Inscribed Square Problem

The source labels AN-22--AN-24 and AN-26--AN-29 as proved at their stated scopes and AN-25 as a proved reduction, yielding little-Besov/VMO regularity, deep packet corridors, a network-or-spiral reduction, and compact target-current/multipole data without global closure.

The source retains an odd relative radial class and a conditional carrier-end cocycle, but B2a and RC-21 still require a proper elegant carrier, a base-times-target three-current, cap/slicing identities, anchor transport, and all boundary faces.

The source reports exact finite-dimensional stress data and model exclusions while retaining source packet-chain localization and propagation under source, frame, phase, and scale control as unresolved.

The source reports a lossless lowering measure and explicit shell currents; B0/B1 must still retain controlled-sector mass and accumulate defects into a valid relation, capping/stress transition, or excluded zero model.

The source separates a boundary-continuum tree, an inherited fixed-action tree and an unmarked detector tree. The boundary tree has an unresolved total-collision or return terminal; the fixed-action tree retains the exact-critical and regular full-tail alternatives; the detector tree needs QSI-1 and an additional positive local-slope input. No exhaustion theorem covers every remaining Jordan curve.

A route to the full target must provide an actual sufficient endpoint under its displayed premises and cover every remaining Jordan curve. Closing a boundary collision subcase, a single charged leaf, a geometric model, or a conditional detector branch is not the missing exhaustion theorem.

The source keeps two distinct upstream interfaces: an independent reconstruction of marked QSL-1C at AI-RISK-1 scope, and implementation of unmarked QSI-1 in the actual twisted periodic category with lifted endpoint, family-Hom, localization and coherent-gauge conditions. The older unqualified statement that the marked-endpoint problem is not controlling must not erase either live interface.

At the source's accepted-root scope, every sufficiently deep state has a pointwise fixed-action consequence. This does not choose a continuous square, a detector, or an all-scale carrier. RSP-1 preserves cofinal exact-critical charged states as an alternative to a regular charged full-tail spine; neither can be discarded in favor of a convenient branch.

The inherited supplied-family arguments retain hits, no-hits, filler channels, source and action rims, both endpoint channels and every selected face. Normalized convergence does not supply physical cap control, source BV, uniform mass, a full positive-wall family map or a terminal for unbounded normalized lens mass.

CIV-1 gives charge collapse only from the stated weighted center/radius variation or physical cap uniform integrability on one full-tail regular spine. Projected inverse concentration, pointwise bounded energy, state-switching examples and tangent models do not supply that premise.

Under the source's square-free Jordan assumption and HUG-WRAP interface, BPG-1 supplies one connected closed configuration set with three actual boundary corners, the missing corner in the interior, the full angle range and retained compact positive-size crossings. It supplies no path, manifold structure, BV or action spine, continuous square selector, or unique endpoint. Rerooting need not preserve the chosen band; balanced compact subsets need not connect the signs.

The source derives strict inradius/radius/hull-area obstructions and a return-degree formula on compact continua. Reverse source order forces a nonzero return degree; positive order can have all degrees zero. Cap hull area and angular advance do not control tangent action or an approximation rate. Return at chord parameter tending to one is terminal only if side length stays nonzero; parameter tending to zero permits narrow return.

For every sufficiently small prescribed eta, BMC-1 uses a compact connected size-floor set through a fixed core to couple a side-eta configuration with another side lambda at least eta whose missing corner lies on the first chord. B4P-1 produces four actual separated boundary points, not the four corners of one square. Both scales can tend to zero; no compatible choice across eta, nonzero final size, or terminal in every ratio/chord-parameter regime is supplied.

AGC-1/2 give sufficient bounds at their stated isotopy, length and incremental-budget scope. UAP-1 defeats the unrestricted universal bounded ambient budget using a square-containing zero-area Jordan example whose smooth approximants have unbounded primitive oscillation. Square-free-conditional and explicitly restricted constructions remain open. C0, Hofer, L2 or finite-q Lorentz strain cannot replace the endpoint cost.

Finite hull-area or stated mixed variation yields source-reported sufficient results only through the selected simple-polygon and spectral-import hypotheses. The sharp half-Hölder primitive threshold is not a sharp square-existence threshold. Infinite local unsigned hull variation does not supply signed charge, smooth models, primitive failure or actual crossings.

Positive all-secants endpoint variation or positive response to a bounded smooth Hamiltonian can force a square under the displayed imports and source interfaces. Full-gradient, common-gauge and actual-detection hypotheses remain explicit. Abstract flat models do not establish realization by a Jordan curve; weak averaging can destroy nonlinear single-square moment equality; positive spectral response and bounded strain are additional inputs.