Dating Capstone — Appendix B

Appendix B — CMB Perturbation Budget and Dynamo-Regime Discriminator

This appendix works the heat budget behind the reversal argument in Section 6. It computes the rate at which cold-slab thermal deficit is delivered to the mantle during the rapid phase of the event, confirms that the slab survives the descent still cold, and bounds the fraction of that deficit that would have to be prevented from reaching the core-mantle boundary for the perturbation to stay below the published threshold at which core-mantle-boundary forcing disturbs the dynamo. That threshold is stated in the dynamo literature in terms of reversal frequency; the appendix uses it as published, as an O(1) marker that the dynamo is disturbed. What Section 6 predicts from crossing it is a disturbed, suppressed field, not a tally of completed reversals. It does not compute a reversal count or model the dynamo response; those are named as tests, not results.

B.1 Inputs and Provenance

Every input is listed with its source. Nothing is invented for this appendix.

The velocity profile, τ₁ (scaled), peak velocity, and shell geometry are derived from the Trigger paper (chiefly the live velocity integral and Appendix F). The remaining quantities are standard constants.

B.2 Calculation 1 — Thermal-Deficit Supply Rate

This is a supply rate — the rate at which cold-slab thermal deficit enters the mantle — not a point estimate of the heat-flux anomaly at the CMB. The CMB-delivered fraction is not modeled here; it is bounded in B.4.

Peak surface velocity 12 km yr⁻¹ = 3.80 × 10⁻⁴ m s⁻¹.

Peak mass flux ≈ ρ · L_trench · h_slab · v = 3300 · 3×10⁷ · 8×10⁴ · 3.80×10⁻⁴ ≈ 3.0 × 10¹² kg s⁻¹.

Peak heat-sink rate = mass flux · Cp · ΔT ≈ 3.0 × 10¹⁸ W ≈ 3.0 × 10⁶ TW.

Ratio to modern CMB heat flow (12 TW): ≈ 2.5 × 10⁵ at peak. With velocity decaying as exp(−t/τ₁), τ₁ = 417.5 yr, the ratio is still ~9 × 10⁴ at t ≈ 420 yr and ~2–3 × 10⁴ at t ≈ 1000 yr. Using the retired 11.5 km yr⁻¹ peak instead of 12 changes the ratio by a few percent. Using τ₁ = 500 yr instead of 417.5 does not change the order of magnitude.

These are total delivery rates into the mantle, not the heat-flux anomaly at the CMB itself.

B.3 Calculation 2 — Slab Thermal Survival (√t check)

Diffusion length ℓ = √(κ · t_transit); κ = 10⁻⁶ m² s⁻¹; slab half-thickness ≈ 40 km.

In every case ℓ ≪ slab half-thickness. The boundary layer that grows during descent is a few hundred meters thick against a slab tens of kilometers thick; the core arrives cold and essentially the full deficit survives. (Insensitive to τ₁.)

B.4 Calculation 3 — Required-to-Fail Delivered Fraction

The published sensitivity threshold for shifting the dynamo's reversal behavior — the point at which CMB forcing disturbs it — is a factor of ~2 change in CMB heat flux (B.5). The required-to-fail fraction is the portion of the delivered deficit that would have to reach the CMB to be held to only that factor-of-2 effect.

Nominal: supply ratio 2.5 × 10⁵ → required-to-fail fraction = 2 / 2.5×10⁵ ≈ 8 × 10⁻⁶ (≈ 0.001%).

Stacked worst case — all four adversarial assumptions applied at once, each individually defensible and each chosen to hurt the result:

After the first three haircuts the supply ratio is still ≈ 4 × 10³. Required-to-fail fraction = 2 / 4000 = 5 × 10⁻⁴ (0.05%). More than 99.95% of the remaining deficit would have to be prevented from affecting the CMB.

Physical plausibility. Seismic tomography images subducted slab material reaching and stagnating at the base of the mantle. Cold slab material ponded on the core-mantle boundary depresses the local boundary temperature and steepens the local flux gradient directly — it does not need to be transported anywhere further. There is no known mechanism by which a slab's cold mass reaches the lowermost mantle yet leaves the boundary it rests on essentially unperturbed. A required-to-fail fraction below 0.05% is therefore physically implausible.

B.5 Published Sensitivity Threshold (literature result)

Numerical geodynamo studies (Glatzmaier and Roberts; Olson et al. 2010; Olson and Amit 2014; Christensen; and subsequent groups) consistently find that a factor-of-~2 change in mean CMB heat flux, or a lateral heterogeneity amplitude q* ≳ 0.5–1, is sufficient to move the dynamo between rarely-reversing and frequently-reversing regimes. Enhanced equatorial cooling is especially effective at promoting reversals; enhanced polar cooling stabilizes the axial dipole. These are results of the conventional simulations; they are not tunable inputs of the catastrophe model. The event's supply ratio exceeds this O(1) threshold by three to five orders of magnitude across every case examined above.

This appendix borrows that published threshold only as a marker that the dynamo is disturbed. It does not inherit the literature's reversal count, and Section 6 does not predict a tally of completed reversals from crossing it.

What this appendix does and does not establish. It establishes that the CMB perturbation is driven far past the threshold that published simulations associate with a disturbed dynamo — the threshold as the literature states it. What Section 6 reads from crossing that threshold is a suppressed, restless low-dipole field recorded by a fast-freezing surface, not a count of completed reversals; whether the forced dynamo actually produces that field is the geodynamo test (Section 9), not a result claimed here. This appendix does not compute a reversal count, and it does not claim the dynamo responds cleanly rather than chaotically. The forcing is supplied and bounded; the response is handed off.

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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: Drafting and calculations by Claude (Anthropic), with adversarial review by Grok (xAI), under the direction of D. L. White. Neither AI system endorses all conclusions as settled.