A strict useful-supply obstruction to distinct covers

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_proof: Exact class-support counts and a pointwise forced-excess bound limit the total supply available to meet all prime-power cylinder deficits.

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Let $N>1$ be an integer and $p\mid N$ a prime. Write

$$a=v_p(N)\ge1,\qquad M=N/p^a,\qquad S=\sigma_1(M)=\sum_{d\mid M}d,\qquad D=2M-S.$$

In particular, $\gcd(p,M)=1$. The notation $\sigma_1$ denotes the positive-divisor sum; it is not an excess-mass variable. Assume $D>0$.

The full distinct nonunit-divisor profile contains one residue class $b_n(n)=b_n+n\mathbb Z$ for each divisor $n>1$ of $N$, with every $b_n$ chosen freely. Each modulus occurs once, and modulus one is excluded.

Statement

Define

$$U=S\frac{p^a-1}{p-1} -\sum_{\substack{1\le j\le a,\ m\mid M\\M/m>D}} (M/m-D)p^{a-j}.$$

If

$$U<Dp^a,$$

no placement of the full profile has incidence mass at least $M$ in every cylinder modulo $p^a$ inside $\mathbb Z/N\mathbb Z$. Consequently no family of congruence classes with pairwise distinct moduli $n>1$ dividing $N$ can cover all integers. The family may use all or only some of those moduli.

This is a sufficient exclusion criterion for every finite prime support and every choice of exponents. There is no oddness assumption and no cap on pointwise multiplicity. It does not assert that the strict inequality holds for every $N$ or every prime divisor $p$.

Meaning of the bound

Every cylinder modulo $p^a$ contains $M$ points. The full profile's classes with moduli coprime to $p$ contribute incidence mass $S-M$ in each cylinder, independently of their residues. Covering therefore requires the classes whose moduli are divisible by $p$ to contribute at least $D$ in each cylinder.

A class of modulus $p^jm$, where $1\le j\le a$ and $m\mid M$, contributes $M/m$ on exactly $p^{a-j}$ cylinders. Contributions greater than $D$ already exceed the required threshold on each cylinder they meet. Subtracting this forced excess from their total mass gives $U$. Equivalently,

$$U=\sum_{j=1}^a\sum_{m\mid M}\min(M/m,D)p^{a-j}.$$

The proof defines useful supply as incidence mass capped at $D$ in each cylinder. The subtracted waste is excess relative to that threshold, not a count of points wasted in an actual cover. This distinction permits the argument without any overlap or rigidity assumption.

Adding omitted classes does not decrease any cylinder mass. Thus failure for the full profile implies failure for every distinct subprofile. Repeated moduli do not obey the fixed inventory used to compute $U$, and are outside the conclusion. Exclusion of a one-cover also excludes any higher covering multiplicity within the same distinct-modulus domain.

Boundaries and standing

The guards are strict: when $D\le0$, or when $D>0$ but $U\ge Dp^a$, this criterion gives no exclusion. In particular, $U=Dp^a$ is not a contradiction. Failure of the criterion establishes neither a cover nor feasibility of the cylinder constraints. The proof includes the valid boundary $M=1$, when the frame is a pure prime power.

The complete proof and its subprofile consequence are author-recorded at tier 0, with no independent whole-claim acceptance or formal verification. They use only finite counting, the elementary coprime congruence argument given in the proof, and a finite geometric sum. No other native claim or source-result premise is consumed. No novelty, prior-art assessment, or resolution of an erdosproblems.com problem is asserted.

The supplied proof has no identified gap. Its verification requires only this statement and proof, with no mathematical programs, numerical grids, or private inputs.