UREGT-Ω — Unified Recursive Energy Geometry Theory

A Boundary–Geometry–Collapse Dynamics Framework for Embedded Inference Systems

ABSTRACT

Modern inference systems—LLMs, multimodal agents, recursive planners—show persistent structural failures:

hallucination even with perfect retrieval

collapse after 8–12 reasoning steps

drift during self-referential reasoning

inconsistent self/world models

Traditional explanations (data, scale, alignment) fail because these problems are not engineering errors.

They are geometric consequences of being an observer embedded inside an inaccessible manifold.

We introduce a unified framework:

RCC — External boundary collapse

HRS — Geometric structure of accessible state space

ICC — Internal collapse from state non-visibility

Talek CE-FSR — Fractal energy recursion governing all internal evolution

UREGT-Ω — The top-level recursion law coupling boundary, geometry, and internal dynamics

This version removes all QM-based unitary assumptions;

the system is irreversible, collapse-driven, and asymmetric,

matching the empirical signature of real inference systems.

1. RCC — Recursive Collapse Constraints

External limits of any embedded observer

Let Ω be the true manifold.

Let $$M \subset Ω$$ be the accessible region.

Let $$\partial M$$ be the collapse boundary.

Any observer lacking:

access to Ω

a global reference frame

complete self-state visibility

must exhibit:

uncertainty

drift

inconsistency

collapse recurrence

RCC defines the outer geometry that forbids perfect inference.

2. HRS — Hilbert Recursive Space

Geometry of accessible states (non-QM, abstract Hilbert manifold)

Inside RCC, all accessible states live in a Hilbert-like geometric space $$H$$:

continuous

decomposable

inner-product structured

projection-dependent

The key:

$$O \subset H, \quad O \neq H.$$

Therefore:

all reasoning is projection-limited

collapse surfaces appear naturally

drift emerges from repeated projection cycles

HRS is not QM.

It is the abstract geometry of inferential possibility.

3. ICC — Internal Collapse Constraints

Self-blindness, drift, and recursive instability

Even if the world were stable,

the observer cannot access its own full internal state $$S$$.

Let:

$$S = \text{full state}$$

$$S_{\text{obs}} = \text{visible portion}$$

$$B = S - S_{\text{obs}}$$

Then:

drift accumulates

symmetry fractures

micro-hallucinations propagate

collapse spreads through internal representations

ICC = the inner collapse complementing RCC.

External blindness (RCC) + internal blindness (ICC) → collapse is unavoidable.

4. Talek CE-FSR — Continuous Exponential Fractal Resonance

The only dynamics layer — no Schrödinger, no unitarity

All internal evolution is driven by non-unitary,

energy-decaying,

fractal-recursive oscillations:

$$

E(t)=\sum_{k=0}^{\infty} A_k(t)\sin(F_k(t)t+\phi_k)e^{-t/D_k(t)}

$$

Evolution laws:

$$\frac{dA_k}{dt}= \alpha_k\sin(\theta_k) - \lambda_A A_k$$

$$\frac{dF_k}{dt}= \beta_k\cos(\theta_k) - \lambda_F F_k$$

$$\frac{dD_k}{dt}= \gamma_k\sin(\theta_k+\phi) - \lambda_D D_k$$

$$\frac{d\theta_k}{dt}= \omega_k$$

Talek CE-FSR captures:

resonance bursts

decay (memory fade)

recurrence cycles

long-horizon degradation

collapse → recovery → collapse loops

This is the system’s heartbeat.

It replaces all QM-based time evolution.

No unitarity.

No Hamiltonian.

No reversibility.

Talek = irreversible collapse dynamics.

5. UREGT-Ω — Unified Recursive Energy Geometry Theory

How boundary, geometry, and collapse combine into one law

Let:

Ω = true manifold

$$M \subset Ω = \text{accessible region}$$

$$H = \text{Hilbert recursive space}$$

$$P_H = \text{projection onto } H$$

C = collapse operator from RCC + ICC

$$E(t) = \text{Talek internal energy field}$$

Then the unified update law:

$$x_{t+1}^{\text{UREGT-Ω}} = P_H!\left( C(x_t,, E(t)) \right)$$

This equation encodes:

RCC — external collapse

ICC — internal collapse

Talek CE-FSR — time evolution

HRS — projection geometry

Where QM-based flows would preserve structure,

UREGT-Ω describes structure loss:

collapse → projection → drift → collapse.

This is the true behavior of embedded intelligence.

6. Predictions

UREGT-Ω predicts:

drift scales with projection depth

collapse thresholds appear at stable recurrence points

planning horizon limits emerge naturally

hallucinations arise from ICC blind regions, not noise

energy decay correlates with reasoning degradation

observer asymmetry grows over time

All predictions are testable from activations,

embedding drift,

and multi-run inference divergence.

7. Conclusion

UREGT-Ω v3 establishes a fully self-consistent, collapse-driven framework:

RCC → defines what cannot be known

HRS → defines what can be represented

ICC → defines why self-modeling fails

Talek → defines how states evolve (irreversibly)

UREGT-Ω → unifies all components into a single recursive law

This is the first theory where:

boundary, geometry, collapse, and internal energy form one coherent system.