Test 09 — b-values shift with depth (Vol XXXII, refinement family)
Supplied working report; original verdicts not validated.
Review status: Needs review. This is an archival source note, not a passed proof.
README.md
# Test 09 — b-values shift with depth (Vol XXXII, refinement family) ## Summary | Prediction | Verdict | |---|---| | b falls with depth (global catalogue) | not supported as stated | | the decline is robust in regional subsets | not supported as stated | | Overall | **not supported as stated** | ## Headline numbers - Global USGS FDSN catalogue, M ≥ 4.0, 2000–2025, depth > 0 km: **354,127 events** in six depth bins. - b (Aki–Utsu maximum likelihood, maximum-curvature Mc + 0.1) by depth bin: **1.087** (0–20 km), **1.114** (20–40), **1.212** (40–70), **1.247** (70–150), **1.360** (150–300), **1.277** (>300 km). - b **rises** with depth, not falls: Spearman rho = **+0.943** between bin-midpoint depth and b (permutation p = 0.018, six bins). - The shallowest bin's 95% CI [1.079, 1.094] and the deepest bin's CI [1.249, 1.305] are separated, but in the opposite direction to the claim. - Japan (M ≥ 4.0): b rises from **1.205** (0–20 km) to **1.550** (70–150 km), rho = +1.0 across four valid bins — the same direction as the global set. ## What this shows On the largest available global catalogue, the posted claim's predicted decline of b with depth is contradicted: b increases systematically from the crust into the sub-crustal bins before dipping slightly in the deepest bin. The claim that "deep and shallow sequences cannot share a statistic" is therefore not supported by these data as stated; if anything, the global pattern is the mirror image of the claim (shallow events have lower b, deep events higher b, in the direction of larger fractions of small events at depth). The regional check that could be run (Japan) agrees with the global direction; the western-US subset could not be analysed because its completeness sits at or below the catalogue truncation (see below). ## What this cannot show This test uses a global bulletin with a fixed magnitude threshold, not high-resolution relocated catalogues. Bin-level b-values can be distorted by regional detection heterogeneity, by the mixture of tectonic settings inside each depth bin (subduction zones contribute most deep events), and by magnitude-type mixing (mb, Ms, mww). The test refutes the claim for the global catalogue as compiled; it does not establish a universal depth–b law, and it does not test whether specific faults show depth-dependent b. ## Limitations 1. **Detection and reporting heterogeneity.** A global bulletin mixes networks with different completeness levels in space and time; the maximum-curvature Mc is a crude per-bin correction for this. Deep bins (150–300, >300 km) are dominated by a few subduction zones (Tonga, Japan, South America), so "deep" is confounded with tectonic setting. 2. **Rival explanation: setting, not depth.** The rise in b with depth may reflect the greater proportion of subduction-zone and intra-slab earthquakes in deeper bins, which have systematically different b than crustal transform events (e.g. oceanic intra-slab seismicity often shows b near or above 1.2). Depth is then a proxy for regime, and the result cannot separate pure depth dependence from composition of the mix. 3. **Western-US subset could not be analysed.** At M ≥ 4.0 the west-US box (lat 25–50, lon −125 to −110, 1,936 events) has its frequency–magnitude peak at the 4.0 truncation, so Mc is unlocatable under the pre-specified rule; the pre-announced M ≥ 3.0 sensitivity (15,514 events) has the same problem (peak at 3.0), so no west-US trend is reported. Completeness in that box sits below both thresholds. 4. **Bootstrap CIs are conditional on the fixed Mc.** The confidence intervals re-sample magnitudes at the observed Mc, so they do not carry the uncertainty of the completeness estimate itself; the true uncertainty in b is larger, though with 12,000–78,000 events per global bin the bin-to-bin differences remain decisive.
hypothesis.md
# Test 09 — b-values shift with depth (Vol XXXII, refinement family) ## Posted claim (verbatim from the notebook) > "The earthquake magnitude-frequency b-value declines with depth, so deep and shallow sequences cannot share a statistic." **Prediction:** b should fall with depth. **Refutation:** constant b across depth. Source: Incoprea Research library, Volume XXXII, "refinement family" theory entry (posted 30 October 2026). ## Pre-specified criteria for this test Written before the data were analysed. Data: USGS FDSN event catalogue, global, 2000-01-01 to 2025-10-01, magnitude ≥ 4.0, events with depth > 0 km. Depth bins: 0–20, 20–40, 40–70, 70–150, 150–300, >300 km. Within each bin, completeness Mc is estimated by maximum curvature plus the standard 0.1 correction, and b is estimated by the Aki–Utsu maximum-likelihood formula b = log10(e) / (mean(M) − (Mc − ΔM/2)), ΔM = 0.1 (the catalogue magnitude rounding). 95% confidence intervals come from a seeded bootstrap (2000 resamples of the bin's events, recomputing Mc and b each time). The same analysis is repeated on two regional subsets: western US (lat 25–50, lon −125 to −110) and Japan (lat 25–45, lon 140–150). Decisive numbers and their reading: - **supported** — b declines monotonically (strictly, across all valid bins) AND the shallowest bin's 95% CI does not overlap the deepest bin's 95% CI, in the global set AND in at least one of the two regional sets; - **partly supported** — a declining trend in the global set (Spearman rho of bin-midpoint depth vs b < 0) but the full "supported" bar is not met (CIs overlap, or the decline is not strictly monotonic, or no region separates); - **not supported** — b is flat or rises with depth in the global set (rho ≥ 0). Bins with fewer than 200 events above Mc, or where Mc cannot be located (maximum curvature at the bin's smallest magnitude), are reported but excluded from the trend computation and flagged in results.json. Supplementary sensitivity, announced before running: if a regional subset is too small at M ≥ 4.0 for its completeness to be located (FMD peak at the query threshold), the same box is re-analysed at M ≥ 3.0 and reported separately as a sensitivity check. The pre-specified decision rule above is applied only to the M ≥ 4.0 analysis.