Dissociated covering attack

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evidence/: Exact arithmetic checks the relations and proper 2-colorings of the 15th roots of unity, a nine-layer two-color certificate, the root spans of arc classes below the rank threshold, signed-pair and signed-triple survival under coordinate twists, the failure of two arcs at 165 and 195, the diverging first moment, and exhaustive survival sets at 105 for arc and layered classes.

fractional_covering: Bourgain's weighted extraction theorem bounds fractional dissociated covering by an absolute multiple of the reciprocal hereditary extraction constant, without symmetry; the remaining problem is an integral covering gap.

layer_race: Coordinate-permutation twists of a proper coloring give layered colorings of the next odd-root grid; robust layering preserves the root-span fraction exactly and three colors win the layer-count race of the roots, but the exact first moment of common values diverges along the odd-prime grids, so deciding the odd-root grids this way needs a rigidity argument for common values of large complexity.

proof_sketch: Editable argument for Problem 774: repeated power-of-two simplexes have uniform hereditary extraction at a precise critical scale, fractional covering is uniformly bounded for every candidate family, and a thirteen-point example crosses the fractional threshold two; arbitrary-color growth remains open, alongside the odd-root rigidity route.

repeated_simplex: Boolean compression proves semistability and uniform hereditary extraction for repeated power-of-two simplexes at k = q(log log q + log log log q + O(1)); entropy excludes larger scales, and flat trades defeat scalar linear fibers, bounded Boolean linear palettes, and every quadratic two-coloring.

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This folder holds a sustained attack on Problem 774, which asks whether every proportionately dissociated set of positive integers is a finite union of dissociated sets. It builds on the inherited research in the E0774 research folder, whose main argument reduces a negative answer to unbounded dissociated coloring of uniformly extracting finite configurations, in particular of the odd-prime grids $S_3\otimes S_5\otimes\cdots$.

The proof sketch states the finite target, exact interfaces, and remaining gaps. The repeated-simplex argument proves semistability and one hereditary extraction constant in L271, tier 0, for a new family with common side $q=2^b$ and about $q(\log\log q+\log\log\log q+c)$ factors, for one fixed $c$. It also excludes larger parameter scales by entropy and constructs flat relations in every scalar linear fiber once the number of factors reaches $q$ (L269, tier 0). Boolean linear fibers of bounded codimension and quadratic two-colorings also fail in their stated ranges. Unbounded coloring under arbitrary palettes remains open.

The fractional-covering formulation shows that every counterexample family must have an integral covering gap, even without symmetry. The exact finite claim L211 has fractional covering number $37/19<2$ and needs three dissociated classes. It is a finite obstruction, not an amplification.

The twisted-layer note retains the odd-root alternative: robust layering preserves the root-span fraction, but a first moment over all common values diverges, leaving a rigidity problem. Its evidence checks the finite cases. None of these arguments resolves Problem 774 or asserts an independent tier for the new work.