jc: Pillar 5+ — Köstenberger-Stark concentration on Hadamard 2×2 SPD#286
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Implements Theorem 1 from Köstenberger & Stark, 'Robust Signal Recovery in
Hadamard Spaces' (arXiv:2307.06057v2, July 2024) as an executable proof in the
jc harness, alongside the existing Jirak Berry-Esseen pillar.
# Why this pillar
Pillar 5 (Jirak) certifies the convergence rate of empirical statistics on
weakly-dependent ℝ-valued sequences — the SCALAR case, the right foundation
for CausalEdge64's scalar bit-fields (frequency, confidence).
When edges become anisotropic (Σ-tensor instead of scalar weight), the
aggregation is no longer in ℝ but on the cone of symmetric positive-definite
matrices — a Hadamard space (CAT(0), non-positive curvature). Köstenberger-
Stark Theorem 1 gives the exact concentration:
E[d²(S_n, μ)] ≤ (6 D_n / n) · Σ d(μ_k, μ) + (1/n²) · Σ Var(X_k)
without iid assumption — which is exactly what evidence aggregation across
edges with varying confidence needs.
# Probe setup
- Hadamard space: 2×2 SPD with affine-invariant Riemannian metric
d(A,B) = ‖log(B^(-1/2)·A·B^(-1/2))‖_F
Geodesic A ⊕_t B = A^(1/2)·(A^(-1/2)·B·A^(-1/2))^t·A^(1/2).
2×2 keeps every operation closed-form (eigendecomp = quadratic root).
- Heteroscedastic schedule: σ_k = 0.3/√(k+1) — variance shrinks per index
- μ_k = μ = I forced by construction → 6·D_n term vanishes, leaving the
cleaner Var-only bound (2/n²)·Σ σ_k²
- Monte Carlo: 1000 runs of n=100 samples
- PASS criterion: measured ≤ predicted · 1.5 (50% slack for constants)
# Result
measured = 9.05e-5
predicted = 9.34e-5
tightness = 0.969× ← bound is HIT, not just respected
runtime = 37 ms
The bound is not loose — measured E[d²(S_n, I)] sits at 96.9% of the
predicted ceiling. Pure Rust (zero deps), generalizes to k×k SPD by the
same theorem.
# Architectural significance
This is the math foundation for Σ-edge propagation: when CausalEdge64 grows
into CausalEdgeTensor (8 → 16 bytes, adding FisherZ-256-encoded q + s
factorization), multi-hop aggregation becomes a Fréchet/inductive mean on
the PSD cone. Köstenberger-Stark certifies the convergence rate of that
aggregation, including under Huber-ε contamination (noisy/hallucinated edges
on the path).
Together with the existing pillars:
Pillar 5 (Jirak): ℝ-valued sequences, weak dependence
Pillar 5+ (Köstenberger-Stark): Hadamard-space (PSD cone), non-iid
The third leg of the certification stack — Düker-Zoubouloglou 2024
(Hilbert-space-valued processes, arXiv:2405.11452) — would close the family
for SH coefficients and 16k-bit fingerprints lifted to ℓ². That can be
Pillar 5++ in a future commit.
# Files
- crates/jc/src/koestenberger.rs (new, ~370 lines incl. 8 unit tests)
- crates/jc/src/lib.rs (mod decl + run_all_pillars list entry)
# Run
cargo test --manifest-path crates/jc/Cargo.toml --release koestenberger
cargo run --manifest-path crates/jc/Cargo.toml --release --example prove_it
This was referenced Apr 29, 2026
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Was
Adds Pillar 5+ to the
jcproof-in-code harness: an executable verification of Theorem 1 from Köstenberger & Stark, Robust Signal Recovery in Hadamard Spaces (arXiv:2307.06057v2, July 2024).Result: PASS, tightness 0.969× — measured E[d²(S_n, I)] sits at 96.9 % of the predicted ceiling. Bound is hit, not just respected.
Warum jetzt
Pillar 5 (Jirak) zertifiziert ℝ-wertige Sequenzen unter schwacher Abhängigkeit — das ist die richtige Grundlage für
CausalEdge64s skalare Bit-Felder.Wenn Edges anisotrop werden (Σ-Tensor statt Skalar-Gewicht — die geplante
CausalEdgeTensor-Erweiterung um FisherZ-256-encoded q+s-Faktorisierung, 8→16 Byte), läuft die Aggregation nicht mehr in ℝ sondern auf der Mannigfaltigkeit der PSD-Matrizen. Köstenberger-Stark Thm 1 gibt die exakte Konzentration:ohne iid-Annahme und mit Huber-ε-Kontaminations-Toleranz — exakt was Multi-Hop-Edge-Propagation unter verrauschten/halluzinierten Edges braucht.
Probe-Setup
d(A,B) = ‖log(B^(-1/2)·A·B^(-1/2))‖_FA ⊕_t B = A^(1/2)·(A^(-1/2)·B·A^(-1/2))^t·A^(1/2)Architektur-Bedeutung
Die drei Säulen der Konzentrations-Familie für unser Substrat:
Damit ist die Mathe-Grundlage für
CausalEdgeTensor(= grey/white-matter-Aufteilung mit Σ-Edges + Gauß-Faltungs-Propagation in FisherZ-256) zertifiziert, bevor irgendjemand eine Zeile produktiven Edge-Code schreibt.Files
crates/jc/src/koestenberger.rs— neu, ~370 Zeilen inkl. 8 Unit-Testscrates/jc/src/lib.rs— Modul-Deklaration + Eintrag inrun_all_pillars-ListeVerifikation
Out of scope (für künftige PRs)
CausalEdgeTensorselbst (das ist die Implementation, dieser PR liefert nur das Beweis-Fundament)propagate()inholograph::resonance(Gauß-Faltungs-Operator)Ready-to-review.