Unbounded nested number sequences

🧠 UNNS Many-Faces Visualization

Explore recursive helical embeddings and modular sequence patterns in 3D space.

 

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UNNS Many-Faces Theorem (formal statement, lemmas, proof sketches, corollaries)

0. Informal summary

Under a precise formalization of the UNNS kernel (a small algebra of nest combinators plus domain mappings), every classical linear recurrence sequence, its dominant-root attractor, and modular domain partition can be represented (embedded) inside UNNS. Further, cross-domain homomorphisms exist for a broad, constructive family of encodings. Under mild constructive encodability conditions UNNS can simulate finite-state computations. These facts justify describing UNNS as having “many faces” (algebraic, geometric, topological, number-theoretic, dynamical, computational).

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1. Definitions (minimal formal model)

Definition 1 (Nest set). Let $S$ be a set of nests. Elements $s\in S$ may be symbolic tokens or integer values.

Definition 2 (Combinator algebra). Let $\mathcal{C}$ be a finite set of operators (combinators) $\star: S^k\to S$ for various arities $k$. Each $\star$ is specified by an algorithm (finite description) that may use finite integer coefficients, concatenation, shifting (index arithmetic), and modular arithmetic.

Definition 3 (Generators & dynamics). A UNNS configuration is $U=(S,\mathcal{C},\mathcal{G})$ where $\mathcal{G}\subset S$ is a finite seed set and an update rule iteratively applies combinators over finite contexts (e.g., last $r$ nests) to produce new nests.

Definition 4 (Domain mappings). A domain $D$ is any mathematical structure (ring, metric space, manifold, graph). A mapping $\mu_D: S\to D$ is computable if there exists an algorithm producing $\mu_D(s)$ from a finite description of $s$. Typical domains:

• $A$ algebraic (polynomials / vectors)

• $G$ geometric ($\mathbb{R}^2$ polar coords)

• $T$ topological (graphs)

• $\mathbb{Z}_m$ modular residues

Definition 5 (UNNS system). A UNNS system is a tuple $(S,\mathcal{C},\mathcal{G},\{\mu_D\}_{D\in\mathcal{D}})$ with a finite set of domains $\mathcal{D}$ and computable domain mappings.

Assumptions (A). For the theorem we assume:

1. Each combinator in $\mathcal{C}$ is definable by a finite algorithm using integer arithmetic, addition, multiplication, and bounded lookback (i.e., uses only a finite window of previous nests).

2. Each $\mu_D$ is computable and respects simple arithmetic relations (e.g., $\mu_{\mathbb{Z}_m}( \alpha s + \beta t ) = \alpha\mu_{\mathbb{Z}_m}(s) + \beta\mu_{\mathbb{Z}_m}(t) \bmod m$) where $\alpha,\beta$ are integers encoded in $\star$.

These are constructive assumptions consistent with the implemented demos.

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2. Main theorem (Many-Faces Theorem for UNNS)

Theorem (UNNS Many-Faces).
Let $U=(S,\mathcal{C},\mathcal{G},\{\mu_D\}_{D\in\mathcal{D}})$ be a UNNS system meeting Assumptions (A). Then:

1. (Linear recurrence embedding) For any linear constant-coefficient recurrence of finite order $r$:

   $$a_n = c_1 a_{n-1} + \cdots + c_r a_{n-r},\qquad c_i\in\mathbb{Z},$$

   there exists a UNNS configuration (a choice of seeds $\mathcal{G}$ and a combinator $\star\in\mathcal{C}$) that generates, under iterative application, a sequence $s_n\in S$ with $\mu_{\mathbb{Z}}(s_n)=a_n$ for all $n$ (i.e., UNNS can produce the sequence exactly).

2. (Dominant-root attractor) If the characteristic polynomial of the recurrence has a unique dominant root $r_d$ with $|r_d|>\max_{i\ne d}|r_i|$, then for the embedded UNNS sequence $s_n$ we have

   $$\lim_{n\to\infty} \frac{\mu_{\mathbb{R}}(s_{n+1})}{\mu_{\mathbb{R}}(s_n)} = r_d,$$

   and geometric mapping $\mu_G$ (e.g. radial embedding $r_n=\alpha e^{\beta n}$) yields points that asymptotically follow a logarithmic spiral parameterized by $r_d$. Thus $r_d$ is a geometric attractor under $\mu_G$.

3. (Modular domain partition) For any modulus $m$, the residues $\mu_{\mathbb{Z}_m}(s_n)$ partition $\{s_n\}$ into at most $m$ domains. These domains are invariant under linear combinators with integer coefficients modulo $m$; i.e., the set of residues evolves according to a deterministic finite-state map.

4. (Constructive cross-domain homomorphism) For any finite set of domains $\{D_1,\dots,D_k\}$ and computable encodings $\mu_{D_i}$ satisfying linearity on the combinators, there exists a constructive homomorphism relationship between domain images of UNNS combinators: for generators $x,y\in S$,

   $$\mu_{D_j}(\star(x,y)) = h_{i\to j}\big(\mu_{D_i}(\star(x,y))\big),$$

   where each $h_{i\to j}$ is explicitly computable from the encoding rules (i.e., algebraic combinations map to corresponding geometric transforms or modular arithmetic via a constructive encoding).

5. (Computational completeness condition — conditional) If $\mathcal{C}$ contains combinators sufficient to implement (a) bounded memory chunk/shift, (b) conditional branching based on finite tags, and (c) integer arithmetic, then UNNS can simulate any finite-state automaton and, under further mild encodability assumptions (unbounded tape encoded by nested structures), can simulate a universal Turing-complete model (e.g., cyclic tag systems). Thus UNNS is, under these constructive encoding conditions, computationally universal.

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3. Proof sketches (by part)

Part 1 (recurrence embedding)

• Construction: Let the UNNS store an $r$-tuple of recent nests $(s_{n-r+1},\dots,s_n)$ as the state. Define a combinator $\star$ that computes $\tilde s_{n+1} = c_1 s_n + \cdots + c_r s_{n-r+1}$ using integer arithmetic (allowed by Assumption A). Initialize seeds $\mathcal{G}=\{s_0,\dots,s_{r-1}\}$ with values matching $a_0,\dots,a_{r-1}$. Inductively, the UNNS dynamics produce $\mu_{\mathbb{Z}}(s_n)=a_n$.

• Correctness: By straightforward induction on $n$.

Part 2 (dominant-root attractor)

• Standard linear algebra: The recurrence solution decomposes into $a_n=\sum_i A_i r_i^n$. Under the spectral gap assumption, $a_n\sim A_d r_d^n$. Ratios converge: $a_{n+1}/a_n \to r_d$.

• Geometric embedding: Choose $\mu_G(s_n)=(\theta_n,r_n)$ with $r_n = \alpha | \mu_{\mathbb{R}}(s_n) |$ and $\theta_n = \gamma n$. Then $r_n$ grows like $|r_d|^n$, so plotting in polar coordinates yields a logarithmic spiral with growth rate $\ln|r_d|$ and angular step $\gamma$. Formalize the asymptotic clustering of points near such a spiral.

Part 3 (modular partition)

• Algebraic closure mod m: Since combinators use integer arithmetic, residues propagate deterministically modulo $m$. The set of possible residues is finite; evolution is a map on a finite set (a deterministic automaton). In particular, residues partition the sequence into classes and cycles; by Pigeonhole, evolution eventually enters cycles — visualizable as modular attractors.

Part 4 (constructive homomorphism)

• Encoding design: Fix encodings: e.g., map integer $n\mapsto (\theta=\alpha n, r=\beta e^{\lambda n})$ for geometry, and polynomial coefficients for algebra. Because combinators are finite linear/polynomial combinations, their images under $\mu_{D_i}$ are computable algebraic or geometric transforms; hence an explicit $h_{i\to j}$ is defined from the encoding formulas. This is constructive: demonstrate on base generators and extend by induction over combinator composition.

Part 5 (computational completeness — conditional)

• Finite-state simulation: Bounded lookback and finite tags simulate a finite automaton. Show an encoding of automaton states and tape segments as nested structures and combinators that implement transition functions.

• Turing completeness (sketch): Encode a cyclic tag system (known to be universal) by representing the tape as nested lists and combinators performing the deletion/appending rules. The formal proof follows the usual reduction used in cellular automata universality proofs (requires careful encoding and unbounded nesting).

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4. Corollaries & Examples

Corollary 1 (Fibonacci & φ). Fibonacci recurrence embedded in UNNS ⇒ ratio of consecutive nest-values converges to φ, producing a geometric attractor spiral with growth parameter $\ln\phi$.

Corollary 2 (Domain coloring). For modulus $m$ and combinators linear over $\mathbb{Z}$, the residue classes form a finite directed graph of transitions; each strongly connected component corresponds to a modular attractor.

Example (Tribonacci). Apply Part 1 with $r=3, c_1=c_2=c_3=1$. Part 2 yields the Tribonacci dominant root ψ and a corresponding spiral growth.

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5. Limitations, precise required conditions, and caveats

• Linearity requirement for some claims. Parts 1–2 use linear constant-coefficient recurrences. Nonlinear recurrences require separate analysis; many behaviors still embedable but proof patterns differ.

• Encoding regularity. Constructive cross-domain homomorphisms require encodings that respect combinator algebra (linearity/polynomial). Arbitrary combinators may break homomorphisms.

• Computational universality is conditional. Full Turing completeness needs unbounded nest size and combinators implementing arbitrary manipulations; for strictly finite $\mathcal{C}$ and bounded nesting, only finite automata are guaranteed.

• Numerical stability. Finite precision in numerical visualizations may obscure asymptotic behavior; proofs are exact algebraic statements, but demos require high precision to visualize deeper tails.

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6. Experimental validation roadmap (reproducible steps)

For each part produce a self-contained experiment:

1. Recurrence embedding test: Provide a UI where user inputs recurrence coefficients $(c_1,\dots,c_r)$ and seed values; UNNS generates sequence and exports it. Validate equality with standard recurrence generator.

2. Attractor visualization: For generated sequence, compute ratios and plot them against $r_d$; polar-plot the points under $\mu_G$; export CSV and PNG.

3. Modular coloring demo: Choose modulus $m$, color spiral points by residue, demonstrate cycles and report cycle lengths.

4. Homomorphism proof-by-example: Given algebraic expression, display algebraic node, geometric node, and compute explicit mapping functions $h_{A\to G}$. Show equality of images on small examples.

5. Universality construction (optional, advanced): Implement a tag-system simulator encoded in UNNS and demonstrate computation of nontrivial behaviors (small universal program).

Make all experiments log seed parameters, exact symbolic definitions of combinators, and exported data so results are auditable.

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7. Next steps toward full formalization

1. Write a formal specification language for UNNS combinators (a small typed language describing chunk/shift/linear/polynomial combinators).

2. Prove a library of embedding theorems: linear recurrences, polynomial recurrences, modular closure, and conditional universality.

3. Publish reproducible artifacts: JS + Python notebooks implementing the constructive encodings and demonstrations.

4. Peer review & stress tests: test prime-filter statistical claims on large datasets and report p-values, confidence intervals, and counterexamples.

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8. Closing statement (mathematical posture)

This Many-Faces Theorem turns the UNNS vision into a provable, constructive mathematical programme: it articulates exactly which faces are provable, what constructive encodings realize them, and what assumptions are required. The theorem is intentionally modular: each part is provable using standard techniques (linear algebra, modular arithmetic, constructive encodings), and the more ambitious computational universality claim is explicitly conditional and constructive.