Aa,b(g)
(ag + b) mod N, with a chosen from the units of ℤN for bijections.
Finite affine systems.
Proved mathematics.
Bounded claims.
VARZIN uses affine maps over finite cyclic spaces as one of its cleanest mathematical and computational layers. This page separates the general affine theorem from construction-specific claims such as Mirror-13.
Core theorem
For Aa,b(g) = (ag + b) mod N, the map is bijective exactly when gcd(a,N)=1. Therefore |Aff(ℤN)| = Nφ(N). For N=12, this gives 48 affine operators.
The affine bijection criterion.
On the cyclic space ℤN, an affine map combines multiplication by a with translation by b. Translation is always invertible; multiplication is invertible exactly when a is a unit modulo N, equivalently gcd(a,N)=1.
Nφ(N)
There are φ(N) admissible multipliers and N possible translations.
ℤ12
The units are 1, 5, 7 and 11, yielding 12 × 4 = 48 affine bijections.
Scope: this is proved mathematics. It does not depend on LUXVAR, semantic interpretation, physical-field claims, or any historical symbolic framing.
How VARZIN uses the affine core.
Finite-state representations
The project uses affine transformations and related finite-state constructions as deterministic objects that can be enumerated, tested, and reproduced.
Mirror-13 is separate
Mirror-13 connectivity, invariant counts, and reported compression or MDL quantities are construction-specific computational or algebraic claims. They are not consequences of the general affine-group counting theorem.
Reproducibility
Current project practice ties computational claims to specific artifacts, assumptions, releases, and deterministic execution records rather than treating a mathematical relation as evidence for unrelated empirical claims.
What this mathematics does not establish.
Supported
- The bijection criterion gcd(a,N)=1.
- The group-size formula |Aff(ℤN)|=Nφ(N).
- The count |Aff(ℤ12)|=48.
- Reproducible enumeration under a fixed implementation.
Not implied
- Independent semantic emergence.
- Universal or non-human origin of a symbolic system.
- Consciousness-field or quantum mechanisms.
- Biological or causal effects of frequency labels.
- That any project-specific construction is uniquely necessary.
Follow the computational record.
The most relevant project-listed software record is VARZIN Level-1 Computational Stack: Finite Affine Core, Genomic Scanner, and Torus Manifold Solver (v2.0.0). Use the release itself for implementation details and assumptions.