The Boundary

3.1 — Why not bounce

You may have read about bouncing universes — black hole interiors that reach extreme density, halt, reverse, and produce an expanding region on the other side. The literature includes torsion-driven bounce solutions from the Einstein-Cartan extensions of general relativity (Popławski 2010, 2016; Lucat and Prokopec 2017; and others in the same lineage).

In a dynamical bounce, the scale factor reaches a minimum at some time t₀. The metric is continuous through t₀. There is a single time coordinate running from before to after. Memory of the pre-bounce phase is carried dynamically into the post-bounce phase. The bounce is something that happens.

The architecture offers a structural alternative. The cosmological identity introduced in AP04 is not a dynamical event. It is a geometric relation between two descriptions of one structure. No single time coordinate runs through both. The two patches are joined at a boundary, not threaded by a worldline.

Two descriptions of one structure. The junction formulation makes this precise.

3.2 — The junction conditions

Let M⁻ denote the collapsing-side manifold patch. Let M⁺ denote the expanding-side patch. Let Σ denote the completion boundary — the hypersurface where the classical description of M⁻ reaches its limit. Curvature invariants reach a completion threshold. No further accessible distinctions in the record algebra. The classical description hands off to Σ.

On Σ, two conditions hold.

Condition J1 (Induced metric continuity).

[γ_ab] = 0 (3.1)

The induced metric on Σ is continuous. Both sides share the same boundary geometry. The 1:1 — the symmetric figure before the break — written as geometry.

Condition J2 (Extrinsic curvature mismatch).

[K_ab] = (ε / ℓ_Σ) · h_ab (3.2)

The way Σ curves into M⁻ and M⁺ respectively has a controlled, small, nonzero mismatch. The ε. The tensor h_ab encodes the asymmetry: anisotropy damping, particle content, torsion corrections from the ECSK completion. The length scale ℓ_Σ is the curvature scale of the completion surface itself; it carries the dimension that the dimensionless ratio ε needs to be multiplied by to match the dimension of the extrinsic curvature [K_ab] (which carries 1/length).

A dimensional note. K_ab carries dimension 1/length; ε is dimensionless (equal to α ≈ 1/137 in the AP06/AP24 identification); h_ab is dimensionless. The ratio ε/ℓ_Σ supplies the 1/length to balance the equation. Without ℓ_Σ the dimensions would not match — the length scale is not optional, it is structural.

Relation of J2’s h_ab to AP04’s L7 perturbation tensor. In AP04’s Conjecture L7 (g⁺ = (φ⁻¹)∗(g⁻) + ε · h, equation 3.3 below), h is a dimensionless perturbation tensor on M encoding the anisotropy difference, the information content, and the initial conditions of expansion. J2’s h_ab is the pullback/restriction of L7’s h to the completion surface Σ — the same dimensionless object, evaluated on the boundary. The two h’s are the same tensor; the appearance in J2 carries an additional ℓ_Σ because J2 is an equation about extrinsic curvature (a 1/length quantity) whereas L7 is an equation about the metric (a dimensionless quantity in the convention used).

You have just seen the axiom written as geometry. J1 is the 1:1. J2 is the ε. The boundary between collapse and expansion is the axiom made manifest in the mathematics of Israel junction conditions — a standard tool of general relativity (Israel 1966), extended to Einstein-Cartan-Sciama-Kibble gravity (Arkuszewski et al. 1975).

The standard junction-condition machinery is the literature framework named here as correspondence. The structural content — that the cosmological identity is a pair of conditions on a hypersurface, that J1 encodes the 1:1 and J2 encodes the ε — comes from the axiom and from AP04, not from the literature framework. The literature provides the formal vocabulary in which the structural fact is most naturally stated; it does not provide the structural fact.

The explicit form of h_ab is determined by the boundary-value problem. Whether the ECSK field equations on the Kantowski-Sachs interior admit a completion across Σ with [γ] = 0 and controlled [K] is the central open problem of the junction formulation. It is named: Open Problem B-JP. See section 0.6 for the full statement of the debt.

3.3 — Relation to AP04

AP04’s Conjecture L7 expresses the structural identity as a global metric relation:

g⁺ = (φ⁻¹)∗(g⁻) + ε · h (3.3)

A global diffeomorphism φ maps the manifold to itself; the relation holds on all of M.

The junction formulation is the boundary-data version of the same claim. J1 constrains the induced metric on Σ. J2 constrains the extrinsic curvature on Σ. Together they determine the relationship between M⁻ and M⁺ at Σ, without requiring a global diffeomorphism to be specified on all of M.

The junction formulation is more conservative. It commits only to boundary data on Σ. It is also more tractable. Israel junction conditions are well-established. ECSK extensions are well-established. The boundary-value problem is a standard kind of problem to attempt.

3.4 — Observational consequences

The bounce formulation predicts smooth evolution through Σ. Continuous correlations across the transition. Memory of pre-bounce perturbations carried dynamically into the post-bounce phase.

The junction formulation entails something different. The post-expansion perturbation spectrum is determined by boundary data on Σ — the extrinsic curvature mismatch (ε/ℓ_Σ) · h — not by dynamical evolution through Σ.

In principle, different observational signatures from bounce cosmologies.

If you could measure those signatures, you could distinguish between the two formulations. That is the point. The architecture commits to something specific enough to be falsified.

The specific empirical mapping from [K] = (ε/ℓ_Σ) · h on Σ to the post-expansion curvature perturbation spectrum P_ℛ(k) is itself an open derivation, held without claiming a separate debt (see section 0.7). Once B-JP is solved, the mapping becomes derivable; once the mapping is derived, KS-B.2 becomes operationally testable against the measured spectrum (CMB anisotropies, near-scale-invariance, tilt n_s ≈ 0.965, running, tensor-to-scalar ratio constraints).

3.5 — Open Problem B-JP

The central open problem of section 3 is named explicitly so the reader knows where the load-bearing derivation gap sits.

Open Problem B-JP. Do the Einstein-Cartan-Sciama-Kibble field equations on the Kantowski-Sachs interior admit a completion across the boundary Σ with conditions [γ_ab] = 0 (induced metric continuity, J1) and [K_ab] = (ε/ ℓ_Σ) · h_ab (controlled extrinsic curvature mismatch, J2), and what is the explicit form of h_ab and the relevant length scale ℓ_Σ determined by the boundary-value problem?

A solution to B-JP would: (a) prove the structural identity exists in the junction formulation (closing KS-B.3); (b) derive the explicit form of h_ab (opening the empirical mapping for KS-B.2); (c) make KS-B.0 operationally testable rather than staged (the universal generator condition becomes a check against derived initial conditions rather than against an undefined map).

The boundary-value problem is well-posed in standard terms. Israel junction conditions are well-established. ECSK extensions are well-established. The Kantowski-Sachs geometry is standard. What remains is the specific existence proof for the matched solution. The 420 Code invites contributions from any reader equipped to make the attempt.