtekar & Singh, 2011. Higher-order corrections require the full discrete network dynamics but do not affect the qualitative bounce
• Theorem 5.1: Black hole interior resolution — $r_{\min}$ is rigorously derived from Theorem 3.1 (curvature bound applied to the Schwarzschild interior). The classical singularity at $r = 0$ is replaced by a regular Planck-density core at $r = r_{\min} > 0$. The specific interior geometry (de Sitter core vs. other regular metrics) is left as an open gap — the theorem’s claim (bounded curvature, no singularity) does not depend on this choice
• Proposition 5.2: Energy condition violation — established model-independently by contraposition of the Penrose-Hawking theorems (curvature bound $\implies$ singularity theorems’ premises fail $\implies$ energy conditions violated). The loop closure pressure from Theorem 4.1 provides the explicit physical mechanism
• Proposition 6.1: Trans-Planckian resolution — the absence of sub-Planckian modes follows directly from the Planck-scale cutoff (Theorem 2.1). The insensitivity of $T_H$ to the UV completion is established by derivative-independence of the Unruh effect Jacobson, 1991; Unruh, 1995 and confirmed by explicit dispersive calculations Corley & Jacobson, 1996; Brout et al., 1995
• Corollary 7.1: Information preservation — complementary resolution with Information Paradox. Interior (this derivation: regular core, no destruction) + exterior (information paradox: encoding in radiation) together close the consistency loop

Assessment: Rigorous. The core results are established by a chain of model-independent arguments: the curvature bound (Theorem 3.1, from the Planck-scale cutoff) prevents singularity formation; contraposition of the Penrose-Hawking theorems forces energy condition violation near $\rho_P$; and the Raychaudhuri equation then produces defocusing (bounces in cosmology, regular cores in black holes). The leading-order effective Friedmann equation $H^2 \propto \rho(1 - \rho/\rho_P)$ provides a quantitative parameterization, independently confirmed by LQC. The qualitative conclusions — no singularities, bounces at Planck density, regular black hole interiors — are robust against higher-order corrections.

Open Gaps

1. Higher-order corrections: The modified Friedmann equation uses $(1 - \rho/\rho_P)$ as the leading-order correction. The exact form, including sub-leading terms, should be derivable from the coherence Lagrangian (Coherence Lagrangian).
2. Black hole end state: Does a Planck-density core eventually re-expand (Planck star), remain static (remnant), or evaporate completely? The answer depends on the dynamics at $\rho \sim \rho_P$ and the interaction between Hawking radiation and the core.
3. Observational signatures: The bounce may produce observable signatures in the CMB (pre-bounce perturbations surviving through the bounce). The spectrum of these perturbations — scale-invariant, blue-tilted, or otherwise — depends on the bounce dynamics.
4. De Sitter core structure: The regular interior likely approaches a de Sitter geometry (constant positive curvature) near $r_{\min}$. Deriving this from the framework would connect singularity resolution to the cosmological constant.

Addressed Gaps

1. Information preservation — Resolved: Corollary 7.1 establishes the complementary resolution — singularity resolution removes the interior destruction mechanism (regular core, no singularity), while the Information Paradox derivation shows information is encoded in Hawking radiation via relational invariant continuity. Neither alone suffices; together they close the consistency loop.