Appendix 2
SQ Wave Function: Fate
- Up (d→d+1, reverse),
- Down (d→d−1, decay under SER),
- Hold (keeping d).
Influence of geometry
The geometric rules for constructing discrete multidimensional space introduce systemic static rules linking cell states across zones and clusters. At any moment:
- Neighbor constraint: the dimensions of neighboring cells differ by no more than 1.
- Face matching: shared faces fully match in size (dimension).
- Dimension staircase: as a consequence of the two axioms above, a dimension transition in gradients is only possible with step 1 (staircase-like).
Iterative mechanism
- Static rules directly imply that:
- Down starts from the most “high-dimensional” cell of a cluster
- Up starts from the least “high-dimensional” cells of a cluster
- The state of the “neighbors” directly permits / prohibits certain outcomes:
- The presence of d−1 neighbors forbids Up
- The presence of d+1 neighbors forbids Down
- Combinatorially, an Up step requires same-d neighbors on all faces of the d-cell (the faces must raise their dimension by 1 simultaneously by “embedding” the neighbors into themselves).
- Consequently, the Down step is combinatorially “easier” (fewer conditions are required).
- A combination of conditions that forbids Up and Down yields Hold (at the same time, Hold is an independent option of Fate).
Dynamic patterns
The same mechanisms allow one to state that:
- Repeating Hold for two ticks in a row is minimally necessary and sufficient to form a zone of “lagging” cells (local inhomogeneities of space).
- Any Up “waits for” the neighborhood to catch up before the next Up.
- Many such local dynamic iterative patterns can exist (and plausibly do exist), with varying complexity (number of steps, a zone of interdependent ticks, etc.). From this one can draw a general conclusion:
- Among several geometrically admissible outcomes, preference goes to those that are non-conflicting for geometry—those that do not violate face-matching rules in the immediate neighborhood,
- Minimum frustration (additional axiom): among all geometrically admissible outcomes, preference goes to the one that minimizes the number of face-matching conflicts at the next tick in the local neighborhood (a local criterion),
- The restructuring of cell dimension in Space proceeds in cascades along permitted geometric patterns [19].
Note 19
I allow that the set of geometrically permitted cascades forms a hierarchy of self-similar structures; in this sense, the dynamics of local dimension patterns may potentially be described using mathematics akin to fractal mathematics.
Macroparameters of Fate – Doom
The QoQ approach assumes that the resulting share of decays (active cells), expressed as the “growth” of Space, is constant at each global SER tick. This sets a frame for the wave function: at each tick, Ψ distributes outcomes across cells and options (Up/Down/Hold) while preserving the overall pace.
Dynamic homeostasis as a whole operates with a zero sum: it redistributes dimension in proportion to the dynamics of substance, while maintaining an overall entropy trend due to the overall “dispersion” of substance.
As a result, these two macro-factors are balanced and, in combination, reflect the main regular systemic outcome of Ψ: maintaining a constant fraction of degrading SQ per unit of global time.
It is also possible that the fixed share of outcomes at each tick is not a manifestation of stochasticity, but a consequence of the mathematics of discrete systems of this type; however, since a full mathematical apparatus for a strict description of such systems and processes is not yet available, this cannot be stated conclusively.
SQ Wave Function: Fate
Appendix 2