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T04 — b-Values Shift With Depth: FINDINGS

Supplied working report; original verdicts not validated.

Review status: Needs review. This is an archival source note, not a passed proof.

FINDINGS.md

# T04 — b-Values Shift With Depth: FINDINGS

**Verdict: REFUTED.** In both subduction zones the b-value **increases** with depth in the
0–300 km range — the opposite of the hypothesis's prediction ("b should fall with depth",
mechanism: "higher confining pressure and temperature raise the stress drop for small events").
The result is statistically clear in Chile and direction-consistent (though individually
marginal) in Japan, and it survives every robustness check (Mc choice, time window,
declustering).

## Headline numbers (primary analysis: 1990–2025, Mc = max-curvature + 0.2 per bin, Aki MLE, 1000 bootstraps)

| Region | Bin (km) | n (total) | n (M≥Mc) | Mc | b | 95% CI | a |
|---|---|---|---|---|---|---|---|
| Japan | 0–30 | 8631 | 3841 | 4.55 | 0.966 | 0.939–0.996 | 7.98 |
| Japan | 30–70 | 13239 | 5180 | 4.55 | 1.083 | 1.057–1.110 | 8.64 |
| Japan | 70–150 | 2792 | 775 | 4.55 | 1.203 | 1.131–1.286 | 8.36 |
| Japan | 150–300 | 1103 | 239 | 4.45 | 1.070 | 0.940–1.228 | 7.14 |
| Chile | 0–30 | 14818 | 3124 | 4.35 | 0.895 | 0.868–0.928 | 7.39 |
| Chile | 30–70 | 12700 | 3230 | 4.45 | 0.954 | 0.927–0.982 | 7.75 |
| Chile | 70–150 | 15476 | 2717 | 4.45 | 1.090 | 1.054–1.130 | 8.29 |
| Chile | 150–300 | 6054 | 2116 | 4.35 | 1.335 | 1.274–1.391 | 9.13 |

Shallow(0–30) minus deep(150–300): **Japan −0.104** (deep b HIGHER; CIs overlap) and
**Chile −0.440** (deep b HIGHER; CIs non-overlapping). A decline anywhere was never observed.

## Support-rule comparison (rule pre-written in hypothesis.md, before fitting)

- SUPPORT required: b decline ≥ 0.15 shallow→deep with non-overlapping bootstrap CIs **in both
  regions**. → **Not met** (no decline in either region).
- NOT SUPPORTED rule ("flat or increasing b with depth in both"): also not the situation, because
  the increase in Chile is significant.
- **REFUTATION rule ("b increasing with depth")**: met — b increases with depth in both regions
  (Japan +0.10, Chile +0.44; Chile's CIs fully separated). Under the programme's rulebook this
  is a refutation of the pressure/temperature stress-drop mechanism.

## Robustness (post-hoc checks)

- **Mc choice** (`results/sensitivity.csv`): with every maximum-curvature-based Mc
  (mode+0.0/+0.2/+0.3/+0.4) the shallow→deep difference is positive in both regions:
  Japan +0.05…+0.29; Chile +0.27…+0.54 (CIs separated in most configurations).
  With fixed low Mc (2.5/3.0) Chile's sign flips slightly negative (−0.06) — this is a
  differential-completeness artifact: small (M2.5–4) deep events are under-reported relative to
  shallow ones, which mechanically depresses the fitted deep-bin b. The mode-based Mc corrects
  for exactly this, and the increase persists.
- **Time windows**: restricted to 2000–2025 or 2010–2025, the increase persists in both regions
  (Japan +0.05…+0.16; Chile +0.27…+0.54 with separated CIs).
- **Declustering** (Gardner–Knopoff, `results/declustered.csv`): absolute b-values drop
  (aftershock-rich shallow bins lose their small-event inflation) but the depth gradient is
  preserved or strengthened — Japan 0–30: 0.70 [0.65–0.75] → 150–300: 0.99 [0.86–1.15]
  (+0.29, separated); Chile 0–30: 0.74 [0.69–0.80] → 70–150: 0.90 [0.85–0.94] (separated),
  150–300: 0.87 [0.78–0.98].
- **Supplementary (post-hoc, outside pre-registered bins)**: Japan 300–700 km (Izu–Bonin deep
  slab, n=1744, b=0.896 [0.841–0.957]) has the LOWEST b of all Japan bins — consistent with the
  literature that the deepest slab has low b. It does not rescue the hypothesis: the depth
  profile is non-monotonic (0.97 → 1.08 → 1.20 → 1.07 → 0.90), and the pre-registered
  0–300 km comparison — where the claim is testable — shows an increase.

## Limitations

1. **Catalogue completeness at depth.** ComCat is a merged global bulletin: for Japan the
   smallest event is M2.7 and the FMD mode is ≈4.3 (small-event reporting is incomplete,
   especially pre-2010; JMA would be far more complete). Mc per bin (max-curvature+0.2)
   compensates, but Mc itself is uncertain; the fixed-Mc=2.5 result shows how sensitive the
   comparison is to this choice. The mode-based Mc is the defensible one, and under it the
   result is unambiguous.
2. **Mc method.** Max-curvature+0.2 is crude; a b-value-stability (McBS) estimator would refine
   per-bin Mc. The consistency across +0.0/+0.2/+0.3/+0.4 corrections suggests the verdict is
   not an Mc artifact.
3. **Subduction vs crustal mixing.** Shallow bins mix crustal and slab seismicity, while the
   mechanism is about confining pressure/temperature. The predominantly slab bins (30–300 km)
   still show the increase in both regions (Japan 30–70: 1.08, 70–150: 1.20; Chile 30–70: 0.95,
   70–150: 1.09, 150–300: 1.33), so slab-only behaviour cannot explain the refutation away.
4. **Bin boundaries.** Fixed at 0/30/70/150/300 by pre-registration; the Chile effect (±0.44)
   is far larger than any plausible boundary sensitivity.
5. **Declustering is post-hoc** and uses Gardner–Knopoff windows (crude but standard); it lowers
   absolute b but preserves or increases the gradient.
6. **Deepest-bin event counts.** Japan 150–300 km has only 239 fitting events (wide CI); the
   Japan result is "same direction, marginal" precisely because of this.
7. **No depth-error weighting**; old (pre-1995) depth determinations are less reliable, but
   restricting to 2000/2010+ changes nothing.

## What would change the conclusion

- A test with genuinely complete small-magnitude coverage at depth (JMA for Japan, CSN for
  Chile) finding a decline ≥ 0.15 with separated CIs in both regions would overturn this
  verdict. The fixed-Mc=2.5 sensitivity shows completeness treatment is decisive — but that
  regime is known-invalid for these catalogs.
- A slab-only, declustered, Mc-stability analysis of the 100–300 km band could sharpen the
  magnitudes of the effect; given every mode-based-Mc check agrees on direction, a reversal is
  unlikely.
- The hypothesis might survive only in the >300 km deepest slab (Japan there shows b=0.90),
  which would narrow the claim to "b declines in the deepest slab" — a different, much smaller
  claim than the published "b declines with depth", and still contradicted in the 0–300 km range
  that contains the overwhelming majority of subduction-zone seismicity.

*Exploratory test, not peer reviewed. Data, code, and full provenance in this folder.*

hypothesis.md

# T04 — b-Values Shift With Depth

**Source:** Incoprea Theory Vol. XXXII #08 — "b-Values Shift With Depth"
https://incoprea.com/research-library/original-theories-vol32-20261030/theories/bvalue-depth.html
Fetched 2026-10-30 (curl, full HTML saved to workspace tmp, text extracted).

## The claim (quoted)

> "The earthquake magnitude-frequency b-value declines with depth, so deep and shallow sequences cannot share a statistic."

## Mechanism (quoted)

> "Higher confining pressure and temperature raise the stress drop for small events."

## Prediction (quoted)

> "b should fall with depth."

## Refutation clause (quoted)

> "Constant b would remove the pressure effect."

## Pre-written support rule (fixed BEFORE fitting)

- **Depth bins** (fixed before fitting, adapted per region to data coverage and documented in the analysis): 0–30 km, 30–70 km, 70–150 km, 150–300 km. Bins with n < 100 (after Mc selection) are documented but flagged as unreliable.
- **Estimators**: Mc per bin via maximum-curvature + 0.2 correction (documented). b via Aki (1965) maximum-likelihood:
  `b = log10(e) / (mean(M) − (Mc − ΔM/2))`, ΔM = 0.1, using only M ≥ Mc events.
  95% CIs from 1000 bootstrap resamples (percentile). a-values also reported.
- **SUPPORT** — b declines with depth by at least 0.15 between the shallowest and deepest usable bins, with non-overlapping bootstrap 95% CIs, in **BOTH** regions, and the shallow→deep difference has the same sign in both regions.
- **CONDITIONAL** — the decline (≥0.15, non-overlapping CIs) appears in only one region, or CIs are only marginally separated in one/both.
- **NOT SUPPORTED** — b is flat or increases with depth in both regions.
- **REFUTATION of the pressure/temperature mechanism** — b increases with depth (sign opposite to prediction, significant in at least one region).

Included code and results