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11.1 Anonymity as Entropy, Not Obscurity

In Abyss, privacy is modeled as entropy in the transaction graph, not as mere absence of labels. The anonymity set represents the set of all plausible deposit commitments that could correspond to a given withdrawal. The larger and more heterogeneous this set, the higher the uncertainty faced by an observer attempting to correlate flows. Entropy increases as:
  • The number of active commitments grows
  • Deposit and withdrawal timing overlaps
  • Withdrawal sizes vary naturally
  • Usage patterns diversify across users and applications
Abyss does not rely on hiding metadata at the consensus layer. Instead, it maximizes entropy at the asset-transition layer, where correlation attacks are most effective.

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11.2 Growth Dynamics of the Anonymity Set

Let: 
C(t) := number of active commitments at time t
The effective anonymity of a withdrawal at time t is bounded by: 
AnonymityEffective(t) ≤ C(t)
Each new deposit strictly increases C(t). Unlike fragmented systems, no deposit is isolated into a sub-pool. This creates a positive-sum privacy dynamic: every participant improves the privacy of every other participant.

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11.3 Temporal Overlap and Mixing

Temporal overlap between deposits and withdrawals is critical. If all users deposit and immediately withdraw, anonymity collapses. Abyss does not enforce delays, but it is designed to support temporal decoupling. Best practices include:
  • Delaying withdrawals
  • Splitting withdrawals across time
  • Using non-uniform amounts
These behaviors increase the number of plausible mappings between deposits and withdrawals.

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11.4 Entropy Degradation Factors

Certain behaviors reduce effective anonymity:
  • Immediate full withdrawals after deposit
  • Unique withdrawal patterns
  • Low pool utilization periods
  • Single-user dominance
Abyss does not prevent these behaviors but does not optimize for them either. Privacy is probabilistic and user-dependent.

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11.5 Infinite Withdrawals and Entropy Preservation

The infinite withdrawal model preserves entropy over time. Unlike note exhaustion systems, commitments remain active even after partial withdrawals. This prevents the anonymity set from shrinking and avoids signaling effects associated with note consumption.

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11.6 Interaction with External Systems

When integrated with merchants, exchanges, or applications, Abyss increases entropy through natural transaction diversity. Payments, settlements, and payouts contribute to organic mixing without requiring user coordination.

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11.7 Summary

Anonymity in Abyss is cumulative and emergent. It improves with adoption, composability, and time. The protocol provides the cryptographic foundation; entropy is produced by real economic activity.