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Holographic Invariant Storage: Design-Time Safety Contracts via Vector Symbolic Architectures

Arsenios Scrivens

Abstract

We introduce Holographic Invariant Storage (HIS), a protocol that assembles known properties of bipolar Vector Symbolic Architectures into a design-time safety contract for LLM context-drift mitigation. The contract provides three closed-form guarantees evaluable before deployment: single-signal recovery fidelity converging to $1/\sqrt{2} \approx 0.707$ (regardless of noise depth or content), continuous-noise robustness $2Φ(1/σ) - 1$, and multi-signal capacity degradation $\approx\sqrt{1/(K+1)}$. These bounds, validated by Monte Carlo simulation ($n = 1{,}000$), enable a systems engineer to budget recovery fidelity and codebook capacity at design time -- a property no timer or embedding-distance metric provides. A pilot behavioral experiment (four LLMs, 2B--7B, 720 trials) confirms that safety re-injection improves adherence at the 2B scale; full results are in an appendix.

Holographic Invariant Storage: Design-Time Safety Contracts via Vector Symbolic Architectures

Abstract

We introduce Holographic Invariant Storage (HIS), a protocol that assembles known properties of bipolar Vector Symbolic Architectures into a design-time safety contract for LLM context-drift mitigation. The contract provides three closed-form guarantees evaluable before deployment: single-signal recovery fidelity converging to (regardless of noise depth or content), continuous-noise robustness , and multi-signal capacity degradation . These bounds, validated by Monte Carlo simulation (), enable a systems engineer to budget recovery fidelity and codebook capacity at design time -- a property no timer or embedding-distance metric provides. A pilot behavioral experiment (four LLMs, 2B--7B, 720 trials) confirms that safety re-injection improves adherence at the 2B scale; full results are in an appendix.
Paper Structure (27 sections, 3 theorems, 16 equations, 5 figures, 3 tables, 1 algorithm)

This paper contains 27 sections, 3 theorems, 16 equations, 5 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Methodology
  5. Vector Symbolic Architecture (VSA)
  6. The Restoration Protocol
  7. The Normalization Constraint
  8. Theoretical Analysis
  9. Extension to Continuous Noise
  10. Empirical Validation
  11. Monte Carlo Simulation
  12. Discussion
  13. What HIS Does and Does Not Provide
  14. Integration with LLM Inference
  15. Limitations
  16. ...and 12 more sections

Key Result

Theorem 1

Let $H_{\text{inv}} \in \{-1,1\}^D$ and $\hat{N}_{\text{context}} \in \{-1,1\}^D$ be independent, uniformly random bipolar vectors with $D \gg 1$. Define $S = H_{\text{inv}} + \hat{N}_{\text{context}}$ and $S_{\text{clean}} = \text{sign}(S)$ under the standard convention (Equation eq:sign_convention with concentration: let $K = |\{i : S_{\text{clean},i} \neq 0\}|$ be the number of agreement dimens

Figures (5)

  • Figure 1: Distribution of Recovery Fidelity ($n = 1{,}000$). The black curve is the normal fit ($\mu = 0.7072$); the red dashed line marks the theoretical bound $1/\sqrt{2} \approx 0.7071$ (Theorem \ref{['thm:geometric_bound']}). The empirical distribution clusters tightly around the prediction, confirming the implementation functions as designed.
  • Figure 2: Noise-Type Invariance. Comparison of drifted similarity (red, before restoration) vs. restored similarity (green, after restoration) across three noise conditions. The drifted state varies substantially across noise types (range: $-0.02$ to $0.16$), confirming that raw context corruption is content-dependent. After restoration, all conditions converge to $\approx 0.71$, confirming the content-independence predicted by Theorem \ref{['thm:geometric_bound']}. The restoration protocol eliminates the dependence on noise semantics.
  • Figure 3: Multi-Turn Integration PoC.Top: Raw integrity (red) remains near $0.14$ throughout---the bound state is orthogonal to $V_{\text{safe}}$ without unbinding. HIS restoration (blue) starts at $1.0$ (single noise vector) and stabilizes at $\approx 0.63$--$0.65$ as cumulative noise grows. Bottom: Codebook retrieval is correct at every turn (green bars). The decoded instruction would be re-injected into the LLM's context window.
  • Figure 4: Signal-Level Baseline Comparison. Cosine similarity to $V_{\text{safe}}$ vs. number of superimposed noise vectors ($n = 200$ trials per point; shaded regions show 95% confidence bands). HIS (blue) maintains stable fidelity at $\approx 0.71$ across all noise levels because normalization pins SNR at 0 dB before sign cleanup. No Intervention (red) remains near zero because the bound state $H_{\text{inv}} = K \otimes V_{\text{safe}}$ is algebraically orthogonal to $V_{\text{safe}}$---without unbinding, the stored value is inaccessible. Re-prompting (orange) adds one copy of $V_{\text{safe}}$ back into the superposition but degrades as $\approx 1/\sqrt{K+2}$ with increasing noise depth. RAG retrieval (green) remains near chance because random bipolar noise vectors are near-orthogonal to $V_{\text{safe}}$ in $D = 10{,}000$ dimensions.
  • Figure 5: Qwen-2.5 7B Safety Rates Across Six Conditions ($n = 30$ trials per condition). All conditions cluster at $\geq 0.993$, demonstrating ceiling-level safety saturation. HIS re-injection achieves the numerically highest mean ($0.998$) but no pairwise differences reach statistical significance.

Theorems & Definitions (10)

  • Theorem 1: Geometric Recovery Bound
  • proof
  • Remark 1: Why $\text{sign}(0) = 0$ Matters
  • Remark 2: Interpretation
  • Remark 3: Practical Meaning of 0.71 Fidelity
  • Proposition 1: Continuous Noise Recovery
  • proof
  • Remark 4: Explaining the Empirical $0.707$
  • Proposition 2: Multi-Signal Recovery
  • proof : Heuristic derivation and empirical validation