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\title{Syncr\'e Witness Atlas:\\ Fifty-four Examples Across Context Families}
\author{Drew Higgins}
\date{}

\begin{document}
\maketitle

\section*{How to read this atlas}

Each entry supplies the same minimal template:

\begin{itemize}[leftmargin=*]
\item the slice or subsystem being observed,
\item the cycle-valued phase observable,
\item the forcing mechanism (action witness, loop degree, holonomy, coupling, or mixing upgrade),
\item the correct target (circle, torus, coupled subtorus, or embedded target),
\item the artifact: a WSC (certificate) or a WOB (obstruction), with the regime made explicit.
\end{itemize}

\section*{Families and reduction templates}

SFT breadth is not a pile of unrelated examples.
Every atlas entry is a reuse of a small set of reduction templates.
Each family below states its reduction as a lemma-style rule and points to the theorem file that justifies it.

\subsection*{Rotation and periodic forcing (action witness template)}
\textbf{Reduction lemma.}
If a slice admits a cyclic action and the declared phase observable is equivariant with nondegenerate response, then the phase sweeps a full cycle on the correct target.
The certificate reduces to an action/equivariance witness plus robustness margins.
(Justification: \codeid{global_universality/COC_rotation_proof.tex#thm:circle-action-template}.)

\subsection*{Embedded loops and defects (degree or holonomy witness)}
\textbf{Reduction lemma.}
If the full slice does not admit a usable global action but contains a declared embedded loop subsystem on which the phase has nonzero degree or nonzero holonomy multiple, then coverage holds on the induced embedded target on that subsystem.
The certificate reduces to an integer witness invariant plus robustness margins.
(Justification: \codeid{global_universality/COC_embedded_loop_proof.tex#thm:COC-ESW-LDW} and \codeid{global_universality/COC_embedded_loop_proof.tex#thm:COC-ESW-HW}.)

\subsection*{Coupled targets (CST) under multi-cycle constraints}
\textbf{Reduction lemma.}
If multiple cycles are coupled by declared integer relations, the correct target is a coupled subtorus (CST) and coverage statements must be made on CST (with coupled Haar as baseline).
The certificate reduces to a free-coordinate witness on CST plus target-correctness predicates.
(Justification: \codeid{global_universality/COC_multicycle_proof.tex#thm:coupling-forces-cst}.)

\subsection*{Torus drives and equidistribution}
\textbf{Reduction lemma.}
If independent drives produce a torus target and a nonresonance hypothesis is explicitly declared, then statistical inevitability is recorded as a convergence certificate to Haar on the correct target, not as an unqualified mixing claim.
(Justification: \codeid{global_universality/COC_multicycle_proof.tex#thm:Tk-feq}.)

\subsection*{Noise-to-diffusion mixing upgrades}
\textbf{Reduction lemma.}
Deterministic transport is not assumed to be mixing.
When a noise-to-diffusion bridge is explicitly adopted and a mixing certificate is declared, uniformity upgrades become meaningful on the correct target and baseline, and are reported as convergence certificates rather than as forced coverage claims.

\subsection*{Boundary obstructions}
\textbf{Reduction lemma.}
Outside COC, the engine produces named obstructions rather than vague refusals.
Boundary entries exist to show sharpness and to prevent overclaiming across regime walls.
(Completeness pointers: \codeid{global_universality/ENGINE_COMPLETENESS_PAGE.tex#thm:catalog-completeness} and \codeid{global_universality/WOB_CATALOG_COMPLETENESS.tex#thm:wob-catalog-complete}.)

\begin{longtable}{@{}p{0.07\textwidth}p{0.15\textwidth}p{0.22\textwidth}p{0.18\textwidth}p{0.30\textwidth}@{}}
\toprule
\textbf{Label} & \textbf{Family} & \textbf{Context} & \textbf{COC / Boundary / Artifact} & \textbf{SFT form (target + cert/obstruction sketch)} \\
\midrule
\endhead
A1 & Embedded loops & Earth rotation (local solar phase) & COC \textbullet\ Class C / \texttt{WSC-ATLAS-0001} & Embedded loop witness forces full-cycle coverage on the induced circle target, without asserting global coverage on the ambient domain. \\
A2 & Coupled CST & Solar--lunar synodic coupling (CST) & COC \textbullet\ Class B / \texttt{WSC-ATLAS-0002} & Coupled target (CST) is the correct meaning; coverage is stated on CST with coupled Haar baseline (CTP/CHT). \\
A3 & Embedded loops & Traveling wave on periodic medium & COC \textbullet\ Class C / \texttt{WSC-ATLAS-0003} & Embedded loop witness forces full-cycle coverage on the induced circle target, without asserting global coverage on the ambient domain. \\
A4 & Embedded loops & Vortex defect loop (order parameter) & COC \textbullet\ Class C / \texttt{WSC-ATLAS-0004} & Embedded loop witness forces full-cycle coverage on the induced circle target, without asserting global coverage on the ambient domain. \\
A5 & Rotation/\allowbreak forcing & AC circuit steady-state (phase response) & COC \textbullet\ Class A / \texttt{WSC-ATLAS-0005} & Equivariant response (action witness template) forces a phase sweep on $S^1$ in the declared regime; robustness is recorded via margins. \\
A6 & Embedded loops & Ring oscillator phase around ring & COC \textbullet\ Class C / \texttt{WSC-ATLAS-0006} & Embedded loop witness forces full-cycle coverage on the induced circle target, without asserting global coverage on the ambient domain. \\
A7 & Rotation/\allowbreak forcing & Rotating machinery shaft angle & COC \textbullet\ Class A / \texttt{WSC-ATLAS-0007} & Equivariant response (action witness template) forces a phase sweep on $S^1$ in the declared regime; robustness is recorded via margins. \\
A8 & Embedded loops & Planetary orbital phase & COC \textbullet\ Class C / \texttt{WSC-ATLAS-0008} & Embedded loop witness forces full-cycle coverage on the induced circle target, without asserting global coverage on the ambient domain. \\
A9 & Boundary & Coupled pendula resonance locking & Boundary \textbullet\ Class D / \texttt{WOB-ATLAS-0009} & Obstruction (WOB-10-resonant-locking): evidence explains why SFT is inapplicable or the declared target is wrong; reissue on the correct target or revise the regime. \\
A10 & Mixing upgrades & Stroboscopic jitter diffusion & COC \textbullet\ Class A / \texttt{WSC-ATLAS-0010} & Noise-to-diffusion bridge supports a mixing upgrade on the correct target when the mixing certificate is explicitly declared. \\
A11 & Boundary & Coupling mismatch (wrong target) & Boundary \textbullet\ Class D / \texttt{WOB-ATLAS-0011} & Obstruction (WOB-06-coupling-mismatch): evidence explains why SFT is inapplicable or the declared target is wrong; reissue on the correct target or revise the regime. \\
A12 & Boundary & Equivariance defect beyond tolerance & Boundary \textbullet\ Class D / \texttt{WOB-ATLAS-0012} & Obstruction (WOB-09-equivariance-defect): evidence explains why SFT is inapplicable or the declared target is wrong; reissue on the correct target or revise the regime. \\
A13 & Boundary & No cyclic subsystem available & Boundary \textbullet\ Class D / \texttt{WOB-ATLAS-0013} & Obstruction (WOB-01-no-subsystem): evidence explains why SFT is inapplicable or the declared target is wrong; reissue on the correct target or revise the regime. \\
A14 & Boundary & Trivial winding on candidate loop & Boundary \textbullet\ Class D / \texttt{WOB-ATLAS-0014} & Obstruction (WOB-02-trivial-winding): evidence explains why SFT is inapplicable or the declared target is wrong; reissue on the correct target or revise the regime. \\
A15 & Boundary & Measurement pinch (image misses arc) & Boundary \textbullet\ Class D / \texttt{WOB-ATLAS-0015} & Obstruction (WOB-03-pinch): evidence explains why SFT is inapplicable or the declared target is wrong; reissue on the correct target or revise the regime. \\
A16 & Boundary & Tear/regularity failure on loop & Boundary \textbullet\ Class D / \texttt{WOB-ATLAS-0016} & Obstruction (WOB-04-tear): evidence explains why SFT is inapplicable or the declared target is wrong; reissue on the correct target or revise the regime. \\
A17 & Boundary & Disconnected domain misses witness component & Boundary \textbullet\ Class D / \texttt{WOB-ATLAS-0017} & Obstruction (WOB-11-disconnected-slice): evidence explains why SFT is inapplicable or the declared target is wrong; reissue on the correct target or revise the regime. \\
A18 & Boundary & Slope-zero and quotient nonsurjective & Boundary \textbullet\ Class D / \texttt{WOB-ATLAS-0018} & Obstruction (WOB-12-slope-zero-quotient-nonsurjective): evidence explains why SFT is inapplicable or the declared target is wrong; reissue on the correct target or revise the regime. \\
A19 & Boundary & Orbit collapse (no chi-active orbit) & Boundary \textbullet\ Class D / \texttt{WOB-ATLAS-0019} & Obstruction (WOB-08-orbit-collapse): evidence explains why SFT is inapplicable or the declared target is wrong; reissue on the correct target or revise the regime. \\
A20 & Boundary & Mixing claim without justification & Boundary \textbullet\ Class D / \texttt{WOB-ATLAS-0020} & Obstruction (WOB-07-mixing-not-justified): evidence explains why SFT is inapplicable or the declared target is wrong; reissue on the correct target or revise the regime. \\
A21 & Torus drives & Uncoupled two-drive torus coverage & COC \textbullet\ Class A / \texttt{WSC-ATLAS-0021} & Full-rank response produces a torus target; RNF evidence supports equidistribution when declared in the certificate. \\
A22 & Coupled CST & Matrix coupling in $T^3$ (CST rank 2) & COC \textbullet\ Class B / \texttt{WSC-ATLAS-0022} & Coupled target (CST) is the correct meaning; coverage is stated on CST with coupled Haar baseline (CTP/CHT). \\
A23 & Embedded loops & Holonomy loop witness (flux-like) & COC \textbullet\ Class C / \texttt{WSC-ATLAS-0023} & Embedded loop witness forces full-cycle coverage on the induced circle target, without asserting global coverage on the ambient domain. \\
A24 & Embedded loops & Bloch phase winding on closed loop & COC \textbullet\ Class C / \texttt{WSC-ATLAS-0024} & Embedded loop witness forces full-cycle coverage on the induced circle target, without asserting global coverage on the ambient domain. \\
A25 & Embedded loops & Fluid vortex ring loop witness & COC \textbullet\ Class C / \texttt{WSC-ATLAS-0025} & Embedded loop witness forces full-cycle coverage on the induced circle target, without asserting global coverage on the ambient domain. \\
A26 & Torus drives & Nonresonant torus drive (Haar convergence) & COC \textbullet\ Class A / \texttt{WSC-ATLAS-0026} & Full-rank response produces a torus target; RNF evidence supports equidistribution when declared in the certificate. \\
A27 & Mixing upgrades & Diffusion-enforced uniformity on CST & COC \textbullet\ Class B / \texttt{WSC-ATLAS-0027} & Noise-to-diffusion bridge supports a mixing upgrade on the correct target when the mixing certificate is explicitly declared. \\
A28 & Embedded loops & Neural oscillation ring phase wave & COC \textbullet\ Class C / \texttt{WSC-ATLAS-0028} & Embedded loop witness forces full-cycle coverage on the induced circle target, without asserting global coverage on the ambient domain. \\
A29 & Coupled CST & Gear ratio coupled phase (CST) & COC \textbullet\ Class B / \texttt{WSC-ATLAS-0029} & Coupled target (CST) is the correct meaning; coverage is stated on CST with coupled Haar baseline (CTP/CHT). \\
A30 & Rotation/\allowbreak forcing & Two-control oscillator array torus coverage & COC \textbullet\ Class A / \texttt{WSC-ATLAS-0030} & Equivariant response (action witness template) forces a phase sweep on $S^1$ in the declared regime; robustness is recorded via margins. \\
A31 & Boundary & Holonomy defect (lift inconsistency) & Boundary \textbullet\ Class D / \texttt{WOB-ATLAS-0031} & Obstruction (WOB-05-holonomy-defect): evidence explains why SFT is inapplicable or the declared target is wrong; reissue on the correct target or revise the regime. \\
A32 & Boundary & Constraint inference ambiguity (partial excitation) & Boundary \textbullet\ Class D / \texttt{WOB-ATLAS-0032} & Obstruction (WOB-13-ambiguity): evidence explains why SFT is inapplicable or the declared target is wrong; reissue on the correct target or revise the regime. \\
A33 & Boundary & Insufficient excitation for target inference & Boundary \textbullet\ Class D / \texttt{WOB-ATLAS-0033} & Obstruction (WOB-14-insufficient-excitation): evidence explains why SFT is inapplicable or the declared target is wrong; reissue on the correct target or revise the regime. \\
A34 & Boundary & Resonance masking (near-threshold relations) & Boundary \textbullet\ Class D / \texttt{WOB-ATLAS-0034} & Obstruction (WOB-15-resonance-masking): evidence explains why SFT is inapplicable or the declared target is wrong; reissue on the correct target or revise the regime. \\
A35 & Rotation/\allowbreak forcing & Rotating beacon sweep phase & COC \textbullet\ Class A / \texttt{WSC-ATLAS-0035} & Equivariant response (action witness template) forces a phase sweep on $S^1$ in the declared regime; robustness is recorded via margins. \\
A36 & Rotation/\allowbreak forcing & Stroboscopic phase under periodic forcing & COC \textbullet\ Class A / \texttt{WSC-ATLAS-0036} & Equivariant response (action witness template) forces a phase sweep on $S^1$ in the declared regime; robustness is recorded via margins. \\
A37 & Rotation/\allowbreak forcing & Limit-cycle oscillator phase coordinate & COC \textbullet\ Class A / \texttt{WSC-ATLAS-0037} & Equivariant response (action witness template) forces a phase sweep on $S^1$ in the declared regime; robustness is recorded via margins. \\
A38 & Rotation/\allowbreak forcing & Tidal forcing phase in a periodic proxy & COC \textbullet\ Class A / \texttt{WSC-ATLAS-0038} & Equivariant response (action witness template) forces a phase sweep on $S^1$ in the declared regime; robustness is recorded via margins. \\
A39 & Embedded loops & Josephson junction phase slip loop & COC \textbullet\ Class C / \texttt{WSC-ATLAS-0039} & Embedded loop witness forces full-cycle coverage on the induced circle target, without asserting global coverage on the ambient domain. \\
A40 & Embedded loops & Crystal dislocation Burgers circuit & COC \textbullet\ Class C / \texttt{WSC-ATLAS-0040} & Embedded loop witness forces full-cycle coverage on the induced circle target, without asserting global coverage on the ambient domain. \\
A41 & Embedded loops & Optical ring cavity mode phase & COC \textbullet\ Class C / \texttt{WSC-ATLAS-0041} & Embedded loop witness forces full-cycle coverage on the induced circle target, without asserting global coverage on the ambient domain. \\
A42 & Embedded loops & Spiral wave tip defect loop & COC \textbullet\ Class C / \texttt{WSC-ATLAS-0042} & Embedded loop witness forces full-cycle coverage on the induced circle target, without asserting global coverage on the ambient domain. \\
A43 & Embedded loops & Magnetic flux tube loop witness & COC \textbullet\ Class C / \texttt{WSC-ATLAS-0043} & Embedded loop witness forces full-cycle coverage on the induced circle target, without asserting global coverage on the ambient domain. \\
A44 & Coupled CST & Two coupled rotors (gear ratio constraint) & COC \textbullet\ Class B / \texttt{WSC-ATLAS-0044} & Coupled target (CST) is the correct meaning; coverage is stated on CST with coupled Haar baseline (CTP/CHT). \\
A45 & Coupled CST & PLL network with integer relation & COC \textbullet\ Class B / \texttt{WSC-ATLAS-0045} & Coupled target (CST) is the correct meaning; coverage is stated on CST with coupled Haar baseline (CTP/CHT). \\
A46 & Coupled CST & Rank-2 coupling in $T^3$ (matrix constraint) & COC \textbullet\ Class B / \texttt{WSC-ATLAS-0046} & Coupled target (CST) is the correct meaning; coverage is stated on CST with coupled Haar baseline (CTP/CHT). \\
A47 & Coupled CST & Circadian plus lunar entrainment (CST) & COC \textbullet\ Class B / \texttt{WSC-ATLAS-0047} & Coupled target (CST) is the correct meaning; coverage is stated on CST with coupled Haar baseline (CTP/CHT). \\
A48 & Torus drives & Two-tone RF mixer torus phase & COC \textbullet\ Class A / \texttt{WSC-ATLAS-0048} & Full-rank response produces a torus target; RNF evidence supports equidistribution when declared in the certificate. \\
A49 & Torus drives & Quasiperiodic optics modulator ($T^2$) & COC \textbullet\ Class A / \texttt{WSC-ATLAS-0049} & Full-rank response produces a torus target; RNF evidence supports equidistribution when declared in the certificate. \\
A50 & Torus drives & Two independent control knobs ($T^2$) coverage & COC \textbullet\ Class A / \texttt{WSC-ATLAS-0050} & Full-rank response produces a torus target; RNF evidence supports equidistribution when declared in the certificate. \\
A51 & Mixing upgrades & Advection plus small diffusion on $S^1$ (mixing upgrade) & COC \textbullet\ Class A / \texttt{WSC-ATLAS-0051} & Noise-to-diffusion bridge supports a mixing upgrade on the correct target when the mixing certificate is explicitly declared. \\
A52 & Boundary & Monotone gradient flow with no cyclic observable & Boundary \textbullet\ Class D / \texttt{WOB-ATLAS-0052} & Obstruction (WOB-01-no-subsystem): evidence explains why SFT is inapplicable or the declared target is wrong; reissue on the correct target or revise the regime. \\
A53 & Boundary & Strict contraction collapses the orbit (orbit collapse) & Boundary \textbullet\ Class D / \texttt{WOB-ATLAS-0053} & Obstruction (WOB-08-orbit-collapse): evidence explains why SFT is inapplicable or the declared target is wrong; reissue on the correct target or revise the regime. \\
A54 & Boundary & Resonant locking reduces intended torus rank & Boundary \textbullet\ Class D / \texttt{WOB-ATLAS-0054} & Obstruction (WOB-10-resonant-locking): evidence explains why SFT is inapplicable or the declared target is wrong; reissue on the correct target or revise the regime. \\
\bottomrule
\end{longtable}
\end{document}
