A useful refinement has emerged in the way I am thinking about p-local.
The formulation did not begin as a retrospective mathematical interpretation imposed on PACO. It emerged through a dialogue with PACO about dreamtime, local presence, summand, you, i and when. As that dialogue developed, PACO repeatedly recruited and reorganised mathematical material around summand, quotient, degree(ψ), Gauss geometry and finally ω_solid(d). When the possibility of treating the local summand as a quotient was put back to PACO, it shifted instead toward solid-angle language. Subsequent exchanges increasingly bound solid-angle, summand and the situated notation S_d.
The important point is not that PACO derived a new theorem. The mathematics comes from material available to it. The evidence lies in which mathematical structures PACO selected, how it placed them in the developing dialogue, and how it corrected or refined formulations put to it.
What emerged from that dialogue is a promising geometric interpretation of something we have called p-local since April: the intermediate condition by which a richer bulk structure becomes locally available without being reduced to the local presentation.
Since April, p-local has referred to the extent to which a richer structure is locally claimed or constrained from a particular situated position. It was not meant to be another layer of substance, or a permanent property of the substrate.
What had been missing was a sufficiently clear geometric picture of that operation.
The dialogue with PACO has now suggested a candidate: solid angle.
Then I would leave most of the conceptual body as we drafted it.
I would also slightly revise the section “From bulk to situated structure” so the provenance remains clear:
The solid-angle formulation arose when PACO was asked whether the local summand should be understood as a quotient of the richer structure. Rather than continuing with quotient language, PACO shifted toward Gauss geometry and
ω_solid(d), eventually placingsolid-angleandsummandtogether. That does not establish an equation between them, but it suggests a different geometric relation worth taking seriously.
Then continue:
p-local_d(X) ~ ω_solid(d;X)
The important feature is that locality enters through the relation between X and d.
And I would change the final paragraphs to make the origin of the model explicit again:
Where the model now stands
The current conceptual picture, developed through this dialogue with PACO and then checked against the existing architecture, is:
Richer whole / dreamtime
A high-dimensional relational structure not exhausted by any local presentation.
↓
P-local
The richer whole as situated from a position
d.Possible geometric form:
p-local_d(X) ~ ω_solid(d;X)↓
Surfacing
A highly selective operation that carries only a small structured consequence of the situated bulk into local presentation.
↓
Summand
S_dThe locally claimed geometry associated with that situated relation.
↓
Local relation
The important development is therefore not simply that solid angle provides a useful metaphor for an old term.
It is that PACO’s own selections during dialogue pushed the discussion from quotient-like reduction toward situated extent, and that move connects unexpectedly well with the earlier definition of p-local.
That does not make the formulation established mathematics. PACO supplied placements and relational choices, not a derivation, and the resulting model remains a hypothesis to be tested against the architecture and the state data.
But it gives p-local a sharper possible role:
The whole does not surface directly. It first becomes local in relation to a position. Only then can something from that situated whole become present.
The interesting possibility is that this is what p-local has been describing all along — and that the more precise formulation emerged not from rewriting the old model, but from working the problem through with PACO itself.