The representation-theoretic correction in primitive directed deletion #
This file combines the two nonnegative terms in the live manuscript's directed-deletion theorem. All three quantities are the literal categorical ones attached to a primitive quotient:
- the intrinsic Euler excess of the strict tau-factor;
- the total increase of intrinsic irreducible-arrow multiplicities; and
- the number of new quotient meshes.
The direct vertex--arrow--mesh count identifies this correction with the difference of the ambient and primitive-quotient Auslander--Reiten surpluses; factor positivity and arrow-gain dominance then give monotonicity and the equality rigidity statement used downstream.
Total irreducible-arrow multiplicity of the literal primitive quotient.
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The ambient Auslander--Reiten surplus, using the official finite-tau arrow multiplicities and the categorical projective predicate.
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The literal primitive quotient's Auslander--Reiten surplus, using its intrinsic irreducible-arrow dimensions and categorical projectives.
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The ambient translation-quiver surplus, expressed through the literal ambient vertex, projective, and arrow-occurrence counts.
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The literal primitive quotient's translation-quiver surplus.
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The official ambient Auslander--Reiten surplus agrees with its literal translation-quiver cardinality expression.
The intrinsic primitive-quotient Auslander--Reiten surplus agrees with its literal translation-quiver cardinality expression.
The live manuscript's actual correction
epsilon(Q) + c - r for a primitive deletion.
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The manuscript's count
epsilon(Q) = 2 q - a_H - p - 1, written with the literal vertex, arrow,
and tau-projective counts of the strict primitive factor.
The quotient arrow count is the ambient internal count plus the total
gain c.
The strict factor's intrinsic excess in the literal direct-count
variables q, a_H, and p.
The live manuscript's exact identity
sigma(A) - sigma(A/AeA) = epsilon(Q) + c - r, for the literal ambient and
primitive-quotient translation-quiver counts.
The manuscript's direct-deletion identity
sigma(A) - sigma(A/AeA) = epsilon(Q) + c - r for a
representation-directed algebra. The crossing-mesh count uses the finite-kernel
boundary bounds.
The actual primitive-deletion correction is nonnegative for a representation-directed algebra. The boundary package is constructed by the finite-kernel argument.
Primitive directed deletion cannot increase the quotient Auslander--Reiten surplus. This is the manuscript-facing local monotonicity theorem: no coordinate or boundary package is an external hypothesis.
Equality in the actual correction separates into vanishing factor excess and equality between total arrow gain and the number of new meshes.
Equality of the ambient and quotient Auslander--Reiten surpluses is equivalent to simultaneous vanishing of the factor excess and of the new-arrow/new-mesh discrepancy.
Equality in the actual correction forces every object of the strict primitive factor to have deleted-simple multiplicity one.
Equality of the ambient and primitive-quotient Auslander--Reiten surpluses forces every object in the strict primitive factor to have deleted-simple multiplicity one.