Original source and historical status labels retained. The current review qualifies model definitions, fixed-space memory, topology, storage and reproducibility. UD_PACHNER_23_HODGE_DIRAC_MOVE_RESPONSE_v0.1 Date: 2026-09-18 Status: EXACT INTERNAL MOVE-RESPONSE THEOREM + WEIGHTED-HODGE CONTROL FAMILY Physical promotion: 0 1. Local 2→3 move Use the common five-vertex ball. Two-tetrahedron side K−K_- K−={0123,0124}. K_-=\{0123,0124\}. Its f-vector is f(K−)=(5,9,7,2). \boxed{f(K_-)=(5,9,7,2)}. Total chain dimension: 5+9+7+2=23. 5+9+7+2=23. The unique internal triangular face is (012). (012). Three-tetrahedron side K+K_+ K+={0134,0234,1234}. K_+=\{0134,0234,1234\}. Its f-vector is f(K+)=(5,10,9,3). \boxed{f(K_+)=(5,10,9,3)}. Total chain dimension: 5+10+9+3=27. 5+10+9+3=27. The new internal edge is (34), (34), with internal triangles (034),(134),(234). (034),(134),(234). 2. Common boundary Both triangulations have exactly the same boundary triangles: (013),(014),(023),(024),(123),(124). \boxed{(013),(014),(023),(024),(123),(124)}. Thus the move changes only the interior triangulation. Euler characteristic is preserved: 5−9+7−2=1, 5-9+7-2=1, 5−10+9−3=1. 5-10+9-3=1. 3. Skew Hodge–Dirac operator With oriented boundary matrices B1:C1→C0,B2:C2→C1,B3:C3→C2, B_1:C_1\to C_0,\quad B_2:C_2\to C_1,\quad B_3:C_3\to C_2, define 𝒜K=(0B100−B1⊤0B200−B2⊤0B300−B3⊤0). \mathcal A_K= \begin{pmatrix} 0&B_1&0&0\\ -B_1^\top&0&B_2&0\\ 0&-B_2^\top&0&B_3\\ 0&0&-B_3^\top&0 \end{pmatrix}. 4. Exact unit-weight spectra For K−K_-, χ−(λ)=λ(λ2+3)4(λ2+5)7. \boxed{ \chi_-(\lambda) = \lambda(\lambda^2+3)^4(\lambda^2+5)^7 }. Hence spec(𝒜−)={0,(±i3)4,(±i5)7}. \boxed{ \operatorname{spec}(\mathcal A_-) = \left\{ 0, (\pm i\sqrt3)^4, (\pm i\sqrt5)^7 \right\}. } For K+K_+, χ+(λ)=λ(λ2+2)2(λ2+5)11. \boxed{ \chi_+(\lambda) = \lambda(\lambda^2+2)^2(\lambda^2+5)^{11} }. Hence spec(𝒜+)={0,(±i2)2,(±i5)11}. \boxed{ \operatorname{spec}(\mathcal A_+) = \left\{ 0, (\pm i\sqrt2)^2, (\pm i\sqrt5)^{11} \right\}. } AUTO_LOCK PACHNER_23_UNIT_HODGE_DIRAC_SPECTRAL_RESPONSE_THEOREM_v0.1 This identifies the previously separate 2323- and 2727-dimensional support spectra as the two sides of one 2↔︎3 Pachner move. 5. Degree-resolved Hodge Laplacians With Δk=(−𝒜K2)|Ck, \Delta_k=(-\mathcal A_K^2)|_{C_k}, the exact spectra are: K−K_- specΔ0={0,3,5,5,5}, \operatorname{spec}\Delta_0=\{0,3,5,5,5\}, specΔ1={3×3,5×6}, \operatorname{spec}\Delta_1=\{3^{\times3},5^{\times6}\}, specΔ2={3×3,5×4}, \operatorname{spec}\Delta_2=\{3^{\times3},5^{\times4}\}, specΔ3={3,5}. \operatorname{spec}\Delta_3=\{3,5\}. K+K_+ specΔ0={0,5×4}, \operatorname{spec}\Delta_0=\{0,5^{\times4}\}, specΔ1={2,5×9}, \operatorname{spec}\Delta_1=\{2,5^{\times9}\}, specΔ2={2×2,5×7}, \operatorname{spec}\Delta_2=\{2^{\times2},5^{\times7}\}, specΔ3={2,5×2}. \operatorname{spec}\Delta_3=\{2,5^{\times2}\}. Thus the move reallocates the nonzero Hodge spectrum across chain degree. 6. Exact spectral response The move removes the unit-weight 3\sqrt3 family and introduces a 2\sqrt2 family while increasing the 5\sqrt5 multiplicity: (±i3)4→(±i2)2, (\pm i\sqrt3)^4 \longrightarrow (\pm i\sqrt2)^2, (±i5)7→(±i5)11. (\pm i\sqrt5)^7 \longrightarrow (\pm i\sqrt5)^{11}. The quadratic spectral moment changes by 118−94=24=23⋅3. 118-94 = \boxed{24} = 2^3\cdot3. The quartic moment changes by 566−422=144=24⋅32. 566-422 = \boxed{144} = 2^4\cdot3^2. These are structural diagnostics, not action costs. AUTO_FREEZE DELTA_TRACE_A2_IS_PHYSICAL_EVENT_ENERGY. 7. Common ambient embedding The union simplex basis has f-vector (5,10,10,5) \boxed{(5,10,10,5)} and total dimension 3030. Embedding both local operators into this common basis gives Δ𝒜=𝒜+∪−𝒜−∪. \Delta\mathcal A = \mathcal A_+^\cup-\mathcal A_-^\cup. Exact isolated-patch rank: rank(Δ𝒜)=18. \boxed{ \operatorname{rank}(\Delta\mathcal A)=18. } For the squared Hodge operator on the same isolated ambient patch, rank[(−(𝒜+∪)2)−(−(𝒜−∪)2)]=19. \boxed{ \operatorname{rank} \left[ (-(\mathcal A_+^\cup)^2)-(-(\mathcal A_-^\cup)^2) \right] =19. } This is a finite-rank local move update on a common chain basis. 8. Positive weighted-Hodge control family Assign positive block weights a,b,c>0 a,b,c>0 to the three adjacent-degree incidence blocks: B1↦aB1,B2↦bB2,B3↦cB3. B_1\mapsto aB_1,\qquad B_2\mapsto bB_2,\qquad B_3\mapsto cB_3. For degree-block Hodge metrics Gk=gkI, G_k=g_kI, the whitened skew operator has a=(g0g1)−1/2,b=(g1g2)−1/2,c=(g2g3)−1/2. a=(g_0g_1)^{-1/2}, \quad b=(g_1g_2)^{-1/2}, \quad c=(g_2g_3)^{-1/2}. These are control variables, not promoted UD coefficients. For K−K_-, χ−(a,b,c)(λ)=λ(λ2+3a2)(λ2+5a2)3(λ2+3b2)2(λ2+5b2)3(λ2+3c2)(λ2+5c2). \boxed{ \chi_-^{(a,b,c)}(\lambda) = \lambda (\lambda^2+3a^2) (\lambda^2+5a^2)^3 (\lambda^2+3b^2)^2 (\lambda^2+5b^2)^3 (\lambda^2+3c^2) (\lambda^2+5c^2) }. For K+K_+, χ+(a,b,c)(λ)=λ(λ2+5a2)4(λ2+2b2)(λ2+5b2)5(λ2+2c2)(λ2+5c2)2. \boxed{ \chi_+^{(a,b,c)}(\lambda) = \lambda (\lambda^2+5a^2)^4 (\lambda^2+2b^2) (\lambda^2+5b^2)^5 (\lambda^2+2c^2) (\lambda^2+5c^2)^2 }. AUTO_LOCK PACHNER_23_WEIGHTED_HODGE_DIRAC_RESPONSE_THEOREM_v0.1 The unit case is a=b=c=1a=b=c=1. AUTO_FREEZE interpreting a,b,ca,b,c as free UD theory constants; fitting a,b,ca,b,c to a desired physical spectrum; calling the control family a physical metric without a bridge. 9. Move-response packet A source-clean event-response object is now ℛ2→3=(Δf,Δ𝒜,Δspeck,ΔTr(−𝒜2),ΔTr((−𝒜2)2)). \boxed{ \mathcal R_{2\to3} = \left( \Delta f, \Delta\mathcal A, \Delta\operatorname{spec}_k, \Delta\operatorname{Tr}(-\mathcal A^2), \Delta\operatorname{Tr}((-\mathcal A^2)^2) \right). } No scalar move cost is selected here. That is the correct input for a later launch/selection law. 10. Governance AUTO_LOCK common-boundary 2↔︎3 identification; exact unit spectral response; exact degree-resolved Laplacian response; isolated common-basis update ranks; positive weighted-Hodge response family. OPEN source-derived scalar move cost; launch/event selector; physical interpretation of move rates; general external-mesh response theorem beyond the local patch. Physical promotion: 0.