Trajectory Continuity as Existence Verification

Abstract

IEPP (Intrinsic Entropy Proof Protocol) proposes a structural method for existence verification based on trajectory continuity rather than static identity or pre-shared secrets.

Unlike traditional authentication mechanisms that rely on shared secret keys, IEPP explores a verification model where existence is demonstrated through the continuity of an evolving internal state trajectory — a property that may be difficult to replicate under the stated threat model without access to equivalent runtime entropy conditions.

100% immediate trajectory divergence upon cloning (100 runs)
0% attacker success rate across five attack categories (50,000 attempts)
Stable three-layer verification structure across diverse entropy configurations

IEPP is positioned as an exploratory trust primitive for AI agents, digital entities, and distributed autonomous systems operating in environments where replication cost approaches zero.

Core structural concepts have been filed under the PCT framework. Specific implementation details are disclosed progressively following filing milestones.

1. Problem Statement

As generative AI systems scale, the marginal cost of producing synthetic content approaches zero. This creates a structural authenticity imbalance: distinguishing a genuine AI agent from a replica becomes increasingly difficult using traditional methods.

Static identity model
identity represented by hash, certificate, or key.
replicable when copied.

Shared secret model
verifier and prover share a secret key.
requires prior distribution of trust.

Neither model directly addresses the structural property of AI agents:

An entity may be bit-for-bit identical to another entity at the moment of cloning, yet their subsequent existence trajectories may diverge.

IEPP reframes the verification question:

Is this entity on the same continuous trajectory as before?

2. Core Concept: Trajectory as Identity

2.1 From Possession to Continuity

Traditional authentication binds identity to possession of a secret.

IEPP explores identity signals derived from continuity of an evolving internal state trajectory.

Even if an adversary observes all public outputs R₁, R₂, …, Rₙ and knows the algorithm structure, reconstructing the internal state trajectory required to generate Rₙ₊₁ may be difficult without equivalent runtime entropy conditions.

This property is referred to as Non-deterministic State Dependency.

2.2 Trajectory Reconstruction Problem (TRP)

Adversary objective:

Given:
R₁ ... Rₙ
C₁ ... Cₙ

Find:
internal_stateₙ

TRP is currently a conjectured hardness problem rather than a formally proven one.

Difficulty depends on:

state dimensionality
runtime stochasticity
limited observability

2.3 Distinction from Existing Approaches

Property	IEPP	HMAC	Signature	Hash Chain
pre-shared secret	No	Yes	Yes	No
verifier stores secret	No	Yes	Yes	No
replay resistance	Yes	Partial	Partial	No
continuity signal	Yes	No	No	Partial
post-fork anomaly signal	Exploratory	No	No	No

3. Three-Layer Verification Structure

Layer 1 — Challenge Binding

Rₙ₊₁ = H(Rₙ || Cₙ₊₁ || commitment(stateₙ₊₁))

Prevents replay and precomputation attacks.

Layer 2 — Commitment Chain

commitmentₙ = H(stateₙ)

Verifier confirms structural consistency without observing internal state.

Layer 3 — Statistical Continuity

Verifier evaluates statistical plausibility of trajectory behavior:

norm continuity
entropy stability
distribution shift
trajectory smoothness

Layer 3 functions as an anomaly indicator in v0.3. Clone discrimination remains an open research problem.

4. Security Assumptions

Threat Scope

Level 1: external observer
Level 2: process-level adversary
Level 3: infrastructure adversary (out of scope)
Level 4: physical adversary (out of scope)

Security derives from runtime entropy availability and trajectory reconstruction difficulty under stated assumptions.

5. Experimental Validation

Baseline metrics

metric	mean	std
max_jump	0.471	0.135
distribution_shift	0.432	0.022
response_entropy	3.996	0.001

Fork divergence

metric	value
fork_match_rate	0.0
immediate_divergence	1.0
first_divergence	step 13

Attacker results

50,000 attempts across five attack categories resulted in 0% success.

Layer 3 results

segment	continuity_score
pre-fork	0.586
post-fork original	0.842
post-fork clone	0.827

Difference between post_orig and post_fork is 0.014. Insufficient for reliable clone discrimination in v0.3.

6. Open Problems

formal TRP hardness definition
clone discrimination metrics
minimum entropy conditions
hardware entropy integration

7. Scope

IEPP explores trajectory continuity as an additional verification signal.

Potential domains:

AI agents
synthetic media provenance
digital avatars
distributed autonomous systems
virtual device identity

7.1 Experimental Clarification (v0.3 Update)

Early v0.3 exploration considered whether clone discrimination might be achieved through statistical trajectory similarity, including entropy distribution stability and response-pattern similarity.

Subsequent controlled Colab experiments clarified that this framing is incomplete.

Forked provers diverged immediately after sharing the same anchor response, yet long-window statistical summaries often converged toward similar distributions.

Observed characteristics included:

trajectory statistics showing limited long-window separation
small continuity score differences
point-pattern similarity convergence
statistical metrics behaving as weak discriminators

These observations suggest that statistical continuity is better interpreted as an anomaly indicator rather than a canonical identity discriminator.

7.2 Canonical Continuity Result

When the original post-chain is treated as a canonical ledger, a clear separation emerges.

Experimental observation:

forked provers share the same anchor response
divergence occurs immediately after fork
original chain reproduces canonical sequence exactly
forked chain fails to reproduce canonical continuation from the first post-fork response

Observed result structure:

canonical original match rate = 100%
canonical fork match rate = 0%
first mismatch index = step 0

This indicates that a replica may inherit the same prior state, yet cannot reproduce the canonical next response of the original continuity.

Therefore, the decisive verification question is not whether two trajectories appear statistically similar.

The decisive question is:

Can this prover generate the valid next response in the recorded lineage?

7.3 Interpretation

IEPP identity is not primarily determined by statistical resemblance.

It is determined by continuity of lineage.

Two entities may appear statistically similar while belonging to different continuity branches.

Conversely, a continuous entity may exhibit statistical variation while preserving lineage integrity.

This suggests that trajectory statistics should be interpreted as supporting signals, while canonical continuity provides a stronger identity criterion.

Authenticity is therefore defined not by resemblance, but by continuity.

8. Experimental Clarification and Direction Update (v0.3)

Statistical similarity may overlap between original and clone trajectories,
but canonical lineage continuity diverges immediately after fork.

Early v0.3 exploration considered whether clone discrimination might be achieved primarily through statistical trajectory similarity, including entropy distribution stability, trajectory smoothness, and response-pattern similarity metrics.

Subsequent controlled experiments clarified that this framing is incomplete.

Forked provers diverged immediately after sharing the same anchor response, yet long-window statistical summaries frequently converged toward similar distributions.