Proof sketch for odd covers along prime-power digits

---

This is the canonical sketch of the odd-covering distortion lead, author-recorded by Plasma AI and unreviewed. The target remains unresolved. The universal small-prime certificate at $138.877$ is false. A more flexible route through the exact-loss tail remains conditional on an unproved measure construction.

Target and common definitions

Problem 7 asks whether finitely many congruence classes with pairwise distinct odd moduli greater than one can cover $\mathbb Z$. The proposed conclusion is that no such family covers. A positive resolution would require an explicit family and a proof covering one full period; no such family is supplied here.

For a common period $N$, the digit page identifies $\mathbb Z/N\mathbb Z$ with the box $Q(N)$ of base-$p$ digits. A modulus $d$ fixes the first $v_p(d)$ digits at each prime. Distinct moduli give distinct prefix shapes. Missing classes modulo the primes can be added, and their first-digit values deleted. The resulting small-prime box $Q_s$ has $p-1$ values at the first digit and $p$ at each later digit, for the odd primes through $73$. Let $R$ be the part of $Q_s$ avoiding all small-prime classes.

For a prefix shape $Y$, its pair weight is

$$W(Y)=\prod_p(2t_p(Y)+1).$$

It counts ordered pairs of prefix shapes whose union is $Y$. Class-mass bounds must cover every such shape, including shapes absent from the exposed family but possible in a later intersection.

A small-prime system $S$ chooses one cylinder $H_b$ of each prefix shape, including $H_\varnothing=Q_s$, and $w_S(x)=\sum_b\mathbf1_{H_b}(x)$. For a probability $P$ on $Q_s$, the system interface uses

$$\sigma(P)=\max_S\mathbb E_Pw_S^2, \qquad \Phi_P^\delta(c)=\frac1{1-\delta}\sup_S \mathbb E_P(cw_S-\delta)_+.$$

The systems in these suprema can be future large-prime projections; they are not restricted to the classes that have already been removed.

Held links

1. Digit reduction. The prime-power digit correspondence turns a distinct odd cover into a nonparallel prefix-cylinder cover. This is Lemma 1 of the digit page, an elementary CRT argument, consistent with the [[../library/covering_systems/berger_1986_necessary_condition_odd_covering_systems/prime_adic_boxes|Berger–Felzenbaum–Fraenkel box correspondence]].
2. Distortion sieve. The digit page derives preservation of old marginals, the pointwise distortion cap, the identity $\mathbb P_k(B_k)=\mathbb E_{k-1}(\alpha_k-\delta_k)_+/(1-\delta_k)$, and survival whenever the summed losses are less than one. It records the extension from the source's $\delta_k\le1/2$ to $\delta_k<1$.
3. Second-moment tail. Lemma 2 of the system-interface page proves, for every $k>21$, the bound $M_k^{(2)}\le\sigma(P)/(p_k-1)^2$ times the product of earlier large-prime distortion factors. It permits arbitrary exponents and residue assignments, and every subsequent legal distortion. For a particular family, its smaller actual-system constant $\kappa^*$ can replace $\sigma(P)$.
4. Exact-loss tail. Lemmas A and B of the exact-loss page give a convex-order comparison for weighted cylinder indicators under sequential conditional-mass bounds. Its Theorem C applies this to every large-prime loss. Corollary D joins finitely many such exact bounds to the source's second-moment recurrence and termination theorem. Its Corollary E, with an outward rational certificate, proves that uniform surviving density at least $1/4$ in $Q_s$ suffices.

The native deductions have author standing only. The imported tail results are from the published BBMST square-free paper, Algebra & Number Theory 15 (2021), and the published density paper, Inventiones 228 (2022), at the versions and read scope identified on their library pages. No conditional, rejected, or pending outside E7 assertion is a premise.

The remaining end-to-end candidate

Given any finite odd family, extract its $73$-smooth classes. For every such family, at every finite depth vector, construct a probability $P$ supported on $R$, an integer $K>21$, and distortions $\delta_k\in(0,1)$ for $21<k\le K$. For independent agreement depths $g_i$ with

$$\Pr(g_i\ge t)=\min\{1,p_i^{-t}/(1-\delta_i)\}\quad(t\ge1), \qquad N_k=\prod_{21<i<k}(g_i+1),$$

establish valid upper bounds on $\Phi_P^{\delta_k}(N_k/(p_k-1))$ giving

$$\mu_K=1-\sum_{21<k\le K} \mathbb E\,\Phi_P^{\delta_k}\!\left(\frac{N_k}{p_k-1}\right)>0.$$

The same definitions must give a bound for

$$f_K=\frac{\sigma(P)}{\mu_K} \prod_{21<i\le K} \left(1+\frac{3p_i-1}{(1-\delta_i)(p_i-1)^2}\right)$$

from which the source's [[../library/covering_systems/balister_2018_erdos_covering_problem_density_uncovered_set/lemma_6_2|Lemma 6.2 recurrence]] reaches the termination criterion of its [[../library/covering_systems/balister_2018_erdos_covering_problem_density_uncovered_set/theorem_6_1|Theorem 6.1]], with positive survival throughout. Subsequent distortions use the source's range $(0,1/2]$. Families ending earlier are handled by survival at their last stage, or by appending free prime coordinates.

Open obligation. Construct this $P$ and these bounds uniformly over all small-prime configurations and depths. The measure and schedule may depend on the configuration. A favorable sample, an exponent cap, or a product model with unspecified support does not prove the obligation.

Established case. If uniform measure $U$ on $Q_s$ has $U(R)\ge1/4$, take $P=U(\,\cdot\mid R)$ and $\delta_k=2/5$ through $p_{2262}=19997$. Corollary E certifies positive mass and $f_{2262}<94000$, below the certified lower bound $101000$ for the source's termination threshold. This includes the concentration example that refutes the old $138.877$ demand. Its 16-class extension has $0<U(R)<1/4$, so this established case does not cover all configurations. For that extension, an adapted product subset has surviving relative density at least $24005/28512$; conditioning its uniform measure on $R$ also satisfies Corollary D, with certified $f_{2262}<21000$.

Composition. If this obligation holds, start the sieve at $P$. All small-prime classes have zero mass. Theorem C bounds the exact phase, and Corollary D supplies positive survival through every remaining prime. The original family misses a residue modulo its period and cannot cover the integers. This is a complete conditional implication with an unproved antecedent; it is not a proof of E7.

The refuted stronger certificate

The stronger Step 3 asks for a small-prime sieve with valid cylinder bounds $c_{21}(Y)$, losses $L_k\ge\mathbb P_k(B_k)$, and

$$\mu_{21}=1-\sum_kL_k>0,\qquad \kappa=\sum_YW(Y)c_{21}(Y),\qquad \kappa/\mu_{21}\le138.877.$$

Here index $21$ denotes the prime $73$, not a digit count. These conclusions would satisfy the retained [[../library/covering_systems/balister_2021_erdos_selfridge_problem_square_free_moduli/corollary_5_2|Corollary 5.2]] tail interface. Conditioning the final measure on $R$ would also give Step 3*: a supported probability with $\sigma(P)\le138.877$.

Both universal demands are false, by the [[research/leads/odd_covering_distortion/support_concentration|support concentration construction]]. A valuation form of Harrington's repeated-3 example supplies a distinct odd family, with primes through 73 and depth 16, whose nonempty uncovered set has first 3-digit fixed. Averaging the point kernel over centres in that branch forces $k(P)>152.14$ for every supported $P$, and $\sigma(P)\ge k(P)$. This is an actual residue configuration, independent of any choice of moment bounds, digit order, or initial block.

The source's square-free certificate remains valid in its own domain: its [[../library/covering_systems/balister_2021_erdos_selfridge_problem_square_free_moduli/lemma_5_3|Lemma 5.3]] and [[../library/covering_systems/balister_2021_erdos_selfridge_problem_square_free_moduli/lemma_5_4|Lemma 5.4]] give $f_{21}\le138.874$. Its [[../library/covering_systems/balister_2021_erdos_selfridge_problem_square_free_moduli/remark_p625|higher-power remark]] permits arbitrary exponents above 73, not at the small primes. The counterexample does not contradict those hypotheses or conclusions.

Failure account and next discriminating question

• Failed link: the universal $138.877$ small-prime certificate and its Step 3* relaxation, with the retained class bounds and pair weights.
• Reason: an explicit distinct family forces survivor concentration at 3; every supported measure exceeds the required system constant. The missing input at that constant is false, not merely unproved here.
• Surviving route: the exact-loss tail and a configuration-dependent supported measure satisfying Corollary D. Its existence is unknown. It includes nonemptiness of the $73$-smooth uncovered set, which is itself part of the target problem.
• Resolved test: uniform conditioning on the concentration example satisfies Corollary D, since its density is at least $773/1920$. The exact certificate bounds the supremum over systems, includes every depth through stochastic domination, and reaches a certified infinite-tail termination criterion. A product measure adapted to the first-digit constraints also handles its 16-class extension, whose uniform density falls below $1/4$.
• Next test: for a configuration with $U(R)<1/4$, construct a supported measure with simultaneous bounds on $\Phi_P$ and $\sigma(P)$, and prove how that construction applies without an exponent or residue restriction. The explicit examples show how product subsets can help, but give no uniform selection rule. A tail calculation on a favorable concentrated system alone does not bound the supremum over systems, and solving one configuration does not prove the universal measure assertion.

The cube-free and depth-four results remain scoped exclusions of complete-prefix moment models. A larger block or a different loss bound can escape those model assumptions, but cannot repair the refuted universal $138.877$ condition with the same definitions. The [[research/leads/odd_covering_distortion/e273_calibration|exact E273 calibration]] is a finite-domain success and supplies no missing E7 premise.

The exact-loss page's floating-point inflation experiments suggest extra room, but do not certify a uniform replacement threshold or a supported measure. They truncate distributions and cover only the prime ranges stated there. A full proof must retain the same loss, system, and distortion definitions at every interface and justify its entire remaining-prime tail.