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Appendix C — Multi-Point Cascade under Cork-Popping Geometry

Calculations by Grok (xAI). Directed by D. L. White and Claude (Anthropic).

This appendix quantifies the global-scale propagation of failure once the passive margins begin to fail. In the asymmetric cork-popping model, the heavy oceanic ring pulls outward on the buoyant continental blocks at the margins. Elastic stress waves from the first rupture trigger simultaneous localization at every major pre-existing weakness around the basin perimeter.

C.1 Elastic Stress-Wave Triggering

When the first margin segment fails and begins runaway descent, the sudden imbalance releases stored elastic strain energy. This generates stress waves that propagate through the lithosphere at compressional wave speeds of 5–8 km/s.

For a 40,000 km basin perimeter, the time for the wave to encircle the entire shell is:

t_wave = 40,000 km / 6 km/s ≈ 1.85 hours

(using a conservative average speed of 6 km/s).

The critical condition is the loading state of the shell at the moment of first failure. In a Pangea configuration with a single large ocean basin, the oceanic lithosphere is approximately uniform in age. Every passive-margin segment, every major transform fault, and every fracture zone along the entire perimeter is loaded to within a few percent of the same critical yield threshold (Appendix A, Section A.6).

A single 750 km segment releasing its stored slab-pull force produces a dynamic overstress pulse of order 0.1–1 MPa at distant sites (after geometric spreading and attenuation in a thin spherical shell). For a shell already at >99% of its yield, this is sufficient to push every major pre-existing weakness past threshold simultaneously.

The result is multi-point initiation at approximately 40–80 sites (one every 500–1,000 km, consistent with observed transform-fault and fracture-zone spacing on the modern ocean floor).

C.2 Parallel Failure: Why Sequential Doesn't Work

Under strictly sequential propagation — segment 1 fails, seeds segment 2, which seeds segment 3 — each segment must wait for its neighbor to complete the incubation before its own runaway begins. With an incubation time of approximately 1,250 years per segment (Appendix B) and 40–80 segments, sequential reorganization would require:

t_sequential = N × t_incubation = 60 × 1,250 ≈ 75,000 years

(using N = 60 as a central estimate). Even with cascade acceleration — each successive segment receiving stronger imposed velocity and incubating faster — the sequential timescale remains tens of thousands of years (Supplementary Material S4). This is too slow for the velocity profile derived in Section 2 of the main text.

The uniform-loading condition eliminates the sequential bottleneck. Because elastic waves trigger all segments within approximately 2 hours, every segment begins its ~1,250-year incubation simultaneously. The total basin reorganization timescale becomes the single-segment incubation time, not the cumulative sum.

C.3 Multi-Point Enhancement

With 40–80 segments incubating in parallel, each segment receives imposed velocity from both neighboring segments (plus the rising background tension from the global reorganization). In the imposed-velocity rheology (Appendix B), the incubation time scales approximately as:

t_incubation ∝ v_imposed^(−β)        where β ≈ 1.2–1.8

(from the grain-size reduction rate, which is proportional to strain rate, which is proportional to imposed velocity).

With velocity imposed from multiple directions, the effective v_imposed per segment is 2–5× higher from the outset than in the single-segment case. This shortens the incubation time by a factor of approximately 3–10 relative to the isolated-segment calculation, compressing the gradual phase to approximately 400–1,000 years.

The total basin reorganization timescale under multi-point parallel initiation is therefore:

t_total ≈ 850–1,550 years    (gradual phase + ~200 yr sudden phase)

C.4 Energy-Release Profile

The total gravitational potential energy released per segment during foundering is:

E_seg ≈ ΔM × g × L_seg × H_eff

where ΔM = 5.46 × 10⁶ kg/m² (excess mass from Appendix A), L_seg ≈ 750 km (segment length), and H_eff ≈ 100–200 km (effective descent depth during the initiation phase — slabs descend approximately one plate thickness before encountering significant mantle resistance, consistent with the L_eff assumption in Appendix A).

E_seg ≈ 5.46 × 10⁶ × 9.81 × 7.5 × 10⁵ × 1.5 × 10⁵ ≈ 6.0 × 10¹⁸ J

For ~60 segments, total energy release is approximately 3.6 × 10²⁰ J.

Present-day global plate-tectonic power dissipation is approximately 10¹² to 10¹³ W. The cascade distributes its total energy over approximately 1,000–1,500 years (3–5 × 10¹⁰ s), giving a time-averaged power of:

P_avg ≈ 3.6 × 10²⁰ / 4 × 10¹⁰ ≈ 10¹⁰ W

However, the energy release is not uniform. Because segments incubate in parallel and reach runaway within a relatively narrow window, the instantaneous power follows a bell-shaped curve:

• Ramp-up (first ~400–600 years): Segments are incubating. Energy release is modest — localization is occurring but runaway has not yet fired. Power rises slowly.

• Peak (~200-year window when most segments reach runaway simultaneously): Maximum number of segments are in catastrophic descent. Peak power reaches approximately 50–200× present-day global tectonic power (10¹⁴–10¹⁵ W).

• Decay (final ~400–600 years): Trailing segments complete their descent. Power falls symmetrically.

The peak is intense but brief, and the total energy is distributed over more than a thousand years rather than concentrated in an instant. This is comparable in total energy to known Phanerozoic boundary events and is consistent with planetary survivability — the planet retains its ocean, atmosphere, and surface conditions sufficient for biological survival.

C.5 Summary

The multi-point parallel cascade is the physically expected outcome for a uniformly loaded shell in a Pangea configuration. It provides the starting condition — simultaneous global margin failure — for the velocity profile constructed in Appendix D. This analysis assumes near-uniform pre-stress; realistic heterogeneity in margin and transform-fault strength (factors of 2–5) would lengthen the reorganization window modestly but would not revert the cascade to the sequential regime, because segments that do not trigger immediately are loaded progressively by their already-failing neighbors (see main text, Section 6, and Appendix E for sensitivity discussion).

All calculations use parameters from Appendix A (driving force, excess mass) and Appendix B (single-segment incubation time, runaway physics). No additional assumptions are introduced. Results are independently reproducible from the values given.

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© 2026 D. L. White. Licensed under CC BY-ND 4.0. https://creativecommons.org/licenses/by-nd/4.0/

AI Collaboration Disclosure: Calculations in this appendix were performed by Grok (xAI), with drafting and integration by Claude (Anthropic), under the direction of D. L. White.