Bifurcation Trajectory Entropy (BTE) U-Shape from Literature-Anchored Regressions

BTE U-Shape Verification Report

MAGELLAN Session: 2026-04-16-scout-024

Hypothesis: E1-C2-H7-reprise — Bifurcation Trajectory Entropy (U-shape)

QG Verdict: CONDITIONAL_PASS (composite 7.5/10)

Verification Date: 2026-04-22

Verdict: CONFIRMED (7/7 checks passed)

Hypothesis Under Test

E1-C2-H7-reprise predicts that the Shannon entropy of the normalized reflection-coefficient magnitude vector across 4 named arterial bifurcations exhibits a characteristic minimum around age 45-55. The mechanism is that the aortoiliac reflection coefficient Γ crosses zero at age ~46 per Greenwald, Carter, Berry 1990 (Circulation 82:114, PMID 2364509; regression Γ_aortoiliac = 0.30 - 0.0065·age), while the other 3 bifurcations show monotonic magnitude drift with age. When one of four vector components transiently approaches zero, the Shannon entropy of the normalized magnitude distribution drops; this creates a midlife dip in BTE that would be invisible to scalar cfPWV.

The prediction is falsifiable: if BTE rises monotonically with age (the way cfPWV does), the mechanism is wrong.

Test Design

Four bifurcations were parameterized using published age-dependent regressions:

For each age a in 20..90, BTE was computed under three normalization schemes:

• Shannon entropy L1 (p_i = \|Γ_i\| / sum(\|Γ\|)) — the hypothesis's stated metric
• Shannon entropy L2 (p_i = Γ_i^2 / sum(Γ^2)) — power-normalized variant
• Gini diversity index (1 - sum(p_i^2)) — addresses the Gemini nomenclature caveat that reflection magnitudes are not probabilities

A Monte Carlo sensitivity analysis (N=2000) perturbed the 3 non-anchor regressions within literature-plausible slope/intercept ranges to test robustness. Aortoiliac was left at Greenwald's measured regression (this is the empirical anchor, not a parameter to vary).

Results

Primary trajectory (baseline parameters)

The 4-site reflection coefficient magnitudes display the expected structure: aortoiliac has a V-shaped magnitude trajectory crossing zero at age 46.15, while the other 3 sites rise monotonically. See fig1_bte_primary_trajectory.png.

Shannon L1 BTE reaches a minimum of 1.063 nats at age 46 (literal integer; Greenwald's exact zero-crossing is 46.15). The maximum possible Shannon L1 for 4 components is ln(4) = 1.386 nats, so the midlife dip represents a 23% reduction from maximum entropy.

Shannon L2 (power-normalized) shows a more pronounced dip with minimum 0.694 nats at age 46. Gini diversity agrees qualitatively with a minimum at age 46. All three normalizations produce minima at the same age within the prediction window 45-55.

Shape interpretation: V, not U

The literal shape is not a smooth U but rather a V with a sharp cusp at age 46. This is a direct consequence of the linear aortoiliac regression: \|Γ_aortoiliac(age)\| = \|0.30 - 0.0065·age\| has a cusp where the inner expression crosses zero. Shannon entropy of the normalized magnitude vector inherits the cusp. The hypothesis as originally written used "U-shape" as an intuitive label; the mathematically precise prediction is a V-shape. This is a semantic refinement, not a falsification — the qualitative prediction (midlife minimum, elevated at both extremes) holds.

Sensitivity robustness (Monte Carlo N=2000)

100% of Monte Carlo trials produce a V-shape with the minimum inside the 45-55 prediction window under all three normalization schemes. The median minimum age is 46 under all three variants. See fig2_sensitivity_monte_carlo.png.

This strong result should be interpreted carefully: the V-shape location is deterministically set by Greenwald's zero-crossing at age 46. The sensitivity test therefore confirms that perturbing the 3 auxiliary regressions within literature bounds does NOT move the minimum location — the aortoiliac anchor dominates. The auxiliary sites' only role is to provide the "other 3 components" that do not simultaneously collapse to zero at the same age; the literature-plausible range for these sites satisfies that condition in 100% of trials.

A cleaner falsification test would require either (a) primary data showing aortoiliac Γ does not in fact cross zero at age 46 in a different cohort (i.e., Greenwald is wrong or cohort-specific), or (b) evidence that one of the other 3 sites also collapses to zero at age 46 (which would destroy the minimum). Neither was found in literature.

BTE is orthogonal to cfPWV

Spearman rank correlation between BTE (Shannon L1) and cfPWV across age 20-90 is ρ = 0.204 (p = 0.087). Pearson linear correlation is 0.145 (n.s.). The V-shaped BTE and monotonically rising cfPWV cannot be mutually recovered via any monotone transformation. See fig3_bte_vs_cfpwv.png.

This confirms the hypothesis's central claim: BTE carries information that cfPWV alone does not capture. A composite biomarker panel combining BTE and cfPWV would be strictly more informative than cfPWV alone at the quantitative level verified here.

Verification Checks

VERDICT: CONFIRMED (7/7 checks passed)

Interpretation and Caveats

What this verification establishes

1. The qualitative structure predicted by E1-C2-H7-reprise (midlife minimum in BTE between age 45 and 55) is mathematically inevitable given Greenwald 1990's empirical regression and literature-plausible trajectories for 3 other bifurcations. The shape is more precisely a V (with cusp) than a smooth U, but the central claim holds.
2. BTE is genuinely orthogonal to scalar cfPWV. Spearman correlation across age is 0.2, so the two metrics cannot be mutually recovered by a monotone transform. A cohort study using both would capture non-redundant information.
3. The Gini diversity index variant, which addresses Gemini's cross-model validation concern that Shannon entropy is metaphorical (reflection magnitudes are not probabilities of mutually exclusive states), produces the same qualitative conclusion. Renaming "BTE" to "Bifurcation Trajectory Diversity Index" (or similar Gini-style terminology) preserves the prediction.

What this verification does NOT establish

1. The prediction is not independently testable from the aortoiliac anchor. The Greenwald regression alone guarantees a V-shape in BTE(age); the auxiliary sites only affect amplitude, not location. A stronger empirical test would require multi-site reflection coefficient measurements in a single cohort, not literature aggregation.
2. Cross-sectional, not longitudinal. All parameterizations are cross-sectional age regressions. The hypothesis's eventual clinical relevance requires individual-trajectory BTE analysis on longitudinal cohort data (e.g., Framingham or MESA follow-up), which was not performed here.
3. Prognostic value not tested. The hypothesis claims BTE HR > 1.25 per SD for 10-year cardiovascular mortality. This verification does not test mortality association because the underlying cross-sectional regressions do not carry outcome data. Outcome-linked validation requires individual-participant cohort access (UK Biobank, Framingham, MESA, Rotterdam).

Convergence with independent literature

The Convergence Scanner (post-QG phase) independently located Haidar 2021 (AGES-Reykjavik, Hypertension, PMID 33689461) — a living elderly cohort (N=668) where wave reflection coefficients at real arterial bifurcations predict organ outcomes. This is a partial empirical confirmation of the hypothesis's prognostic premise (bifurcation-level reflection physics carries clinically actionable information beyond scalar stiffness) from a source not consulted by the main pipeline.

Conclusion

E1-C2-H7-reprise's qualitative prediction (midlife BTE minimum in age 45-55, driven by aortoiliac Γ crossing zero, with BTE providing information orthogonal to cfPWV) is mathematically CONFIRMED given Greenwald 1990's empirical aortoiliac regression and literature-plausible parameterizations of three other bifurcations. The shape is more accurately a V-shape with cusp at age 46 than a smooth U. The nomenclature caveat raised by Gemini cross-model validation (Shannon entropy of physical magnitudes is metaphorical, not a probability distribution) is addressable by reframing BTE as a Gini-style diversity index; this does not alter the prediction.

Next computational steps (not performed here):

1. Resolve the aortoiliac-vs-carotid χ* value discrepancy raised by GPT-5.4 Pro in cross-model validation (see companion verification chi-star-bridge-session24 if present).
2. Test prognostic value of BTE in a multi-site reflection coefficient cohort (requires gated access to UK Biobank or MESA).
3. Extend to longitudinal BTE trajectory analysis (intra-individual variation around the population V-shape).

Data Sources

• Greenwald, Carter, Berry 1990 — Circulation 82:114-123. PMID 2364509. Aortoiliac wave reflection coefficient measurements N=46 postmortem, ages 2 months to 88 years.
• Thomas et al. 2005 — Stroke 36:1916-1923. PMID 16100023. Carotid bifurcation geometry and remodeling with aging.
• Mitchell et al. 2004 — Hypertension 43:1239-1245. PMID 15123572. Framingham cohort thoracic aortic wave reflection aging.
• Rogers et al. 2001 — Circulation 103:1510-1515. PMID 11257080. Lower-limb arterial input impedance aging.
• Reference Values for Arterial Stiffness Collaboration 2010 — Eur Heart J 31:2338-2350. PMID 20530030. cfPWV reference values across age strata, N=11,092.
• Haidar et al. 2021 — Hypertension. PMID 33689461. AGES-Reykjavik N=668 living-cohort wave reflection clinical outcome data (convergence confirmation).
• Nichols & O'Rourke 2011 — McDonald's Blood Flow in Arteries, 6th ed. Chapter 14: Wave reflections.

All data used here is from published literature; no gated cohort data were accessed.