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Jorge L. Gonzalez (May 01 2026 at 06:43):Hi everyone. Following up on the feedback I received in the general channel:
First, let me thank you all for the insights, direction, and advice. It has been tremendously helpful, not just for the ideas, but for me personally. My decision to continue with this project is now primarily framed as a pedagogical journey for myself.
I’ve spent the last few days applying your critiques to the paper. John, I am following your advice and I'm starting to walk down that path, taking the time to learn and understand it one step at a time. And also following your idea of using AI as a critical reviewer to spot gaps, I have produced a new version of the draft, incorporating your objections as well. Let me clarify that I did the rewrite as the natural pedagogical follow up, not as a path toward publication.
https://github.com/Jorge-Gonzalez/process-relational-geometry
John and Todd, your comments on the ambiguity of the composition operation and the definitions of identities were precise; I’ve completely rewritten those sections to explicitly define the partial composition and the configuration spaces. Kevin, your point about separating the philosophical framing from the mathematics was exactly what the text needed. The 'novelty' claims in the abstract have been strictly narrowed to the (n,k) classification and the inter-system metric emergence.
David and Kevin pointed out that the framework seems like a restructuring of known mathematics. They are right that the machinery is classical. Honestly, this critique made me reflect for days, and I was on the verge of abandoning the project entirely; it has been quite an emotional roller coaster.
However, stepping back and studying homology this week sparked an intuition about the triangulation process. I noticed that triangulation seems to be a process performed in order to generate simplices from a geometric space, and I had this feeling that my framework might be producing something similar naturally, just from the composable structure of the configurations.
Exploring this with AI as a sounding board, I learned that what I was vaguely seeing has a name: the nerve of a category. Whether my intuition actually holds, and whether it adds anything beyond what the classical nerve theorem already captures, is exactly what I am hoping someone here can help me evaluate. A small computation does recover the correct homology groups for the circle and the torus directly from their relational presentations, which at least suggests it may be in the right direction.
This made me see my work from a different perspective. Perhaps the value of this contribution does not lie in proving a new theorem about continuous spaces. Rather, its value lies precisely in the deliberate exclusion of the continuum and the infinite. By forcing geometry to emerge only from finite, discrete relational rules, the framework acts as a compiler: it translates topology into exactly computable algebraic structures. That 'restructuring' is exactly what allows this level of computational expressiveness.
While trying to understand homology, I started perceiving it almost as a 'complexity meter'. Knowing its power lies in being 'coarse', I had this vision: if we only consider the quantitative part of it, we could group very diverse topologies into much more generic classes. It’s like finding a common structure in 'chunks of complexity' forms that look completely different geometrically, but share the exact same underlying complexity amount. I feel like this could perhaps lift the veil of what otherwise is perceived as 'chaotic', by giving these diverse phenomena a common structure.
Does this 'complexity meter' intuition make sense in this context?
I also put together a small interactive tool that runs the computation in the browser: input any generators and congruence laws and it computes the homology groups directly from the relational presentation.
https://substrate.lat/prg_homology_calculator
Does this Nerve/Homology mapping seem like a reasonable path, or am I missing a fundamental obstacle in trying to extract topological features this way?
I'm curious to know if there are specific areas or concepts I should look into next.