UNNS Discipline Manifesto

Invariants, Constants, Thresholds

Reframing Unbounded Nested Number Sequences as a structured mathematical discipline with rigorous foundations

πŸ—οΈ

1. Core Objects

UNNS Sequences: Recursive nests with integer or algebraic integer coefficients

UNNS Lattices: Embeddings into structured integer rings

UNNS Fields: Edge/face potentials from DEC/FEEC links

βš–οΈ

2. Invariants

P(x): Characteristic polynomial

Ξ±: Dominant root (asymptotics)

D: Nest depth (recurrence order)

R_UNNS: Coefficient ring (cyclotomic)

πŸ”’

3. Constants

Limit ratios: lim u_{n+1}/u_n = Ξ±

Gauss/Jacobi constants: From character sums

Edge constants (c₁, cβ‚‚): UNNSβ†’DEC convergence

⚠️

4. Thresholds: UPI

UNNS Paradox Index: CFL-like stability threshold

Spectral ratio: Ξ»_self / Ξ»_damp

Predicts coercivity collapse in FEEC/DEC

UNNS Lattice Embedding

β„€
βŠ‚
β„€[i]
βŠ‚
β„€[Ο‰]
βŠ‚
...
Integers Gaussian Eisenstein Extended

🎯 UNNS Paradox Index (UPI)

UPI = (D Γ— R) / (M + S)

The fundamental stability threshold for UNNS systems

🟒 SAFE
UPI < 1
Exponentially stable
🟑 TRANSITIONAL
1 ≀ UPI ≀ 3
Marginal stability
πŸ”΄ UNSTABLE
UPI > 3
Paradox-prone

πŸŽ›οΈ Interactive UPI Calculator

0.33
SAFE - Exponentially Stable
Spectral ratio: Ξ»_self / Ξ»_damp β‰ˆ 0.33

πŸ”¬ Theoretical Grounding

UPI β‰ˆ Ξ»_self / Ξ»_damp (spectral ratio)
D, R, M, S β†’ functions on recurrence matrices over algebraic integer rings
FEEC/DEC: UPI β‰₯ 1 coincides with coercivity collapse

The UPI threshold provides a rigorous stability criterion analogous to CFL conditions in PDEs

πŸŒ‰ FEEC/DEC Bridge: UNNS β†’ Discrete Edge Potentials

Revolutionary Discovery: UNNS sequences can be rigorously interpreted as discrete 1-forms on nested mesh hierarchies, bridging abstract number theory with computational geometry.

πŸ“ Fundamental Lemma (UNNS β†’ Discrete Edge Potentials)

Setup: Let {T_h}_{hβ†’0} be a nested family of oriented simplicial meshes on domain Ξ© βŠ‚ ℝ³

UNNS Data: For each refinement level h, we have finite list U^{(h)} = {u_e^{(h)} : e ∈ E_h}

πŸ”‘ Edge Consistency Condition
A_h(e) = βˆ‘_{e'βŠ‚e} A_h'(e')

Coarse edge value equals oriented sum of refined edge values

⚑ Key Proof Results

🎯
Discrete 1-Forms
A_h: E_h β†’ ℝ given by A_h(e) := u_e^{(h)} defines discrete 1-form (edge cochain)
πŸ”„
Face Consistency
F_h = d_h A_h (oriented face sums) obey refinement-consistency telescoping
πŸ“ˆ
FEEC Convergence
||F - F_h||_{LΒ²(Ξ©)} ≀ C h^p ||A||_{H^{p-1}} with O(h^p) convergence rate
πŸ”’
Gauge Uniqueness
Discrete potentials A_h unique up to discrete closed 1-form (harmonic cochain)

πŸ› οΈ Practical Significance

Computational Bridge

UNNS provides a systematic way to assign edge values on nested meshes while preserving consistency

Maxwell Compatibility

The consistency condition ensures discrete Maxwell equations remain valid across refinement levels

FEEC Integration

Direct pathway for UNNS sequences to integrate with finite element exterior calculus frameworks

Simulation Ready

Provides runnable Python code for constructing UNNS-based nested meshes and computing discrete fields

πŸ”— The Complete Connection
UNNS Sequences β†’ Edge Consistency β†’ Discrete 1-Forms β†’ FEEC Convergence β†’ Maxwell Simulation

"Abstract number theory becomes computational electromagnetism"

Ο€-Analog

lim u_{n+1}/u_n = Ξ±
Fundamental ratio constant

e-Analog

Gauss/Jacobi Constants
From character sum theory

Edge Constants

c₁, cβ‚‚
UNNS→DEC convergence rates

🌊 Discipline Flow Architecture

Core Objects
β†’
Invariants
β†’
Constants
Constants
β†’
UPI Thresholds
β†’
Stability Analysis
Theoretical Grounding
β†’
FEEC/DEC Link
β†’
Strategic Outcome
🎯 Strategic Outcome

By codifying invariants, constants, and thresholds:

β€’ UNNS becomes a discipline with structure and laws
β€’ UPI acts as stability constant, like CFL in PDEs
β€’ Signals rigor and systematic scope for UNNS
β€’ Bridges discrete and continuous mathematics

Experience the complete UNNS Discipline through interactive exploration above ↑

The mathematical substrate of reality awaits your discovery! πŸŒ€βœ¨