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A mathematically valid equation is not automatically a valid description of every physical regime.
Tensor calculus gives physics a powerful coordinate-independent language. But its use still assumes that the quantities and relationships being represented remain physically meaningful across the domain being studied.
This paper asks:
When does a tensor description remain structurally faithful to the system it describes?
I propose a validity domain:
Ω_valid = {x ∈ X | P(x) ≥ P_c and R_c(x) > R_crit}
where:
P(x) represents structural persistence, and R_c(x) represents relational coherence.
Within this domain, the relevant variables and relations remain sufficiently stable for the tensor description to retain its physical meaning.
At a regime boundary, the equations may remain formally writable even as the correspondence between the mathematical description and the physical organization begins to fail.
That is not a claim that tensor calculus is wrong.
It is a narrower claim:
Coordinate invariance does not, by itself, guarantee unlimited physical applicability.
A boundary of description is not necessarily a boundary of reality.
Relational Coherence and the Domain of Validity of Tensor Descriptions