Balanced additive complements

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collision_density: Finite-bin difference counting forces a quantitative collision density.

counting_regularity: Elementary consequences of termwise balance and bounded cross representations.

monotone_path_shells: Flexible path shells give balanced cofinite complements and uniform bounds for every block pair, but only a logarithmic bound for the full unions.

nonconstant_representations: Every bounded representation function of a balanced pair differs from each nonzero periodic pattern on a set of positive lower density.

prefix_resonance: An arbitrary large path can be matched by admissible smaller paths so that one target has logarithmically many distinct representations. The all-prefix estimate needs coordination across scales; the local increment restrictions and pairwise bounds do not supply it.

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This folder's notes that cite no source are author-recorded research, not claims.

The linked notes are retained here; the notes for the routes named without links are not part of this repository, so their results are recorded here without proofs.

Where things stand

Unresolved. No complete proof or counterexample to Problem 1145 is established. The target is Problem 1145, whose Current assessment records the status search this folder attacks. A positive answer includes the diagonal Erdős–Turán problem. No retained construction has both proved cofinite coverage and a uniform full representation bound. Every theorem in this folder is author-recorded; none has an independent verification record.

Necessary conditions on a counterexample

A hypothetical bounded counterexample has $A(x),B(x)=\Theta(\sqrt x)$, $A(x)-B(x)=o(\sqrt x)$, and square-root local continuity (counting regularity). Repeated sums have positive lower density (collision density), and the representation function differs from every fixed nonzero periodic pattern on a set of positive lower density (nonconstant representations). These restrictions do not exclude a bounded, persistently varying function.

• Both unphased parity-cover routes are excluded. Initial births force logarithmically growing allocation discrepancy. Terminal births also exclude a full forward odd list: its endpoint $t_{2q}-1$ accumulates in one residue class of later prefixes. Even alternating the full forward parity cannot avoid the fixed residue-capacity bound. The unphased asymmetric-cover route assumed that a full odd-row list stays available; the terminal-birth residue-capacity bound shows that it cannot.
• Two generic integer-code repairs fail under explicit hypotheses. For integer-carry encodings, an injective $(\mathbb Z/p\mathbb Z)^d$-encoding with $K$ ambient additive defects satisfies $p^d\le\binom{K+p-2}{p-1}$; a fixed list of global carry offsets is impossible for fixed $p$ as $d\to\infty$. Carry-free radix padding instead loses square-root lower density. These restrictions do not exclude selected-pair or branch-dependent corrections. The tested polynomial encodings already fail through imbalance, gaps, or same-shell carry collisions.
• Zero microscopic variance has strong, but still conditional, consequences. Marked-fiber localization forces a purely singular common measure, integer conditional sum and difference fibers, and even limiting sum multiplicities. It excludes the deterministic relational pattern, not all equality-case measures. The topological check concerns unique geometric sums only: conditional counts do not provide the needed bound on every support representation.

Useful approaches and their exact gaps

Nonlinear path shells have termwise balance, cofinite coverage, and $r(n)=O(\log n)$. The missing condition is a uniform convolution bound against the entire opposite old prefix. The exact-resonance construction, recorded in L126 at tier 0, shows that admissible paths can realize logarithmic peaks, even with the large path prescribed. Thus local increment restrictions alone cannot provide that uniform estimate; coordinated choices remain open. The stated affine, shared-prefix, and independent-path subclasses are excluded; arbitrary coordinated nonlinear paths are not. The scale-gap formulation states a sufficient bounded-overlap invariant, but its geometric size budget is not a Cartesian construction or a carry-correct decoder.

Residue routing and controlled parity allocation preserve balance, residue margins, and uniform bounds for individual shell pairs. Ordered-routing lemmas give conditional cyclic coverage from full row-class lists or constant allocation discrepancy; neither premise can persist for unphased forward parity sorting. Moving rank phases, or a four-class rule with an interior forward class full, evade the stated endpoint obstructions. No compatible indefinite allocation, ordinary covering lift, or whole-prefix bound is known. Fractional routing does bound whole-prefix convolutions, but constant-loss integral rounding and a covering invariant are missing.

Cyclic-prefix covers and clone extensions have finite witnesses and specific dead ends, not an infinite extension theorem. Any retained-prefix example needs persistent branching irregularity; eventually subcritical child products are excluded by a supermartingale argument. The randomized relational relaxation still lacks consistent shared-vertex choices.

Finite-field blocks solve local covering problems, not infinite gluing. The specified partial-mirror offsets are excluded, but arbitrary length-dependent offsets remain outside that proof; no carry-correct scale-tag decoder has been established. The existing energy and row-profile relaxations remain feasible at fixed bounds. No argument forces their microscopic variance to vanish.

Provisional directions: phased or interior-class routing still needs an actual extension theorem. Scale-gap coding would have to realize the bounded-overlap invariant with exact integer carries while retaining coverage and balance. Neither is established as a feasible route.