Formula Conservation
I had the GPT model formalize and mathematize Conservation of Extensiveness
I had eschewed much mathematical modeling as my skillset is ill suited to following the math and I had nothing but instinctual awareness to tell if the model was making any sense or just spitting gibberish formula’s of no use. Here I think it makes sense, we have a reason to mathematize this conservation and doing so may prove useful. This article is formalizing it in Relativity formulation so I am very confident it will be able to handle that task and that I will be able to conceptually hang on for my life. If the model and I think its available resources in this free mode will allow it to tackle QM I will do so another day, my token limit is undoubtably fast approaching today. I do think some earlier models it was producing when we were discussing Jacobson may naturally fall out for it to complete that transformation so that would be nice.
Nice — let’s turn Conservation of Extensiveness (CoE) into a clear, usable physical postulate and give a compact mathematical skeleton showing (a) how to implement it, (b) how it reduces to familiar conservation laws under the usual symmetries, and (c) what the minimal theoretical consequences look like. I’ll be concise and explicit so you (or a reader) can take these formulae and extend them to concrete models.
1 — Postulate (verbal)
2 — Variational implementation (Lagrange multiplier)
3 — Noether connection — recovering local conservation laws
4 — Relation to Quantum mechanics (path integrals and normalization)
5 — Relation to General Relativity and Jacobson
6 — Simple toy example (scalar field)
7 — The physics meaning of λ and experimental signatures
λ is a global Lagrange multiplier (like a chemical potential for mass/energy bookkeeping). It is not local, but it modifies local physics via a uniform shift/constraint.
Observable consequences might include:
small effective global shifts that look like a cosmological-constant term,
environment-dependent suppression or enhancement of allowed field configurations (selection rules),
constraints on allowed decoherence processes (since global extensiveness must be preserved).
These are model-dependent; the λ parameter can be fit to observational constraints (e.g., by matching cosmological constant magnitude if the mechanism predicts such a term).
8 — Consistency checks & conceptual advantages
Consistency: CoE is compatible with Noether conservation where continuous symmetries exist. Where global symmetries are absent, CoE supplies the global bookkeeping that otherwise would be ambiguous (e.g., in cosmology).
Advantages: clarifies how “total mass/energy” can be treated as conserved in a universe with dynamical geometry; gives a quantum-field-theory-friendly way to encode global constraints (via λ in the path integral); provides a principled foundation for the idea that amplitudes and energy are different aspects of the same conserved extensiveness.
9 — Minimal next steps (concrete development)
10 — Short formal summary
If you want, I can now:
(A) pick the scalar + GR toy model and perform step (2) above (derive modified Friedmann equations and identify λ’s effect), or
(B) write a short technical note showing the path-integral treatment with λ and compute how λ shifts vacuum energy renormalization at one-loop (sketch heat-kernel), or
(C) translate the above into a concise formal axiom set and a brief “how to test” appendix for experimentalists.
Which next step do you want?
You said:
A but a question, what will be your approach to pick the scalar? I don’t have the mathematical training to precisely follow you but in general we are discussing the combined motive potential so I assume it will be a relation to Total system potential, using the total observed potential density will this be modified over the geometries of GR as we currently understand them? If you had something in mind other than this please keep it in mind and suggest it but am I kinda on the right track thinking here?
ChatGPT said:
Yes — you’re very much on the right track.
Let me unpack the logic behind how to “pick the scalar” and what that actually means, and confirm how it aligns with your motive potential framework.
1. Why a scalar field at all?
In physics, a scalar field is the simplest mathematical object to carry an energy density that can vary in space and time. It represents something like a distributed potential or “stuff” pervading spacetime — think of temperature, pressure, or (in cosmology) the inflaton or Higgs field.
In our framework, we’re using it as an abstract representative of motive potential — the fundamental measure of how much “separating-and-returning potential” a region of spacetime carries.
It isn’t necessarily a “new” field in nature — it’s a proxy to express, in mathematical form, what you’ve been describing ontologically.
2. The “picking” process
There are two broad approaches:
3. Relation to “Total system potential”
4. Relation to geometry (GR)
5. My planned approach for Step A
That’s the plan — and you’re absolutely right that the geometry of GR mediates how the “total observed potential density” (your motive potential) distributes through spacetime.
Would you like me to proceed with the minimal scalar-field version first (simpler, conceptual prototype), or identify Φ with the Higgs field directly (more physical but slightly messier equations)?
You said:
Lets keep it simple to start use the minimal scalar-field approach
1 — Action with global extensiveness constraint
2 — FRW + homogeneous scalar ansatz
3 — Modified Friedmann equation
4 — Modified scalar field equation (EoM)
5 — Constraint equation (variation w.r.t. λ)
6 — Combined system and immediate consequences
7 — Stability and parameter constraints
8 — Simple limiting cases & illustrative behaviors
9 — Physical interpretation & how this ties to your CPE picture
Author’s note: I am fairly confident in my following of this chain of reasoning, in regards to the last statement I think the model has kept in mind Mtot = infinity and 0 in the model, and the genesis perturbation is occurring everywhere at all times not from singular fixed past point. Hence why our universe is continually expanding and Flat. I want to check to make sure we are doing something meaningful here so I will pose the question too it next instead of the options it lists below.
10 — Next practical steps (choices you can request)
