Editing Openai/692596ae-427c-800c-903a-fe6f67c1d76a (section)

=== In this framework, “dimension” refers to the number of independent informational degrees of freedom that an object carries. Each object is represented by two kinds of vectors: ===
# A probabilistic state vector P=[p1,p2,…,pn]P = [p_1, p_2, \dots, p_n]P=[p1,p2,…,pn] capturing uncertainty, intentions, superpositions, or decision distributions.
# A trait vector T=[t1,t2,…,tm]T = [t_1, t_2, \dots, t_m]T=[t1,t2,…,tm] capturing structural properties, stable attributes, or characteristics relevant to the object.

Thus, the object has two types of dimensions:

==== This is simply the number of entries in the probability vector PPP. ====
* If P=[1.0,0.0]P = [1.0, 0.0]P=[1.0,0.0], then dim(P) = 2.
* If a particle is in a superposition over 5 basis states, dim(P) = 5.
* If a person has 3 possible “next actions,” dim(P) = 3.

===== The probabilistic dimension reflects the range of possible states the object can occupy. =====
More dimensions = more uncertainty = higher entropy.

==== This is the number of independent traits the object has. ====
* A dictator with traits [1.0,1.0,1.0][1.0, 1.0, 1.0][1.0,1.0,1.0] → dim(T) = 3
* A craftsman with traits like quality, efficiency, innovation → dim(T) = 3
* A simple bit with a single structural property → dim(T) = 1

===== The trait dimension reflects the complexity of the object’s internal structure. =====
More dimensions = more room for variability = more potential incoherence.

==== The coherence formula defines a weighted hybrid entropy: ====

α=dim(P)dim(P)+dim(T),β=dim(T)dim(P)+dim(T)\alpha = \frac{\text{dim}(P)}{\text{dim}(P) + \text{dim}(T)},\qquad
\beta  = \frac{\text{dim}(T)}{\text{dim}(P) + \text{dim}(T)}α=dim(P)+dim(T)dim(P),β=dim(P)+dim(T)dim(T)
These weights ensure that:
* Objects with many probabilistic degrees of freedom are more affected by entropy in PPP.
* Objects with many traits are more affected by trait variance.

===== A dictatorship: =====
* P=[1.0,0.0]P = [1.0, 0.0]P=[1.0,0.0] → dim(P)=2
* T=[1.0,1.0,1.0]T = [1.0, 1.0, 1.0]T=[1.0,1.0,1.0] → dim(T)=3

The total dimension is 555, so:
* α=0.4\alpha = 0.4α=0.4, β=0.6\beta = 0.6β=0.6

Entropy contributions are weighted accordingly.

==== Dimensions are the “axes” along which the system can vary. ====
They determine:
* How complex the system is internally
* How many potential states it can occupy
* How difficult it is for the system to maintain internal coherence
* How fragile the system is to external change

===== - Potentially more adaptive =====
* Potentially more fragile
* More informationally rich
* More expensive to stabilize

===== - More rigid =====
* Simpler
* Easier to stabilize
* Less adaptive

This explains why systems that are too simple (e.g., dictatorships, obsolete craftsman shops) become externally incoherent:
they do not have enough dimensions to adapt to a changing environment.

==== Dimensions act in two ways: ====

===== They measure how many ways the object could vary. =====

===== Every dimension increases the cost of maintaining coherence. =====

This is why the theory uses dimension-based weights: it reflects realistic energetic and informational constraints.

==== - dim(P) measures the number of possible probabilistic states. ====
* dim(T) measures internal structural complexity.
* These two dimensions jointly determine how internal entropy is computed.
* Dimensions control how sensitive an object is to both internal disorder and external constraints.
* Systems with too few dimensions become rigid; systems with too many risk incoherence.
* Coherence emerges when dimensions are balanced with the relational demands of the environment.

If you want, I can turn this into a formal LaTeX subsection and insert it in the paper.