Manuscript XXII-C@1 · XXII-C.tex · section · b9a05a24

Event and cone extensions

Continuum, refinement, and evidence discipline › Event and cone extensions · lines 1209–1238

XXII-C@1#eq:extension-faithfulness

If a later construction replaces $E_{D}$ by an extended carrier $E_{D}$ , it must provide a declared map

j_{D} : E_{D} ⟶ E_{D}

and state whether it is injective. If the old comparisons are claimed to survive faithfully, it must prove

x ⪯_{D} y ⟺ j_{D} (x) ⪯_{D} j_{D} (y) .

Every new event and comparison must have its declared operational or limiting meaning. A completion is not licensed merely by drawing a continuum around the countable experiment family.

For a cone limit, finite-dimensional comparison spaces and maps must be supplied before statements such as $K_{n} \to K$ have a meaning. The chosen cone topology must be stated, and the limiting faithfulness and generation equations must be proved. An arbitrary norm used to formulate a mathematical convergence criterion is not a supplied spacetime metric, but neither does it provide an operational identification by itself.

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unlabeled · line 1213
$j_{D} : E_{D} ⟶ E_{D}$
XXII-C@1#eq:extension-faithfulness · line 1219
$x ⪯_{D} y ⟺ j_{D} (x) ⪯_{D} j_{D} (y) .$

Event and cone extensions

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