Twistor Configuration Geometry: A Structural-State Review at Four-Arc Completeness

This is a structural-state review of the DAEDALUS / Twistor Configuration Geometry (TCG) corpus at the moment when its four-arc named-residual pattern is complete. The 36-paper TCG construction proceeds from a fixed active postulate ledger (P_0–P_4, P_{5'}, P_6, P_7, P_{H'}, P_{SO(10)}) that has been stable since the 2026-05-01 framework closure verdict; on top of this ledger, three within-TCG structural arcs — electron P_4, gauge envelope, hadronic P_{H'} — have each been brought to theorem-level obstruction with a named residual placed explicitly outside the active ledger. The 2026-06-15 substrate-arc pair (Papers #37, #38, DOIs:10.5281/zenodo.20709751 and 10.5281/zenodo.20709846) adds the fourth arc, one structural level below the corpus's CP^3 starting datum: a substrate-level obstruction theorem on minimal twistor-incidence data, paired with an Atiyah–Hitchin–Singer (AHS) S^4-anchor conditional-closure construction test.

All four arcs now carry named residuals classified as labeled successor-construction targets outside the active ledger; the four-arc named-residual pattern is the central organizing observation consolidated here. A second consolidation observation is the obstruction-then-construction pattern shared by the hadronic and substrate arcs: name a residual at theorem level, propose a structural input, verify conditional closure of one sub-residual, name any new sub-residual the input introduces. The two instantiations of this pattern differ in closure type — the hadronic-arc cohomological closure (Paper #36, DOI:10.5281/zenodo.20264444) is internal to the established TCG/FPA combinatorial machinery, whereas the substrate-arc AHS-S^4 closure (Paper #38) imports an external four-manifold anchor — and this maturity-register asymmetry is itself structural content, not a flaw to flatten.

The CP^3 starting datum is protected by two complementary defenses: the configurable framing of Paper #16 (DOI:10.5281/zenodo.20077307) dissolves the substrate question by declining its presupposition; the substrate-derivation framing of Paper #37 takes the question seriously and proves it obstructed. The review consolidates external positioning against pre-geometric quantum gravity programs (Quantum Graphity, Group Field Theory, Causal Set Theory, twistorial loop quantum gravity, Wolfram Physics) and adjacent geometric-physics programs (AdS/CFT and tensor-network emergence, Migdal Geometric QCD). The forward-falsifiable content of the framework remains the spin-1 fifth-force prediction α_Y ≈ 1.88 × 10^4 in the surviving window m ≳ 20–40 meV, λ ≲ 5–10 μm, currently ~500× below the binding short-range experimental sensitivity.

This review introduces no new theorem, postulate, residual, or empirical prediction. Its only new content is organizational: it names the structural state of the already-published corpus at four-arc completeness, compares maturity registers across arcs, and consolidates external positioning. Active TCG/τCG postulate ledger UNCHANGED: P_0–P_4, P_{5'}, P_6, P_7, P_{H'}, P_{SO(10)}; the 2026-05-01 framework closure verdict is preserved.

Five failure modes F1-F5: do not claim universal no-go from minimal-data results; do not license substrate-derivation for other TCG primitives without independent obstruction analysis; do not conflate maturity registers across arcs; do not merge the two-defense protections of CP^3; do not claim TCG correctness from structural-completeness. Maturity register: review-level synthesis, distinct from the construction notes, obstruction notes, τCG specification, obstruction-then-construction pairs, or empirical review of Paper #14.

The strongest revised thesis: the 38-paper DAEDALUS / TCG corpus has reached a stable named-residual state. All four structural arcs (electron, gauge, hadronic, substrate) are terminated by theorem-level obstruction or conditional construction with a named residual outside the active ledger. The active postulate ledger is unchanged. The remaining work is organized as named successor-construction targets rather than unresolved ambiguities. Internal mathematical progress within the bounds of present methods has hit theorem-level limits; further progress along the mainline requires either fundamentally new theoretical input or experimental confirmation of the spin-1 fifth-force prediction (currently ~500× below the binding short-range experimental sensitivity, treated in the monitoring notes as a possible multi-iteration optomechanical vector-sensing target rather than a present capability).