Technical Notes
The Technical Notes (TN) series provides the formal closure layer of the Constraint–Surface Dynamics programme.
While Papers A–D establish the core theoretical structure, the Technical Notes:
supply missing derivations and constructions
formalise operational components
close finite-dimensional gaps
provide reusable mathematical templates
They are not standalone theories, but supporting technical documents that complete and stabilise the framework.
All notation follows TN0: Canonical Notation and Conventions. (See: https://doi.org/10.5281/zenodo.19334769)
Role within the Programme
The CSD programme is structured as follows:
Core papers (A–D) → conceptual and structural foundation
Technical notes (TN0–TN6) → mathematical and operational closure
Companion papers (c1–c2) → consistency with no-go theorems
Bridge papers (c3, Σ-series) → scaling and continuum structure
The Technical Notes sit between foundation and application.
They ensure that all claims made in the core papers are explicit, testable, and internally consistent.
Technical Notes
TN0 — Canonical Notation and Conventions
Defines the standard notation, symbols, and conventions used across the programme.
Ontic and epistemic variable definitions
Measure conventions
Projection map
Operator and probability conventions
Paper classification and naming scheme
Purpose: ensure consistency and readability across all documents
See:https://doi.org/10.5281/zenodo.19334769
TN1 — Measure-Bridge Support Note
Provides formal support for the measure bridge between ontic and epistemic levels.
SU(N)-invariant measure structure
Pushforward behaviour
Identification with Fubini–Study measure
Purpose: supports Paper B and the Born-weight structure
TN2 — Two-Qubit Singlet from Deterministic Volume Geometry
Derives singlet correlations in the finite-dimensional composite setting.
Symmetry-based construction
Context-dependent outcome regions
Recovery of correlations
Purpose: demonstrates compatibility with Bell-type experiments
TN3 — Constructive Measurement Model for the Singlet
Provides an explicit system–apparatus Hamiltonian model.
Deterministic evolution of composite system
Local parameter dependence
No-signalling structure
Purpose: moves from abstract construction to physical modelling
TN4 — Mixed States, POVMs, and Reduction
Completes the operational layer beyond pure states.
Mixed states as epistemic ensembles
POVM construction
Reduced states and marginalisation
Purpose: establishes full finite-dimensional measurement framework
TN5 — Worked Example Template
Defines the canonical template for finite-dimensional models.
Truncated harmonic oscillator
Explicit outcome probabilities
Discrete-to-continuous diagnostics
Purpose: provides a reusable modelling framework
TN6 — Outcome Sectors and Sequential Measurement
Closes the sequential measurement problem.
Deterministic post-measurement update
Outcome-sector restriction
Multi-step experiment structure
Purpose: ensures operational closure of the theory
How to Use the Technical Notes
For readers new to CSD:
Start with Paper D for conceptual overview
Read TN0 to fix notation
Use TN1 + TN4 to understand probabilities and measurement
Refer to TN5 for concrete examples
Use TN6 for sequential experiments
For technical validation:
TN1, TN3, and TN4 are the most critical
TN5 and TN6 support modelling and applications
Status and Scope
All Technical Notes are written within a finite-dimensional framework
Continuum and field-theoretic extensions are deferred
Several results remain conditional or regime-dependent, particularly:
measure bridge outside symmetry sector
classification of measurement partitions
derivation of the quantum-effective sector
These are addressed in later bridge papers.
Access
All Technical Notes will be available via Zenodo:
TN0–TN6 (Technical Notes series)
Each note is versioned and may be updated as the programme evolves.