Angell Framework, Volume 2, Part III: Visual Atlas of Bounded Transcendence & Nr-Indexed Dynamics

Description

Series information

MIT LICENSE

Copyright (c) 2025 Nicholas Reid Angell
Permission is hereby granted, free of charge, to any person obtaining a copy of this software
and associated documentation files (the “Software”), to deal in the Software without restriction,
including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense,
and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do
so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all copies or sub-
stantial portions of the Software.

ATTRIBUTION AND CITATION NOTICE (published alongside the MIT License above):
Any redistribution of this document or substantial portions of it must retain the copyright notice,
the MIT License text, and the attribution line below. For academic or technical publications that
materially use this work, cite:

Angell, N. R. (2025). Angell Framework, Vol. 2 Part III: Visual Atlas of Bounded Transcendence & Nr-Indexed Dynamics. December 2025. DOI: doi.org/10.5281/zenodo.17979196

THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EX-

PRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MER-
CHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT.
IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFT-
WARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

Series information

AI Assistance Acknowledgment This document was prepared with computational assistance from multiple generative AI systems including Claude Sonnet 4.5 (Anthropic), Grok(XAI), ChatGPT-5 (OpenAI), Gemini 3.0 Pro (Google), and Microsoft Copilot (Microsoft). These systems assisted with mathematical formalization, LaTeX formatting, notation standardization, and document organization. The author reviewed, edited, and validated all mathematical content and takes full responsibility for the accuracy and originality of this work.

Technical info

ℕʳ

Not to be mixed up with N, the "Nr" always being referenced in all of work is a canonical term ℕʳ which refers to new cananocial unit the "Nick"

Construction of `\(\mathbb{N}^{R}\)`

Carrier set and interpretation

• Definition:\mathbb{N}^{R}=\{(n,r)\mid n\in\mathbb{N},\ r\in\mathbb{R}^{+}\}

where \(n\) is a natural magnitude and \(r\) is a positive recursion depth (metadata).

• Interpretation:• Magnitude: \(n\) counts discrete steps or event cardinality.

• Depth: \(r\) records recursion intensity (e.g., nesting level, escape time, loop rank).

Canonical projections and sections

• Projections:

\(\pi_{\mathbb{N}}(n,r)=n,\quad \pi_{\mathbb{R}^{+}}(n,r)=r\).

• Sections:

For any fixed \(r\), \(\sigma_{r}(n)=(n,r)\) embeds \(\mathbb{N}\) as an iso-depth slice.

Algebra on `\(\mathbb{N}^{R}\)`

Merge operators for depth

• Sum-merge:r\oplus_{\text{sum}} s = r+s

• Max-merge:r\oplus_{\text{max}} s = \max\{r,s\}

• Axioms: Both \(\oplus\) choices are associative and commutative on \(\mathbb{R}^{+}\), with identities \(0\) (sum) and \(0\) (max), and idempotence for max.

Lifted addition and multiplication

• Recursive addition:(n,r)\ \widehat{+}\ (m,s)=(n+m,\ r\oplus s)

• Recursive multiplication:(n,r)\ \widehat{\times}\ (m,s)=(nm,\ r\oplus s)

• Powers (depth accumulation): For \(k\in\mathbb{N}\),(n,r)^{\widehat{k}}=\underbrace{(n,r)\ \widehat{\times}\cdots \widehat{\times}\ (n,r)}_{k\ \text{times}}

• With sum-merge: \((n,r)^{\widehat{k}}=(n^{k},\ kr)\).

• With max-merge: \((n,r)^{\widehat{k}}=(n^{k},\ r)\).

Basic properties

• Closure: \(\widehat{+}\) and \(\widehat{\times}\) map \(\mathbb{N}^{R}\times\mathbb{N}^{R}\to\mathbb{N}^{R}\).

• Commutativity: Holds for both operations under either merge (\(\mathbb{N}\) ops commutative; \(\oplus\) commutative).

• Associativity: Holds because \(\mathbb{N}\) ops associative and \(\oplus\) associative.

• Identities:• Addition: \((0,0)\) under sum-merge; \((0,0)\) under max-merge.

• Multiplication: \((1,0)\) for both merge choices.

• Distributivity:(n,r)\ \widehat{\times}((m,s)\ \widehat{+}\ (p,t))

= (nm+np,\ r\oplus (s\oplus t))

matches \(((n,r)\ \widehat{\times}\ (m,s))\ \widehat{+}\ ((n,r)\ \widehat{\times}\ (p,t))\) because \(\oplus\) is associative and commutative.

Metric and topology

Weighted product metric

• Definition: For \(\lambda>0\),d_{\lambda}((n,r),(m,s))=|n-m|+\lambda\,|r-s|

• Interpretation: \(\lambda\) sets relative sensitivity to recursion depth versus magnitude.

Metric axioms (proofs)

• Non-negativity and identity of indiscernibles:

\(|n-m|\ge 0,\ |r-s|\ge 0\) with equality iff \(n=m\) and \(r=s\). Hence \(d_{\lambda}\ge 0\) and \(d_{\lambda}=0\iff (n,r)=(m,s)\).

• Symmetry:d_{\lambda}((n,r),(m,s))=|n-m|+\lambda|r-s|

=|m-n|+\lambda|s-r|

=d_{\lambda}((m,s),(n,r))

• Triangle inequality: For \((a,u),(b,v),(c,w)\in\mathbb{N}^{R}\),d_{\lambda}((a,u),(c,w))

=|a-c|+\lambda|u-w|

\le |a-b|+|b-c|+\lambda(|u-v|+|v-w|)

= d_{\lambda}((a,u),(b,v)) + d_{\lambda}((b,v),(c,w))

using the triangle inequality in \(\mathbb{R}\) for each coordinate.

Iso-depth slices and product structure

• Iso-depth contour:\mathbb{N}^{R}_{r}=\{(n,r)\mid n\in\mathbb{N}\}

Each \(\mathbb{N}^{R}_{r}\) is isometric to \((\mathbb{N},|\cdot|)\).

• Product view: \((\mathbb{N},|\cdot|)\times(\mathbb{R}^{+},\lambda|\cdot|)\) with \(\ell^{1}\)-metric.

Embeddings and invariants

Forgetful and enrichment functors

• Forgetful to naturals:U:\mathbb{N}^{R}\to\mathbb{N},\quad U(n,r)=n

• Enrichment by fixed depth:E_{r}:\mathbb{N}\to\mathbb{N}^{R},\quad E_{r}(n)=(n,r)

• Invariant under recursion orthogonality: If a process modifies \(n\) without changing the recursion mechanism class, depth comparison across variants remains meaningful because \(r\) is recorded independently of \(n\).

Depth-sensitive observables

• Definition (observable):

An observable is any \(F:\mathbb{N}^{R}\to\mathbb{R}\) where depth affects evaluation.

• Recursive ratio:R_{\delta}((n,r);F)=\frac{F(n,r+\delta)}{F(n,r)}

measures sensitivity to marginal depth change \(\delta>0\).

Worked examples:

Example 1: Sum-merge arithmetic

• Setup:• Elements: \((2,1.0)\) and \((3,2.0)\).

• Addition:(2,1.0)\ \widehat{+}\ (3,2.0)=(5,3.0)

• Multiplication:(2,1.0)\ \widehat{\times}\ (3,2.0)=(6,3.0)

• Powers:(4,0.5)^{\widehat{3}}=(64,1.5)

• Distance with \(\lambda=0.5\):d_{0.5}((1,2.0),(4,3.0))=|1-4|+0.5|2.0-3.0|=3+0.5=3.5

Example 2: Max-merge bottleneck

• Setup:• Elements: \((2,1.0)\) and \((3,2.0)\).

• Addition:(2,1.0)\ \widehat{+}\ (3,2.0)=(5,2.0)

• Multiplication:(2,1.0)\ \widehat{\times}\ (3,2.0)=(6,2.0)

• Powers:(4,0.5)^{\widehat{3}}=(64,0.5)

Example 3: Iso-depth contour and projection

• Iso-depth \(r=2\):\{(0,2),(1,2),(2,2),(3,2),\ldots\}

• Projection to naturals: \(\pi_{\mathbb{N}}\) maps this contour to \(\mathbb{N}\) isometrically.

Experimental protocol for scientists

Data generation

• Grid construction:• Magnitude range: choose \(n\in\{0,1,\ldots,N\}\).

• Depth levels: choose \(r\in\{r_{0},r_{1},\ldots,r_{K}\}\subset\mathbb{R}^{+}\).

• Dataset: \(\mathcal{X}=\{(n,r)\}\) with Cartesian product.

Metric verification

• Triangle inequality test:• Procedure: Sample triplets \((a,u),(b,v),(c,w)\in\mathcal{X}\), verifyd_{\lambda}((a,u),(c,w))\le d_{\lambda}((a,u),(b,v))+d_{\lambda}((b,v),(c,w)).

• Outcome: Record violations; for exact arithmetic there should be none.

Algebraic property checks

• Associativity:• Addition: Check \((x\ \widehat{+}\ y)\ \widehat{+}\ z = x\ \widehat{+}\ (y\ \widehat{+}\ z)\).

• Multiplication: Check similarly.

• Commutativity:

Confirm \(x\ \widehat{+}\ y = y\ \widehat{+}\ x\) and \(x\ \widehat{\times}\ y = y\ \widehat{\times}\ x\).

• Distributivity:

Verify \(x\ \widehat{\times}\ (y\ \widehat{+}\ z)=(x\ \widehat{\times}\ y)\ \widehat{+}\ (x\ \widehat{\times}\ z)\).

Observable sensitivity

• Choose observable \(F\):• Examples: \(F(n,r)=n+r\), \(F(n,r)=n\cdot r\), domain-specific escape-time estimators.

• Measure sensitivity:

For each \((n,r)\), compute \(R_{\delta}((n,r);F)\) for small \(\delta\), analyze monotonicity and bounds.

Reproducible code (Python)

from dataclasses import dataclass

from typing import Callable, List

import random

import math

@dataclass(frozen=True)

class NR:

    n: int

    r: float

def merge_sum(r: float, s: float) -> float:

    return r + s

def merge_max(r: float, s: float) -> float:

    return r if r >= s else s

def add(x: NR, y: NR, merge: Callable[[float, float], float]) -> NR:

    return NR(x.n + y.n, merge(x.r, y.r))

def mul(x: NR, y: NR, merge: Callable[[float, float], float]) -> NR:

    return NR(x.n * y.n, merge(x.r, y.r))

def pow_hat(x: NR, k: int, merge: Callable[[float, float], float]) -> NR:

    depth = x.r

    for _ in range(k - 1):

        depth = merge(depth, x.r)

    return NR(x.n ** k, depth)

def d_lambda(x: NR, y: NR, lam: float) -> float:

    return abs(x.n - y.n) + lam * abs(x.r - y.r)

def verify_triangle(samples: List[NR], lam: float, trials: int = 1000) -> int:

    violations = 0

    for _ in range(trials):

        a, b, c = random.sample(samples, 3)

        left = d_lambda(a, c, lam)

        right = d_lambda(a, b, lam) + d_lambda(b, c, lam)

        if left > right + 1e-12:  # numerical tolerance

            violations += 1

    return violations

Example usage:

grid = [NR(n, r) for n in range(0, 10) for r in (0.0, 1.0, 2.0, 3.0)]

assert verify_triangle(grid, lam=0.5, trials=10000) == 0

Pedagogical integration

Curriculum mapping

• Discrete mathematics:• Topic fit: Product sets, metrics, algebraic structures, proofs of axioms.

• Exercises: Prove distributivity; explore iso-depth slices; compare \(\oplus_{\text{sum}}\) and \(\oplus_{\text{max}}\).

• Numerical methods:• Topic fit: Weighted metrics, sensitivity analysis, reproducibility practices.

• Exercises: Implement \(d_{\lambda}\); analyze rounding effects vs. tolerance.

• Dynamical systems:• Topic fit: Encoding escape time, loop depth, or nesting with \(r\).

• Exercises: Map iteration processes into \(\mathbb{N}^{R}\); study depth observables.

Assessment-ready problems

• Problem 1: Show \((\mathbb{N}^{R},\widehat{+},\widehat{\times})\) with sum-merge forms a semiring; identify identities.

• Problem 2: With max-merge, show addition is idempotent on depth but not on magnitude; discuss modeling implications.

• Problem 3: Prove that \(d_{\lambda}\) is the \(\ell^{1}\)-product metric for \((\mathbb{N},|\cdot|)\) and \((\mathbb{R}^{+},\lambda|\cdot|)\).

• Problem 4: For \(F(n,r)=n\cdot r\), analyze \(R_{\delta}\) as \(\delta\to 0^{+}\), and interpret in sensitivity terms.

Appendix: Formal statements and quick proofs

Semiring (sum-merge)

• Claim: \((\mathbb{N}^{R},\widehat{+},\widehat{\times})\) with sum-merge is a commutative semiring.

• Proof sketch:• Additive monoid: Associativity and commutativity inherited from \(\mathbb{N}\) and \(\mathbb{R}^{+}\) under sum; identity \((0,0)\).

• Multiplicative monoid: Associativity and commutativity inherited; identity \((1,0)\).

• Distributivity: Follows from distributivity in \(\mathbb{N}\) and associativity/commutativity of sum on depth.

• No additive inverses: As in \(\mathbb{N}\); thus semiring (not ring).

Idempotent addition (max-merge)

• Claim: With max-merge, depth addition is idempotent: \((n,r)\ \widehat{+}\ (m,r)=(n+m,r)\).

• Proof: \(\max\{r,r\}=r\); magnitude addition unaffected.

Power law on depth

• Claim: With sum-merge, \((n,r)^{\widehat{k}}=(n^{k},kr)\); with max-merge, \((n^{k},r)\).

• Proof: Induction on \(k\) using associativity and the definition of \(\widehat{\times}\).

---

Notation and glossary

• Carrier: \(\mathbb{N}^{R}\) — pairs of magnitude \(n\) and depth \(r\).

• Merge: \(\oplus\) — rule for combining depths (sum or max).

• Recursive ops: \(\widehat{+},\widehat{\times}\) — lifted algebra on pairs.

• Metric: \(d_{\lambda}\) — weighted \(\ell^{1}\) product distance.

• Iso-depth: set of elements with fixed \(r\).

• Observable: \(F:\mathbb{N}^{R}\to\mathbb{R}\) — depth-sensitive measurement.

• Sensitivity: \(R_{\delta}\) — ratio under marginal depth change.