Manuscript XVI-M@1 · XVI-M.tex · theorem · 7346c909

Theorem: Ambient freedom versus graph-visible collision freedom

M.3 Contact space, graph maps, and invisible directions · lines 961–979

XVI-M@1#thm:M3

These are reported exact ranks, cross-checked mod $p$ (Lemma lem:M3comp):

Map / variant	$dim U_{N}$	$dim im$	$dim ker$	on $C_{reg}$ : im / ker
$G_{formal}$	$(1, 4, 10, 20, 35)$	$3321$	$0$	$3150/0$
time-only (formal)	$(1, 4, 10, 20, 35)$	$3321$	$0$	$3150/0$
$G_{background}$	$(1, 4, 10, 20, 10)$	$1571$	$1750$	$1400/1750$
time-only + background	$(1, 4, 10, 16, 10)$	$1431$	$1890$	$1260/1890$
plateau ( $J_{3}$ only)	$(0, 0, 10, 20, 35)$	$3300$	$21$	$3150/0$
$G_{fixed}$	—	—	—	Criteria only; Unresolved (Stage 2)

Injectivity is claimed only for $G_{formal}$ (and its time-only variant).

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