Substrate-Coupling Pathway

Coupling pathway methodology

Your covariance matrix has zeros where the substrate measures real coupling. When the next fragmentation event propagates along these pathways — Firefly Alpha 2nd stage at 369 km retrograde, coupled to 9,556 active fleet members today — the assets you priced as spatially independent fail together, because they were never independent in the substrate. The reserves you hold against that scenario were calibrated against a spatial-independence assumption the physics empirically violates. We measure the off-diagonal block, name the objects driving it, and compute the closed-form Δ⟨P_σ⟩_LEO under any removal scenario you specify. Closed-form algebra over L_signed = ρ · σ_mag. Not a heuristic — exact arithmetic against the snapshot SHA, reproducible bit-for-bit.

Definitions: D_mom(i, j) is the momentum-aware edge weight in the catalog's conjunction graph; L_signed(j) = ρ(j) · σ_mag(j) is the foreign object's per-substrate leverage; the product D_mom · |L_signed| is per-pair coupling strength. Equation (3) applies the tumbling-state correction α_j to objects in the chaotic-tumble sector (see methodology section below).

Tumbling-state correction (Cauchy / SO(3))

Tumbling state is a topological observable on SO(3). A flat-spinner explores a 1-dimensional orbit in SO(3); a chaotic tumbler explores a 3-dimensional region. They are topologically distinct sectors — rotational analog of the substrate's Klein-4 σ-parity. For any convex body tumbling ergodically, Cauchy's 1841 surface-area theorem gives the orientation-averaged projected cross-section:

Your reserves price the leverage-dominant cohort — KH-8 reconnaissance cylinders, Firefly Alpha second stage, Delta II second stages, Shavit upper stages — at the catalog's xSectAvg. The catalog publishes the wrong number for tumbling assets. A 184-year-old theorem (Cauchy 1841: ⟨A⟩_iso = S/4) says the orientation-averaged cross-section of a chaotically tumbling convex body is exactly S/4 — and the leverage-dominant cohort under-counts against it by +15.84% leverage-weighted. That is tail risk your model did not see. The substrate computes the exact correction in closed form, per object, against the snapshot SHA. You pay for it now, or the next cascade confirms it for you.

Tumbling-state cohort — α per object

Cohort spans cylinders (Cauchy exact) and paneled satellites (convex-hull approximation flagged). KH-8 (Gambit-3) reconnaissance bus dominates the leverage-weighted aggregate at 46% cohort weight × +20.1% T_state = +8.77 percentage points of the +15.84% total. Iridium-1 −1.0% is the methodology detecting a real geometric distinction: panel-broadside-dominant geometry puts orbit-averaged major-axis spin marginally above Cauchy-ergodic. The tumbling-state correction is not universally drag-enhancing; the methodology resolves the geometric distinction rather than fitting all objects to a positive correction. Audit charter: T1 / T2 / T3 PASS at v0.4.1.

What you receive

• Per-fleet substrate-coupling pathway report: named foreign objects, per-pair coupling strengths, threshold-based fleet counts. D_mom edge weights derived from per-object DISCOSweb cross-sections (xSectAvg) rather than unit-normalized geometry.
• Tumbling-state-corrected L_signed per object: closed-form α from Cauchy/SO(3) on the asset's manufacturer geometry, applied multiplicatively to the geometric leverage for objects in the chaotic-tumble sector.
• Atomic intervention prescriptions: per-pathway counterfactuals showing the per-asset Δ⟨P_σ⟩ delta if a named foreign object were removed. Closed-form arithmetic over the leverage map; reproducible bit-for-bit against the snapshot SHA.

Deliverable: an audit JSON pinned to the snapshot SHA, plus a rendered PDF. Same form as the public artifacts in the audit charter. Engagements are direct; per-pathway named-target detail is delivered only on engagement, while aggregate findings publish openly.