From 5d6a4765fceedd4a6d0a00b9c5b8f280be907e52 Mon Sep 17 00:00:00 2001 From: allaun Date: Sun, 21 Jun 2026 01:02:13 -0500 Subject: [PATCH] feat(docs): architecture and speculative materials updates - ARCHITECTURE.md: update level-0 job counts (8332 full workspace, 3314 Compiler surface, 0 errors). - PyrochloreSidonBridge.md: point to current pyrochlore_sidon_receipt_v2.json (S=1) and note removal of stale S=5/2 receipt. - fiedler_non_identifiability.md: new note on Fiedler non-identifiability. --- .../docs/fiedler_non_identifiability.md | 150 ++++++++++++++++++ .../PyrochloreSidonBridge.md | 2 +- ARCHITECTURE.md | 4 +- 3 files changed, 153 insertions(+), 3 deletions(-) create mode 100644 6-Documentation/docs/fiedler_non_identifiability.md diff --git a/6-Documentation/docs/fiedler_non_identifiability.md b/6-Documentation/docs/fiedler_non_identifiability.md new file mode 100644 index 00000000..13191524 --- /dev/null +++ b/6-Documentation/docs/fiedler_non_identifiability.md @@ -0,0 +1,150 @@ +# Fiedler Non-Identifiability Under k-Hop Projection + +## Setup + +Let $G = (V,E)$ be a graph, $L$ its Laplacian, and $v_2$ the Fiedler vector +(second eigenvector of $L$). Define the Fiedler sign $\phi(x) = \operatorname{sign}(v_2(x))$. + +Define a $k$-hop sampling operator $\mathcal{P}_k : G \to \tilde{G}_k$ that: +- samples seeds uniformly from $V$, +- expands each seed by $k$ hops of neighborhood closure, +- induces the subgraph on the resulting node set (edges preserved iff both endpoints are sampled). + +$\mathcal{P}_k$ is **not spectrum-preserving**: it applies a local density filter +to the eigenspace before eigenvalue decomposition. + +## Empirical Finding (cit-HepPh, $k=2$) + +On the 30,985-node 2-hop induced subgraph of cit-HepPh (sampled from 34,546 +nodes, 421,578 edges): + +- $\phi$ is constant on 94.62% of computed nodes +- $\operatorname{Var}(\phi) \approx 0.0509$ (essentially collapsed) +- PR-kNN recovers $\phi$ at 94.40% accuracy — equal to the trivial majority + predictor (94.56%) +- The minority class (B, 5.38%) has 99.7% PR-kNN error: the feature geometry + contains no separable decision boundary for it +- $\lambda_2(\tilde{L}_2) \approx 0.20$, well-separated from $\lambda_1 \approx 0$, + but the eigenvector aligns with a local density mode, not a global partition + +## Non-Identifiability Theorem + +The Fiedler sign $\phi$ is **non-identifiable** under $\mathcal{P}_k$ when +three conditions hold simultaneously: + +### Condition A — Boundary dilution + +$$ +P_{\mathcal{P}_k}(x \in \partial C) \ll P(x \in C) +$$ + +Boundary nodes (where $\phi$ changes sign) are under-sampled because they +occupy bridge positions with lower $k$-hop closure probability. A bridge +node has $O(1)$ neighbors in each community, so its $k$-hop expansion +captures only one side of the partition. + +### Condition B — Degree-conditioned closure + +$$ +\mathbb{E}[d(x) \mid x \in \tilde{G}_k] \not\approx \mathbb{E}[d(x)] +$$ + +The sampled subgraph over-represents high-degree hubs whose neighborhoods +are dense. Low-degree bridges are systematically excluded. Since degree +correlates with community core membership, the sampling distribution is +biased toward the dominant community's interior. + +### Condition C — Spectral gap collapse under restriction + +$$ +\lambda_2(\tilde{L}_k) \ll \lambda_2(L) +$$ + +or equivalently the second eigenvector of $\tilde{L}_k$ aligns with the +constant vector: the restricted Laplacian's second mode describes local +density variation, not global partition structure. Operationally: + +$$ +\langle \tilde{v}_2, \mathbf{1} \rangle \approx \|\tilde{v}_2\| +$$ + +### Theorem (informal) + +If (A) boundary dilution and (B) degree-conditioned closure both hold, then +$\tilde{L}_k$ almost surely admits a Fiedler vector whose sign structure +converges to a constant on the sampled support: + +$$ +\tilde{v}_2(x) \approx c \quad \forall x \in \tilde{G}_k +$$ + +and therefore $\phi$ is unrecoverable from $\tilde{G}_k$: + +$$ +\phi \not\approx \operatorname{sign}(\tilde{v}_2) +$$ + +*Proof sketch.* Under (A), the $k$-hop subgraph contains predominantly +interior nodes of the dominant community. Under (B), the spectral mass is +concentrated in high-degree regions. The Cheeger inequality gives: + +$$ +\lambda_2(\tilde{L}_k) \leq 2h(\tilde{G}_k) +$$ + +where $h$ is the Cheeger constant. With boundary dilution, $h(\tilde{G}_k)$ +approaches the trivial cut (one community empty), driving $\lambda_2$ toward +the conductance of a single cluster's internal expansion rather than the +inter-cluster bottleneck. The corresponding eigenvector therefore tracks +local density rather than global division. $\square$ + +## Interpretation + +The $k$-hop projection pipeline does not "lose signal" — it enters a regime +where the second eigenmode of the induced measure space is no longer the +second eigenmode of the original. + +**Why PR and HITS survive.** PageRank and HITS operate on first-order +stationary mass flow: +- PR ≈ Perron eigenvector of the transition matrix (stationary distribution) +- HITS ≈ leading singular vectors of the adjacency matrix + +These are stable under $\mathcal{P}_k$ because they depend on *aggregate +mass flow* through the $k$-hop subgraph, which approximates the global +stationary distribution under ergodic assumptions. The Fiedler vector +($\lambda_2$) is a *second-order* property describing *difference* between +communities, and collapses when boundary structure is suppressed. + +## Regime Classification + +| Regime | Fiedler behavior | Detection | +|--------|-----------------|-----------| +| (1) True spectral | Bimodal sign, meaningful partition | $\operatorname{Var}(\phi) \approx 0.25$ | +| (2) Sampling collapse | Unimodal sign, density mode | $\operatorname{Var}(\phi) \ll 0.1$ | +| (3) Propagation fake agreement | Majority prior amplified | PR-kNN accuracy ≈ majority baseline | + +The cit-HepPh cross-validation under $\mathcal{P}_2$ operates in regimes +(2) → (3). + +## Operational Rule + +If on the computed subgraph: + +$$ +\frac{\max(n_{\text{neg}}, n_{\text{pos}})}{n_{\text{total}}} > 0.90 +\quad\text{and}\quad +\operatorname{Var}(\phi) < 0.1 +$$ + +flag Fiedler as **non-informative** under the current sampling regime and +exclude from any scoring or purity metric. + +## References + +- von Luxburg, U. (2007). A tutorial on spectral clustering. *Statistics and + Computing*, 17(4), 395–416. — Fiedler vector as global partition signal. +- Chung, F. (1997). *Spectral Graph Theory*. CBMS Regional Conference Series. + — Cheeger inequality and $\lambda_2$ as conductance. +- This repo: `snap_pist_spectral_crossval.py` — empirical collapse detection. +- This repo: `docs/specs/DP_RRC_RECEIPT_ENCODING_SPEC.md` — broader receipt + encoding context for spectral observables. diff --git a/6-Documentation/docs/speculative-materials/PyrochloreSidonBridge.md b/6-Documentation/docs/speculative-materials/PyrochloreSidonBridge.md index c7573fe9..e39fd8c1 100644 --- a/6-Documentation/docs/speculative-materials/PyrochloreSidonBridge.md +++ b/6-Documentation/docs/speculative-materials/PyrochloreSidonBridge.md @@ -2,7 +2,7 @@ **Status:** SPECULATIVE_MATERIALS_RECEIPT_VERIFIED **Claim level:** formal isomorphism with two experimental cross-checks -**Receipts:** `pyrochlore_sidon_receipt.json` (S=5/2), `pyrochlore_sidon_receipt_v2.json` (S=1) +**Receipts:** `pyrochlore_sidon_receipt_v2.json` (S=1) — current. The historical S=5/2 analytic receipt (`pyrochlore_sidon_receipt.json`) was removed from the working tree because it had no active generator and was stale; the classical KMMC analytic values are still documented below. ## The claim diff --git a/ARCHITECTURE.md b/ARCHITECTURE.md index b2dd3a2c..1a489fa7 100644 --- a/ARCHITECTURE.md +++ b/ARCHITECTURE.md @@ -24,7 +24,7 @@ Level 0 — Primordial PIST/DIAT Shells │ Q16.16 Arithmetic │ Br | Level | Domain | Key Substrates | Invariant | |-------|--------|----------------|-----------| -| 0 (Primordial) | Pure math | PIST/DIAT shells, Q16.16, BraidField, BracketedCalculus, EigensolidConvergence, Geometry Emergency Boot Witness | mass = t·(2k+1-t), arithmetic totality; 746 modules, 3529 jobs, 0 errors | +| 0 (Primordial) | Pure math | PIST/DIAT shells, Q16.16, BraidField, BracketedCalculus, EigensolidConvergence, Geometry Emergency Boot Witness | mass = t·(2k+1-t), arithmetic totality; 8332 jobs full workspace, 3314 jobs Compiler surface, 0 errors | | 1 (Geometric) | Shape-aware | GWL rotational coupling, toroidal shells, torsion quaternions, GWL throat | dE/dt ≤ 0, no zero-mass singularities | | 2 (Biological) | Life-aware | 64 codon tables, Izhikevich spiking neurons, STDP plasticity | codon validity, spike threshold v < 30mV | | 3 (Thermodynamic) | Energy-aware | Trixal state (thermal/work/irreversibility), homeostatic governor, HyperFlow NS-on-shells | irreversibility < threshold, |γ+s'(p*)| < 1 | @@ -181,7 +181,7 @@ GCCL = Law Stack (what must be preserved) | `docs/` | Research papers, specs, roadmaps (aliased into 6-Documentation) | | `4-Infrastructure/infra/ene-rds/` | ENE RDS Rust workspace (8 crates, replaces Python RDS stack; Axum API on :3000) | | `4-Infrastructure/storage/` | restic + Garage S3 + rclone storage stack; `storage_agent.py` observer | -| `.devcontainer/` | NixOS devcontainer flake (OpenGL/X11, Lean, Python science stack, MCP: Notion + AWS) | +| `.devcontainer/` | NixOS devcontainer flake (OpenGL/X11, Lean, Python science stack, MCP: Notion + Headroom) | | `shared-data/` | Databases, golden traces, artifacts | | `workspace-config/` | IDE and environment settings | | `scratch/` | Experimental code |