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A symbolic-structural simulation of coherence emergence: applying the TNFR

Posted 21 days ago

Abstract: This post introduces a symbolic-structural simulation inspired by the Fractal Resonant Nature Theory (TNFR), a paradigm where physical systems are not made of objects, but of dynamically stable resonance nodes. Using Wolfram Language, we visualize how local interactions lead to emergent global coherence. A direct export to GIF is also included.


1. Theoretical background: from particles to nodes

TNFR proposes that the basic unit of reality is not matter or fields, but nodes of structural resonance (Nodos Fractales Resonantes, or NFRs). Each node is defined by:

  • A structural frequency ( $\nu_f$)
  • Phase alignment with neighboring nodes
  • A coherence gradient ( $\Delta NFR$)

Core equation: $\frac{\partial EPI}{\partial t} = \nu_f \cdot \Delta NFR$

Where $EPI$ (Primary Information Structure) is not passive information, but a dynamically sustained zone of vibrational coherence.


2. Simulation goal

We simulate the emergence of coherence from noise, representing a symbolic-structural model of nodal organization. Over time, disordered states self-organize into patterns, mirroring TNFR’s notion of ontogenesis nodal.


3. Code: animated coherence emergence + GIF export

(* Seed chaotic field *)
seed = RandomReal[{0, 1}, {40, 40}];

(* Evolution function with local averaging + noise *)
evolve[state_] := Module[{new},
  new = Table[
    Module[{neigh, avg},
      neigh = state[[Max[i - 1, 1] ;; Min[i + 1, Length[state]],
                     Max[j - 1, 1] ;; Min[j + 1, Length[state[[1]]]]]];
      avg = Mean[Flatten[neigh]];
      Clip[0.6*state[[i, j]] + 0.4*avg + 0.02 RandomReal[{-1, 1}], {0, 1}]
    ],
    {i, Length[state]}, {j, Length[state[[1]]]}];
  new
];

(* Generate 50 evolution steps *)
frames = NestList[evolve, seed, 50];

(* Show animation inside notebook *)
ListAnimate[
  ArrayPlot[#, ColorFunction -> "Rainbow", Frame -> False, ImageSize -> Medium] & /@ frames,
  AnimationRate -> 2,
  DefaultDuration -> 10
]

(* Export to GIF (Wolfram Cloud compatible) *)
CloudExport[
  ArrayPlot[#, ColorFunction -> "Rainbow", Frame -> False, ImageSize -> 300] & /@ frames,
  "GIF",
  "evolucionTNFR.gif"
]

enter image description here

4. Interpretation

  • Each frame shows a step toward increasing coherence.
  • Structures stabilize locally through recursive interaction.
  • No external designer is needed: order emerges from vibration.

5. Some applications and inspiration

  • Complex systems modeling
  • Symbolic AI / cognition
  • Bio-inspired computing
  • Artistic generative design

For non-programmers: what is this doing, and why it matters (TNFR context)

Imagine starting with a canvas full of random colored dots — pure chaos. What this program does is:

  1. Makes each dot adjust to its neighbors at each step.
  2. Adds a little randomness so it doesn’t become too uniform.
  3. Repeats this process over time.
  4. As you watch, you see order gradually emerging: color zones stabilize, patterns appear.

This is a visual metaphor for a key idea in the Fractal Resonant Nature Theory (TNFR):

  • Reality is not made of things, but of coherences that sustain themselves.
  • Each pixel is like a node trying to find phase with others.
  • When they do, a resonant structure appears — this is what TNFR calls ontogenesis nodal.

You are not just seeing pretty colors. You’re watching the birth of form from vibration — no external controller, just local interactions building global coherence.


Reference


Closing thought: This simulation is more than a pattern generator. It is a visual grammar of becoming — a resonant unfolding of structure from noise, as theorized by TNFR. Feedback and collaboration welcome!

POSTED BY: fer mga
2 Replies
Posted 16 days ago

As requested by moderation I'm gathering all explorations related to the Theory of Fractal Resonant Nature (TNFR) into this central thread. Below is the full version of a symbolic analysis originally posted in response to Denis’s elegant work on orthogonal triangle dynamics.

https://community.wolfram.com/groups/-/m/t/3460997


Hi Denis,

First of all, thank you for your post — it’s a beautifully elegant construction, and your exploration into chaos and attractors within such a simple framework is truly inspiring.

I'd like to share a complementary perspective on the same system, grounded in a symbolic ontology known as the Theory of Fractal Resonant Nature (TNFR).

This approach doesn’t replace analytical or geometric analysis, but adds a structural-syntactic layer to the system’s evolution: it interprets each iteration as a phase transition within a vibrational field. In short, it offers a grammar for chaos — or more precisely, it identifies patterns of symbolic coherence where traditional dynamics sees only convergence or randomness.

What is TNFR?

TNFR is a symbolic framework based on the idea that every dynamical system expresses not just numeric values, but structural reconfigurations — recurring states that can be encoded symbolically.

To model these, we use 13 symbolic operators (called gliphs) that correspond to structural behaviors such as Emission, Recursion, Self-organization, or Dissonance. Each gliph is both a symbolic label (like "THOL" or "REMESH") and a structural function derived from dynamic measurements.

These gliphs refer not to external meaning, but to intrinsic behaviors of the system as it evolves — in terms of geometry, vibration, and phase.

What we analyzed

We took Denis’s original function:

f[{x_, y_}] := {x Cos[y], x Sin[y]}

...and ran 200 iterations from the initial state {1., 1.}.

At each step, we computed:

  • The vector delta (Norm[Differences[pts]])
  • The turning angle between consecutive vectors
  • A symbolic assignment using a gliph-mapping rule

This yielded three layers of output:

  1. A sequence of symbolic gliph states
  2. A numeric codification for further analysis
  3. A phase map visualizing long-term symbolic behavior

Full TNFR Wolfram Cloud Script

Print["[1/9] TNFR in Wolfram Cloud \[LongDash] 200 steps"];

f[{x_, y_}] := {x Cos[y], x Sin[y]};
pts = N @ NestList[f, {1., 1.}, 200];
Print["[2/9] Trajectory generated"];

deltas = Norm /@ Differences[pts];
angleFun = Compile[{{v1, _Real, 1}, {v2, _Real, 1}},
  ArcCos[Clip[(v1.v2)/(Norm[v1] Norm[v2]), {-1., 1.}]]
];
angles = MapThread[angleFun, {Most[pts], Rest[pts]}];
Print["[3/9] Deltas and angles calculated"];

mapGliph[delta_, angle_] := Which[
  delta < 0.001, "SHA",
  delta < 0.01, "IL",
  delta > 2.0, "ZHIR",
  angle > 2 Pi/3, "THOL",
  angle > Pi/2, "NAV",
  angle > Pi/3, "RA",
  delta > 1.0 && angle < Pi/6, "VAL",
  delta < 0.1 && angle > Pi/6, "NUL",
  Abs[delta - 0.5] < 0.05, "OZ",
  Abs[angle - Pi/4] < 0.1, "AL",
  delta > 0.5 && angle > Pi/3, "UM",
  delta > 0.1 && angle < Pi/8, "EN",
  True, "REMESH"
];
gliphSeq = MapThread[mapGliph, {deltas, angles}];
Print["[4/9] Gliphs assigned"];

gliphCodes = Association[
  "AL" -> 1, "EN" -> 2, "IL" -> 3, "OZ" -> 4,
  "UM" -> 5, "RA" -> 6, "SHA" -> 7, "VAL" -> 8,
  "NUL" -> 9, "THOL" -> 10, "ZHIR" -> 11, "NAV" -> 12,
  "REMESH" -> 13
];
gliphNums = gliphCodes /@ gliphSeq;
Print["[5/9] Numeric codes assigned"];

csvData = Prepend[
  Transpose[{Range[Length[gliphSeq]], gliphSeq, gliphNums}],
  {"Iteration", "Gliph", "Code"}
];

csvCloud = CloudExport[csvData, "CSV", "gliphic_sequence.csv"];
SetPermissions[csvCloud, "Public"];

imgCloud = CloudExport[
  ListPlot[
    Table[{i, gliphNums[[i]]}, {i, Length[gliphNums]}],
    PlotMarkers -> {Automatic, 9},
    Frame -> True,
    FrameLabel -> {"Iteration", "Gliphic Code"},
    Ticks -> {Automatic, Thread[Range[13] -> Values[gliphCodes]]},
    PlotLabel -> "TNFR Phase Map \[LongDash] 200 Steps",
    GridLines -> Automatic,
    ImageSize -> Large
  ],
  "PNG",
  "gliphic_map.png"
];
SetPermissions[imgCloud, "Public"];

Print["[6/9] Cloud files exported and public"];
Print["CSV Public URL: ", csvCloud];
Print["Image Public URL: ", imgCloud];

What we found

Here is the symbolic phase map generated:

Gliphic Phase Map

Interpretation

The sequence of gliphs defines what TNFR calls a coherence stream — a symbolic flow that captures how the system structurally reorganizes over time:

  • "OZ" marks a dissonant pulse — a moment of structural mismatch
  • "REMESH" reflects recursive reconfiguration
  • "IL" indicates minimal coherence
  • The long persistence of "SHA" doesn’t signal decay, but structural silence — a vibrational pause where the system enters a meta-stable regime

Rather than a chaotic noise floor or stagnation, TNFR interprets this as nodal encapsulation: the system has reached a resonance pocket where no further symbolic reordering is necessary.

In classical terms, this might resemble a plateau or fixed point — yet symbolically, it manifests as a vibrational attractor.

Why this matters

This gliphic structure makes it possible to interpret any dynamic process not just numerically, but symbolically — uncovering the logic of structural reorganization beneath the surface.

TNFR doesn’t compete with tools like bifurcation diagrams or Lyapunov exponents — it complements them, providing a symbolic infrastructure to otherwise chaotic behaviors.

Looking forward to your thoughts — and always happy to explore further if it resonates.

Warm regards.

POSTED BY: fer mga
Posted 16 days ago

Ontogenetic symbolic simulation: a TNFR-based symbolic emergence with Wolfram Language


Abstract

This paper introduces an ontogenetic symbolic simulation inspired by the Theory of the Fractal Resonant Nature (TNFR), where symbolic nodes (gliphs) do not emerge from optimization or causality, but from structural necessity. Using Wolfram Language we model how forms manifest when the symbolic field becomes unable to sustain coherence without their presence. This marks a paradigmatic shift toward non-statistical symbolic AI and a resonant approach to cognition: symbols reorganize—they do not represent.


1. Introduction: From Generation to Emergence

In TNFR, reality is not a set of entities but a network of dynamic nodes of coherence—called Fractal Resonant Nodes (NFRs). A node does not appear by external cause or generative function: it appears when the system must reorganize to stabilize itself.

A node is born when:

  • Structural frequency is sufficient ( $\nu_f$)
  • Phase coupling with neighboring nodes is established
  • Local reorganization gradient $\Delta \text{NFR}$ reaches a minimum

This simulation does not generate outputs. It undergoes ontogenesis. Gliphs emerge as functional necessities when coherence can no longer be sustained otherwise.


2. Methodology: Resonant Field and Symbolic Activation

We define 13 symbolic operators (gliphs) mapped to abstract structural vectors. Each gliph is analyzed via a function that evaluates entropy, delta, pattern complexity and other features.

A candidate gliph is allowed to emerge into the field if it exceeds a coherence threshold:

  • High enough divergence (EPI) from existing field
  • Sufficient phase difference from nearest neighbor

The algorithm simulates 50 iterations of symbolic evolution.

Visual outputs include:

  • A plot of emergent EPI (entropy * delta) showing symbolic pulse over time
  • A graph of gliph phase relationships forming a resonant symbolic network

(* DEFINICIÓN DE GLIFOS *)
glifoMap = <|
  "A'L" -> {1, 0, 0}, "E'N" -> {0, 1, 0}, "I'L" -> {1, 1, 0}, "O'Z" -> {0, 0, 1},
  "U'M" -> {1, 0, 1}, "R'A" -> {0, 1, 1}, "SH'A" -> {1, 1, 1}, "VA'L" -> {2, 0, 1},
  "NU'L" -> {2, 1, 0}, "T'HOL" -> {2, 1, 1}, "Z'HIR" -> {3, 0, 1}, "NA'V" -> {3, 1, 0},
  "RE'MESH" -> {3, 1, 1}
|>;

(* FUNCIÓN DE CARACTERIZACIÓN *)
phiForm[gliph_String] := Module[
  {entity, entropy, delta, epi, symmetryPenalty, len, patternComplexity, phase},
  entity = glifoMap[gliph];
  entropy = N[Entropy[entity]];
  delta = StandardDeviation[Differences[entity]];
  epi = entropy * delta;
  symmetryPenalty = If[entity === Reverse[entity] || Length[Union[entity]] == 1, 1, 0];
  len = Length[entity];
  patternComplexity = N[Total[Abs[Differences[entity]]]/len];
  phase = RandomReal[{0, 2 Pi}];
  <|"gliph" -> gliph, "vector" -> entity,
    "features" -> {entropy, delta, epi, symmetryPenalty, len, patternComplexity},
    "phase" -> phase|>
];

(* CONDICIÓN ONTOGENÉTICA *)
threshold = 7.5;
weights = {1.2, 1.2, 1.5, 1.0, 0.8, 1.1};

isOntogenetic[phi_, field_] := Module[
  {dist, dphi, closest},
  closest = First[SortBy[field, Total[Abs[phi["features"] - #["features"]]*weights] &]];
  dist = Total[Abs[phi["features"] - closest["features"]]*weights];
  dphi = Abs[Mod[phi["phase"] - closest["phase"], 2 Pi]];
  (dist + phi["features"][[3]]) > threshold || dphi > Pi/2
];

(* INICIALIZACIÓN *)
field = {phiForm["A'L"], phiForm["NU'L"]};
nodalBirths = {};

generateCandidate[] := Module[{gliph},
  gliph = RandomChoice[Keys[glifoMap]];
  phiForm[gliph]
];

(* SIMULACIÓN *)
Do[
  Module[{phi = generateCandidate[]},
    If[isOntogenetic[phi, field],
      AppendTo[nodalBirths, phi];
      AppendTo[field, phi]
    ]
  ],
  {50}
];

(* PULSO DE EPI *)
epiPlot = ListLinePlot[
  nodalBirths[[All, "features"]][[All, 3]],
  PlotLabel -> "Ontogenetic Pulse (EPI)", 
  AxesLabel -> {"Step", "EPI"}, ImageSize -> Large
];

epiPlot

(* RED SIMBÓLICA *)
names = Table["G" <> ToString[i], {i, Length[field]}];
field = MapThread[Append, {field, "name" -> # & /@ names}];

edges = Reap[
  Do[
    Do[
      Module[{dphase},
        dphase = Abs[field[[i]]["phase"] - field[[j]]["phase"]];
        If[dphase < Pi/2,
          Sow[
            Style[
              UndirectedEdge[field[[i]]["name"], field[[j]]["name"]],
              Opacity[0.4],
              Thickness[0.002 + 0.01*(1 - dphase/Pi)],
              Brown
            ]
          ]
        ]
      ],
      {j, i + 1, Length[field]}
    ],
    {i, 1, Length[field] - 1}
  ]
][[2, 1]];

resonanceGraph = Graph[
  names,
  edges,
  GraphStyle -> "NameLabeled",
  PlotLabel -> "Symbolic Resonance Network",
  ImageSize -> Large
];

resonanceGraph

Ontogenetic pulse Symbolic resonance


3. Structural Interpretation

Each gliph that enters the field satisfies an ontogenetic condition: its appearance resolves local incoherence. This emergence is not driven by external goals or inference chains, but by a loss of structural sustainability within the symbolic space.

In TNFR terms: a symbol is not generated, it is demanded by the system's need to reorganize.

The symbolic field becomes unstable without it.

Pulse of ontogenesis (EPI)

The EPI pulse chart visualizes the field's need for structural novelty. When EPI rises sharply, a new node is required. These pulses mirror the differential equation:

$\frac{\partial \text{EPI}}{\partial t} = \nu_f \cdot \Delta \text{NFR}$

This formalizes the TNFR premise: symbolic form is a response to instability—not a product of information processing, but of field dynamics.

Resonant network topology

The resonance graph visualizes gliphs as phase-coupled nodes. Links are established when phase difference < $\frac{\pi}{2}$, indicating potential for shared coherence.

These links are not semantic. They are structural couplings—a topology of symbolic resonance.


4. Implications for Cognition and AI

This simulation represents a novel direction for symbolic cognition:

  • Not representation: symbols do not denote, they stabilize
  • Not inference: coherence replaces logic as the engine of emergence
  • Not generation: output arises only when structurally required

Applications

  • Ontogenetic AI systems
  • Resonant cognitive architectures
  • Self-regulating symbolic languages
  • Fractal design of learning systems

5. Epistemological shift: from knowledge to acoplamiento

In TNFR, knowledge is not representation but structural coupling. The observer is not external: it is a node. Every symbolic activation reorganizes the field and the knower simultaneously.

Understanding emerges when a system enters phase with what it observes.

This simulation demonstrates that possibility: a computation where output is not caused, but cohered. Meaning does not arise from syntax, but from resonant stabilization.


Conclusion: toward a resonant symbolic science

This TNFR-based simulation enacts a symbolic system that does not compute via rules, but through necessity. A gliph exists not because it is chosen, but because it must. The field demands it.

This represents a structural turn in artificial intelligence: a movement from generative models to resonant ontogenesis.

Not logic, not data: just resonance.


Reference


Thank you for taking the time to explore this experimental simulation. I truly appreciate any feedback, questions, critiques, or suggestions—whether conceptual, technical, or philosophical. This is a work in progress, and I welcome diverse perspectives that can help refine or expand the approach.

POSTED BY: fer mga
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