Odd covers by the distortion sieve on prime-power digits

../

cube_free_obstruction: Exact bounds rule out the complete-prefix first/second-moment LPs with consecutive digit order: initial blocks through 7 for any class-mass bounds, through 17 for the capped Lemma 3.3 recursion, and through 11 when the weighted recursion is also allowed.

deep_digit_obstruction: The inflation-free floor of the cube-free test, repeated at uniform depth D at every odd prime up to 73, exceeds the tail threshold for every initial block through {3,...,17} once D >= 4, in three digit orders and at every larger depth. In consecutive order the block {3,...,19} is not excluded at any depth.

digit_sieve: Base-p digits turn a distinct-moduli cover into a hyperplane cover of a digit box with prefix-shaped fixed sets. The distortion sieve then runs digit by digit, with a (t+1)-set loss bound, a pigeonhole step, two class-mass recursions and the tail interface at the prime 73.

e273_calibration: The digit-coordinate distortion sieve proves, with an exact certificate, that the 149 moduli p-1 with 5 <= p <= 877 cover the integers for no choice of residues: the 7-smooth block loses at most 115/126 and every later digit loses nothing.

evidence/: Exact rational checks for the lead: the Problem 273 calibration certificate, a cross-check of the sieve recursions against actual distorted measures, cube-free LP failure bounds with exact logarithmic weak-duality margins, the depth-four floor in three digit orders, and a realizable support-concentration counterexample to the universal tail threshold, and an exact-loss tail certificate from uniform density 1/4.

exact_loss_tail: A convex-order bound for weighted indicator sums, applied coordinate by coordinate, bounds each exact large-prime loss of the source's tail by an aligned model with independent dominated agreement depths and the worst small-prime system. An outward rational certificate accepts any uniform surviving density at least 1/4 in the reduced small-prime box through 73, including a configuration that defeats the 138.877 moment interface.

point_kernel: The source's tail needs at each large prime only the small-prime projections of the classes present. Bounding their pair sums directly replaces the class-mass constant by an actual-system constant. Its universal worst-system bound dominates the point-kernel value of the small-prime measure. The demand for a supported measure with this bound at most 138.877 is false: a realizable residue configuration forces a value above 152.14. The actual-system tail implication remains valid.

sketch: Canonical conditional route for distinct odd coverings. The universal 138.877 small-prime certificate is false by an explicit concentration example. The exact-loss tail remains valid; the open obligation is a supported measure satisfying its system-functional bounds at every small-prime depth and residue assignment.

support_concentration: A valuation form of Harrington's repeated-3 construction gives a distinct odd family whose survivors have one fixed first 3-digit. At depth 16 through 73, every supported measure has point-kernel value above 152.14. Thus the universal 138.877 small-prime certificate is false, independently of its digit order, class-mass recursion, or initial block.

This lead's results that are not drawn from a cited source are Plasma AI's, credited to Plasma AI. All of them are author-recorded at tier 0, and no independent review has examined the lead.

Target and route

The target is the negative answer to Problem 7: no finite family of congruence classes with pairwise distinct odd moduli greater than one covers the integers. The problem page's current assessment records the problem as open, and this lead does not change that.

The route extends the square-free proof of Balister, Bollobás, Morris, Sahasrabudhe and Tiba along each $p$-adic tree. Base-$p$ digits make every congruence class a hyperplane whose fixed coordinates form an initial segment of each prime's digits, and the distortion sieve then runs digit by digit. The large primes are handled by the source's tail, so the whole difficulty lies in the odd primes up to $73$ with arbitrary exponents.

Pages

The proof sketch is the canonical account of the route, with its held steps, its gap and its failure account. The [[research/leads/odd_covering_distortion/digit_sieve|digit page]] derives the digit form of the sieve: the reduction, a loss bound over $(t+1)$-sets of completed classes with a pigeonhole case, two class-mass recursions, the prefix pair count and the tail interface.

The [[research/leads/odd_covering_distortion/e273_calibration|calibration page]] proves with an exact certificate that the $149$ moduli $p-1$ with $5\le p\le877$ cannot cover the integers, a finite statement about Problem 273. The [[research/leads/odd_covering_distortion/cube_free_obstruction|cube-free obstruction]] proves that specified moment LPs cannot certify the cube-free case of Problem 7. They keep the full prefix inventory, charge all single and pair terms without using residue compatibility, and process each prime's two digits consecutively in increasing prime order. Initial blocks through $7$ are excluded for any valid class-mass bounds, and those through $17$ for the capped source recursion. The [[research/leads/odd_covering_distortion/deep_digit_obstruction|depth-four obstruction]] repeats the inflation-free floor at every uniform depth and in three digit orders, and excludes every initial block through $17$ for any valid class-mass bounds once the depth is at least four.

The [[research/leads/odd_covering_distortion/point_kernel|point-kernel page]] derives the tail with the pair sums of the classes actually present, with the point kernel giving a lower bound for the universal system constant. The [[research/leads/odd_covering_distortion/support_concentration|support concentration construction]] gives an actual distinct odd family whose survivors have one fixed first 3-digit. Every supported measure then has system constant above $152.14$, refuting both the universal $138.877$ demand of Step 3* and the stronger sieve certificate of Step 3.

The exact-loss tail proves a convex-order bound for exact large-prime losses, in terms of the worst small-prime system. Its Corollary D allows a different interface and is not refuted by the concentration argument. Corollary E and its outward rational certificate accept uniform surviving density at least $1/4$, which handles that example. An explicit extension has density below $1/4$, so this sufficient condition is not a universal substitute for the missing measure.

Why the lead is blocked

The universal $138.877$ certificate is false with its stated definitions. A different initial block, digit order, or class-mass recursion cannot fix it. The cube-free and depth-four moment-model exclusions retain their narrower scope, but no longer supply the decisive failure account for that threshold.

The remaining route asks for a measure on the uncovered set of every family of $73$-smooth classes, at arbitrary depth, satisfying Corollary D of the exact-loss tail. That condition needs bounds on the system count's upper tail under a supported measure. Its universal existence is unknown and includes the nonemptiness of the $73$-smooth uncovered set itself. The full sketch states its quantifiers and how it would compose to a negative answer to Problem 7.

Evidence

The evidence program checks the calibration certificate, the pigeonhole steps, the implementation of the recursions against actual distorted measures on small domains, the cube-free bounds and the depth-four bounds, and the explicit concentration construction and kernel bound, all in exact rational arithmetic. It also certifies the exact-loss tail from density $1/4$ and its handoff to the source's termination theorem. The checks take a few seconds:

uv run --no-sync python wiki/research/leads/odd_covering_distortion/evidence/main.py