Constraint-Weighted State Selection: When Geometry and Memory Actually Shape Which States Get Realized

A minimal, mathematically grounded extension to entropy-driven models that makes constraint structure an active player instead of a passive boundary.

By Sal Attaguile

Independent Systems Research

Zenodo Preprint (v5.1)

DOI: https://doi.org/10.5281/zenodo.19629245

Standard statistical mechanics and probabilistic inference models weight states by entropy (or energy). Constraints are treated as hard walls: they exclude the impossible, but every allowed state inside the wall is still chosen according to the same entropy gradient.

That framing is clean. It is also incomplete.

What if constraint geometry and accumulated history actively bias which of the allowed states actually get realized?

That is the question this work asks — and the answer is a compact extension called Constraint-Weighted State Selection (CWSS).

The Core Idea in One Equation

The probability of realizing state ( i ) at time ( t ) becomes:

[
P_i(t) \propto e^{-S_i} \cdot e^{-\alpha K_i} \cdot e^{-\beta K_i C_L(t)}
]

• ( S_i ): the usual entropy (or energy) term — the Boltzmann baseline.
• ( K_i ): the constraint cost of that state — how geometrically expensive it is (distance to the nearest boundary plus local curvature).
• ( \alpha ): instantaneous geometric suppression.
• ( \beta ): memory coupling — the cost gets amplified by history.
• ( C_L(t) ): accumulated constraint load — the dynamical memory variable.

The first exponential is the model you already know.

The second and third are the extension.

Together they turn constraint from a static wall into a time-dependent, geometry-aware filter that shapes the realized distribution.

Memory Dynamics (Non-Markovian by Design)

The load updates as a simple recurrence:

[
C_L(t+1) = C_L(t) + a \langle K \rangle_t - b C_L(t)
]

where ( \langle K \rangle_t ) is the expected constraint cost under the current probabilities.

High-cost selections increase future suppression of similar states. Damping keeps the system bounded. The feedback loop is negative and self-stabilizing — until load crosses a threshold ( \zeta ), at which point a partial reset and coupling shift occur.

The Observable Bridge: Effective Disorder

The MRML effective disorder functional gives us something we can actually measure:

[
S_{\rm eff}(t) = w_1(1-C(t)) + w_2 D(t) + w_3 {\rm depth}(t)
]

In the stationary regime the two quantities are linked by an exact, parameter-explicit coupling constant:

[
c = \frac{a K_{\rm norm}}{b w_1}
]

So accumulated constraint load ( C_L ) can be read directly from observable disorder components (coherence, drift, recursive depth). No black-box fitting required.

What the Model Actually Predicts (and What Standard Models Cannot)

Four signatures distinguish CWSS from pure entropy-driven or Markovian models:

1. Geometric bias — States with identical entropy but different ( K ) are realized at measurably different rates. The ratio depends on history through ( C_L^* ).

2. History-dependent drift — Early high-cost selections produce a persistent bias toward low-cost states whose autocorrelation decays exactly as ( e^{-b\tau} ).

3. Threshold redistribution — When ( C_L \ge \zeta ), expected constraint cost ( \langle K \rangle ) drops discontinuously. The size of the drop is ( \langle K \rangle_\beta - \langle K \rangle_{\beta'} ).

4. Disorder-load correspondence — A sustained rise in measurable ( S_{\rm eff} ) predicts a future rise in constraint pressure of magnitude ( c\beta ) per unit disorder. Falsifiable from sequence traces alone.

Simulation Evidence (20-State Periodic Manifold)

Across 25 parameter configurations the stationary coupling holds to within 1.1 % deviation.

A single injected load spike triggers the threshold transition:

• Probability mass concentrates on the low-( K ) half of state space (0.808 pre-crossing vs. uniform 0.5).
• After the partial reset, ( \langle K \rangle ) visibly drops and the system settles into a new stationary regime.

The figures in the preprint show the load trajectory, residual ( \varepsilon(t) ), and occupation shift side-by-side.

Why This Matters for Builders

Most of us are already fighting drift, hallucination, and incoherent multi-agent outputs.

CWSS does not replace your model or your prompt strategy — it gives you a lightweight, computable layer that makes geometry and memory structural rather than accidental.

The math is minimal.

The observable (disorder functional) is already computable in any system that can track coherence, drift, and depth.

The predictions are falsifiable from the logs you already have.

If you are building long-horizon agents, governed reasoning pipelines, or any system where history and constraint geometry should matter, this is a framework worth stress-testing.

Read the full preprint (v5.1)

Constraint-Weighted State Selection: Geometry, Memory, and Thresholded Disorder in State Realization

It is deliberately scoped to finite discrete state spaces and stationary behavior. Continuous manifolds, non-stationary constraints, and quantum-compatible forms are left as open extensions.

I wrote it to be attacked.

If the model fails under your conditions, the failure will be visible and diagnostic — exactly how it should be.

What do you think?

Drop your critique, your parameter regime, or the system you want to test it on in the comments. I’m reading every one.

— Sal