If Spatiality is necessarily instantiated ([§7.3](<7.3 Space.md>)), a state of zero relational extension is a state in which Spatiality fails to be instantiated at all. By the *Triconditional Cross-Entailment* established in [§2.6.7](<2.6.7 Triconditional Cross-Entailment.md>) (the result that the three roles Temporality, Spatiality, and Physicality mutually require each other), the absence of any one of them is the absence of all three. A condition of permanent zero extension is therefore absolute nothingness under cosmological description: not an extreme configuration the universe might occupy, but the point at which the structure for any configuration is absent. It is not a condition the universe can persist in. A stable Singularity (maximum density, zero extension) is therefore modally precluded.

The prediction from logic alone is that total compression cannot persist as a terminal state. The argument does not preclude *asymptotic approach* to such a state; it precludes *permanent arrival*. The constraint depends on Modal Invariance ([§2.1.7](<2.1.7 Two Readings.md>)), the requirement that the structural conditions of Nature hold across all configurations the universe can occupy, not merely the ones the current cycle has visited: if invariance fails, the preclusion fails with it. And though General Relativity's "singularity" is not literally a point at zero extension but a boundary at which the manifold's description breaks down, the modal constraint applies to the *outcome* of the approach: wherever the approach happens, at a cosmological collapse (should the universe ever recontract) or at a black hole's interior.

The prediction, then, is that no physical framework can produce a stable zero-extension state. Physics reports exactly that instability, not as six independent confirmations but as several directions that sort into three kinds of pressure ([§7.6.2](<7.6.2 What is Required.md>) Requirement), with the mathematical breakdown the explanandum itself rather than a separate witness.

*The mathematics breaks down.* The mathematics already shows it. Newton's law for the gravitational force between two masses is

$$
F = \frac{G\, m_1 m_2}{r^2}
$$

As the separation r shrinks toward zero, the denominator vanishes and the force F runs to infinity. That is not a physical prediction: it is the formula reporting its own breakdown. At zero separation, the quantity the formula was written to compute is not defined. General Relativity delivers the same verdict in sharper form. Curvature invariants such as the *Kretschmann scalar* (a coordinate-invariant measure of tidal curvature, formed by contracting the Riemann curvature tensor with itself, which detects real curvature near a black hole even where the simpler Ricci scalar vanishes)

$$
K = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} = \frac{48\, G^2 M^2}{c^4\, r^6}
$$

also diverge as r \to 0, and geodesics (the curves a freely falling test particle traces through spacetime, the relativistic generalization of straight-line motion) cannot be continued past the singularity (Misner, C. W., K. S. Thorne, and J. A. Wheeler, 1973); (Hawking, S. W., and G. F. R. Ellis, 1973). The mathematical description does not produce a state there. It fails to produce one. Divergence and geodesic incompleteness are the signatures, within the mathematics, of the modal constraint.

*Why nothing classical halts the collapse.* The same formula, read dynamically, shows why no classical description supplies a stopping point short of its own breakdown. Gravity exists between any two masses separated by any nonzero distance: the Newton formula F = Gm_1m_2/r^2 is defined and nonzero for every r > 0. As long as any separation remains, there is a distance across which gravity can act, so collapse progresses, structure compresses, and accumulated potential energy deepens *precisely because* r > 0. The point is about the classical description, not a full account of black-hole dynamics (which accretion, merger, and the general-relativistic horizon govern, and which dark-energy-driven expansion eventually isolates rather than feeds): the classical law has no last step, and runs the separation down toward the zero at which it fails.

What classical physics cannot describe is the terminus of that compression. As r approaches zero, the descriptions break down, not because a physical event 'at r = 0' has been observed, but because the mathematical structure they rely on no longer holds. The classical descriptions' failure at the boundary is the empirical signature of the modal constraint: a stable zero-extension state is not anywhere the universe can end up. What replaces the failed description (bounce, repulsion, branching, conformal rescaling) is the empirical question [§7.6.2](<7.6.2 What is Required.md>) surveys. The modal argument entails the failure of any classical terminus; it does not pick the mechanism.

*A falsifiable consequence.* The structural argument generates pressure toward information preservation across the boundary, not a strict entailment from the modal Triconditions. Distinction requires Spatiality; permanent destruction of all information at a locus would require a persisting region in which no distinction is available, which under local description is the structural condition the modal preclusion forbids. The structural step here (that permanent loss of all distinguishability *is* permanent loss of Spatiality at that locus) is a structural expectation rather than a logical equivalence; a defender of information destruction could in principle locate the loss in a regime that does not register as zero-extension under the relevant description. The prediction, scoped accordingly, is that information must be preserved as a structural expectation downstream of the modal preclusion, not as a deductive consequence of it.

The *black hole information paradox* (the question of whether information that falls into a black hole is destroyed or preserved) has divided physics for fifty years. Hawking -(Hawking, S. W., 1976) argued it is destroyed. Quantum mechanics, whose laws forbid the destruction of information (only its scrambling), says it must be preserved. Recent work on the *Page curve*, the specific entropy trajectory an evaporating black hole has to follow if information is in fact preserved, in which the entropy of the outgoing radiation rises through the first half of evaporation, peaks at the "Page time," then falls back to zero as the hole finishes evaporating (Page, D. N., 1993), together with the *quantum extremal surface* prescription (Penington, G., 2020); (Almheiri, A., T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, 2020), a semiclassical method for locating the entanglement boundary whose area reproduces the Page curve, has moved physical consensus toward preservation. The structural account does not adjudicate the mechanism. The structural expectation it generates, downstream of the modal preclusion rather than as a strict deductive consequence of it, is that information survives: total loss of all distinguishability at a locus would coincide with absolute nothingness there, and absolute nothingness is not anywhere a universe can end up. A philosophical argument derived from the meaning of *Change* still yields a concrete, falsifiable expectation about black holes (abductive in status, not a deductive consequence of the modal core). If it fails, the argument's abductive reach into this regime fails with it, while the strictly modal Singularity Preclusion result stands untouched.


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← [7.6.2 What is Required](<7.6.2 What is Required.md>) | [7.6.4 What is Confirmed](<7.6.4 What is Confirmed.md>) →
