Routes

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adaptive_determinant_support: Binary colorings split every determinant support from prescribed relation menus, even with same-color anchored factors; this does not exclude color-adaptive selection.

anchor_gluing: Small-loss gluing improves three-color obstructions, but the Grow source has a checked no-anchor obstruction at every simplex arity, including the boundary.

anchored_phase_gluing: A phase correction gives an analytic gluing loss quadratic in the reciprocal number of blocks; diagonal padding shows that small loss does not ensure anchor propagation.

angular_colorings: Coloring the roots of odd squarefree order by arcs of argument bounds every prime grid by the least degree of a flat cyclotomic multiple, so unbounded coloring of the grids needs signed vanishing sums in arbitrarily short arcs.

axis_line_coloring: Three-direction connectivity and an entropy lower bound are proved, but unbounded dissociated coloring of L(3,4,...,t+2) remains open; full tensor products lose uniform extraction.

axis_line_extraction: Target-independent root-polygon rounding gives a direct uniform Sidon estimate for full axis-line indicators, including sides d_i=i+2; this does not inherit extraction from a full simplex tensor or prove unbounded coloring.

axis_line_quotient: Quotienting the point lattice by all axis lines produces full simplex tensors with an explicit unbounded coloring lower bound; lifting the forced flat point relations through actual line colors remains unproved.

axis_line_slice: Full axis-line indicators have an exact saturated integral presentation and give a smaller coloring target; direct extraction permits linearly growing sides.

boolean_moments: Critical Boolean moments have uniform hereditary full-dissociation extraction; entropy and restricted-color results do not yet prove unbounded actual coloring.

bounded_height_flattening: A fixed palette forces uniformly bounded-height relations; Hall-tight determinant supports flatten coefficients safely, but forcing the needed monochromatic support is unproved.

coefficient_growth_flattening: Weighted-simplex subset counting gives a sufficient flattening criterion and a necessary double-exponential height escape; actual-color focusing still lacks the required growth control.

difference_towers: Global seam deletion gives a near-lossless extraction theorem for cyclic difference families; all stages halve integer-relation height, but arbitrary-color growth remains unproved.

field_tower_lines: Field towers, linear flags, and a nested polynomial model retain uniform extraction; exact integer and Fourier formulas and an explicit mixed trade leave arbitrary-color growth unresolved.

fixed_height_focusing: Profile focusing works at one fixed coefficient height with unbounded chromatic number; the proved palette-dependent height theorem does not supply that quantifier.

generic_voltage_coloring: Independent incidence variables give flat cofactor relations at Hall deficiency and one finite voltage cover excludes every affine homomorphic q-coloring when the base has more than q times as many columns as rows; arbitrary coloring remains open.

hadamard_gluing: Saturated Hadamard feedback gives cheap chromatic jumps from a fixed anchored source, but sufficiently large order or diagonal padding loses every next-color anchor.

incidence_cover_extraction: Arbitrary finite covers of all common row duplications of linear-side axis-line matrices retain one uniform full-dissociation extraction constant; integer relations lift to ternary ones, but no color-unavoidable cover theorem is proved.

mobile_anchor_gluing: Triangular buffers give exact saturation and small analytic loss; for finite-threshold sources, buffers above an explicit source-dependent length prevent chromatic increase.

moment_boundary_extraction: Semistability and the weak stable-lattice Gaussian bound prove the fixed-scale theta estimate and one hereditary full-dissociation extraction constant at the rounded critical Boolean-moment boundary; coloring remains open.

moment_gaussian_coloring: The logarithmic-scale reverse-Minkowski argument gives actual two-colorings away from the boundary; boundary extraction is proved by a separate argument, while its arbitrary-color growth remains open.

moment_phase_entropy: A finite Gaussian-width extraction criterion, satisfied at the critical moment boundary by the weak stable-lattice Gaussian theorem.

moment_phase_lattice: The exact phase-basis criterion remains valid; its basis premise is still open, but a separate weak stable-lattice theta argument proves critical moment extraction.

parameter_word_limits: A finite coloring makes every monochromatic parameter-word image rationally independent under any fixed-alphabet coordinatewise encoding into conductor-disjoint odd-root groups.

partial_box_compression: Good monochromatic cells obey a quantitative compression bound; every fixed disjoint-box complexity has a uniform finite coloring, with 16 colors sufficient for two boxes.

polynomial_fiber_flattening: Prouhet identities flatten bounded coefficients on a profile-monochromatic progression, but that fiber already has its own ternary relation and full polynomial fibers lose uniform extraction; a sparse signed-span target is exactly height-three coloring.

prime_reset_plateau: Root-span coverage resets maximally at S3 times S5 times S7, whose separation threshold is 3 or 4; ordinary coloring stays exactly three through prime 19.

prime_tensor_extraction: Explicit elementary prime estimates extend the tensor theorem to every squarefree prime grid and to all roots of unity additively; unbounded dissociated coloring remains unproved.

recurrent_high_height_roots: Dissociated root sets can have product-recurrent tails of increasing height independence and infinite rational coloring number; piecewise syndeticity gives an exact stronger reformulation of the unresolved premise.

resolvable_matchings: The phase induction works for arbitrary signed linear direction matchings of sizes i+2; resolvable supports have explicit common covers with boxes, but coloring growth remains unproved.

root_profile_compression: Finite prime-prefix colorings obey a uniform profile-deficit bound; global colorings have summable deficits, but arbitrary distinct profiles remain unexcluded.

roots_coloring_structure: Finite dissociated coloring of all roots is equivalent to one prime-fiber transversal; equivariant partitions have an exact exponential obstruction that does not cover arbitrary colorings.

shift_simplicial_gain_checks: Full shift supports and gain-graph stars have bounded colorings under stated hypotheses; simplicial boundaries at n/k asymptotic to C log n have uniform extraction but an open actual-coloring gap.

simplicial_extraction: The exact exterior-root relation lattice is semistable by an elementary irreducibility proof, giving uniform full-dissociation extraction for n/k asymptotic to C log n; actual coloring growth remains unproved.

tensor_coloring: Signed-span intersections characterize coloring after adding a simplex factor, isolating the missing ingredient in the uniform-extraction construction.

tensor_extraction: A randomized lattice-rounding argument gives one uniform full dissociated extraction constant for an explicit tensor family; unbounded dissociated coloring is not proved.

tensor_holes: A proved root-coverage recurrence amplifies separation thresholds below a critical level, but an explicit arbitrarily large chromatic jump resets coverage maximally.

tensor_schedule_constraints: A small-column entropy argument rules out uniform proportional dissociation when retained tensor sides satisfy p_i≤Ci for a fixed C; a lattice-volume bound explains the phase-lift cost.

tensor_separation_lifts: Exact constructions bound separation thresholds after large tensor factors, identifying coefficient doubling and a coarse fixed-base saturation bound.

translation_cover_automata: For a fixed one-dimensional voltage symbol, full ternary coloring is recognized by finite residual-state machines; infinite proper colorings periodicize, cyclic admissible degrees are ultimately periodic, and a sharp factor-two large-degree bridge holds.

transversal_gadgets: Arbitrary row twists preserve the phase mechanism; pure-transversal gluing has a two-witness reset under compatible next-step twists, while matching decorations retain uniform extraction and force three colors but no unbounded amplification.

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The prime-tensor construction supplies the extraction half of the finite-block target. Its dissociated coloring number remains unresolved; the arc colorings bound it by the least degree of a flat cyclotomic multiple, so unbounded coloring needs signed vanishing sums in arbitrarily short arcs. The Grow anchor source is excluded at every arity; different sources remain subject to the primitive-width and mobile-buffer restrictions recorded here.