Global arrow gains under primitive deletion #
The manuscript counts ordered pairs of surviving indecomposables whose
irreducible-arrow multiplicity increases after passing from A to A/AeA.
Both arrow multiplicities are defined intrinsically as dimensions of
Irr = rad / rad², using the literal common label type supplied by the
primitive-quotient skeleton.
Ambient irreducible-arrow multiplicity between surviving labels.
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Irreducible-arrow multiplicity inside the literal primitive quotient.
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The multiplicity gained by an ordered pair after primitive deletion.
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Every endomorphism of a quotient-skeleton representative is scalar. This is transported from the literal ambient representative through the linear quotient equivalence.
The Hoshino quotient mesh equipped with the ambient middle term's chosen decomposition, now relabeled by literal surviving quotient labels.
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The same Hoshino quotient mesh equipped as a minimal left almost-split decomposition at its literal source label.
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At a new quotient mesh, the intrinsic quotient Irr dimension is the
existing relative middle-term multiplicity.
Reading the same quotient mesh from its left side identifies the intrinsic arrow multiplicity out of its source with the same relative middle multiplicity.
The intrinsic irreducible-Hom dimension agrees with the official finite-tau arrow multiplicity whenever the source endomorphisms are scalar. This is the occurrence-basis comparison, including the projective boundary mesh.
At every ambient endpoint, including the projective boundary, the
intrinsic ambient Irr dimension is the manuscript's ambient arrow
multiplicity.
The manuscript's gaining pairs: ordered surviving labels whose arrow multiplicity strictly increases in the primitive quotient.
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The manuscript's gaining pair selected by a positive new mesh.
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Distinct positive new meshes select distinct gaining pairs, since the second coordinate is the mesh endpoint.
Positive new meshes inject into the manuscript's global gaining-pair set.
The manuscript's gaining pair selected from a new mesh whose contragredient new mesh is positive. Its source is the original new-mesh source; its target is the dual of the positive source selected on the opposite side. This is the numerical core of the negative-mesh case, kept separate from the marker-complement theorem that supplies dual positivity.
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The negative primitive new meshes in the manuscript's sign convention.
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The negative-side gaining-pair assignment. Marker complement converts the negative mesh to a positive dual mesh before the numerical construction above is applied.
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Distinct contragredient-positive meshes select distinct negative-side gaining pairs, since their first coordinates are the original mesh sources.
Negative new meshes inject into the global gaining-pair set.
The sign partition of the manuscript's primitive new meshes.
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Every new mesh belongs to exactly one of the positive and negative parts.
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The manuscript's gaining-pair assignment on the disjoint sign partition.
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Positive and negative meshes cannot select the same gaining pair: the first entry of a positive pair has a nonzero map from its inverse translate to the deleted simple, whereas the first entry of a negative pair has none.
The manuscript's gaining-pair assignment on the original, unsigned new mesh type.
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Distinct new meshes receive distinct gaining pairs.
The number of new meshes is at most the number of gaining pairs.
The manuscript's total arrow gain c. The primitive quotient algebra is
finite-dimensional, so the routine noetherian instance is constructed
internally rather than exposed by this numerical invariant.
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Every gaining pair contributes at least one to the total arrow gain.
The manuscript's inequality c ≥ r: total arrow gain dominates the
number of primitive new meshes.