Level 0 Validation Layer (Operational Specification)

Near-Parameter-Free Network Coherence Validation

author: U. Warring affiliation: Institute of Physics, University of Freiburg version: 0.1.0 last_updated: 2025-12-18 license: CC BY 4.0 Disclaimer – This document specifies a statistical validation gate for multi-link timing-network residuals. It is intended as an operational, auditable precondition for downstream inference or control protocols. It does not establish causality, attribution, or steering laws. A positive coherence detection indicates statistically significant shared structure under the stated model class and assumptions; it does not, by itself, imply a physical mechanism or causal direction. A null result is informative and bounds shared processes below detectability at the tested scales.\

How to Cite This Specification

Stable citation format:

Warring, U. (2025). Level 0 — Parameter-Free Network Coherence Validation (Version 0.1.0). Ordinans Series.

Version-specific: Always cite the version number. Thresholds and schemas are locked within a version.

For specific results: Reference section numbers (e.g., “Section 5.2, robust noise-floor estimation”).

---

0. Summary (Operator View)

Purpose: Decide whether observed cross-link correlations are distinguishable from finite-sample noise before any causal inference or steering is attempted.

Inputs: comparison streams (z(t)) and network topology (H)

Outputs: level0_certificate.json containing spectral test results, kill flags, and provenance hashes

Hard stops (“kill conditions”): autocorrelation, conditioning, block count, and MP-validity regime checks.

---

1. Scope and Guardrails

1.1 In scope

• Statistical detection of shared structure in residuals via covariance spectra

• Multi-(\tau) execution contract with trial-factor correction

• A machine-readable certificate binding results to topology and data provenance

1.2 Out of scope (explicit)

• Causality (“who drives whom”), attribution, and physical interpretation

• Feedback laws, feedforward filters, closed-loop control

• Relativistic modelling, oscillator physics, or platform-specific noise models

Operational invariant No downstream protocol may run unless a valid Level 0 certificate exists for the relevant (\tau)-range and topology hash.

---

2. System Model

2.1 Topology

A timing network with (N) nodes and (M) comparison links is encoded by the incidence matrix [ H \in \mathbb{R}^{M\times N}. ] Each row corresponds to a signed link measurement.

2.2 Observations

Measured link streams: [ z(t) = H,x(t) + \eta(t), ] where (x(t)) are latent node states and (\eta(t)) is link-local noise.

Only (z(t)) and (H) are required.

---

3. Input/Output Contracts

3.1 Input data format (hard gate)

File format: HDF5

Required structure

/comparisons/link_001 (T,)

/comparisons/link_002 (T,)

…

/comparisons/link_M (T,)

Required HDF5 attributes

• sampling_interval : float (seconds)

• unit : one of seconds, phase_cycles, fractional_frequency

• link_topology_sha256 : hex digest binding data to the topology file

Kill: missing any required attribute → input_nonconformant

3.2 Topology file format

File: network.incidence_matrix.csv

Header:

link_id,node_i,node_j,sign_i,sign_j

Row convention: a single link between nodes (i) and (j) is represented by a row with (+1) at node (i) and (-1) at node (j). Any equivalent signed convention is admissible if it is consistent and hashed.

---

4. Residual Formation (Gauge Projection)

4.1 Projection matrix construction (explicit)

Gauge freedom (absolute time) is removed by projection onto the row space of (H): [ P = H,H^{+}, ] where (H^{+}) is the Moore–Penrose pseudoinverse.

Implementation note: use a numerically stable pseudoinverse for rank-deficient matrices.

Connectivity validation: for a connected network, [ \mathrm{rank}(P) = N-1. ]

4.2 Residual definition

Residuals: [ r(t) = P,z(t). ]

---

5. Blocked Covariance Estimation

5.1 Blocking

For a chosen block duration (\tau), segment (r(t)) into (B) non-overlapping blocks, yielding the matrix [ R(\tau)\in\mathbb{R}^{M\times B}. ]

5.2 Sample covariance with regularisation

[ \widehat{C}(\tau) = \frac{1}{B}R(\tau),R(\tau)^{\mathsf T} + \varepsilon_{\mathrm{reg}} I, \qquad \varepsilon_{\mathrm{reg}} = 10^{-12},\mathrm{Tr}!\left(\frac{1}{B}R R^{\mathsf T}\right). ]

---

6. Spectral Decision Logic (RMT Gate)

Let ({\lambda_k}_{k=1}^{M}) denote eigenvalues of (\widehat{C}(\tau)), sorted descending.

6.1 MP boundary

Define the aspect ratio [ q = \frac{M}{B}. ] The upper Marčenko–Pastur edge is [ \lambda_{\mathrm{MP}} = \sigma^2(1+\sqrt{q})^2, ] where (\sigma^2) is the noise-floor variance.

6.2 Noise-floor estimation (robust)

Iterative median clipping (max 3 iterations)

1. Initialise (\hat{\sigma}^2 \leftarrow \mathrm{median}({\lambda_k}))

2. Compute (\lambda_{\mathrm{MP}}(\hat{\sigma}^2))

3. Remove all (\lambda_k > \lambda_{\mathrm{MP}})

4. Recompute (\hat{\sigma}^2) on remaining eigenvalues

5. Repeat until convergence or iteration limit

Kill: if more than (M/2) eigenvalues are clipped → noise_floor_contaminated

---

7. Coherence Metrics

7.1 Effective rank

[ \widehat{R}{\mathrm{eff}}(\tau) = #{k : \lambda_k > \lambda{\mathrm{MP}}}. ]

7.2 Coherence index

[ \widehat{\kappa}(\tau) = \frac{\sum_{k=1}^{\widehat{R}{\mathrm{eff}}}\lambda_k}{\sum{k=1}^{M}\lambda_k}. ]

7.3 Uncertainty on (\widehat{\kappa}) (explicit)

Estimate (\Delta\widehat{\kappa}) via block bootstrap:

• resample blocks of (R(\tau)) with replacement

• (L=50) blocks per replicate

• (10^{4}) replicates

[ \Delta\widehat{\kappa} = \mathrm{std}\left({\widehat{\kappa}^{(b)}}\right). ]

---

8. Hypothesis Test

8.1 Hypotheses

• (\mathcal{H}_0): residuals consistent with finite-sample noise (no detectable shared mode)

• (\mathcal{H}_1): at least one shared mode exceeds the noise bulk

8.2 Decision rule (hard)

Reject (\mathcal{H}0) at scale (\tau) if [ \widehat{R}{\mathrm{eff}}(\tau)\ge 1 \quad\land\quad \lambda_1(\tau) > \lambda_{\mathrm{MP}}(\tau). ]

---

9. Kill Conditions (Hard Stops)

Condition	Threshold	Meaning
Autocorrelation	\(|\rho_1| < 0.1\)	Blocks sufficiently independent
Conditioning	\(\mathrm{cond}(\widehat{C}) < 10^8\)	Numerical stability (no near-singularity)
Block count	\(B \ge 200\)	RMT convergence regime
MP validity	\(M/B \le 0.1\)	Do not apply MP threshold outside regime
Noise-floor contamination	\(>\!M/2\) clipped eigenvalues	Median estimator invalid (dense signal)
Input conformant	Required HDF5 attrs present	Provenance and schema integrity

---

10. Multi-Scale Execution Contract (Multi-(\tau))

10.1 Sweep definition

Choose (\tau)-grid: [ \tau \in {\tau_{\min}, 2\tau_{\min}, 4\tau_{\min}, \dots, \tau_{\max}}. ]

10.2 Trial-factor correction

If testing (N_\tau) scales, apply Bonferroni correction: [ \alpha \rightarrow \alpha/N_\tau. ]

10.3 Integration rule

Report [ \widehat{R}{\mathrm{eff}} = \max{\tau}\widehat{R}_{\mathrm{eff}}(\tau), ] and store all per-(\tau) results in the certificate under "multi_tau_results".

---

11. Certificate Artifact (Machine-Readable Output)

11.1 Required fields

{
  "protocol_version": "0.1.0",
  "execution_timestamp": "2025-12-18T00:00:00Z",
  "topology_hash": "sha256:…",
  "data_provenance": {
    "source_file": "…",
    "link_topology_sha256": "…",
    "preprocessing_hash": "sha256:…"
  },
  "parameters": {
    "M": 12,
    "tau_grid": [1000, 2000, 4000, 8000, 16000],
    "B_min": 200,
    "alpha": 0.01,
    "bonferroni_alpha": 0.002
  },
  "results": {
    "R_eff_max": 1,
    "tau_at_R_eff_max": 8000,
    "multi_tau_results": [
      {
        "tau": 8000,
        "B": 500,
        "lambda_MP": 1.23,
        "top_eigenvalues": [2.45, 1.18, 1.12],
        "R_eff": 1,
        "kappa": 0.034,
        "dkappa": 0.008,
        "decision": "reject_H0"
      }
    ]
  },
  "kill_flags": {
    "input_nonconformant": false,
    "autocorr_violation": false,
    "condition_violation": false,
    "block_count_violation": false,
    "mp_validity_violation": false,
    "noise_floor_contaminated": false
  }
}

11.2 Contract with higher levels

• If any kill_flags.* == true → downstream modules must not run

• If R_eff_max == 0 → downstream modules must not run

• If decision == "reject_H0" → downstream modules may proceed only on certified subspace

---

12. Figures

Figure 0-A — Validation Flow (Mermaid)

graph LR subgraph L0[Level 0: Validation Loop] direction LR Z[z(t)] -->|Projection P| R[r(t)] R -->|Block into B| B1[Blocks] B1 -->|Covariance Ĉ(τ)| C[Ĉ] C -->|Eigenvalues| L[λ_k] L -->|MP Threshold| MP[λ_MP] L -->|Compute| K[κ̂, R̂_eff] K --> D{Valid?} D -- Yes --> Cert[Issue Certificate] D -- No --> Kill((Kill Stop)) end

Figure 0-B — Spectral Gate Intuition (Operator Schematic)

Interpretation: in (\mathcal{H}_0), all eigenvalues remain within the MP bulk. A detectable shared mode produces a top eigenvalue (\lambda_1) separating from the bulk.

eigenvalue density ^ | * | * * signal spike (λ1) | * * | * * * * * * * * * noise bulk (MP) +----------------------------------------------> λ λ_MP

13. Interpretation Rules (Guardian)

• (\widehat{\kappa} > 0) does not imply causality or a specific mechanism

• A null result ((\widehat{R}_{\mathrm{eff}}=0)) is informative and bounds shared coupling below detectability

• MP threshold use is void outside (M/B \le 0.1)

• Any post-hoc threshold modification requires a new version

---

References

1. Marčenko, V. A. & Pastur, L. A. (1967). Distribution of eigenvalues for some sets of random matrices. Math. USSR-Sb. 1, 457–483.

2. Baik, J., Ben Arous, G. & Péché, S. (2005). Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33, 1643–1697.

3. Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Stat. 29, 295–327.

4. Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis (3rd ed.). Wiley.

5. Efron, B. & Tibshirani, R. (1993). An Introduction to the Bootstrap. Chapman & Hall/CRC.

6. Ledoit, O. & Wolf, M. (2004). A well-conditioned estimator for large-dimensional covariance matrices. J. Multivar. Anal. 88, 365–411.

---

Version History

Version	Date	Changes
0.1.0	2025-12-18	Initial operational specification: explicit projection, robust noise-floor estimation, multi-\(\tau\) contract, schema + certificate, Mermaid figures, locked kill conditions

```