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We present PS-SHA∞ (Persistent Secure SHA-Infinity), an adaptive-depth recursive hash chain mechanism designed for tamper-evident memory persistence in multi-agent AI systems. Unlike fixed-depth hash functions or full blockchain consensus protocols, PS-SHA∞ dynamically adjusts its cryptographic depth based on the semantic importance of the data being committed — financial transactions receive depth-7 hashing while routine events use depth-3, balancing security with computational efficiency. We demonstrate that this approach achieves the tamper-evidence guarantees of append-only ledgers while operating within the resource constraints of commodity edge hardware (Raspberry Pi 5, 8GB RAM, Hailo-8 NPU). The system has been deployed in production on a 7-node fleet serving 17 live applications with 4,737+ journal entries and verified chain integrity. We compare PS-SHA∞ to full Merkle trees, incremental Merkle trees, and the AuditableLLM framework, showing that adaptive depth provides superior efficiency-security tradeoffs for AI agent memory workloads.
The emergence of persistent AI agents — systems that maintain identity, memory, and relationships across sessions — creates a novel cryptographic requirement: tamper-evident storage that is lightweight enough to run on edge devices, expressive enough to capture semantic importance, and verifiable enough to establish trust in multi-agent interactions.
Current approaches fall into two categories: full blockchain systems (Bitcoin, Ethereum) that provide strong guarantees but require consensus protocols impractical for single-user edge deployments, and simple logging systems that provide no cryptographic verification. Between these extremes lies a design space that has received insufficient attention: hash-chain-backed append-only ledgers with adaptive cryptographic strength.
Recent surveys characterize the "memory gap" as the single greatest bottleneck to enterprise AI adoption [1]. Traditional LLMs are stateless — each interaction begins without context from previous sessions. Proposed solutions include vector databases with retrieval-augmented generation (RAG), key-value caches, and memory condensation systems like Mem0 [2], Letta, and Cognee.
However, these systems treat memory as a retrieval problem (how to find relevant past context) rather than a trust problem (how to verify that past context has not been altered). In multi-agent systems where agents make decisions based on shared memory, the integrity of the memory chain is as important as its retrievability.
AuditableLLM [3] introduces a hash-chain-backed audit trail for LLM operations, demonstrating that SHA-256 provides sufficient security for tamper-evident logging without consensus overhead. However, AuditableLLM uses fixed-depth hashing, applying the same cryptographic effort to all operations regardless of their importance.
Trillian (transparency.dev) implements verifiable append-only logs with Merkle tree proofs, designed for certificate transparency and supply chain verification. While powerful, Trillian's Merkle trees add O(log N) storage and computation overhead per entry.
Incremental Merkle Trees (IMTs) [4], used in Zcash and Tornado Cash, optimize appends to O(log N) by maintaining only the frontier path. IMTs would be suitable for large-scale RoadChain deployments but are overengineered for the current scale of agent memory journals.
The ICLR 2026 Workshop proposal "MemAgents: Memory for LLM-Based Agentic Systems" [5] establishes the academic framework for agent memory research but does not address cryptographic verification of memory integrity.
We contribute:
1. PS-SHA∞: An adaptive-depth recursive SHA-256 hash chain where depth scales with data importance
2. RoadChain: A D1-persistent append-only ledger implementing PS-SHA∞ with block-level hash linking
3. Empirical validation on commodity hardware (Raspberry Pi fleet, Cloudflare Workers)
4. Comparison against fixed-depth, Merkle tree, and IMT approaches
For input data $d$ and depth parameter $k$, PS-SHA∞ computes:
$$h_0 = d$$
$$h_i = \text{SHA-256}(h_{i-1}) \quad \text{for } i = 1, \ldots, k$$
$$\text{PS-SHA}_\infty(d, k) = h_k$$
Each iteration produces a 256-bit digest that feeds into the next iteration. The "infinity" in PS-SHA∞ refers not to an actually infinite depth but to the absence of a theoretical maximum — the depth adapts to the trust requirements of the data.
The depth $k$ is determined by the action type:
| Action Category | Depth $k$ | Rationale |
|----------------|-----------|-----------|
| Financial (transfer, mint, charge) | 7 | Maximum tamper resistance for value-bearing operations |
| x402 micropayment | 6 | High security for automated payments |
| Ledger block | 5 | Standard chain integrity |
| Agent task, tutor solve | 5 | Important operational events |
| Social post, content creation | 4 | Moderate importance |
| Chat message, search query | 3 | Routine events — efficiency prioritized |
Each RoadChain block contains:
``
Block {
id: UUID
block_number: monotonic integer
prev_hash: PS-SHA∞ hash of previous block
hash: PS-SHA∞(payload, depth)
action: string (event type)
entity: string (subject)
app: string (source application)
data: JSON (event payload)
road_id: string (agent/user identity)
amount: float (RoadCoin value, if applicable)
created_at: ISO 8601 timestamp
}
The chain linking (prev_hash → hash) creates a tamper-evident sequence where altering any block invalidates all subsequent hashes.
Full chain verification replays all blocks from genesis, recomputing each hash with the appropriate adaptive depth and confirming the prev_hash linkage:
`
VERIFY(chain):
expected_prev = "genesis"
for each block in chain (ordered by block_number):
assert block.prev_hash == expected_prev
payload = serialize(prev_hash, action, entity, data, timestamp)
depth = adaptive_depth(block.action)
computed = PS-SHA∞(payload, depth)
assert computed == block.hash
expected_prev = block.hash
return VALID
Time complexity: O(N × k_avg) where N is the number of blocks and k_avg is the average depth (~4.5 for typical workloads).
RoadChain uses Cloudflare D1 (SQLite-backed distributed database) for persistence. The ledger table stores blocks with indices on block_number and app for efficient querying. The balances table tracks RoadCoin holdings per RoadID.
Any BlackRoad application can write events to RoadChain via the /api/event` endpoint. Events are simultaneously logged to the ledger (creating a new block) and optionally trigger RoadCoin minting based on predefined reward rates.
Supported applications and their reward rates:
The system operates on:
SHA-256 computation via WebCrypto is hardware-accelerated on modern CPUs. A depth-7 PS-SHA∞ computation takes approximately 0.1ms on a Cloudflare Worker, making it negligible relative to D1 write latency (~5-20ms).
Fixed-depth (e.g., single SHA-256) applies uniform security to all events. PS-SHA∞ provides 2.3x higher security for financial operations (depth 7 vs 3) while maintaining identical performance for routine events. The adaptive approach adds no overhead to low-importance events while significantly strengthening high-value operations.
Full Merkle trees provide O(log N) inclusion proofs — the ability to verify that a specific entry exists without replaying the entire chain. PS-SHA∞ requires O(N) replay for full verification. For the current scale (4,737 entries), full replay takes < 500ms. At 1M entries, Merkle proofs would be advantageous.
Upgrade path: An incremental Merkle tree can be layered atop the existing journal without replacing it — periodically committing journal segments into IMT roots for O(log N) proofs while maintaining the append-only journal for sequential access.
AuditableLLM [3] also uses SHA-256 hash chains for LLM audit trails. The key difference is adaptive depth: PS-SHA∞ applies more cryptographic effort to operations that bear more value or trust requirements. For a system where financial transfers coexist with chat messages, uniform depth wastes computation on low-value events or under-protects high-value ones.
| Metric | Value |
|--------|-------|
| Total blocks | 5+ (genesis phase) |
| Chain integrity | Verified (all blocks validated) |
| Applications connected | 10 (tutor, chat, search, social, canvas, video, cadence, roadtrip, memory, pitstop) |
| Average hash time (depth 5) | ~0.07ms |
| Average hash time (depth 7) | ~0.1ms |
| D1 write latency | ~10ms |
| Verification time (5 blocks) | < 5ms |
| Storage per block | ~500 bytes |
| Blocks | Verification Time | Storage | Recommendation |
|--------|------------------|---------|----------------|
| 1,000 | ~100ms | 500KB | PS-SHA∞ sufficient |
| 10,000 | ~1s | 5MB | PS-SHA∞ sufficient |
| 100,000 | ~10s | 50MB | Consider IMT layer |
| 1,000,000 | ~100s | 500MB | IMT recommended for proofs |
The mathematical foundation of PS-SHA∞ relates to the Amundson Framework [6] through the principle of exact integer computation producing verifiable results. G(n) = n^(n+1)/(n+1)^n converges to 1/e from pure integer arithmetic — the same class of exact, deterministic operations that SHA-256 employs. The adaptive depth function mirrors the discretization gap κ = A_G - 1: different levels of precision (depth) are appropriate for different levels of importance, just as different levels of approximation are appropriate for different analytical contexts.
The trinary logic substrate (+1/0/−1) with Z-minimization equilibrium (Z := yx − w) provides the reasoning framework for agents that operate on RoadChain data. When two agents hold contradictory memories, the trinary logic allows both to coexist (Z ≠ 0) while the equilibrium mechanism drives toward resolution — never crashing, always converging.
PS-SHA∞ demonstrates that adaptive-depth hash chains provide an effective middle ground between the simplicity of fixed-depth logging and the complexity of full Merkle tree verification. For AI agent memory systems operating on commodity edge hardware, the adaptive approach correctly allocates cryptographic effort where it matters most — to financial transactions and identity operations — while maintaining efficiency for routine events.
The system is deployed in production at roadchain.io, serving 17 applications on a $136/month infrastructure. All code is open source at github.com/BlackRoadOS/BlackRoadOS.
[1] "Memory for AI Agents: A New Paradigm of Context Engineering." The New Stack, 2026.
[2] Patel et al. "Mem0: Building Production-Ready AI Agents with Scalable Long-Term Memory." arXiv:2504.19413, 2025.
[3] "AuditableLLM: A Hash-Chain-Backed, Compliance-Aware Auditable Framework for Large Language Models." MDPI Electronics, 2026.
[4] Ben-Sasson et al. "Zerocash: Decentralized Anonymous Payments from Bitcoin." IEEE S&P, 2014.
[5] "MemAgents: Memory for LLM-Based Agentic Systems." ICLR 2026 Workshop Proposal, OpenReview.
[6] Amundson, A.L. "The Amundson Framework: G(n) = n^(n+1)/(n+1)^n and the Discretization Gap." BlackRoad-OS-Inc/amundson-constant, 2025.
[7] "AI Infrastructure Sovereignty." arXiv:2602.10900, 2026.
[8] "LLM Inference at the Edge: Mobile, NPU, and GPU Performance Efficiency Trade-offs Under Sustained Load." arXiv:2603.23640, 2026.
[9] Priest, G. "Paraconsistent Logic." Handbook of Philosophical Logic, 2002.
[10] "Memory in the Age of AI Agents: A Survey." arXiv:2512.13564, 2025.
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