Theorem: Equivariant contact assignments

Write ${\mathrm{Hom}}_G({\mathcal{H}}_{4+N},W_N)$ for $\mathfrak{so}(1,3)$-equivariant contact maps at grade $N$.

1. Abstract count. ${\mathrm{Poly}}_N={{⨁}_{k=0}}^{\lfloor N/2\rfloor }{H}_{N-2k}$ with $\dim H_m=(m+1{)}^2$, $W_N\cong {\mathrm{Poly}}_{\le N}$ as representations (via the intertwiner of (LC2)), and

   $$\dim {\mathrm{Hom}}_G({\mathcal{H}}_{4+N},W_N)=\frac{(\lfloor N/2\rfloor +1)(\lfloor N/2\rfloor +2)}{2}.$$

   Explicit basis: $C_{k,j}=P\circ {\phi }_{k,j}$ with ${\phi }_{k,j}:p\mapsto Q^j{\square }^{k}p$, $k=0,\dots ,\lfloor N/2\rfloor$, $j=0,\dots ,k$; in words, $Q^j{\square }^{k}p$ read as a polynomial in ${\partial }^{\mu }$ acting on $\delta$.

2. Coordinate verification. Each $C_{k,j}$ satisfies $C{\rho }_N(X)={{\rho }^{\delta }}_N(X)C$ exactly for all six generators and all $N\le 4$. The uncorrected maps ${\phi }_{k,j}$ used previously fail exactly the three boost equations whenever $P$ acts nontrivially on their image, i.e.\ for $(N,k,j)\in \{(1,0,0),(2,0,0),(3,0,0),(3,1,0),(3,1,1),(4,0,0),(4,1,0),(4,1,1)\}$; the six maps whose image lies at $|\gamma |$ even with $P=+1$ ($N=0$; $(2,1,\cdot )$; $(4,2,\cdot )$) were unaffected. For $N\le 2$ the full constraint map was additionally solved by an exact verified kernel computation.

3. Graded compatibility matrix. Slots $=\dim {\mathcal{H}}_{4+N}\cdot \dim W_N$; the constraint map is $C\mapsto (C{\rho }_N(X)-{{\rho }^{\delta }}_N(X)C{)}_X$ with codomain ${⨁}_{6\text{ generators}}\mathrm{Hom}({\mathcal{H}}_{4+N},W_N)$ of dimension $6\cdot$slots;\ rank $=$ slots $-\dim$ sol.\ $=$ codimension of the solution space; coker $=6\cdot$slots $-$ rank.

   $N$	$\dim H$	$\dim W$	slots	rank (=codim)	$\dim$ codomain	coker	$\dim$ sol./block
   0	1	1	1	0	6	6	1 / K
   1	4	5	20	19	120	101	1 / K
   2	10	15	150	147	900	753	3 / K
   3	20	35	700	697	4200	3503	3 / ${\mathcal{C}}_{\mathrm{reg}}$
   4	35	70	2450	2444	14700	12256	6 / ${\mathcal{C}}_{\mathrm{reg}}$

   The cokernel is dominated by the redundancy of the six generator equations and is not, by itself, an obstruction count. Total equivariant freedom: $14=5$ (K-block, $N\le 2$)$+9$ (${\mathcal{C}}_{\mathrm{reg}}$, $N\ge 3$) (unchanged by the repair, since $P$ is an isomorphism).

4. Separation: (a) ${\mathcal{C}}_{\mathrm{reg}}$ solutions ($N\ge 3$) are strict $K$-regression-preserving ($\Delta {\mathfrak{R}}_{\mathrm{L}}{|}_{{\mathcal{V}}_{\mathrm{K}}}=0$); (b) K-block solutions ($N\le 2$) modify ${\mathfrak{R}}_{\mathrm{L}}{|}_{{\mathcal{V}}_{\mathrm{K}}}$ and break XVI-K regression if used.

5. The inhomogeneous ${\mathrm{Obs}}_L({\mathfrak{R}}_{\mathrm{L}})$ is Unresolved.

Proof. Schur count from the harmonic decomposition; linear independence of $\{C_{k,j}\}$; equivariance against ${\rho }^{\delta }$ on all six generators; $N\le 2$ exact kernel of the full constraint map (m4_common.block_kernel, verified on all rows). For $N=3,4$ the rank follows from rank–nullity with the abstract kernel dimension and the explicit verified basis. Certificates: covariance_matrix.txt, covariance_certificates.txt, covariance_summary.json, covariance_exact_certificates.json.