The Ladder of Truncation L1: Triangulated Realization of Cohomological Carriers—Euler-Factor Defect Chain Complex

This paper treats the first layer of the problem of making rigorous the cohomological carrier on which the registration of truncation coordinates at the numbertheoretic layer depends: without relying on any analytic input, we construct apurely algebraic chain complex whose generators are prime local factors and whosedifferential is the defect map of a local factor relative to the unit. The complex hasas its degree-0 object the monoid ring A of the multiplicative monoid N+, as itsdegree-1 object the free Abelian group X on the set of primes, and as its degree(−1) object the ring of integers Z. On prime basis elements the defect differentialis ι(ep) = [p] − [1], and the augmentation is ε([n]) = 1. We prove that this construction satisfies ε ◦ ι = 0, that ι is injective, that the zeroth homology has as afree Z-basis all composite indices, and that the first and all remaining homologyvanish. On this basis we incorporate the multiplicative structure of A: we provethat A is isomorphic as a Z-algebra to the polynomial ring Z[xp : p ∈ P] in several variables, give a multiplicative identity for defect elements, and distinguishthe trivialization of the A-linear closure from the generation gap of the Z-linearversion. We further define a valuation shadow ρs and an index-type linear functionλ, prove that ρs(ι(ep)) = Yps − 1 gives a formal counterpart of the local Eulerfactor defect, and prove the Z-module direct-sum decomposition of H0 accordingto the zero/nonzero pattern of λ: nontrivial prime-power indices give a free directsummand, while composite indices that are not prime powers span the kernel ofλ. All proved conclusions are borne by proofs in the full text, while finite-windowmachine checks verify only implementation consistency. Structural-layer readingsprovide only positive positional correspondences and are not upgraded to analytictheorems; the second-layer carrier problem is registered as an open node, and itssurvival criterion is not settled.