From fee78933d0d3a45174ea393c05ca5206b68539d9 Mon Sep 17 00:00:00 2001 From: Johan Mathe Date: Sat, 18 Apr 2026 05:37:41 +0000 Subject: [PATCH 1/2] Remove octa on octa proof --- paper/paper.tex | 347 ++---------------------------------------------- 1 file changed, 11 insertions(+), 336 deletions(-) diff --git a/paper/paper.tex b/paper/paper.tex index 5214adc..0cc1f28 100644 --- a/paper/paper.tex +++ b/paper/paper.tex @@ -90,13 +90,11 @@ drop-in differentiable \texttt{nn.Module}s. The selective bispectrum reduces invariant cost from $O(|G|^2)$ to $O(|G|)$ while remaining complete for generic signals. -Building the library required two theoretical contributions: -(i)~an augmented selective $\SO(3)$ invariant on $S^2$ achieving +Building the library required one theoretical contribution: +an augmented selective $\SO(3)$ invariant on $S^2$ achieving $\Theta(L^2)$ coefficients (the information-theoretic lower bound) instead of $O(L^3)$ for the full bispectrum -(completeness proved for $4 \le L \le 100$), and -(ii)~diagnosing a previously undocumented inversion failure for the -octahedral group. +(completeness proved for $4 \le L \le 100$). Across three benchmarks ($C_8$, octahedral, $\SO(3)$), the library delivers near-exact rotation invariance ($\sigma_{\mathrm{rot}} < 10^{-3}$) and a data-efficiency advantage of @@ -131,12 +129,10 @@ \section{Introduction} inverses. The library supports cyclic, dihedral, torus, disk, sphere, and octahedral settings as differentiable \texttt{nn.Module}s intended to plug into standard ML workflows. -Beyond the software contribution, the paper makes two technical contributions +Beyond the software contribution, the paper makes one technical contribution that arose during implementation: an augmented selective $\SO(3)$ bispectrum on $S^2$ with $O(L^2)$ coefficients -(Section~\ref{sec:so3-selective}), and an analysis of -the bootstrap inversion failure~\citep{mataigne2024selective} for the -octahedral group together with a partial workaround (Section~\ref{sec:octa}). +(Section~\ref{sec:so3-selective}). \paragraph{Contributions.} \begin{enumerate} @@ -157,10 +153,6 @@ \section{Introduction} via a combination of exact rational certificates and a closed-form linear bootstrap family (Appendix~\ref{app:so3-proof}); the verification extends to any desired~$L$ by re-running a single script. - \item \textbf{Octahedral inversion analysis.} An analysis of why the standard - selective-bispectrum bootstrap inversion fails for the octahedral group, - together with a bootstrap + Levenberg--Marquardt approach that achieves - local convergence but not reliable global recovery. \item \textbf{Empirical validation.} A task-driven evaluation of the library as an invariant pooling primitive across three group/domain settings, showing that complete bispectral invariants provide a @@ -402,28 +394,6 @@ \subsection{Augmented selective $\SO(3)$ bispectrum on $S^2$} machine epsilon); they can be upgraded to interval arithmetic if desired. The full proof is in Appendix~\ref{app:so3-proof}. -\subsection{Octahedral inversion as an implementation case study} -\label{sec:octa} - -The bootstrap inversion of \citet{mataigne2024selective} recovers -$\hat{f}(\rho_1)$ up to an orthogonal ambiguity~$Q$, then extracts -higher Fourier blocks via CG matrices. -For $C_n$ and $D_n$ the ambiguity is low-dimensional and absorbed by -the CG structure, but for the octahedral group~$O$ the ambiguity is -$Q \in \SO(3)$ and the CG matrices develop $O(1)$ cross-block leakage -for generic $Q \notin O$, corrupting higher blocks. -The selective $O$-bispectrum \emph{is} complete (172 coefficients -determine the signal up to $O$-action); the failure is purely algorithmic. -We partially address it by combining bootstrap initialisation with -Levenberg--Marquardt refinement of -$\min_f \norm{\beta^{\mathrm{sel}}(f) - - \beta^{\mathrm{sel}}_{\mathrm{target}}}^2$, -using multi-start randomisation over~$Q$. -The LM residual converges in $<$50 iterations, but the overall signal -recovery rate remains modest (5.7\% with 4~restarts), reflecting the -difficult landscape created by CG leakage. -The full analysis is in Appendix~\ref{app:octa-convergence}. - \subsection{Computational benchmarks} \label{sec:benchmarks} @@ -464,7 +434,7 @@ \subsection{Computational benchmarks} \texttt{DnonDn} & $D_{32}$ & 64 & 245 & -- & 2.52 & -- & 42.33 \\ \texttt{SO2onDisk} & $\mathrm{SO}(2)$ & $\infty$ & 411 & -- & 1.94 & -- & 17.69 \\ \texttt{SO3onS2}\rlap{$^*$} & $\mathrm{SO}(3)$ & $\infty$ & 430 & -- & 26.54 & -- & -- \\ - \texttt{OctaonOcta} & $O$ & 24 & 172 & -- & 0.70 & -- & 2172.76 \\ + \texttt{OctaonOcta} & $O$ & 24 & 172 & -- & 0.70 & -- & -- \\ \bottomrule \end{tabular} \end{table} @@ -479,8 +449,7 @@ \subsection{Computational benchmarks} $C_{128}$ uses 128 coefficients vs.\ 8{,}256 full (64$\times$ reduction); $\mathbb{T}^2$ at $|G|{=}1{,}024$ uses 1{,}024 vs.\ 524{,}800 full (512$\times$). Inversion is fast for abelian groups -(9--68\,ms) and practical for the octahedral group (2.17\,s via -Levenberg--Marquardt). GPU throughput scales linearly with batch size, +(9--68\,ms). GPU throughput scales linearly with batch size, exceeding $10^7$~samples/s for $C_n$ and $\mathbb{T}^2$ (Appendix~\ref{app:bench}). @@ -774,11 +743,9 @@ \section{Discussion and Conclusion} We have presented \texttt{bispectrum}, a PyTorch library that makes selective $G$-bispectral invariants practical for machine learning. The library covers seven group/domain pairs with a uniform \texttt{nn.Module} interface and -required two technical contributions during implementation: an augmented +required one technical contribution during implementation: an augmented selective $\SO(3)$ bispectrum on $S^2$ achieving the optimal $O(L^2)$ -coefficient count (Theorem~\ref{thm:so3-selective}), and an analysis of -the octahedral bootstrap inversion failure with a partial Levenberg--Marquardt -workaround (Section~\ref{sec:octa}). Table~\ref{tab:benchmarks} confirms that the +coefficient count (Theorem~\ref{thm:so3-selective}). Table~\ref{tab:benchmarks} confirms that the selective forward pass adds negligible overhead to training pipelines ($<$1\,ms for all finite-group modules on a single GPU). Across three experimental settings, bispectral pooling provides numerically exact rotation @@ -808,7 +775,6 @@ \section{Discussion and Conclusion} measure-zero algebraic variety) and the bootstrap rank verification covers $L \le 100$; a structural all-$L$ proof and a full characterisation of the degenerate set remain open. -The octahedral inversion achieves only a 5.7\% signal recovery rate. Our experimental comparisons do not include tensor-product architectures (e3nn, MACE) or wavelet scattering, which target different input assumptions and would require non-trivial adaptation @@ -1947,23 +1913,6 @@ \subsection{Complexity in pixel space} ${\sim}4\mathrm{K}$ scalars at ${\sim}260\mathrm{K}$~ops---a $65\times$ reduction. -\subsection{Relation to the inversion gap} - -For finite groups ($O$, $D_n$), the ``inversion gap'' arises when the -bootstrap ambiguity $Q$ is not absorbed by the Clebsch--Gordan -structure (see Appendix~\ref{app:octa-convergence}). For $\SO(3)$ on -$S^2$, the situation differs fundamentally: -\begin{enumerate}[nosep] - \item The ambiguity in $\mathbf{F}_1$ is an $\SO(3)$ rotation $g$ - (3~DOF). - \item We fix $g$ explicitly by gauge-choosing $\mathbf{F}_1$'s - direction (2~DOF) and $\mathbf{F}_2$'s azimuthal phase (1~DOF). - \item At $\ell \ge 4$, no ambiguity remains---the linear system has - a unique solution. -\end{enumerate} -This avoids the finite-group problem entirely: we never need the CG -matrices to ``absorb'' a continuous ambiguity. - \subsection{Limitations and open questions} \begin{enumerate} @@ -1984,278 +1933,6 @@ \subsection{Limitations and open questions} $\ell \ge 4$: linear solve (pseudoinverse). \end{enumerate} -\section{Octahedral Inversion Gap -- Full Analysis} -\label{app:octa-convergence} - -This appendix provides the complete analysis of why the standard -bootstrap inversion algorithm fails for the octahedral group, and how -we resolve it. The material is complementary to -Section~\ref{sec:octa} in the main text. - -\subsection{The bootstrap strategy (recap)} - -All inversion algorithms in \citet{mataigne2024selective} share the -same skeleton: -\begin{enumerate}[nosep] - \item Recover $\hat{f}(\rho_0)$ from - $\beta_{\rho_0,\rho_0} = \hat{f}(\rho_0)^3$. - \item Recover $\hat{f}(\rho_1)$ from - $\beta_{\rho_0,\rho_1}/\hat{f}(\rho_0) - = \hat{f}(\rho_1)^\dagger \hat{f}(\rho_1)$, - up to a unitary/orthogonal ambiguity $Q$. - \item Extract higher Fourier coefficients from subsequent bispectral - entries using CG matrices to block-diagonalise tensor products. -\end{enumerate} -Step 2 always introduces an ambiguity: -$\hat{f}(\rho_1) = S \cdot Q$ where -$S = (\hat{f}(\rho_1)^\dagger \hat{f}(\rho_1))^{1/2}$ is determined, -but $Q$ is a free orthogonal matrix. For the algorithm to work, this -ambiguity must not corrupt step~3. - -\subsection{Why it works for $C_n$} - -The irreps are 1D: $\rho_k(g) = e^{2\pi i k g / n}$. The ambiguity in -$\hat{f}(\rho_1)$ is a phase $e^{i\varphi}$, -$\varphi \in [0, 2\pi)$. Step~3 computes: -\[ - \hat{f}(\rho_{k+1}) = \left(\frac{\beta_{\rho_1, \rho_k}}{ - \hat{f}(\rho_1) \hat{f}(\rho_k)}\right)^\dagger. -\] -The phase $e^{i\varphi}$ propagates coherently through all coefficients -as $e^{i\varphi k}$, corresponding to a cyclic shift -$\Theta(g) \to \Theta(g + \varphi)$. The paper correctly resolves this -by finding the unique $\varphi$ that makes the reconstructed signal -real. - -\subsection{Why it works for $D_n$} - -The generating irrep $\rho_1$ is 2D. Recovery gives -$\hat{f}(\rho_1) = V\Lambda^{1/2}V^\dagger U$ with -$U \in \OG(2)$. Step~3 extracts higher coefficients via: -\[ - \bigoplus_{\rho \in \rho_1 \otimes \rho_{k-1}} \hat{f}(\rho) - = \left(C_{\rho_1,\rho_{k-1}}^\dagger - \left[\hat{f}(\rho_1) \otimes \hat{f}(\rho_{k-1})\right]^{-1} - \beta_{\rho_1,\rho_{k-1}} - C_{\rho_1,\rho_{k-1}}\right)^\dagger. -\] -The 2D irreps of $D_n$ are restrictions of $\SO(2)$ representations -to $\mathbb{Z}/n\mathbb{Z}$. The CG matrices for $D_n$ therefore -block-diagonalise $\rho(U)$ for \emph{all} $U \in \OG(2)$, not just -$U \in D_n$. Concretely, $C^\dagger(U \otimes U)C$ remains -block-diagonal for any $U \in \OG(2)$. The $\OG(2)$ ambiguity passes -cleanly through the CG extraction. - -\subsection{Why it fails for $O$} - -For the octahedral group, $\rho_1$ is the standard 3D representation. -From $\beta_{\rho_0, \rho_1}$ we recover -$\hat{f}(\rho_1)^\top \hat{f}(\rho_1) = -\beta_{\rho_0,\rho_1}/\hat{f}(\rho_0)$. -Taking the symmetric square root gives -$S = (\hat{f}(\rho_1)^\top \hat{f}(\rho_1))^{1/2}$, with -$\hat{f}(\rho_1) = Q \cdot S$ for unknown $Q \in \SO(3)$. - -The next step extracts $\hat{f}(\rho_2)$ and $\hat{f}(\rho_3)$ -from $\beta_{\rho_1, \rho_1}$ using the CG matrix $C_{11}$ that -block-diagonalises $\rho_1 \otimes \rho_1$: -\[ - C_{11}^\top \cdot \beta_{\rho_1,\rho_1} \cdot (S \otimes S)^{-1} - \cdot C_{11} - = C_{11}^\top (Q \otimes Q) C_{11} - \cdot \bigoplus_k \hat{f}(\rho_k)^\top. -\] - -The CG matrix $C_{11}$ block-diagonalises -$\rho_1(g) \otimes \rho_1(g)$ for $g \in O$: -\[ - C_{11}^\top (\rho_1(g) \otimes \rho_1(g)) C_{11} - = \rho_0(g) \oplus \rho_1(g) \oplus \rho_2(g) \oplus \rho_3(g) -\] -with dimensions $1 + 3 + 3 + 2 = 9$. But $Q$ is \textbf{not} -constrained to be an element of $O$. It is an arbitrary element of -$\SO(3)$, which is a continuous group with 3 parameters, while $O$ has -only 24 elements. For a generic $Q \in \SO(3) \setminus O$, the matrix -$C_{11}^\top (Q \otimes Q) C_{11}$ is \textbf{not block-diagonal}. - -\paragraph{Cross-block leakage.} -The Kronecker decomposition of $\rho_1 \otimes \rho_1$ for $O$ gives -blocks: - -\begin{center} -\small -\begin{tabular}{llcl} - \toprule - Block & Rows & Irrep & Dim \\ - \midrule - 1 & 0 & $\rho_0$ (trivial) & 1 \\ - 2 & 1--3 & $\rho_1$ (standard 3D) & 3 \\ - 3 & 4--6 & $\rho_2$ (product 3D) & 3 \\ - 4 & 7--8 & $\rho_3$ (2D) & 2 \\ - \bottomrule -\end{tabular} -\end{center} - -When $Q = \rho_1(g)$ for $g \in O$, the off-diagonal blocks (e.g., -rows 4--6 $\times$ columns 7--8) are zero---exactly what the CG -matrix guarantees. When $Q$ is a generic rotation, e.g., -$Q = R_x(0.7)$, then -$\abs{[C_{11}^\top (Q \otimes Q) C_{11}]_{4:7,\, 7:9}}_\infty -\approx 0.788$. -This is not a small perturbation---it is $O(1)$. The recovered -$\hat{f}(\rho_2)$ is a mixture of $\rho_2$ and $\rho_3$ contributions. - -\paragraph{Why this doesn't happen for $D_n$.} -The 2D irreps of $D_n$ are restrictions of $\SO(2)$ irreps. The CG -decomposition of $\rho_j \otimes \rho_k$ for $D_n$ inherits from the -continuous group: $\rho_j \otimes \rho_k = \rho_{j+k} \oplus -\rho_{|j-k|}$ holds for all of $\SO(2)$, not just $D_n$. The CG -matrices are inherited from $\SO(2)$ and respect arbitrary rotations. -For $O \subset \SO(3)$, the irreps of $O$ are \emph{not} restrictions -of $\SO(3)$ irreps (except $\rho_1$), so the CG matrices are specific -to the 24-element group and do not block-diagonalise arbitrary -$\SO(3)$ rotations. - -\subsection{Connection to the paper's own remarks} - -\citet{mataigne2024selective} acknowledge a related issue in the -commutative group setting (Appendix C, around Eq.~10): -``\emph{Again, the indeterminacy factor $h_l$ is not restricted to -$\mathbb{Z}/n_l\mathbb{Z}$ but can belong to $[0, n_l]$. We will -have to solve this issue further.}'' -This is the same phenomenon in a different guise. For commutative -groups, the continuous indeterminacy is scalar (a phase) and can be -resolved by enforcing real-valuedness. For $O$, the continuous -indeterminacy is a $3 \times 3$ rotation matrix $Q \in \SO(3)$, and -there is no analogous closed-form trick. - -\subsection{Working alternative} - -The selective $O$-bispectrum \emph{is} complete (172 coefficients -uniquely determine the signal up to $O$-action). The issue is purely -algorithmic. We implement a working inversion using: -\begin{enumerate}[nosep] - \item \textbf{Bootstrap initialisation}: recover $\hat{f}(\rho_0)$ - exactly, $\hat{f}(\rho_1)$ up to $Q$, and approximate - $\hat{f}(\rho_2), \hat{f}(\rho_3), \hat{f}(\rho_4)$ from diagonal - blocks (ignoring leakage). - \item \textbf{Levenberg--Marquardt correction}: use forward-mode - autodiff through the bispectrum computation to solve - $\norm{\beta(f_{\mathrm{rec}}) - - \beta_{\mathrm{target}}}^2 = 0$ via damped least-squares steps. - Each step is a single linear solve, not iterative optimisation. - Jacobians are computed via \texttt{torch.func.jacfwd} composed with - \texttt{torch.vmap} for batched forward-mode AD. - \item \textbf{Multi-start}: randomise the initial $Q$ across several - restarts to escape local minima. -\end{enumerate} -This converges reliably in $<$50 iterations and is fully -differentiable. - -\subsection{Quantitative inversion results} - -Table~\ref{tab:inversion-stats} and -Figure~\ref{fig:inversion-convergence} report quantitative inversion -results over 1{,}000 random signals on the octahedral group (4 restarts, -10 LM corrections per restart, on a single GPU). - -\begin{table}[h] - \centering - \caption{Octahedral bispectrum inversion statistics - ($N=1{,}000$, 4 restarts $\times$ 10 LM steps).} - \label{tab:inversion-stats} - \begin{tabular}{lcc} - \toprule - Metric & Median & P95 \\ - \midrule - Bisp.\ residual $\|\beta(\hat{f}) - \beta^*\|$ & $5.7 \times 10^{-5}$ & $1.5 \times 10^{-4}$ \\ - Recovery error (mod $O$) & $1.03$ & $1.17$ \\ - Wall-clock time (ms) & 18.4 & 30.1 \\ - \midrule - Success rate (error $<0.1$) & \multicolumn{2}{c}{5.7\%} \\ - \bottomrule - \end{tabular} -\end{table} - -\begin{figure}[h] - \centering - \includegraphics[width=0.85\linewidth]{figures/inversion_convergence.pdf} - \caption{Bispectral residual vs.\ LM iteration for octahedral inversion. - The residual drops rapidly, but the signal recovery rate remains low - (5.7\%), indicating that the optimisation landscape contains many - spurious minima where the bispectrum is matched without recovering - the correct orbit representative.} - \label{fig:inversion-convergence} -\end{figure} - -The LM procedure reliably drives the bispectral residual to near-zero -(median $5.7 \times 10^{-5}$), confirming that local convergence to a -bispectrum-matching signal is fast. However, the signal recovery rate -(5.7\% at tolerance $<0.1$) shows that the inversion landscape has many -spurious local minima. This reflects the inherent difficulty of the -$O$-inversion problem noted in Section~\ref{sec:octa}: the bootstrap -initialisation provides a poor starting point due to CG leakage, and -the 4-restart multi-start strategy is insufficient to reliably escape -the resulting basin structure. Increasing the number of restarts or -employing global optimisation would improve recovery at the cost of -additional compute. We leave such improvements to future work. - -\subsection{Summary} - -\begin{table}[h] - \caption{Bootstrap inversion across group families.} - \label{tab:app-bootstrap-summary} - \centering - \small - \begin{tabular}{lcccl} - \toprule - Group & Irrep dim & Ambiguity & CG absorbs? & Bootstrap? \\ - \midrule - $C_n$ & 1 & $e^{i\varphi} \in \SO(2)$ & N/A (scalar) & Yes \\ - $D_n$ & 2 & $U \in \OG(2)$ & Yes (CG from $\SO(2)$) & Yes \\ - $O$ & 3 & $Q \in \SO(3)$ & \textbf{No} & \textbf{No} \\ - \bottomrule - \end{tabular} -\end{table} - -\subsection{Numerical disproof of CG absorption for $O$} - -The following code demonstrates the cross-block leakage: - -\begin{lstlisting} -import torch -from bispectrum.octa_on_octa import OctaonOcta - -bsp = OctaonOcta() -C11 = bsp._cg_11.to(torch.float64) - -# Generic SO(3) rotation (not in O) -theta = 0.7 -Q = torch.tensor([ - [1.0, 0.0, 0.0], - [0.0, torch.cos(torch.tensor(theta)), - -torch.sin(torch.tensor(theta))], - [0.0, torch.sin(torch.tensor(theta)), - torch.cos(torch.tensor(theta))], -], dtype=torch.float64) - -QQ = torch.kron(Q, Q) -M = C11.T @ QQ @ C11 - -# rho2 block: rows 4:7, rho3 block: rows 7:9 -cross = M[4:7, 7:9] -print(f"Cross-block max: {cross.abs().max():.6f}") -# Output: ~0.788 -- far from zero - -# Compare with a group element -rho1_g4 = bsp._get_irrep_mats(1)[4].to(torch.float64) -QQ_grp = torch.kron(rho1_g4, rho1_g4) -M_grp = C11.T @ QQ_grp @ C11 -cross_grp = M_grp[4:7, 7:9] -print(f"Cross-block for g in O: {cross_grp.abs().max():.6f}") -# Output: ~0.0 (machine precision) -\end{lstlisting} - \section{Extended Benchmark Results} \label{app:bench} @@ -2294,10 +1971,8 @@ \subsection{GPU throughput and inversion} $5.7 \times 10^6$~samples/s. All modules scale linearly with batch size. \textbf{(b)}~Inversion wall-clock time. Abelian groups ($C_n$, - $D_n$, $\SO(2)$/disk) scale roughly linearly. The octahedral group - requires ${\sim}$580\,ms due to Levenberg--Marquardt optimisation - over 5 irreducible representations. $\SO(3)$ on $S^2$ inversion is - not yet implemented.} + $D_n$, $\SO(2)$/disk) scale roughly linearly. $\SO(3)$ on $S^2$ + inversion is not yet implemented.} \label{fig:gpu-scaling} \end{figure} From 938752d72513954f5da36fe98406631e81e4fc6e Mon Sep 17 00:00:00 2001 From: Johan Mathe Date: Sat, 18 Apr 2026 06:04:35 +0000 Subject: [PATCH 2/2] Update --- paper/paper.pdf | Bin 458099 -> 423314 bytes paper/paper.tex | 519 +++++++++++++++++++++++++++--------------------- 2 files changed, 297 insertions(+), 222 deletions(-) diff --git a/paper/paper.pdf b/paper/paper.pdf index 4c6526e21976f0b8d85b518f41cc327f902df765..fdd5f9268eb23456f739fc00a96b911a17b5b120 100644 GIT binary patch delta 258328 zcmY&fLvSVxkd1BIwrxyo+qRuA$;7s8+qP|UV%y37ReRa3>h5#jUO)8X2X5jeHWIm_ z*dKak26hWKwQqZb$ip3AXz}d@q91J?oj;B79UhrhO>dp?6c%@YFh+HOQ+>~- 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z?U!#Cy`w2MlWDwT`I0&Lna!*>-f8(J**hO)Gg-#Vm+u3;i(lHz2IF0pe`|PGezTd4 z#w%zcnd`5wWRvl3T(j|J 0$ by the gauge condition. -Then $v^2 = S - u^2$, giving $|v|$ but not $\sgn(v)$. -The fibre at degree~$2$ is generically $\{(y,x,u,v),\; (y,x,u,-v)\}$. -\end{lemma} - -\begin{lemma}[Degree-3 recovery: finite fibre]\label{lem:app-ell3} -Given $\mathbf{F}_2 = (y,x,u,\pm v)$, the degree-3 parameters -$\mathbf{F}_3 = (t, p_1, q_1, p_2, q_2, p_3, q_3)$ are constrained by the -augmented entries: -\begin{itemize}[nosep] -\item $\beta_{1,2,3}$, $\beta_{1,3,2}$, $\beta_{2,3,1}$: bilinear in - $\mathbf{F}_2 \otimes \mathbf{F}_3$. -\item $\beta_{0,3,3} = A_0\norm{\mathbf{F}_3}^2$: gives the norm. -\item $\beta_{3,3,2}$: quadratic in $\mathbf{F}_3$, linear in $\mathbf{F}_2$. -\item $\beta_{2,3,2}$, $\beta_{2,2,3}$: odd parity with repeated indices, - vanish identically for real signals (Proposition~\ref{prop:app-odd-vanishing}). -\item $P_{1,2,1}$ (from degree-2, Lemma~\ref{lem:app-ell2}), - $P_{1,3,2}$, $P_{2,3,1}$, $P_{2,3,2}$, $P_{3,3,2}$: five CG power - entries (the last four quadratic in $\mathbf{F}_3$). -\end{itemize} -The full augmented seed system uses $13$ nonzero entries for $11$ unknowns. -The bispectral entries achieve rank~$6$; each of the five -CG power entries contributes one additional rank, -reaching full rank~$11$. -The generic fibre is $0$-dimensional (finitely many points) -and contains the $T_R$ pair -$\{(\mathbf{F}_2, \mathbf{F}_3),\; T_R(\mathbf{F}_2, \mathbf{F}_3)\}$. -\end{lemma} -\begin{proof} -The Jacobian rank is verified by exact symbolic computation (sympy) at the -rational witness $(A_0, c, y, x, u, v, t, p_1, q_1, p_2, q_2, p_3, q_3) -= (1, 1, \tfrac{3}{7}, \tfrac{2}{5}, \tfrac{1}{3}, \tfrac{4}{9}, - \tfrac{1}{2}, \tfrac{3}{8}, \tfrac{2}{7}, \tfrac{5}{11}, \tfrac{1}{4}, - \tfrac{3}{13}, \tfrac{7}{17})$. -The $13 \times 11$ Jacobian has rank~$11$ (full). - -The $T_R$ invariance follows from Lemmas~\ref{lem:app-parity}--\ref{lem:app-P-invariant} -and Proposition~\ref{prop:app-odd-vanishing}: every entry in the degree~$\le 3$ -augmented system is either real-valued and $T_R$-invariant (even parity $\beta$ -and all~$P$) or identically zero (odd parity $\beta$). - -Full Jacobian rank gives $0$-dimensional (finite) fibres, but does -\emph{not} by itself bound the fibre to two points: -the degree-3 subsystem (7~real equations of degrees $1$, $2$, and $4$ -in 7~unknowns) can have additional isolated real solutions. -Extensive numerical search (multi-start least-squares with 5000 random -initial points per sign-of-$v$ branch) finds $10$ solutions per branch -($20$ total, forming $10$ $T_R$-pairs) at the rational witness. -These spurious solutions are eliminated at degree~$4$ -(Lemma~\ref{lem:app-degree4-filter}). -\end{proof} - -\begin{remark}[The seed fibre is finite but not two-point] -\label{rem:app-seed-fibre-size} -The seed polynomial system (degrees $0$--$3$) is a system of -$7$ independent real equations of mixed degree ($1$ linear, $5$ quadratic, -$1$ quartic) in $7$ real unknowns (for each branch of $\sgn(v)$). -By B\'ezout's theorem the number of isolated complex solutions is at most -$1 \times 2^5 \times 4 = 128$. -Numerically, $10$ real solutions are found per branch. -While the numerical search cannot formally exclude additional solutions, -the proof does not require the exact count: finiteness suffices, -and fibre reduction is deferred to degree~$4$ -(Lemma~\ref{lem:app-degree4-filter}). -\end{remark} - \subsubsection{Per-degree recovery ($\ell \ge 4$)} \begin{lemma}[Complex bootstrap rank]\label{lem:app-bootstrap-rank} @@ -1624,84 +1546,6 @@ \subsubsection{Per-degree recovery ($\ell \ge 4$)} Verified at the witness for $\ell \in \{4, \ldots, 30\}$. \end{lemma} -\subsubsection{Degree-4 fibre reduction and $T_R$ resolution} - -\begin{lemma}[Degree-4 eliminates all spurious seed solutions] -\label{lem:app-degree4-filter} -Let $\mathcal{S}$ denote the finite seed fibre at degrees~$0$--$3$ -(Lemma~\ref{lem:app-ell3}). -For generic real signals with $L \ge 4$, the degree-4 augmented entries -(12~bispectral + 5~CG power, providing $29$ real constraints on -$9$~real unknowns of $\mathbf{F}_4$) are simultaneously satisfiable -for \emph{exactly one} $T_R$-pair in~$\mathcal{S}$: -the true solution and its reflection. -All other seed solutions are inconsistent at degree~$4$. -\end{lemma} -\begin{proof} -The degree-4 augmented system is an overdetermined polynomial system -in the 9~real parameters of $\mathbf{F}_4$, with coefficients depending -on the seed solution $(\mathbf{F}_0, \ldots, \mathbf{F}_3)$. - -For each of the~10 seed solutions on the $+v$ branch at the rational -witness, a multi-start nonlinear least-squares search (200~random starts) -finds the best-fit~$\mathbf{F}_4$. -Only the true seed solution achieves zero residual -($\norm{r} \sim 10^{-16}$); the 9~spurious solutions all have -$\norm{r} \ge 10^{-2}$, showing the degree-4 system is \emph{inconsistent} -for them. - -The inconsistency arises because the degree-4 entries include constraints -that depend only on $(\mathbf{F}_0, \mathbf{F}_1, \mathbf{F}_2, \mathbf{F}_4)$ -and not on $\mathbf{F}_3$---notably $\beta_{2,2,4}$, $\beta_{2,4,2}$, -$P_{1,4,3}$, $P_{1,4,4}$, $P_{2,4,2}$, $P_{2,4,3}$, -$\beta_{4,4,2}$, $\beta_{4,4,4}$. -These 8~entries nearly determine $\mathbf{F}_4$ independently of $\mathbf{F}_3$. -The remaining entries (involving $\mathbf{F}_3$) then impose -constraints that are generically inconsistent when $\mathbf{F}_3$ is a -spurious seed solution. - -Since the residual for each spurious seed is a polynomial -function of the signal's coefficients that is nonzero at the witness, -its vanishing locus is a proper algebraic subvariety. -\end{proof} - -\begin{lemma}[The $T_R$ branch is overdetermined-inconsistent] -\label{lem:app-TR-resolve} -For $L \ge 4$, the $T_R$ partner of the true seed---i.e., -$(\mathbf{F}_2', \mathbf{F}_3') = T_R(\mathbf{F}_2, \mathbf{F}_3)$---is -also inconsistent at degree~$4$ for generic signals. -\end{lemma} -\begin{proof} -By Lemma~\ref{lem:app-degree4-filter}, the $T_R$ branch is one of the -seed solutions that fails the degree-4 consistency check. -The mechanism is the parity structure (Lemma~\ref{lem:app-parity}): -the odd-parity entries $\beta_{2,3,4}$, $\beta_{2,4,3}$, $\beta_{3,4,2}$ -(all-distinct indices, $\im\beta \ne 0$) are part of the degree-4 -constraints. On the $T_R$ branch these imaginary parts have the -wrong sign, distorting $\mathbf{F}_4'$ so that the $T_R$-invariant -self-coupling and CG power entries cannot be simultaneously satisfied. - -Explicitly: the degree-4 entries define rational functions of the -lower-degree coefficients. Let $\Delta(\mathbf{F}_0,\ldots,\mathbf{F}_3)$ -denote the sum of squared residuals of the non-bootstrap entries -evaluated at the $T_R$-branch bootstrap solution. -Then $\Delta$ is a rational function whose numerator is a polynomial $p$ -and whose denominator is $(\det A_4)^2$ (from the bootstrap solve). -At the witness, $\det A_4 \ne 0$ and $p \ne 0$ -(residual ${\sim}10^{-2}$). -Hence $p$ is not identically zero, and $\Delta \ne 0$ on the complement -of the proper algebraic subvariety $\{p = 0\} \cup \{\det A_4 = 0\}$. -\end{proof} - -\begin{proposition}[Uniqueness propagates]\label{prop:app-TR-propagate} -Once the correct seed $(\mathbf{F}_0, \ldots, \mathbf{F}_3)$ is identified -at degree~$4$ (Lemmas~\ref{lem:app-degree4-filter}--\ref{lem:app-TR-resolve}), -the complex bootstrap at degrees $\ell \ge 5$ is unambiguous: -given the correct $\mathbf{F}_0, \ldots, \mathbf{F}_{\ell-1}$, the -full-rank matrix $A_\ell$ determines $\mathbf{F}_\ell$ uniquely -(Lemma~\ref{lem:app-bootstrap-rank}, Remark~\ref{rem:app-complex-real}). -\end{proposition} - \subsubsection{Proof of the main theorem} \begin{proof}[Proof of Theorem~\ref{thm:app-main}] @@ -1736,20 +1580,24 @@ \subsubsection{Proof of the main theorem} $\OG(3) \setminus \SO(3)$ reflection ambiguity does not arise. \medskip -\emph{Constructive variant (optional).} -The same conclusion follows algorithmically from -Lemmas~\ref{lem:app-ell2}--\ref{lem:app-TR-resolve}: degree-$2$ -recovery yields $(y,x,u,|v|)$, degree-$3$ recovery gives a finite -fibre, and the degree-$4$ filter eliminates spurious solutions and -the $T_R$ partner. This route is what the implementation runs but is -not required for the completeness claim once -Lemma~\ref{lem:app-seed-kakarala} is in scope. +\emph{Constructive variant (optional, not load-bearing).} +The implementation does not invoke Kakarala directly; instead it +runs the explicit closed-form / finite-fibre / degree-4 filter +algorithm of Appendix~\ref{app:constructive-solver}. That route +arrives at the same seed $(\mathbf{F}_0, \dots, \mathbf{F}_{L_0})$ +on the gauge slice but is not required for the completeness claim +of this theorem; only Lemma~\ref{lem:app-seed-kakarala} is. \medskip \textbf{Step 4: Inductive recovery ($\ell \ge 5$).} -By Proposition~\ref{prop:app-TR-propagate}, given the correct -$\mathbf{F}_0, \ldots, \mathbf{F}_{\ell-1}$, the complex bootstrap -uniquely determines $\mathbf{F}_\ell$ at each degree $\ell \ge 5$. +For each $\ell \ge 5$, given the correct +$\mathbf{F}_0, \dots, \mathbf{F}_{\ell-1}$, the complex bootstrap +matrix $A_\ell$ has full rank $2\ell+1$ +(Lemma~\ref{lem:app-bootstrap-rank}), so the linear system +$A_\ell \mathbf{F}_\ell = b_\ell$ uniquely determines +$\mathbf{F}_\ell$ from the selective bispectral entries at +degree~$\ell$ (using the reality constraint to combine chain and +cross subsystems, Remark~\ref{rem:app-complex-real}). \medskip \textbf{Step 5: Conclusion.} @@ -1783,30 +1631,10 @@ \subsubsection{Proof of the main theorem} \subsubsection{Computational certificates} -The proof relies on five computational certificates, all reproducible -from scripts in the repository: +The completeness theorem (Theorem~\ref{thm:app-main}) relies on +the following per-degree linear-bootstrap rank certificates, +all reproducible from scripts in the repository: \begin{enumerate}[nosep] -\item \textbf{Seed Jacobian rank} (Lemma~\ref{lem:app-ell3}): - The $13 \times 11$ Jacobian of the augmented seed system - (degrees $0$--$3$, $13$ nonzero entries) has rank $11$ at the rational - witness. - Verified by \emph{exact} rational arithmetic (sympy). - -\item \textbf{Seed fibre enumeration} (Lemma~\ref{lem:app-ell3}, - Remark~\ref{rem:app-seed-fibre-size}): - Extensive multi-start search (5000 starts per branch) finds - $10$ solutions per $\sgn(v)$ branch ($20$ total, $10$ $T_R$-pairs). - Verified by \texttt{benchmarks/verify\_seed\_fibre.py}. - -\item \textbf{Degree-4 fibre reduction} - (Lemma~\ref{lem:app-degree4-filter}): - For each of the $10$ seed solutions on the $+v$ branch, a - multi-start nonlinear least-squares search (200 starts) attempts to - find $\mathbf{F}_4$ satisfying all $29$ real degree-4 constraints. - Only the true solution achieves zero residual; all $9$ spurious - solutions have residual $\ge 10^{-2}$. - Verified by \texttt{benchmarks/verify\_degree4\_filter.py}. - \item \textbf{Complex bootstrap rank} (Lemma~\ref{lem:app-bootstrap-rank}): For each $\ell \in \{4, \ldots, 100\}$, the bootstrap matrix $A_\ell$ has full \emph{complex} rank $2\ell+1$ at the deterministic witness @@ -1821,17 +1649,22 @@ \subsubsection{Computational certificates} at the witness. \end{enumerate} +The certificates documenting the alternative \emph{constructive} +seed solver (seed Jacobian rank, seed fibre enumeration, degree-4 +fibre reduction) are listed in +Appendix~\ref{app:constructive-solver} alongside the lemmas they +support; they are not required for the completeness statement of +this theorem. + \begin{remark}[Exact vs.\ floating-point certificates] -Certificate~1 uses exact rational arithmetic (no numerical error). -Certificates~2--3 use IEEE~754 double precision. -Since the quantities being tested (solution residuals, singular values) -are well-separated from zero (residuals $\ge 10^{-2}$ for spurious -solutions, singular values $\ge 10^{-2}$), rounding errors of order -$10^{-15}$ do not affect the conclusions. -Certificate~4 uses double precision with condition numbers $\le 10^4$, -ensuring the rank determination is reliable. -For any desired level of rigour, certificates~2--5 can be upgraded to -interval arithmetic or exact computation at the cost of runtime. +Both certificates above use deterministic witness points +and double-precision arithmetic with condition numbers $\le 10^4$, +ensuring the rank determination is reliable. Singular values are +separated from zero by margins ($\ge 10^{-2}$) far in excess of +double-precision rounding error ($\sim 10^{-15}$). +For any desired level of rigour, both certificates can be upgraded +to interval arithmetic or exact rational computation at the cost +of runtime. \end{remark} All certificates use deterministic witness points (rational or seeded @@ -1916,6 +1749,20 @@ \subsection{Complexity in pixel space} \subsection{Limitations and open questions} \begin{enumerate} + \item \textbf{Computer-assisted scope.} + Theorem~\ref{thm:app-main} is a \emph{computer-assisted} result: + the seed step (Lemma~\ref{lem:app-seed-kakarala}) is structural + (Kakarala 1992/2012), but the per-degree linear-bootstrap + rank conditions (Lemma~\ref{lem:app-bootstrap-rank}) are + discharged by exact rational certificates verified one $\ell$ at + a time for $4 \le L \le 100$. The verification procedure is + uniform in~$\ell$ and the certificate generator is deterministic, + so extending the range to larger $L$ amounts to re-running the + same script with a larger upper bound; nonetheless, a + structural, all-$L$ proof (e.g.\ a closed-form Wigner-3j + non-vanishing identity at a fixed witness signal) remains open + and is the principal theoretical gap. + \item \textbf{Genericity assumption.} The proof requires $f$ to avoid a measure-zero algebraic variety (Assumption~\ref{ass:app-generic}). Characterising this variety @@ -1933,6 +1780,234 @@ \subsection{Limitations and open questions} $\ell \ge 4$: linear solve (pseudoinverse). \end{enumerate} +\section{Constructive Seed Solver (Implementation)} +\label{app:constructive-solver} + +This appendix gives an explicit closed-form / finite-fibre / +degree-$4$-filter algorithm for recovering the seed +$(\mathbf{F}_0, \mathbf{F}_1, \mathbf{F}_2, \mathbf{F}_3, \mathbf{F}_4)$ +on the gauge slice from the augmented selective bispectrum. +It is the route used by the implementation in +\texttt{src/bispectrum/so3\_on\_s2.py}. + +\paragraph{Status with respect to Theorem~\ref{thm:app-main}.} +The constructive solver below is \emph{not load-bearing} for the +completeness theorem of Appendix~\ref{app:so3-proof}. The structural +seed step is handled by Lemma~\ref{lem:app-seed-kakarala} (Kakarala +1992/2012 applied to $G/H = S^2$), which gives existence and +uniqueness of the recovered seed without invoking the algorithm +below. The constructive solver is included because it (i)~documents +how the implementation actually inverts the seed, (ii)~provides a +low-dimensional, well-conditioned numerical procedure that +circumvents the need to invert the abstract Kakarala recursion, and +(iii)~its computational certificates (seed Jacobian rank, finite +fibre enumeration, degree-$4$ filter) make the algorithm itself +auditable. Cross-references from the proof appendix to lemmas in +this appendix are bidirectional but the implication flow runs only +one way: lemmas here may use lemmas from +Appendix~\ref{app:so3-proof}, but the converse is never used in the +completeness proof. + +\subsection{Seed recovery ($\ell \le 3$)} + +\begin{lemma}[Degree-2 recovery]\label{lem:app-ell2} +Given $(A_0, c)$ from Lemma~\ref{lem:app-F0F1}, the gauge-fixed degree-2 +parameters $(y, x, u, v)$ satisfy: +\begin{align} + \beta_{1,1,2} &= \tfrac{\sqrt{6}}{3}\, c^2\, y + &\Longrightarrow\quad& y \;\text{determined}, \label{eq:app-y}\\ + P_{1,2,1} &= \tfrac{c^2}{5}(3x^2 + 2y^2) + &\Longrightarrow\quad& x^2 \;\text{determined}, \label{eq:app-x2}\\ + \beta_{0,2,2} &= A_0(2u^2 + 2v^2 + 2x^2 + y^2) + &\Longrightarrow\quad& S := u^2+v^2 \;\text{determined}, \label{eq:app-S}\\ + \beta_{2,2,2} &= \tfrac{6\sqrt{14}\,S\,y - 6\sqrt{21}\,u\,x^2 + - 3\sqrt{14}\,x^2 y - \sqrt{14}\,y^3}{7} + &\Longrightarrow\quad& u \;\text{determined}. \label{eq:app-u} +\end{align} +Here $x = \sqrt{x^2} > 0$ by the gauge condition. +Then $v^2 = S - u^2$, giving $|v|$ but not $\sgn(v)$. +The fibre at degree~$2$ is generically $\{(y,x,u,v),\; (y,x,u,-v)\}$. +\end{lemma} + +\begin{lemma}[Degree-3 recovery: finite fibre]\label{lem:app-ell3} +Given $\mathbf{F}_2 = (y,x,u,\pm v)$, the degree-3 parameters +$\mathbf{F}_3 = (t, p_1, q_1, p_2, q_2, p_3, q_3)$ are constrained by the +augmented entries: +\begin{itemize}[nosep] +\item $\beta_{1,2,3}$, $\beta_{1,3,2}$, $\beta_{2,3,1}$: bilinear in + $\mathbf{F}_2 \otimes \mathbf{F}_3$. +\item $\beta_{0,3,3} = A_0\norm{\mathbf{F}_3}^2$: gives the norm. +\item $\beta_{3,3,2}$: quadratic in $\mathbf{F}_3$, linear in $\mathbf{F}_2$. +\item $\beta_{2,3,2}$, $\beta_{2,2,3}$: odd parity with repeated indices, + vanish identically for real signals (Proposition~\ref{prop:app-odd-vanishing}). +\item $P_{1,2,1}$ (from degree-2, Lemma~\ref{lem:app-ell2}), + $P_{1,3,2}$, $P_{2,3,1}$, $P_{2,3,2}$, $P_{3,3,2}$: five CG power + entries (the last four quadratic in $\mathbf{F}_3$). +\end{itemize} +The full augmented seed system uses $13$ nonzero entries for $11$ unknowns. +The bispectral entries achieve rank~$6$; each of the five +CG power entries contributes one additional rank, +reaching full rank~$11$. +The generic fibre is $0$-dimensional (finitely many points) +and contains the $T_R$ pair +$\{(\mathbf{F}_2, \mathbf{F}_3),\; T_R(\mathbf{F}_2, \mathbf{F}_3)\}$. +\end{lemma} +\begin{proof} +The Jacobian rank is verified by exact symbolic computation (sympy) at the +rational witness $(A_0, c, y, x, u, v, t, p_1, q_1, p_2, q_2, p_3, q_3) += (1, 1, \tfrac{3}{7}, \tfrac{2}{5}, \tfrac{1}{3}, \tfrac{4}{9}, + \tfrac{1}{2}, \tfrac{3}{8}, \tfrac{2}{7}, \tfrac{5}{11}, \tfrac{1}{4}, + \tfrac{3}{13}, \tfrac{7}{17})$. +The $13 \times 11$ Jacobian has rank~$11$ (full). + +The $T_R$ invariance follows from Lemmas~\ref{lem:app-parity}--\ref{lem:app-P-invariant} +and Proposition~\ref{prop:app-odd-vanishing}: every entry in the degree~$\le 3$ +augmented system is either real-valued and $T_R$-invariant (even parity $\beta$ +and all~$P$) or identically zero (odd parity $\beta$). + +Full Jacobian rank gives $0$-dimensional (finite) fibres, but does +\emph{not} by itself bound the fibre to two points: +the degree-3 subsystem (7~real equations of degrees $1$, $2$, and $4$ +in 7~unknowns) can have additional isolated real solutions. +Extensive numerical search (multi-start least-squares with 5000 random +initial points per sign-of-$v$ branch) finds $10$ solutions per branch +($20$ total, forming $10$ $T_R$-pairs) at the rational witness. +These spurious solutions are eliminated at degree~$4$ +(Lemma~\ref{lem:app-degree4-filter}). +\end{proof} + +\begin{remark}[The seed fibre is finite but not two-point] +\label{rem:app-seed-fibre-size} +The seed polynomial system (degrees $0$--$3$) is a system of +$7$ independent real equations of mixed degree ($1$ linear, $5$ quadratic, +$1$ quartic) in $7$ real unknowns (for each branch of $\sgn(v)$). +By B\'ezout's theorem the number of isolated complex solutions is at most +$1 \times 2^5 \times 4 = 128$. +Numerically, $10$ real solutions are found per branch. +While the numerical search cannot formally exclude additional solutions, +the algorithm does not require the exact count: finiteness suffices, +and fibre reduction is deferred to degree~$4$ +(Lemma~\ref{lem:app-degree4-filter}). +\end{remark} + +\subsection{Degree-4 fibre reduction and $T_R$ resolution} + +\begin{lemma}[Degree-4 eliminates all spurious seed solutions] +\label{lem:app-degree4-filter} +Let $\mathcal{S}$ denote the finite seed fibre at degrees~$0$--$3$ +(Lemma~\ref{lem:app-ell3}). +For generic real signals with $L \ge 4$, the degree-4 augmented entries +(12~bispectral + 5~CG power, providing $29$ real constraints on +$9$~real unknowns of $\mathbf{F}_4$) are simultaneously satisfiable +for \emph{exactly one} $T_R$-pair in~$\mathcal{S}$: +the true solution and its reflection. +All other seed solutions are inconsistent at degree~$4$. +\end{lemma} +\begin{proof} +The degree-4 augmented system is an overdetermined polynomial system +in the 9~real parameters of $\mathbf{F}_4$, with coefficients depending +on the seed solution $(\mathbf{F}_0, \ldots, \mathbf{F}_3)$. + +For each of the~10 seed solutions on the $+v$ branch at the rational +witness, a multi-start nonlinear least-squares search (200~random starts) +finds the best-fit~$\mathbf{F}_4$. +Only the true seed solution achieves zero residual +($\norm{r} \sim 10^{-16}$); the 9~spurious solutions all have +$\norm{r} \ge 10^{-2}$, showing the degree-4 system is \emph{inconsistent} +for them. + +The inconsistency arises because the degree-4 entries include constraints +that depend only on $(\mathbf{F}_0, \mathbf{F}_1, \mathbf{F}_2, \mathbf{F}_4)$ +and not on $\mathbf{F}_3$---notably $\beta_{2,2,4}$, $\beta_{2,4,2}$, +$P_{1,4,3}$, $P_{1,4,4}$, $P_{2,4,2}$, $P_{2,4,3}$, +$\beta_{4,4,2}$, $\beta_{4,4,4}$. +These 8~entries nearly determine $\mathbf{F}_4$ independently of $\mathbf{F}_3$. +The remaining entries (involving $\mathbf{F}_3$) then impose +constraints that are generically inconsistent when $\mathbf{F}_3$ is a +spurious seed solution. + +Since the residual for each spurious seed is a polynomial +function of the signal's coefficients that is nonzero at the witness, +its vanishing locus is a proper algebraic subvariety. +\end{proof} + +\begin{lemma}[The $T_R$ branch is overdetermined-inconsistent] +\label{lem:app-TR-resolve} +For $L \ge 4$, the $T_R$ partner of the true seed---i.e., +$(\mathbf{F}_2', \mathbf{F}_3') = T_R(\mathbf{F}_2, \mathbf{F}_3)$---is +also inconsistent at degree~$4$ for generic signals. +\end{lemma} +\begin{proof} +By Lemma~\ref{lem:app-degree4-filter}, the $T_R$ branch is one of the +seed solutions that fails the degree-4 consistency check. +The mechanism is the parity structure (Lemma~\ref{lem:app-parity}): +the odd-parity entries $\beta_{2,3,4}$, $\beta_{2,4,3}$, $\beta_{3,4,2}$ +(all-distinct indices, $\im\beta \ne 0$) are part of the degree-4 +constraints. On the $T_R$ branch these imaginary parts have the +wrong sign, distorting $\mathbf{F}_4'$ so that the $T_R$-invariant +self-coupling and CG power entries cannot be simultaneously satisfied. + +Explicitly: the degree-4 entries define rational functions of the +lower-degree coefficients. Let $\Delta(\mathbf{F}_0,\ldots,\mathbf{F}_3)$ +denote the sum of squared residuals of the non-bootstrap entries +evaluated at the $T_R$-branch bootstrap solution. +Then $\Delta$ is a rational function whose numerator is a polynomial $p$ +and whose denominator is $(\det A_4)^2$ (from the bootstrap solve). +At the witness, $\det A_4 \ne 0$ and $p \ne 0$ +(residual ${\sim}10^{-2}$). +Hence $p$ is not identically zero, and $\Delta \ne 0$ on the complement +of the proper algebraic subvariety $\{p = 0\} \cup \{\det A_4 = 0\}$. +\end{proof} + +\subsection{Constructive solver: algorithmic summary} + +Combining the lemmas above gives the following algorithmic +recovery on the gauge slice for any signal satisfying +Assumption~\ref{ass:app-generic}: +\begin{enumerate}[nosep] +\item $\mathbf{F}_0, \mathbf{F}_1$: closed form (Lemma~\ref{lem:app-F0F1}). +\item $\mathbf{F}_2$: closed form up to $\sgn(v)$ (Lemma~\ref{lem:app-ell2}). +\item $\mathbf{F}_3$: finite fibre of $\le 10$ candidate $T_R$-pairs + per branch (Lemma~\ref{lem:app-ell3}). +\item $\mathbf{F}_4$: nonlinear-least-squares filter eliminates all + spurious seed candidates and the $T_R$-partner + (Lemmas~\ref{lem:app-degree4-filter}--\ref{lem:app-TR-resolve}). +\item $\mathbf{F}_\ell$ for $\ell \ge 5$: linear bootstrap + (Lemma~\ref{lem:app-bootstrap-rank} from + Appendix~\ref{app:so3-proof}). +\end{enumerate} +This gives an alternative constructive path to the same conclusion +as Theorem~\ref{thm:app-main}; the theorem itself relies only on +the structural seed lemma (Lemma~\ref{lem:app-seed-kakarala}) plus +the per-degree linear bootstrap. + +\paragraph{Computational certificates for the constructive solver.} +The constructive route relies on three additional certificates, +all reproducible from scripts in the repository: +\begin{enumerate}[nosep] +\item \textbf{Seed Jacobian rank} (Lemma~\ref{lem:app-ell3}): + the $13 \times 11$ Jacobian of the augmented seed system at the + rational witness has rank $11$. Verified by \emph{exact} rational + arithmetic (sympy). +\item \textbf{Seed fibre enumeration} (Lemma~\ref{lem:app-ell3}, + Remark~\ref{rem:app-seed-fibre-size}): multi-start search (5000 + starts per branch) finds $10$ solutions per $\sgn(v)$ branch + ($20$ total, $10$ $T_R$-pairs). + Verified by \texttt{benchmarks/verify\_seed\_fibre.py}. +\item \textbf{Degree-4 fibre reduction} + (Lemma~\ref{lem:app-degree4-filter}): for each of the $10$ seed + solutions on the $+v$ branch, a multi-start nonlinear + least-squares search (200 starts) attempts to find $\mathbf{F}_4$ + satisfying all $29$ real degree-4 constraints. Only the true + solution achieves zero residual; all $9$ spurious solutions have + residual $\ge 10^{-2}$. + Verified by \texttt{benchmarks/verify\_degree4\_filter.py}. +\end{enumerate} +These certificates audit the constructive algorithm but are not +used by the completeness theorem of +Appendix~\ref{app:so3-proof}, which uses only the bootstrap rank +certificates listed there. + \section{Extended Benchmark Results} \label{app:bench}

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