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.

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.

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
 

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 |