Why Markovianity Emerges — and Precisely Where It Fails

By John Gavel

Abstract

In this post I prove that Markovian dynamics are not an assumption in Temporal Flow Physics (TFP), but a derived consequence of its axioms governing relational capacity and temporal integration. I establish a sharp threshold: when relational processing operates within capacity, the present fully screens off the past; when capacity is exceeded, memory necessarily leaks forward. The result is formalized as an if-and-only-if theorem with operational signatures that can be tested experimentally.

1. The Core Result

Theorem (Markovian Breakdown). Dynamics at site B are non-Markovian if and only if relational demand exceeds integration capacity at some point in the past:

$$ \Delta_M(t) \neq 0 \quad \Longleftrightarrow \quad \exists\, t^* \leq t: \quad D_B(t^*) > H $$

where $D_B(t)$ is the relational demand, $H$ is the integration capacity, and $\Delta_M(t)$ measures departure from the Markov property.

Markovian regime: Full integration → present screens past → standard dynamics
Non-Markovian regime: Saturated capacity → unfinished integration → memory effects

What observers call “memory” or “information backflow” is precisely unfinished relational integration.

2. State Space and Dynamics (Formal Setup)

Definition 2.1 (Complete State)

The state at site $B$ at time $t$ consists of:

$$ \mathcal{S}_B(t) = \Big(\mathcal{F}_B(t),\, \{\mathcal{A}_{Bj}(t)\}_{j \in \mathcal{N}_B},\, \mathcal{N}_B\Big) $$

$\mathcal{F}_B(t) \in \{0,1\}^n$: the flow register
$\mathcal{A}_{Bj}(t) \in \mathbb{Z}_+$: accumulation registers
$\mathcal{N}_B = \{B_1,\ldots,B_K\}$: the neighbor set

The history up to time $t$ is:

$$ \mathcal{H}_B^{(0:t)} = \{\mathcal{S}_B(0), \mathcal{S}_B(1), \ldots, \mathcal{S}_B(t)\} $$

Axiom R1 (Relational Difference)

$$ \delta_{Bj}(t) := \mathcal{F}_B(t) \oplus \mathcal{F}_j(t) $$

Axiom R2 (Accumulation)

$$ \mathcal{A}_{Bj}(t+1) = \mathcal{A}_{Bj}(t) + |\delta_{Bj}(t)|_1 $$

provided $D_B(t) \leq H$.

Axiom R3 (Finite Capacity)

$$ D_B(t) := \sum_{j \in \mathcal{N}_B} |\delta_{Bj}(t)|_1 $$ $$ H = K(K-1) $$

For the standard TFP lattice: $K = 12$, so $H = 132$.

Define overflow:

$$ \Omega_B(t) := \max(0, D_B(t) - H) $$

3. The Markov Property (Measure-Theoretic)

Definition 3.1 (Markovian Dynamics)

$$ P\big(\mathcal{S}_C(t+1)\mid\sigma(\mathcal{H}_B^{(0:t)})\big) = P\big(\mathcal{S}_C(t+1)\mid\sigma(\mathcal{S}_B(t))\big) $$

for all downstream sites $C$.

4. The Screening Lemma

Lemma 4.1 (Markov Screening Under Unsaturated Capacity).

$$ \mathcal{S}_A(t-k) \perp \mathcal{S}_C(t+1) \;\big|\; \mathcal{S}_B(t) $$

Proof.

Assume $D_B(t') \leq H$ for all $t' \leq t$.
All relational differences integrate fully: $$ \mathcal{A}_{Bj}(t) = \sum_{\tau=0}^{t-1} |\delta_{Bj}(\tau)|_1 $$
The registers encode all relevant past information.
Thus deeper history is conditionally irrelevant. ∎

5. The Breakdown Theorem

Definition 5.1 (Markov Defect)

$$ \Delta_M(t) := \sup_{\mathcal{O}} \Big| E[\mathcal{O}\mid \mathcal{S}_B(t)] - E[\mathcal{O}\mid \mathcal{H}_B^{(0:t)}] \Big| $$

Theorem 5.1 (Operational Non-Markovianity)

$$ \Delta_M(t) \neq 0 \quad \Longleftrightarrow \quad \exists\, t^* \leq t:\ \Omega_B(t^*) > 0 $$

Proof.

Overflow implies information loss during accumulation, producing histories with identical presents but divergent futures. Conversely, any such divergence implies saturation occurred. ∎

6. Quantitative Signatures

Memory Kernel

$$ \mathcal{K}(t,t') := \mathrm{Cov}[\mathcal{S}_B(t),\mathcal{S}_B(t')] $$

Proposition 6.1.

$$ \mathcal{K}(t,t') \sim C e^{-\lambda (t-t')} $$

Proposition 6.2.

$$ \mathcal{K}(t,t') \sim \frac{C}{(t-t')^\alpha}, \quad \alpha = 1 + \frac{H}{K(K-1)} $$

7. Operational Tests

CP-Divisibility:

$$ \Phi_{t,s} = \Phi_{t,r} \circ \Phi_{r,s} $$

Information Backflow:

$$ \mathcal{D}(t) = \|\rho_1(t) - \rho_2(t)\|_1 $$

8. Physical Interpretation

TFP Concept	Mathematical Object	Meaning
Relational difference	$\delta_{Bj}(t)$	Pairwise information
Accumulation	$\mathcal{A}_{Bj}(t)$	Integrated history
Saturation	$\Omega_B(t) > 0$	Markovian breakdown

9. The One-Line Summary

$$ \text{Markovianity} \iff \text{Unsaturated Relational Capacity} $$

10. Why This Matters

Memory is unfinished integration.
Markovianity is structural, not statistical.
Non-Markovianity is diagnostic.
TFP yields an exact if-and-only-if criterion.

— John Gavel

Dist Babble

htmljava