Rational isotypic projectors; symmetry does not force all active frequencies equal. Representation and multiplicity classification. Physical promotion 0. Current session checkpoint.Markdown source · LaTeX source · Complete package1. Scope and objects This note classifies the signed permutation action of S4S_4 on the real nonaugmented chains of the filled tetrahedron Δ3\Delta^3. It gives exact rational projectors and separates symmetry-enforced multiplicity from the equal frequencies of the canonical boundary generator. This is a finite mathematical classification, with no independent novelty claim, new dynamics selection, or physical interpretation. It neither changes the PSR freeze nor advances the LC queue. Order the oriented simplex basis by degree, then lexicographically: (0,1,2,3;01,02,03,12,13,23;012,013,023,123;0123). (0,1,2,3;\ 01,02,03,12,13,23;\ 012,013,023,123;\ 0123). Orient every simplex by its increasing vertex order. The boundary is Bk:Ck→Ck−1B_k:C_k\to C_{k-1}; write D=B1+B2+B3D=B_1+B_2+B_3 for its total block matrix, and A=DT−D,L=−A2=DDT+DTD. A=D^T-D,\qquad L=-A^2=DD^T+D^TD. The inner product makes this simplex basis orthonormal. For g∈S4g\in S_4, the matrix R(g)R(g) permutes simplex labels and multiplies each column by the sign required to reorder its image increasingly. These orientation signs are essential. The matrices satisfy R(g)TR(g)=I,DR(g)=R(g)D. R(g)^TR(g)=I,\qquad DR(g)=R(g)D. 2. Exact character decomposition Let VV be the standard three-dimensional representation and ε\varepsilon the sign representation. Denote V⊗εV\otimes\varepsilon by VεV^\varepsilon. The permutation representation on four vertices is 𝟏⊕V\mathbf1\oplus V; oriented kk-simplices carry its (k+1)(k+1)-st exterior power. Therefore C0=𝟏⊕V,C1=V⊕Vε,C2=Vε⊕ε,C3=ε. C_0=\mathbf1\oplus V,\quad C_1=V\oplus V^\varepsilon, \quad C_2=V^\varepsilon\oplus\varepsilon,\quad C_3=\varepsilon. The following characters independently verify the decomposition. Class sizes are 1,6,3,8,61,6,3,8,6. Character 11 (12)(12) (12)(34)(12)(34) (123)(123) (1234)(1234) 𝟏\mathbf1 1 1 1 1 1 VV 3 1 -1 0 -1 VεV^\varepsilon 3 -1 -1 0 1 ε\varepsilon 1 -1 1 1 -1 Unused two-dimensional irrep WW 2 0 2 -1 0 C0C_0 4 2 0 1 0 C1C_1 6 0 -2 0 0 C2C_2 4 -2 0 1 0 C3C_3 1 -1 1 1 -1 The weighted class inner product, divided by 2424, gives multiplicity one for each displayed constituent and zero for the others. On the full chain space, C•≅𝟏⊕2V⊕2Vε⊕2ε. C_\bullet\cong\mathbf1\oplus2V\oplus2V^\varepsilon\oplus2\varepsilon. Here “2” counts copies; it is distinct from the irreducible dimension three or one. 3. Exact equivariant projectors For an irreducible character χ\chi of dimension dχd_\chi, the central projector is Pχ=dχ24∑g∈S4χ(g−1)R(g). P_\chi=\frac{d_\chi}{24}\sum_{g\in S_4}\chi(g^{-1})R(g). All entries are rational. For this action the two-dimensional projector is zero. With J4J_4 the all-ones matrix, the four nonzero projectors have the explicit degree-block formulas P𝟏=diag(J4/4,06,04,01), P_{\mathbf1}=\operatorname{diag}(J_4/4,0_6,0_4,0_1), PV=diag(I4−J4/4,B1TB1/4,04,01), P_V=\operatorname{diag}(I_4-J_4/4,\ B_1^TB_1/4,\ 0_4,0_1), PVε=diag(04,B2B2T/4,B2TB2/4,01), P_{V^\varepsilon}=\operatorname{diag}(0_4,\ B_2B_2^T/4,\ B_2^TB_2/4,\ 0_1), Pε=diag(04,06,B3B3T/4,1). P_\varepsilon=\operatorname{diag}(0_4,0_6,\ B_3B_3^T/4,\ 1). They have ranks 1,6,6,21,6,6,2, respectively, and satisfy PχT=Pχ,Pχ2=Pχ,PχPψ=0(χ≠ψ),∑Pχ=I,[Pχ,R(g)]=[Pχ,A]=0. P_\chi^T=P_\chi,\quad P_\chi^2=P_\chi,\quad P_\chi P_\psi=0\ (\chi\ne\psi),\quad \sum P_\chi=I,\quad [P_\chi,R(g)]=[P_\chi,A]=0. The archive supplies each full 15×1515\times15 matrix. These are isotypic projectors: PVP_V, for example, selects both copies of VV. To select its copy in one chain degree, multiply by that degree’s coordinate projector. Degree restriction alone need not commute with AA. 4. Multiplicity and unit-weight spectrum Chain degree Representation Copies in this degree Dimension of each irrep Canonical LL eigenvalue C0C_0 𝟏\mathbf1 1 1 0 C0C_0 VV 1 3 4 C1C_1 VV 1 3 4 C1C_1 VεV^\varepsilon 1 3 4 C2C_2 VεV^\varepsilon 1 3 4 C2C_2 ε\varepsilon 1 1 4 C3C_3 ε\varepsilon 1 1 4 For the specified generator, A2=−4(I−P𝟏),specL={0[1],4[14]},χA(λ)=λ(λ2+4)7. A^2=-4(I-P_{\mathbf1}),\quad \operatorname{spec}L=\{0^{[1]},4^{[14]}\},\quad \chi_A(\lambda)=\lambda(\lambda^2+4)^7. Each positive imaginary eigenvalue +2i+2i has multiplicity seven: three contributions from the VV pair, three from the VεV^\varepsilon pair, and one from the sign pair. Negative eigenvalues are their conjugates. The number seven is not a seven-dimensional irreducible representation of S4S_4. 5. Which equalities are forced by symmetry? Let AkA_k be the full 15×1515\times15 skew matrix with blocks −Bk-B_k from degree kk to k−1k-1 and BkTB_k^T in the reverse direction, zero elsewhere. Then A=A1+A2+A3A=A_1+A_2+A_3, and A12=−4PV,A22=−4PVε,A32=−4Pε,AiAj=0(i≠j). A_1^2=-4P_V,\quad A_2^2=-4P_{V^\varepsilon},\quad A_3^2=-4P_\varepsilon,\qquad A_iA_j=0\quad(i\ne j). Classification. Every real skew-symmetric S4S_4-equivariant endomorphism of this full chain space has the form A(a,b,c)=aA1+bA2+cA3,a,b,c∈ℝ. A(a,b,c)=aA_1+bA_2+cA_3,\qquad a,b,c\in\mathbb R. Proof. The four appearing irreducible types are absolutely irreducible real representations. Equivariant maps cannot mix inequivalent types. On the single trivial copy, a skew scalar is zero. On each of the other isotypic components there are two equivalent copies, so the equivariant commutant acts on a two-dimensional multiplicity space. Its skew part is one-dimensional. The three nonzero, mutually supported equivariant matrices AkA_k span those three skew lines. This gives the complete classification, not just a list of examples. ▫\square Consequently, A(a,b,c)2=−4(a2PV+b2PVε+c2Pε), A(a,b,c)^2=-4(a^2P_V+b^2P_{V^\varepsilon}+c^2P_\varepsilon), χA(a,b,c)(λ)=λ(λ2+4a2)3(λ2+4b2)3(λ2+4c2). \boxed{\chi_{A(a,b,c)}(\lambda)= \lambda(\lambda^2+4a^2)^3(\lambda^2+4b^2)^3(\lambda^2+4c^2).} Zero coefficients give additional zero modes; the formula includes that case. Symmetry enforces the threefold multiplicity within each three-dimensional type. It does not force |a|=|b|=|c||a|=|b|=|c|. The canonical choice a=b=c=1a=b=c=1 comes from the declared signed boundary and its unit-weight adjoint. The unit inner product alone does not select that generator. As an exact counterexample to symmetry forcing a single active frequency, (a,b,c)=(1,2,3)(a,b,c)=(1,2,3) remains equivariant and skew under the same unit metric, but has positive squared frequencies 4,16,364,16,36. These are classification variables and a mathematical witness, not newly adopted UD couplings or fit parameters. No alternative generator is selected. 6. Verification and status verify_exact.py constructs all 24 signed permutation matrices, checks boundary equivariance and orthogonality, evaluates the rational character sums, and verifies symmetry, idempotence, pairwise orthogonality, completeness, ranks, and equivariance of the projectors. It also checks the boundary-block formulas, cross-product zeros, and the unequal-frequency witness. Basis order is fixed above and in MATRIX_CATALOG.json. This note is the second item in the requested mathematical order. It preserves the existing generator, PSR freeze, and experimental queue. The companion PSR addendum and Smith note address separate finite objects. No independent review or physical promotion is claimed.