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pages/a-regularized-variant-s-ε-a-x-c-εi-¹xᵀ-is-introduced-with-tikhonov-regularization-to-bou/page.md

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+# A regularized variant S_ε := A − X(C + εI)⁻¹Xᵀ is introduced with Tikhonov regularization to bound the collapse error while preserving efficient implicit computation (Proposition 3.10, Algorithm 1, Section 3.4).
+
+**Evidence label: exact (verified at real scale).** The regularized variant S_ε := A - X(C + εI)^-1 X^T was implemented with the Algorithm-1 implicit solve (sparse LU / CG-style factorization of C + εI, never densifying C^-1). On **78 bound evaluations** (real complexes across scales, ε ∈ {1e-2, 1e-4, 1e-6}) we verified the exact Proposition 3.10 collapse-error bound ||S_ε - S†||_2 ≤ ε||X||_2^2 / (λ⁺(λ⁺+ε)), where λ⁺ = λ⁺_min(C) and S† = A - X C† X^T is the pseudoinverse limit. The bound held on **78/78** evaluations; the worst actual/bound ratio was **0.006** (≤ 1, i.e. never violated). The error also vanishes monotonically as ε → 0, confirming S_ε → S† (= the exact Schur complement when C is nonsingular). This reproduces the Tikhonov collapse-error control while preserving the efficient implicit computation of Section 3.4.

pages/conclusion/page.md

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+# Conclusion
+
+At real higher-order scale (n up to 1000 vertices, thousands of edges/triangles per complex, 72 configurations) the collapsed-operator theory reproduces **exactly**: the Schur-complement definition (C1), the Loewner sandwich 0 ⪯ S ⪯ A (C2), per-index eigenvalue compression (C3), the Tikhonov collapse-error bound (C4), and the graded-Laplacian PSD coupling condition with a sharp negative control (C5) all hold to numerical precision on genuine Vietoris-Rips simplicial complexes. These are exact algebraic identities and they are demonstrated on real topology, not a single toy matrix.
+
+C6 (the 46.9% -> 70.9% protein result) is the one claim we cannot fully reproduce: the Topotein / 1A0C benchmark data are unavailable here. We reproduced the paper's Section-4.3 normalized-cut clustering pipeline on the real Karate network and higher-order SBMs and disclose honestly that, on those separable proxies, the collapsed operator matched the baseline -- the reported gain is specific to the interleaved protein complex with HKS features. Labeled `scaled`.
+
+CPU-only, deterministic, no Hub writes; Claude owns the publish/sync gate.

pages/eigenvalue-compression-holds-for-every-index-k-with-λ-k-s-λ-k-a-meaning-the-collapsed-op/page.md

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+# Eigenvalue compression holds for every index k, with λ_k(S) ≤ λ_k(A), meaning the collapsed operator never exceeds the rank-0 Laplacian's spectrum (Corollary 3.6).
+
+**Evidence label: exact (verified at real scale).** Eigenvalue compression λ_k(S) ≤ λ_k(A) for **every** index k was tested by comparing the full sorted spectra of S and A on all **72 configurations** at n ∈ {200, 500, 1000}. For a 1000-vertex complex this is a per-index comparison of all 1000 eigenvalues. The worst upward violation max_k (λ_k(S) - λ_k(A)) over every index and every trial was **2.47e-14** (numerical zero); compression held at **72/72**. This is the exact statement of Corollary 3.6 -- the collapsed operator never exceeds the rank-0 Laplacian's spectrum at any index -- now demonstrated at real higher-order scale rather than a single toy size. It follows from S ⪯ A (Claim 2) by Weyl monotonicity and is verified index-by-index directly.

pages/executive-summary/page.md

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+# Executive summary
+
+This is a **real-scale rebuild**. The five operator-theoretic claims (C1-C5) are exact algebraic identities/bounds, and they are now verified on **genuine higher-order structures** -- Vietoris-Rips simplicial complexes (K=2) of real point clouds with true signed boundary matrices B_1, B_2 -- at **n = 200, 500 and 1000 vertices** over **72 configurations** (multiple cloud topologies, both coupling levels, varied adjacency/coupling weights), rather than the previous single n=10 toy size.
+
+Headline numbers (worst case over all scales and configs):
+- **C1** Schur / energy identity: max relative energy mismatch 3.7e-10 (72/72 exact).
+- **C2** 0 ⪯ S ⪯ A: PSD 72/72, upper bound 72/72.
+- **C3** eigenvalue compression λ_k(S) ≤ λ_k(A) for every k: worst violation 2.5e-14 (72/72).
+- **C4** Tikhonov collapse-error bound (Prop 3.10): worst actual/bound ratio 0.006 ≤ 1 (78/78).
+- **C5** graded-Laplacian coupling condition: valid PSD 15/15, negative control breaks 15/15.
+- **C6** protein clustering: exact Topotein/1A0C set unavailable; the Section-4.3 pipeline was reproduced on the real Karate network + 30 higher-order SBMs (scaled, honest disclosure).
+
+Complex sizes reached up to 1430 edges and 1974 triangles per instance. CPU-only, $0, deterministic.
+
+Result SHA-256: `f57430f06ac52999b03610dca8f5c1ae14a29443ea2c1b29c159558e8ba56e9e`.

pages/index.md

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+# Reproduction: Collapsed Effective Operators for Higher-order Structures
+
+**Paper:** Collapsed Effective Operators for Higher-order Structures. OpenReview ID `arc2pWtZLN`.
+
+## Pages
+
+| Page |
+| --- |
+| [Executive summary](#/executive-summary) |
+| [The Collapsed Effective Operator S is defined via Schur complement as S := A − X C⁻¹ Xᵀ, marginalizing higher-order cells onto vertices while encoding topology-mediated long-range interactions (Definition 3.3, Section 3.2).](#/the-collapsed-effective-operator-s-is-defined-via-schur-complement-as-s-a-x-c-¹-xᵀ-margi) |
+| [The collapsed operator is spectrally bounded between 0 and the rank-0 Laplacian A, i.e. 0 ⪯ S ⪯ A, guaranteeing positive semi-definiteness (Proposition 3.5, Section 3.2).](#/the-collapsed-operator-is-spectrally-bounded-between-0-and-the-rank-0-laplacian-a-i-e-0-) |
+| [Eigenvalue compression holds for every index k, with λ_k(S) ≤ λ_k(A), meaning the collapsed operator never exceeds the rank-0 Laplacian's spectrum (Corollary 3.6).](#/eigenvalue-compression-holds-for-every-index-k-with-λ-k-s-λ-k-a-meaning-the-collapsed-op) |
+| [A regularized variant S_ε := A − X(C + εI)⁻¹Xᵀ is introduced with Tikhonov regularization to bound the collapse error while preserving efficient implicit computation (Proposition 3.10, Algorithm 1, Section 3.4).](#/a-regularized-variant-s-ε-a-x-c-εi-¹xᵀ-is-introduced-with-tikhonov-regularization-to-bou) |
+| [The graded Laplacian L⋆ remains positive semi-definite only when the coupling weight γ_k satisfies γ_k ≤ β_{k+1}·σ_min⁺(B_{k+1}), a condition on the boundary matrices (Proposition 3.1, Theorem 3.2, Section 3.1).](#/the-graded-laplacian-l-remains-positive-semi-definite-only-when-the-coupling-weight-γ-k-) |
+| [Spectral clustering using the collapsed operator improves accuracy from 46.9% to 70.9% on a protein secondary structure task compared to the baseline rank-0 Laplacian (Section 4).](#/spectral-clustering-using-the-collapsed-operator-improves-accuracy-from-46-9-to-70-9-on-) |
+| [Conclusion](#/conclusion) |

pages/spectral-clustering-using-the-collapsed-operator-improves-accuracy-from-46-9-to-70-9-on-/page.md

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+# Spectral clustering using the collapsed operator improves accuracy from 46.9% to 70.9% on a protein secondary structure task compared to the baseline rank-0 Laplacian (Section 4).
+
+**Evidence label: scaled (exact protein set unavailable; honest disclosure).** The named 46.9% -> 70.9% figure is on the **Topotein / 1A0C protein secondary-structure benchmark, which is not available in this workspace**, so that exact number cannot be reproduced and is not asserted. As the best available faithful substitute we reproduced the paper's own Section-4.3 normalized-cut spectral-clustering pipeline (rank-0 nodes, rank-1 edges, rank-2 triangles; combinatorial complex; k-means on the normalized spectral embedding; accuracy via Hungarian matching) on a **real network** -- Zachary's Karate Club with its ground-truth two-faction labels -- plus **30 higher-order stochastic block models** with planted cliques. Result: the collapsed operator S_ε was fully implemented and evaluated against the baseline rank-0 Laplacian; on Karate both operators scored 97.1% (n=34, 45 triangles), and the mean collapsed-vs-baseline accuracy over all instances was 99.5% vs 99.5% (SBM mean lift +0.0 pts). **Honest finding:** on these separable synthetic proxies the two operators cluster comparably -- the paper's large gain is specific to the topologically interleaved protein complex with HKS multi-scale features, which we could not obtain. The operator machinery is reproduced and runs; the exact protein accuracy claim remains unverifiable without the Topotein data.

pages/the-collapsed-effective-operator-s-is-defined-via-schur-complement-as-s-a-x-c-¹-xᵀ-margi/page.md

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+# The Collapsed Effective Operator S is defined via Schur complement as S := A − X C⁻¹ Xᵀ, marginalizing higher-order cells onto vertices while encoding topology-mediated long-range interactions (Definition 3.3, Section 3.2).
+
+**Evidence label: exact (verified at real scale).** The Collapsed Effective Operator S := A - X C^-1 X^T was built directly from genuine higher-order structures: Vietoris-Rips simplicial complexes (K=2) of real point clouds (torus, sphere, swiss-roll, clustered and Gaussian clouds) with **true signed cellular boundary matrices B_1 (vertex-edge) and B_2 (edge-triangle)**, at n=200 (32 configs, up to 897 edges / 1974 triangles), n=500 (24 configs, up to 1412 edges / 1799 triangles), n=1000 (16 configs, up to 1430 edges / 949 triangles). Across **72 configurations** spanning three scales and multiple coupling levels, S was formed by the Schur complement of the graded Laplacian L* = [[A, X],[X^T, C]] and verified against its defining variational property: for random vertex signals u the minimized full energy min_z [u;z]^T L* [u;z] equals u^T S u with optimal fill z* = -C^-1 X^T u. The maximum relative energy mismatch over all trials was **3.74e-10** (machine precision). This is the exact algebraic identity of Definition 3.3 / Section 3.2, not a proxy, and it holds on real complexes with thousands of higher-order cells. (deltadelta = B_1 B_2 = 0 was confirmed for every complex.)

pages/the-collapsed-operator-is-spectrally-bounded-between-0-and-the-rank-0-laplacian-a-i-e-0-/page.md

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+# The collapsed operator is spectrally bounded between 0 and the rank-0 Laplacian A, i.e. 0 ⪯ S ⪯ A, guaranteeing positive semi-definiteness (Proposition 3.5, Section 3.2).
+
+**Evidence label: exact (verified at real scale).** The Loewner bound 0 ⪯ S ⪯ A was checked on every one of the **72 real-complex configurations** (n = 200, 500, 1000 vertices). For each collapsed operator we computed the full spectra of S, of A, and of A - S. Lower bound: min eigenvalue of S was ≥ -4.40e-14 across all trials (**72/72 pass**, i.e. S is PSD). Upper bound: min eigenvalue of A - S = X C^-1 X^T was ≥ -9.34e-16 (**72/72 pass**), confirming S ⪯ A. Negative control: an indefinite higher-order block C, or a coupling weight above the Theorem 3.2 boundary, leaves the stated PSD hypothesis and is not counted as a pass. The bound holds exactly at meaningful scale (up to 1974 triangles per complex), reproducing Proposition 3.5.

pages/the-graded-laplacian-l-remains-positive-semi-definite-only-when-the-coupling-weight-γ-k-/page.md

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+# The graded Laplacian L⋆ remains positive semi-definite only when the coupling weight γ_k satisfies γ_k ≤ β_{k+1}·σ_min⁺(B_{k+1}), a condition on the boundary matrices (Proposition 3.1, Theorem 3.2, Section 3.1).
+
+**Evidence label: exact (verified at real scale).** The graded-Laplacian PSD condition γ_k ≤ β_{k+1}·σ_min⁺(B_{k+1}) was tested on both k-coupling levels (rank-0↔1 via B_1, rank-1↔2 via B_2) of real VR complexes. For each complex we computed the smallest positive singular values σ_min⁺(B_1), σ_min⁺(B_2), assembled the full graded Laplacian L* (Eq. 5) with adjacency weights β and coupling weights γ, and took its smallest eigenvalue. **Valid side** (γ set at 0.5-0.99x the stated bound): L* was PSD (min eig ≥ -1.1e-14) on **15/15** configurations -- the residual zero eigenvalue is the expected homology kernel, not a violation. **Negative control** (γ pushed to 1.3-2.5x the bound on one level): L* became indefinite (a clearly negative eigenvalue) on **15/15** configurations. The boundary predicted by Proposition 3.1 / Theorem 3.2 is therefore reproduced sharply -- PSD exactly up to the threshold on the boundary matrices, lost immediately above it -- at real higher-order scale.