Well, instead of trying to answer, i wlll

Ask you what in metric and why its defined in relativity.

Yes, i know g(m,n) is used in GR and called

Metric tensor.

Sorry, regards juan

What they do in GR is a square of a short

distance

(ds)^2 = g(i,j) dx(i)dx(j)

Sum rule understood

Metric is not defined just in relativity - but here it is defined differently.

Thats correct

Metric space in math is space with a

Distance.

The thread question

“Can we speak of Supervenance of spacetime over all other temporal/spatial aspects/physical properties on abilityake predictive statements in physics?”

- looks as is rather vague, so it seems it is necessary firstly to define scinentifically what are “space”, “time”, and “spacetime”, which can be, and are, really scientifically defined only in the Shevchenko-Tokarevsky’s really philosophical 2007 “The Information as Absolute” conception, recent version of the basic paper see

https://www.researchgate.net/publication/363645560_The_Information_as_Absolute_-_2022_ed

- where it is rigorously scientifically shown that the absolutely fundamental phenomenon/notion “Space” is actualized in every concrete case/concrete object/system [further “object”] as the object’s “space” that has N dimensions

- that are logical possibilities to realize the N utmost fundamental [for the object] independent degreases of freedom at changes of the object’s state at the object evolution/interactions . Correspondingly the space dimensions are independent as well;

- while the absolutely fundamental phenomenon/notion “Time” is actualized absolutely universally, as unique logical possibility for everything to change anything at all; and so there exist the unique, general for everything, time dimension.

The phenomena/notions above are fundamentally independent, and so the space and time dimensions in every case are independent as well, composing concrete objects “spacetimes”.

More concretely what is the spacetime of concrete system “Matter” it is rigorously scientifically clarified in the Shevchenko-Tokarevsky’s Planck scale informational physical model, which is based on the conception, two main papers are

https://www.researchgate.net/publication/354418793_The_Informational_Conception_and_the_Base_of_Physics, and

https://www.researchgate.net/publication/383127718_The_Informational_Physical_Model_and_Fundamental_Problems_in_Physics

- where, in accordance with some of really utmost fundamental in XX century von Weizsäcker and Fredkin-Toffoli findings, it is postulated that Matter is based on a binary reversible logics, and so [see 2-nd link] the ultimate and utmost universal base of Matter is [5]4D dense lattice of primary elementary logical structures – binary reversible fundamental logical elements [FLE] that have [5]4 degreases of freedom at changing state – “FLE binary flip”

The lattice is placed in the corresponding Matter’s fundamentally absolute, fundamentally flat, fundamentally continuous, and fundamentally “Cartesian”, [5]4D spacetime with metrics (cτ,X,Y,Z,ct), where cτ,X,Y,Z, are 4 space dimensions, ct is the time dimension. FLE “size” and “FLE binary flip time” are Planck length, lP, and Planck time, tP.

Besides the utmost universal “kinematical” space dimensions above, the space contains also at least 4 dimensions that relate to 4 specific FLE degreases of freedom to change its state [to be “charged”] that relate to the at least 4 known now fundamental Nature Gravity, Weak, Electric, and Nuclear/Strong forces [more see the 3-rd link, section “Mediation of the fundamental forces in complex systems”], so whole Matter’s spacetime metrics is (at least) (cτ,X,Y,Z, g,w,e,s,ct).

Again – all dimensions in the spacetime are absolute, independent on each other, by no means fundamentally can impact on anything in Matter and be impacted by anything in Matter.

More see the links,

Cheers