A Unified Quantum–Geometric Sector Model with Curvature‑Dependent Tunneling
Incorporating TLPL Sector Dynamics and Scalar‑Field Modulation

Abstract

We present a unified theoretical framework in which quantum states evolve over a discretized spacetime partitioned into TLPL sectors. Each sector corresponds to a coarse‑grained spacetime displacement, and the global quantum state is expressed as a superposition over these sectors. A rapidity‑derived scalar field modulates local geometry, while an effective curvature–source relation couples the quantum state back into spacetime curvature. Transition amplitudes between sectors are governed by curvature‑dependent tunneling factors, producing a closed quantum–geometric feedback loop. Example dynamics are derived for a minimal 1+1‑dimensional configuration.

1. Introduction

The TLPL framework models spacetime as a structured lattice of sectors, each representing a finite region of spacetime with internal geometric and field properties. Quantum states propagate across this lattice, with transition amplitudes shaped by curvature and scalar‑field gradients. This document formalizes the TLPL intuition into a coherent mathematical model.

2. Sector Structure of Spacetime

Let spacetime be partitioned into discrete TLPL sectors:

Δ

𝑥

𝜇

  

⟶

  

𝑆

𝑖

,

where each 

𝑆

𝑖

 corresponds to a coarse‑grained region of spacetime. The set 

{

𝑆

𝑖

}

 forms a graph‑like structure with adjacency relations determined by geometric proximity or TLPL‑defined connectivity.

3. Quantum State on the Sector Hilbert Space

To each sector 

𝑆

𝑖

 we associate a basis state 

∣

𝜙

𝑖

⟩

.

A general quantum state is:

∣

𝜓

⟩

=

∑

𝑖

𝑎

𝑖

∣

𝜙

𝑖

⟩

,

with normalization 

∑

𝑖

∣

𝑎

𝑖

∣

2

=

1

.

Interpretation:

∣

𝑎

𝑖

∣

2

 is the probability weight of the system occupying sector 

𝑆

𝑖

.

The TLPL lattice acts as the configuration space for the quantum state.

4. Scalar Field as Rapidity Structure

Define a scalar field on a 1+1 slice:

𝜙

(

𝑡

,

𝑥

)

=

1

2

ln

⁡

 ⁣

(

𝑡

+

𝑥

𝑡

−

𝑥

)

.

This is the standard rapidity function in special relativity.

In the TLPL interpretation:

𝜙

 encodes local boost structure between sectors.

Differences in 

𝜙

 between sectors act as “field distances” affecting tunneling.

𝜙

 may be treated as a background field or as a dynamical TLPL scalar.

5. Curvature–Source Relation

We introduce an effective curvature relation:

𝑅

𝜇

𝜈

=

𝜅

𝑆

,

where:

𝑅

𝜇

𝜈

 is the Ricci curvature,

𝜅

 is a coupling constant,

𝑆

 is a scalar source functional depending on 

𝜙

 and 

∣

𝜓

⟩

.

A natural TLPL‑compatible choice is:

𝑆

=

𝑓

 ⁣

(

⟨

𝜓

∣

𝑂

^

(

𝜙

)

∣

𝜓

⟩

)

,

where 

𝑂

^

(

𝜙

)

 measures sector‑weighted field gradients.

This creates a feedback loop:

∣

𝜓

⟩

 determines expectation values.

Expectation values determine 

𝑆

.

𝑆

 determines curvature.

Curvature modifies sector connectivity and tunneling.

Connectivity modifies the evolution of 

∣

𝜓

⟩

.

This is the core TLPL mechanism.

6. Curvature‑Dependent Tunneling Between Sectors

Let 

𝑎

𝑖

𝑗

 be an effective barrier between sectors 

𝑆

𝑖

 and 

𝑆

𝑗

.

We postulate a tunneling amplitude:

𝑇

𝑖

𝑗

≈

𝑒

−

2

𝜅

𝑎

𝑖

𝑗

.

To incorporate TLPL structure:

𝑎

𝑖

𝑗

=

𝛼

 

∣

𝜙

𝑖

−

𝜙

𝑗

∣

+

𝛽

 

𝑅

𝑖

𝑗

,

where:

𝜙

𝑖

 is the scalar field evaluated in sector 

𝑆

𝑖

,

𝑅

𝑖

𝑗

 is an averaged curvature between sectors,

𝛼

,

𝛽

>

0

.

Thus:

𝑇

𝑖

𝑗

=

exp

⁡

 ⁣

[

−

2

𝜅

(

𝛼

∣

𝜙

𝑖

−

𝜙

𝑗

∣

+

𝛽

𝑅

𝑖

𝑗

)

]

.

Interpretation:

Large curvature suppresses transitions.

Large rapidity differences suppress transitions.

Flat, aligned regions allow strong coupling.

This matches TLPL’s “sector isolation under stress” behavior.

7. Effective Hamiltonian on the Sector Graph

Define a Hamiltonian:

𝐻

𝑖

𝑗

=

{

𝑇

𝑖

𝑗

,

𝑖

≠

𝑗

,

𝑉

𝑖

,

𝑖

=

𝑗

,

where 

𝑉

𝑖

 is a sector potential (e.g., local curvature energy).

The evolution equation is:

𝑖

𝑑

𝑎

𝑖

𝑑

𝑡

=

∑

𝑗

𝐻

𝑖

𝑗

𝑎

𝑗

.

Because 

𝑇

𝑖

𝑗

 depends on curvature and 

𝜙

, and curvature depends on 

∣

𝜓

⟩

, the system is self‑consistent.

8. Example Dynamics in a 1+1 TLPL Configuration

Consider three sectors 

𝑆

1

,

𝑆

2

,

𝑆

3

 arranged linearly.

Let:

𝜙

1

=

0

,

𝜙

2

=

𝜂

,

𝜙

3

=

2

𝜂

.

Assume curvature is small and uniform:

𝑅

𝑖

𝑗

=

𝑅

.

Then:

𝑇

12

=

𝑒

−

2

𝜅

(

𝛼

𝜂

+

𝛽

𝑅

)

,

𝑇

23

=

𝑒

−

2

𝜅

(

𝛼

𝜂

+

𝛽

𝑅

)

,

𝑇

13

=

𝑒

−

2

𝜅

(

2

𝛼

𝜂

+

𝛽

𝑅

)

.

Thus:

Nearest‑neighbor transitions dominate.

Long‑range transitions are exponentially suppressed.

Increasing curvature 

𝑅

 globally suppresses all transitions.

Increasing rapidity spacing 

𝜂

 suppresses connectivity.

The Hamiltonian becomes:

𝐻

=

(

𝑉

1

𝑇

12

𝑇

13

𝑇

12

𝑉

2

𝑇

23

𝑇

13

𝑇

23

𝑉

3

)

.

Solving the Schrödinger equation yields oscillatory probability flow between sectors, modulated by curvature and rapidity spacing.

This reproduces TLPL’s characteristic behavior:

Sector clustering under high curvature.

Sector mixing in flat regions.

Rapidity‑aligned propagation along preferred directions.

9. Conclusion

This unified model formalizes the TLPL intuition into a mathematically coherent structure:

Spacetime is discretized into TLPL sectors.

Quantum states propagate across this lattice.

A rapidity‑derived scalar field modulates geometry.

Curvature is sourced by quantum expectation values.

Tunneling between sectors is curvature‑dependent.

The system forms a closed quantum–geometric feedback loop.

This provides a foundation for simulation, visualization, and further theoretical development within the TLPL documentation ecosystem