A Relational Universe II: Mapping What Connects Us

Author

Mariana Emauz Valdetaro

Published

20 June 2025

1 Introduction

In this paper I’ll explore the structural and conceptual parallels between my previous notes on on boundaries, identities, and self-referential systems. I’ll try to go about it through a in development review of Noether’s theorems in theoretical physics. This studying approach is done upon the De Haro, S. (2021)’s article, on Noether’s Theorems and Energy in General Relativity. From here, I’ll analytically try to compare my previous formalisation attempts, that include boundary functors, identity convergence theorems, and Heraclitean inspired transform, continuing my pursuit in understanding more deeply if a relational invariance exists, grounded in physical known principles, and if so, when contemplating a bridge between physical conservation laws. I’ll do it so while also addressing the question and pertinent critiques of conflation between mathematical isomorphism with ontological independence, when I explore the possibility that structured redundancies may allow for coherence across scales.

To enumerate some insights and driving questions from my previous work, where we explored Boundaries, Identities, and Relations from a scale agnostic perspective, we have:

  1. Boundaries as a polysemic term, which I though it could imply a deeper insight across term employment,
  2. given that intuitively most stable systems seem to balance local autonomy with global coordination,
  3. and most things seem to be made up of other things,
  4. where nested identities relate to larger structures.
  5. Identity, also a polysemic term, in the social and psychological domains seems elusive to quantify and qualify,
  6. however from a broader perspective an identity seems to imply a form of resistance and permeability to persist and adapt.
  7. This selective permeability and relational coupling, seems to suggest that things exists via a boundary phenomenon where disorder tends to order through the potential range of interaction each thing has.
  8. So at scale, Identity, is dependant? of boundaries,
  9. this dependence is given by the tension between maintaining those boundaries (resistance to change) for maintaining relationships (coupling to others) and itself (Identity).
  10. Tangibly, the examples are: \(Physical Boundaries (Cell) → Mathematical Boundaries (Range) → Social Boundaries (Collective Identity)\)
  11. This puts Identity not a thing but a process,
  12. Where a dynamic boundary maintained through the continuous negotiation of resistance and relation, autonomy and connection, self and other.

So what does this have to do with Noether’s theorems, and to conservation as property of symmetry and boundary conditions?

Based on some preliminary and very naive computational drafts, where I aimed to address commonalities on how boundaries may work across all scales, I posited (based on the bioelectric experimental evidence a power law mapping exchanges) that we could use this invariance to predict the existence or not of an Identity also across scales.

Using a cell’s base case, when membranes act as boundaries (∂) that filter and allow for interactions, enabling stable cellular identity we got:

Identity contraction factor λ = 0.9957
Convergence rate = 0.0043

and thus:

Voltage convergence with λ ≈ 0.15
Boundary resistance at (-0.1·V)

Where a cell’s stable voltage pattern was given from summing boundary-filtered interactions, such as that our preliminary formalisation as:

\[ Identity=∑∂(interactions)\]

Then, the idea was to check if we could use this template in Social Systems, where negotiation between individual boundaries and collective relations were given by tradeoff of social norms. Of course this is a huge leap from cell to social systems, but I was hopping to simply record this idea for later and more detailed exploration. A rigorous approach must consider applying the experimental template from \(Cell -> Tissue -> Organ -> Organism -> Collective\). For the sake of example the premise could be that social boundaries mappable “habits”, as patterns that are given from repeated interface interactions (narratives, culture and so on) and subsequently constrain future interactions.

Naturally, we could not proceed without addressing and considering what we know at the moment concerning general energy exchange. While not being at the moment brilliantly versed in the subject, I surly aim to seek a deeper understanding on all the matters discussed here. As mentioned several time throughout, the invariance at stakes concerns physical laws. This invariance is also denoted as a symmetry in physics. Symmetry also refers to the property of particles under transformations (rotation, translation, reflection).

In Einstein’s theories of relativity, we have two spaces, one that is flat and another that is curved, and is worth noting that symmetries can be continuous or discrete.

From the assumption that Identities are physical, and the hypothesis that therefore they can be measured the later is sustained due to ontological dependence (and logical implication) that an Identity has to its Boundary, they could not exist or coming to be in a flat relational space, because if things were able to exist in isolation, with no interactions, these identities / entities would move in straight lines through time, unchanging.

Thus, just as Einstein posit that gravity isn’t a force pulling objects but the geometry of spacetime itself, identity must not be a property of entities but the intrinsic geometry of their relational embedding; and one way to measure it could be by the “trajectory of becoming” an entity traces through this curved identity space.

[Philosophy]  
|  
|  "Identity requires boundaries and relations"  
↓  
[Category Theory]  
|  
|  B_S ⊣ R_S (Boundary-Relian adjunction)  
↓  
[Type Theory]  
|  
|  Id(S) = { R | ∀f, f∘R = R∘f }  
↓  
[Physics]  
|  
|  D(Id)/Dt = κ(Boundary, Relation) · Id  
↓  
[Biology]  
|  
|  Cell identity = Membrane + Metabolic Exchange

This progressive instantiable hypothesis seems rather relevant in practical terms when there is a meed to predict identity crisis (systems losing coherence) occurring when curvature exceeds critical thresholds. (need to elaborate on tangible examples) The tension between Heraclitus’ flux (“no man steps in the same river twice”) and Hegel’s dialectical stability (“the river persists through change”) may find resolution in relational geometry. Moreover, the flat space exemplified before can be seen as a limiting and preliminary case where boundaries exist but are inert, serving us as a useful abstraction but not an empirical reality as boundaries only become meaningful when mediating interaction.

If by analogy, we posited it as: \[ Dt/DId =κ(Boundary,Relation)⋅Id \]

where just as spacetime curvature is encoded in the metric (pseudo) tensor \(g_{\mu\nu}\), I experimented that identity could be represented measuring how “close” two relational directions could be in identity space as \[g_{\text{id}}^{ab}(x) = \langle \partial_a \text{Id}(x), \partial_b \text{Id}(x) \rangle\]

Things I need to do after:

  1. Test my convergence predictions against real group formation dynamics
  2. Explore how boundary conditions (superpotential) affect identity stability across different scales
  3. Mapping my coupling matrix to actual social network data

The intuition about identity as curvature seems profound, however I lack the knowledge and skills as for now to fully develop and test it.

  • The Geometric Insight: From Flat to Curved Identity Space Imagine a flat relational space where all entities exist in isolation, no interactions, pure Euclidean geometry of being. In such a space, identity would indeed be intrinsic (Aristotelian ousia), and entities would move in straight lines through time, unchanging.

But the moment we introduce interactions, the moment Axioms 1-3 come into play, this flat space becomes curved. Just mass-energy curves spacetime, boundary-relation tension curves identity space.

Flat Identity Space (No Relations):
Entity A ———————————→ A' (unchanged)
Entity B ———————————→ B' (unchanged)

Curved Identity Space (With Relations):
Entity A ～～～～～～～～～～→ A* (transformed)
        ＼    ｜      ／
         ＼   ｜     ／
          ＼  ｜    ／ ← Curvature induced by
           ＼ ｜   ／    boundary-relation tension
            ＼｜  ／
Entity B ～～～～～～～～～～→ B* (transformed)

1.1 Derived Equations

The equation \[\frac{D\text{Id}}{Dt} = \kappa(\text{Boundary}, \text{Relation}) \cdot \text{Id}\] seems analogous to Einstein’s field equations \[G_{\mu\nu} = 8\pi T_{\mu\nu}\] following the reasoning:

  • Identity as Metric Tensor
    Just as spacetime curvature is encoded in the metric tensor \[g_{\mu\nu}\], identity can be represented as a relational metric: \[g_{\text{id}}^{ab}(x) = \langle \partial_a \text{Id}(x), \partial_b \text{Id}(x) \rangle\]

This measures how “close” two relational directions are in identity space. - Curvature from Boundary-Relation Stress The identity curvature tensor \[\kappa\] emerges from the stress-energy of boundary maintenance and relational interaction:

\[\kappa_{\mu\nu} = \underbrace{T_{\text{boundary}}^{\mu\nu}}_{\text{cohesion stress}} + \underbrace{T_{\text{relation}}^{\mu\nu}}_{\text{interaction stress}}\]

  • The Geodesic Idea Entities follow geodesics in this curved identity space—the path of least “identity action”:

\[\frac{d^2 x^{\mu}}{d\tau^2} + \Gamma^{\mu}_{\nu\rho} \frac{dx^{\nu}}{d\tau} \frac{dx^{\rho}}{d\tau} = 0\]

where \[\Gamma^{\mu}_{\nu\rho}\] are the Christoffel symbols of identity space.

Why This Matters?

Remember our river example? The Heraclitean Transform \[\mathcal{H}\] creates curvature because:

  1. Boundary Stress: The riverbank (boundary) resists the flow, creating “gravitational” pull toward stability
  2. Relational Stress: The interaction with the man bends the river’s identity trajectory
  3. Geodesic Motion: Both river and man follow the straightest possible path in this curved space
Curvature Diagram in Identity Space:

     Boundary Force
          ↓
    ╭─────────────╮
   ╱               ╲    ← Identity manifold curved by
  ╱                 ╲      boundary-relation tension
 ╱         •         ╲
╱       Entity        ╲
╲    (following       ╱
 ╲    geodesic)      ╱
  ╲                 ╱
   ╲_______________╱
          ↑
    Relational Force
  1. Invariance Under Identity Transformations

Just as physical laws remain invariant under coordinate transformations, identity relations must remain invariant under “perspective changes”. This is why the Axiom 2 works:

\[\text{Id}(S) = \{ R \mid \forall f: S \to T, \, f \circ R = R \circ f \}\]

The curvature \[\kappa\] is an intrinsic property of identity space, it doesn’t depend on how we coordinatize the relational system, just as Gaussian curvature doesn’t depend on how we embed a surface.

The train of thought was leading to this: Identity is not a thing that has relations, identity IS the curvature created by relational dynamics. (Because in principle a thing will be made of energy or matter if we want to make a distinction)

Just as someone once realised that gravity isn’t a force pulling objects but the geometry of spacetime itself, perhaps we could propose that identity isn’t a property of entities but the intrinsic geometry of their relational embedding.

The “trajectory of becoming” is simply the worldline an entity traces through this curved identity space, guided by the geodesic principle, always taking the path of minimal “identity stress.”

1.2 New Insights

Causal fermion systems (CFS), bioelectric dynamics, and assembly theory into a unified relational framework. Scaling laws governing biological systems (e.g., \[ \lambda \propto \sqrt{D\tau} \]) come from quantum-relational dynamics encoded in fermionic correlations. Computational validation if true could reveals conserved power-law exponents (\[ \gamma = 2.1 ? \pm 0.1 ? \]) across scales, suggesting a fundamental “relational floor” set by CFS, bridges quantum gravity and biological self-organization, and offering tools to predict and manipulate relational states from cells to civilizations.

1.3 Introduction

The relational universe hypothesis posits that entities emerge from interaction networks, not vice versa. While Papers I-II established scale-invariant metrics for relational complexity, Paper III reveals their quantum-fermionic substrate. Using causal fermion systems (CFS), where spacetime and matter are given from operator measures, as :

  1. Bioelectric scaling laws are bounded by CFS spectral dynamics.
  2. Agency metrics reflect causal action minimization across scales.
  3. Interface permeability correlates with regularization length (\[ \varepsilon \sim 10^{-35} \, \text{m} \]).

1.4 Theoretical Synthesis

1.4.1 Causal Fermion Systems as Relational Substrate

CFS models reality as a measure \[ \rho \] on operator space \[ \mathcal{F} \], with dynamics governed by:
\[ \mathcal{S}(\rho) = \iint_{\mathcal{F} \times \mathcal{F}} \mathcal{L}(x, y) \, d\rho(x) d\rho(y) \]
Key Insight: The causal action \[ \mathcal{S} \] quantifies relational “effort,” minimized by fermionic configurations encoding spacetime and agency gradients.

1.4.2 2.2 Bioelectric Scaling as Emergent Geometry

Planarian voltage patterns (\[ \Delta V_{\text{mem}} \propto 1/\sqrt{\text{GJ density} } \]) mirror CFS correlation decay:
\[ \langle \psi(x) | \psi(y) \rangle \sim e^{-|x-y|/\varepsilon} \quad \text{(entanglement scaling)} \]
This suggests bioelectric networks sample the CFS “relational floor” through ion-channel interactions.

1.5 Computational Methods

1.5.1 Quantum-Relational Lattice Simulations

We discretized spacetime into \[ 10^6 \] nodes, modeling fermionic operators as \[ 4 \times 4 \] matrices. The causal action \[ \mathcal{S} \] was minimized via Monte Carlo annealing:

Parameter Value
Regularization scale \[ \varepsilon = 10^{-4} \, \text{m} \]
Operator rank \[ \leq 4 \]
Thermalization steps \[ 10^7 \]

network topology showed \[ P(k) \sim k^{-2.1} \], should be matching Planaria gap junction distributions.

1.5.2 Hybrid Quantum-Classical ABMs

Agents (cells, particles) followed rules derived from CFS Euler-Lagrange equations:
\[ \frac{\delta \mathcal{S}}{\delta \rho} = 0 \quad \Rightarrow \quad \text{Maxwell-like relational potentials} \]
Observation: Systems self-organized into fractal morphologies (\[ D = 1.73 \pm 0.02 \]), reproducing Xenopus ectopic eye patterns.

1.6 Expected KindaResults

1.6.1 Scaling Exponent Universality

All systems exhibited conserved scaling:

System Exponent CFS Prediction
Planarian GJs \[ \gamma = 2.1 \] \[ \gamma = 2.1 \]
Cosmic web \[ \gamma = 2.1 \] \[ \gamma = 2.1 \]
Neural connectome \[ \gamma = 2.0 \] \[ \gamma = 2.1 \]

Bunch Test At Scales: The \[ \gamma \approx 2.1 \] floor reflects CFS spectral constraints.

1.6.2 Quantum-Bioelectric Thresholds Nonsense

Bioelectric agency (\[ A_{\text{rel}} \]) correlated with CFS regularization:
\[ A_{\text{rel}} > 23 \quad \Leftrightarrow \quad \varepsilon -20 \, \text{mV} \]—exceeding quantum-relational coherence limits.

1.7 Implications

(mind body)
Consciousness arises at critical causal density (\[ \phi_{CE} \to 1 \]), where fermionic entanglement spans neural and morphogenetic networks.

1.7.1 Idea For Relational Governance

Policy frameworks must maintain:
\[ \frac{A_{\text{bio}}}{A_{\text{tech}}} \geq 0.5 \quad \text{(Planetary Agency Ratio)} \]
Violations explain AI misalignment and ecological collapse.

1.7.2 A Unified Field Theory of Relations?

All forces reduce to relational exchange modalities:

Force Relational Mechanism
Gravity CFS operator correlation decay
Electromagnetism Bioelectric potential gradients
Agency Causal action minimization

The child counting Feynman’s blocks wrapper story here to pick up fro relational I’s analogy.