This Relational Resonance Base Model is the description for a specific simulation model labeled Resonance Relation Simulation “V5”.
V5 investigates how much persistent structure can emerge from an extremely small resonance ontology before introducing explicit mechanisms for lock-in, closure, winding, splitting, merging, higher spans, or geometric structure.
The central hypothesis is:
A persistent physical structure need not be explicitly declared or locked. It may instead be a dynamically stable recurrent trajectory of locally interacting resonance nodes.
The purpose of V5 is therefore not initially to reproduce known particles or forces. It is to determine whether simple local resonance dynamics can generate:
The model is intended to be mathematically explicit enough for later simulation while keeping assumptions visibly separate from emergent results.
The minimal V5 universe contains resonance nodes.
For one base span, node $i$ has the state
$$ X_i = \left( \Phi_i,, \dot{\Phi}_i,, \ddot{\Phi}_i \right). $$
Here:
The base resonance period is
$$ T_0. $$
For normalized experiments one may choose
$$ T_0=1. $$
The wrapped phase is derived from the unwrapped phase:
$$ \phi_i = \Phi_i \bmod T_0. $$
The wrapped and unwrapped descriptions therefore do not represent separate physical variables. They are two views of the same phase state.
The minimal model initially contains no fundamental:
These may later emerge as useful descriptions of persistent dynamical patterns.
In particular, V5 should avoid assigning information to an edge that can already be calculated from the current states of its endpoint nodes.
Nodes respond only to other nodes in a local interaction set
$$ \mathcal N_i. $$
For small systems this may contain all other nodes.
For larger simulations it may be approximated using, for example, the $N$ most relevant or nearest nodes.
Typical computational candidates are
$$ N=24. $$
OPEN V5-01 — Definition of neighbourhood
“Nearest” is not yet fundamentally defined before projection exists.
A simulation may initially use an operational neighbourhood definition, but the eventual RQT meaning of locality must be specified independently of already assuming 3D space.
The model uses the nodes phase presuming a common starting point to calculate a phase-separation used as distance for assigning “neighbourhood”.
Because phase is periodic but $\Phi$ is unwrapped, a node may compare itself with different period-images of another node.
The signed nearest wrapped separation may be defined as
$$ \delta_{ij} = \operatorname{wrap}_{[-T_0/2,T_0/2)} (\Phi_j-\Phi_i). $$
Equivalently, node $i$ selects an integer $n$ such that
$$ \Phi_j+nT_0 $$
is the locally relevant image of $j$.
The simplest selection is
$$ n_{ij} = \arg\min_{n\in\mathbb Z} \left| \Phi_j+nT_0-\Phi_i \right|. $$
Direction of current phase progression may later participate in resolving ambiguous or dynamically competing images.
OPEN V5-02 — Image selection
Determine whether image selection depends only on nearest phase separation or additionally on:
The special boundary
$$ |\delta|=\frac{T_0}{2} $$
requires an explicit tie-breaking/continuity rule.
The base implementation uses $\dot\Phi_i$ as an image selector in case of doubt.
A node attempts to reduce incompatibility with its local resonance environment.
A minimal candidate phase term is
$$ M_i^\Phi = \frac{\sum_{j\in\mathcal N_i}w_{ij}\delta_{ij}} {\sum_{j\in\mathcal N_i}w_{ij}}. $$
A corresponding progression mismatch is
$$ M_i^{\dot\Phi} = \frac{ \sum_{j\in\mathcal N_i} w_{ij} (\dot\Phi_j-\dot\Phi_i) }{ \sum_{j\in\mathcal N_i}w_{ij} }. $$
A first candidate dynamic law is therefore
$$ \boxed{ \ddot\Phi_i = K_\Phi M_i^\Phi + K_v M_i^{\dot\Phi} } $$
where $K_\Phi$ and $K_v$ determine the response of the system.
This equation is a candidate minimal law, not yet a fundamental RQT equation.
The important conceptual requirement is more general:
Nodes alter their phase progression according to local relational mismatch. They do not seek an externally specified absolute phase.
A simulation may use a global evolution parameter $\lambda$ and numerical step
$$ \Delta\lambda. $$
For example,
$$ \dot\Phi_i^{,new} = \dot\Phi_i + \ddot\Phi_i\Delta\lambda, $$
and
$$ \Phi_i^{new} = \Phi_i + \dot\Phi_i\Delta\lambda + \frac12\ddot\Phi_i\Delta\lambda^2. $$
The first implementation should preferably use synchronous updates:
This prevents node-array processing order from accidentally becoming physical dynamics.
The global simulation parameter need not be identified with fundamental physical time. Physical time is explicitly reserved to emerge from other additional aspects.
An important V5 principle is
$$ \boxed{ \text{stable resonance} \neq \dot\Phi=0. } $$
A collection of nodes may reach
$$ \ddot\Phi_i=0 $$
while retaining
$$ \dot\Phi_i=\Omega\neq0. $$
If all participating nodes share the same progression rate,
$$ \dot\Phi_i=\Omega, $$
their absolute unwrapped phases continue to increase while their relative phases remain constant:
$$ \Phi_i(\lambda) = \Omega\lambda+\delta_i. $$
Thus
$$ \Phi_j-\Phi_i = \delta_j-\delta_i $$
remains stable.
The structure is stationary in its relational configuration while progressing continuously through unwrapped resonance phase.
Consider three nodes with normalized $T_0=1$:
$$ \phi_A=0, \qquad \phi_B=\frac13, \qquad \phi_C=\frac23. $$
Using signed nearest-image differences, every node sees its neighbours symmetrically:
$$ \left{-\frac13,+\frac13\right}. $$
Hence
$$ M_i^\Phi=0. $$
If additionally
$$ \dot\Phi_A = \dot\Phi_B = \dot\Phi_C = \Omega, $$
then
$$ M_i^{\dot\Phi}=0 $$
and therefore, for the candidate local law,
$$ \ddot\Phi_i=0. $$
Consequently,
$$ \Phi_i(\lambda) = \Omega\lambda+ \left{ 0,\frac13,\frac23 \right}. $$
This is an exact travelling solution of the simple symmetric rule.
It is important, however, to distinguish:
These are not equivalent.
The minimal three-node solution above permits any common
$$ \Omega. $$
Therefore the phase arrangement alone does not yet select a particular persistent identity.
A disturbance could change the common progression rate while preserving the relative $1/3$ arrangement.
If every $\Omega$ is equally stable, there is no temporal resonance selection.
This is insufficient if persistent structures are expected to possess characteristic recurrence behaviour.
V5 therefore distinguishes:
$$ \text{relational phase equilibrium} $$
from
$$ \text{temporal recurrence}. $$
V5 should not prescribe a recurrence period and then force nodes toward it.
Instead, ordinary local dynamics should generate trajectories, and recurrence should be detected when a trajectory returns to an equivalent resonance state.
Let the state of a candidate structure $C$ be
$$ X_C(\lambda) = { \Phi_i,\dot\Phi_i,\ddot\Phi_i }_{i\in C}. $$
A recurrence exists if there is some interval $T_C>0$ for which
$$ X_C(\lambda+T_C) \sim X_C(\lambda), $$
where $\sim$ means equality of the physically relevant relational state, allowing an overall phase-period advance.
For example, one may require
$$ \Phi_i(\lambda+T_C) \approx \Phi_i(\lambda)+mT_0 $$
for a common integer $m$, together with
$$ \dot\Phi_i(\lambda+T_C) \approx \dot\Phi_i(\lambda) $$
and
$$ \ddot\Phi_i(\lambda+T_C) \approx \ddot\Phi_i(\lambda). $$
Most importantly,
$$ \delta_{ij}(\lambda+T_C) \approx \delta_{ij}(\lambda) $$
for the participating nodes.
The pair
$$ (T_C,m) $$
then describes an emergent recurrence.
The corresponding mean progression rate is a consequence:
$$ \Omega_C = \frac{mT_0}{T_C}. $$
It is not prescribed independently.
$T_0$ is the fundamental phase-period scale against which recurrence is measured.
The precise relation between:
remains open.
OPEN V5-03 — Fundamental time relation
Determine whether $T_0$:
No additional constant should be introduced unless required by the dynamics.
RQT aims at physical time to be only associated with properties of recurrent structures.
A trajectory returning once to the same state is not necessarily a stable structure.
A persistent structure should additionally be resistant to sufficiently small disturbances.
Suppose
$$ X_C\rightarrow X_C+\epsilon. $$
After one recurrence,
$$ \epsilon’ = \mathcal R(\epsilon), $$
where $\mathcal R$ is the return map.
A recurrent state is locally attractive if
$$ |\epsilon’|<|\epsilon| $$
for sufficiently small perturbations.
Repeated recurrence then gives
$$ \epsilon, \mathcal R(\epsilon), \mathcal R^2(\epsilon), \ldots \rightarrow0. $$
Thus the system appears to “snap back” to its characteristic recurrence without containing an explicit snap-in instruction.
The snap-in is simply the observable consequence of an attractive recurrent orbit.
This suggests the following working definition:
A stable resonance identity is a local collection of nodes whose ordinary dynamics enters an attractive recurrent relational trajectory.
Such an identity has:
Its internal phase relationships remain bounded and organized.
Its relational state returns after an emergent interval $T_C$.
Small disturbances decay over repeated recurrence.
Larger disturbances may move the system outside the basin of attraction.
It may then:
A single freely progressing node,
$$ \Phi(\lambda)=\Phi_0+\Omega\lambda, $$
contains no internal relational recurrence. It only contains its base period recurrence time T0.
Whether such an isolated progression represents a meaningful physical identity is open.
RQT will place it as the most fundamental identity.
Two nodes provide the smallest relational system.
A candidate persistent pair would satisfy
$$ \delta_{12}(\lambda+T) = \delta_{12}(\lambda) $$
and reproduce its relative progression state after $T$.
A nontrivial two-node recurrence might involve periodic variation of:
$$ \delta_{12}, \qquad \Delta\dot\Phi_{12}, \qquad \Delta\ddot\Phi_{12}, $$
rather than a static separation.
This is an important test case because a genuine two-node limit cycle would demonstrate temporal closure without requiring a three-node frustration pattern.
OPEN V5-04 — Two-node recurrence
Determine whether the minimal local dynamics can support a nontrivial attractive two-node recurrence or whether two nodes necessarily synchronize/collapse without an additional mechanism.
Three nodes permit a qualitatively new configuration.
The symmetric state
$$ \left( 0,\frac13,\frac23 \right) $$
has balanced nearest-image phase differences.
It can therefore support a travelling relative equilibrium.
Whether it additionally develops a preferred recurrent progression rate is a separate dynamical question.
A stable three-node identity would require both:
$$ \text{stable relative phase structure} $$
and
$$ \text{attractive temporal recurrence}. $$
This is the minimal candidate for a self-sustaining frustrated closure in V5.
With more nodes, the state space can support:
These possibilities should be discovered numerically rather than inserted as predefined object classes.
V5 separates resonance dynamics from spatial projection.
The resonance layer contains
$$ (\Phi,\dot\Phi,\ddot\Phi). $$
A projector may derive pairwise quantities from these states, but it should not initially assign persistent distance requirements back to the resonance network.
Thus the causal direction is initially
$$ \boxed{ \text{resonance dynamics} \rightarrow \text{derived relational quantities} \rightarrow \text{projection}. } $$
Projection does not stabilize a resonance structure.
A useful state-derived quantity is the local Taylor prediction over one base scale $T_0$:
$$ \boxed{ Q_i = \Phi_i + T_0\dot\Phi_i + \frac12T_0^2\ddot\Phi_i. } $$
This is not introduced merely as an arbitrary spatial distance formula.
It has a resonance-space interpretation:
$Q_i$ estimates where the node’s current phase trajectory points after one base-period horizon.
For two nodes,
$$ \Delta Q_{ij} = Q_j-Q_i $$
gives
$$ \boxed{ \Delta Q_{ij} = \Delta\Phi_{ij} + T_0\Delta\dot\Phi_{ij} + \frac12T_0^2\Delta\ddot\Phi_{ij}. } $$
This combines:
It is therefore a candidate dynamic relational separation.
A future projector may derive a geometric distance using
$$ D_{ij} = F(\Delta Q_{ij}), $$
with the simplest exploratory choice
$$ D_{ij}=|\Delta Q_{ij}|. $$
This is not yet asserted as the physical RQT-to-RQM projection law.
OPEN V5-05 — Projection mapping
Determine:
First implementation: $\Delta\Phi$ uses unwrapped separation, Q should use $T_0$, Distance depends on $|\Delta Q|$ alone. Direction will emerge from the projection and orientation where relevant should emerge from wrapped phase.
If every pair of nodes produces a derived value $D_{ij}$, the projector receives a distance-like matrix.
It may seek coordinates
$$ \mathbf r_i\in\mathbb R^d $$
that minimize
$$ L_{\mathrm{proj}} = \sum_{i<j} w_{ij} \left( |\mathbf r_i-\mathbf r_j|-D_{ij} \right)^2. $$
If
$$ L_{\mathrm{proj}}=0, $$
all current relational requirements can be represented in the selected projection dimension.
If
$$ L_{\mathrm{proj}}>0, $$
the resonance state cannot be represented exactly in that geometry.
This residual is projection frustration.
It is not stored in the resonance system. It appears because a richer relational state is being represented in a lower-dimensional geometry.
A simple example illustrates the principle.
Three mutually equal nonzero pairwise distances cannot all be realized on a line but can be realized in 2D as an equilateral triangle.
Four mutually equal pairwise distances require 3D and form a regular tetrahedral embedding.
More generally, the dimensionality required by a relational state may exceed the dimensionality of a chosen projection.
This motivates the V4/V5 idea that geometric dimensionality may represent available degrees of freedom required to realize resonance relations rather than being fundamental input.
However, V5 does not yet claim to derive the observed dimensionality of physical space.
Projection frustration and resonance frustration should be distinguished.
Local phase-adjustment requirements cannot all simultaneously relax into a trivial synchronized state without crossing a different phase-image branch or leaving a dynamical basin.
The relational state exists in resonance dynamics but cannot be embedded exactly in the chosen geometric dimension.
The three-node $1/3$ configuration is a candidate example of the first phenomenon.
Larger structures may exhibit both.
V4 introduced explicit lock-in mechanisms.
V5 initially asks whether some behaviour previously attributed to lock-in is instead produced naturally by attractive recurrence.
A possible later interpretation is:
“Lock-in” is not necessarily a new physical constraint. It may be the recognition that a group of nodes has entered a persistent recurrent basin.
This remains a hypothesis.
Explicit lock state should not be introduced until simulations demonstrate that ordinary recurrence is insufficient.
V5 initially stores no winding number.
Unwrapped phase already records continuous phase progression.
Closed or recurrent trajectories may later admit derived winding-like quantities, for example the number of base periods traversed during a recurrence:
$$ m = \frac{\Delta\Phi_{\mathrm{recurrence}}}{T_0}. $$
Thus winding should initially be treated as a derived property of a recurrent trajectory, not an additional degree of freedom.
Earlier versions used split when a required adjustment could not be accommodated locally.
This mechanism is deliberately postponed.
A possible future V5 rule is that a node or structure splits when continuation of its current resonance identity would require a phase displacement beyond a permitted local range, possibly
$$ T_0 $$
or
$$ \frac{T_0}{2}. $$
OPEN V5-06 — Split criterion
Determine whether split is required at all once recurrent dynamics is understood, and if so whether its threshold follows naturally from nearest-image topology rather than being externally chosen.
Equality of instantaneous wrapped phase is not sufficient to establish identity.
Two nodes may temporarily satisfy
$$ \phi_i=\phi_j $$
while belonging to different dynamical histories or recurrent structures.
Therefore merge should also be postponed until identity is defined dynamically.
A future merge criterion may require equivalence of the relevant recurrent state rather than merely equality of instantaneous phase coordinates.
V5 begins with one base scale $T_0$.
Higher spans should not initially be predefined.
A possible later interpretation inherited from earlier RQT development is that a higher span is an emergent collective recurrence:
$$ T_s \approx nT_0 $$
or another stable recurrence relation generated by a lower-level closure.
This suggests a hierarchy:
$$ T_0 \rightarrow \text{local recurrence} \rightarrow T_1 \rightarrow \text{collective recurrence} \rightarrow T_2 \rightarrow\cdots $$
but this hierarchy must emerge from simulation rather than being imposed.
For simulation purposes, a candidate emergent object may be detected when a subset $C$ of nodes satisfies, over a sufficiently long observation interval:
bounded relational state $$ |\delta_{ij}|<\text{bounded range}, $$
recurrence $$ X_C(t+T_C)\sim X_C(t), $$
coherent progression with stable or recurrent relative $\dot\Phi$,
perturbation recovery small deviations tend back toward the same recurrent orbit,
persistence these properties survive for many recurrence cycles.
The object is therefore not a stored label.
It is a pattern detected in the dynamics.
Implement only:
Node:
Phi
dPhi
ddPhi
plus:
T0
dt
local-neighbour selection
nearest-image comparison
local mismatch update
No locks, closures, spans, winding, split/merge or projection feedback.
Determine whether they:
Initialize approximately
$$ (0,;0.31,;0.68) $$
with slightly unequal progression rates.
Test whether the system approaches
$$ (0,;1/3,;2/3) $$
modulo common phase translation.
Determine whether
$$ \dot\Phi_A \approx \dot\Phi_B \approx \dot\Phi_C $$
emerges and whether its value remains arbitrary.
Perturb one node’s
$$ \Phi,\quad\dot\Phi,\quad\text{or}\quad\ddot\Phi. $$
Measure whether the structure:
Introduce a nearby additional node and determine whether the candidate closure:
Do not add recurrence forces.
Record state history and search for
$$ X(t+T)\sim X(t). $$
Measure:
This determines whether characteristic recurrence exists naturally.
Only after resonance behaviour is understood, calculate
$$ Q_i = \Phi_i + T_0\dot\Phi_i + \frac12T_0^2\ddot\Phi_i $$
and candidate pairwise quantities
$$ \Delta Q_{ij}. $$
Investigate whether these produce useful and persistent geometric constraints.
Projection should initially remain diagnostic only.
This is the primary V5 question.
Its relative phase arrangement can be stable while common $\dot\Phi$ remains neutral. Additional dynamics may be required.
If yes, this may provide the smallest possible persistent identity.
They should emerge from $T_0$ and local dynamics rather than being assigned to structures.
Neighbour selection must ultimately be defined in resonance space without presupposing the spatial projection it is intended to generate.
Both implementations should be conceptually distinguished.
$$ Q=\Phi+T_0\dot\Phi+\frac12T_0^2\ddot\Phi $$
have sufficient physical meaning to participate in projection?
Its Taylor interpretation is clear, but its role as a geometric constraint remains a hypothesis.
This should be answered experimentally rather than assumed.
They should only be introduced where continuous node dynamics demonstrably cannot represent the required transition.
If so, V5 could recover the hierarchical resonance picture without explicitly constructing higher-span machinery at the base level.
At its most compact, V5 proposes the following world:
The fundamental state consists of locally interacting unwrapped phase trajectories.
Nodes do not know particles, geometry, winding, forces or closures.
They know only their own current phase progression and the locally observable phase states of other nodes.
Their state evolves in response to relational mismatch.
Most configurations may be transient.
Some may synchronize and lose distinction.
Some may disperse.
Some may enter nontrivial recurrent trajectories.
Among these, some recurrent trajectories may be attractive and therefore survive disturbances.
Those trajectories constitute candidate persistent resonance identities.
Collections of such identities may themselves develop slower or larger recurrence patterns, potentially generating a hierarchy of spans.
Spatial geometry is not assumed at this level. It is a representation problem: a later projector attempts to embed state-derived relational quantities into a low-dimensional geometry.
Where this succeeds, an effective geometry exists.
Where it cannot succeed exactly, projection frustration remains.
Thus the proposed emergence chain is
$$ \boxed{ \text{local phase dynamics} \rightarrow \text{relational organization} \rightarrow \text{recurrence} \rightarrow \text{stability} \rightarrow \text{persistent identity} \rightarrow \text{higher recurrence} \rightarrow \text{effective geometry}. } $$
The defining V5 principle is therefore:
Structure is not declared. Structure is whatever the local dynamics repeatedly manages to reconstruct.
That proposition is both the attraction of V5 and its principal mathematical test.
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