Response-frontier checkpoint immediately preceding the action–phase and rank-2 control updates. Tetrahedral Packet Complexes: Hodge–Dirac Transport, Projective Receipts, and Retriangulation Benjamin Walker Mayes Manuscript state: LEAN JOURNAL REFACTOR v2.13 — RESPONSE FRONTIER CHECKPOINT Source scientific baseline: UD JOP v1.9 + provenance ledger v0.44 Scientific change: APPEND-ONLY CONSOLIDATED RESPONSE-FRONTIER CHECKPOINT / NO NEW RESPONSE CLAIM Abstract We study a finite relational dynamics on tetrahedral packet complexes while separating exact discrete mathematics from physical interpretation. The local carrier is the tetrahedral graph K4K_4 with automorphism group S4S_4, together with an opposite-edge binary fiber Θ≅𝔽23\Theta\cong\mathbb F_2^3. On the filled tetrahedron we use the oriented simplicial chain complex of dimensions 4,6,4,14,6,4,1 and a real skew Hodge–Dirac generator. The single-cell carrier satisfies the exact identity 𝒜2=−4(I−PH0), \mathcal A^2=-4(I-P_{H_0}), which yields one harmonic zero mode, fourteen nonzero directions at frequency magnitude 22, and exact conservation of the quadratic chain support. On glued tetrahedral complexes we derive oriented Hasse-current balance laws and analyze a finite 2↔32\leftrightarrow3 Pachner relay. The current A/B/C relay has exact spectra with B/C signed-permutation covariance and no finite nonzero common recurrence. The central new observable-facing result is a Projective Support Receipt (PSR). We prove that current receipt summaries do not identify the chain ray, derive a sufficient PSR consisting of normalized simplex supports, forest-relative signs, and one root sign per support component, and prove exact projective reconstruction and event-level covariance through the current relay. For the fully supported filled tetrahedron, the Hasse graph provides an intrinsic 28=14+1428=14+14 reconstruction/held-out split. A coherent photonic map is given as a faithful representation of the support and bilinear-current observables; its held-out experimental test remains pending. No identification with physical spacetime, energy, elementary particles, or empirical coupling constants is claimed. 1. Introduction and contribution boundary This paper isolates a finite mathematical core of the Unified Dynamics (UD) research program. The object studied here is not assumed to be physical spacetime, a quantum field, or a completed unification theory. The aim is narrower: define a finite tetrahedral carrier, derive its transport and receipt structure exactly, identify what information is and is not reconstructible, and specify a falsifiable observable realization without importing downstream physical constants. The construction uses standard ingredients from simplicial homology, Hodge/Dirac-type finite operators, finite group actions, and finite-dimensional operator theory. Those ingredients are background. The claimed contribution is their specific organization on the declared tetrahedral packet carrier together with the following results: a symmetry-resolved S4×ΘS_4\times\Theta finite exposure architecture with Θ≅𝔽23\Theta\cong\mathbb F_2^3; exact single-cell and glued-carrier support transport identities on the declared chain complex; exact finite retriangulation constraints, including B/C signed-permutation covariance and a no-common-recurrence result for the current A/B/C relay; receipt-conditioned Θ\Theta transport identities and an exact source-complete relative-frame extractor; a chain-ray nonidentifiability theorem for the pre-PSR receipt set; a sufficient PSR and exact projective reconstruction theorem; PSR forest-choice gauge equivalence and A/B/C event covariance; an intrinsic 28=14reconstruction+14heldout28=14_{\mathrm reconstruction}+14_{\mathrm heldout} falsification design on the fully supported filled tetrahedron. The main unresolved scientific issue is constitutive: the finite transport architecture does not by itself determine which event occurs. Source-complete frame extraction is available; general event polarity/multiplicity remains conditional on additional source receipts. This constitutive gap is treated as an explicit open problem rather than repaired by fitting. The remainder of the paper is mathematical. Physical correspondence is discussed only as an open map. 2. Tetrahedral carrier and oriented chain complex 2.1 Tetrahedral carrier and opposite-edge fiber Let V(K4)={1,2,3,4},E(K4)={12,13,14,23,24,34}. V(K_4)=\{1,2,3,4\}, \qquad E(K_4)=\{12,13,14,23,24,34\}. Then |V(K4)|=4,|E(K4)|=(42)=6,Aut(K4)≅S4. |V(K_4)|=4, \qquad |E(K_4)|=\binom42=6, \qquad \operatorname{Aut}(K_4)\cong S_4. The three perfect matchings are MA={12,34},MB={13,24},MC={14,23}. M_A=\{12,34\},\qquad M_B=\{13,24\},\qquad M_C=\{14,23\}. Assign one binary coordinate to each opposite-edge class: Θ:=𝔽23. \Theta:=\mathbb F_2^3. The Hamming observable wH(θ)=θA+θB+θC w_H(\theta)=\theta_A+\theta_B+\theta_C has shell multiplicities 1,3,3,1. 1,3,3,1. The product carrier 𝒳1=S4×Θ \mathcal X_1=S_4\times\Theta has cardinality |𝒳1|=24⋅8=192. |\mathcal X_1|=24\cdot8=192. This is a finite carrier count only. 2.2 Filled tetrahedral chain carrier Use the ordered vertices 0<1<2<3, 0<1<2<3, edge basis (01,02,03,12,13,23), (01,02,03,12,13,23), face basis (012,013,023,123), (012,013,023,123), and oriented 3-cell 01230123. The chain dimensions are dimC0=4,dimC1=6,dimC2=4,dimC3=1. \boxed{ \dim C_0=4,\qquad \dim C_1=6,\qquad \dim C_2=4,\qquad \dim C_3=1. } With the canonical orientation, the boundary maps satisfy B1B2=0,B2B3=0. \boxed{ B_1B_2=0, \qquad B_2B_3=0. } Their ranks are rankB1=3,rankB2=3,rankB3=1, \operatorname{rank}B_1=3,\qquad \operatorname{rank}B_2=3,\qquad \operatorname{rank}B_3=1, so the filled tetrahedron has dim(H0,H1,H2,H3)=(1,0,0,0). \boxed{ \dim(H_0,H_1,H_2,H_3)=(1,0,0,0). } The explicit matrices are retained in Appendix A. 3. Hodge–Dirac transport and support continuity 3.1 Skew transport and conserved support On 𝒞=C0⊕C1⊕C2⊕C3, \mathcal C=C_0\oplus C_1\oplus C_2\oplus C_3, define 𝒜=(0−B100B1T0−B200B2T0−B300B3T0). \boxed{ \mathcal A= \begin{pmatrix} 0&-B_1&0&0\\ B_1^T&0&-B_2&0\\ 0&B_2^T&0&-B_3\\ 0&0&B_3^T&0 \end{pmatrix}. } Then 𝒜T=−𝒜. \mathcal A^T=-\mathcal A. For ċ=𝒜c, \dot c=\mathcal A c, the quadratic support Sμ(c)=cTc S_\mu(c)=c^Tc obeys dSμds=0. \boxed{ \frac{dS_\mu}{ds}=0. } This is finite norm conservation. No physical energy interpretation is assumed. 3.2 Exact single-cell spectrum and decomposition Let PH0P_{H_0} be the orthogonal projector onto the uniform vertex harmonic mode. The canonical tetrahedral incidence matrices satisfy 𝒜2=−4(I−PH0),𝒜3+4𝒜=0. \boxed{ \mathcal A^2=-4(I-P_{H_0}), \qquad \mathcal A^3+4\mathcal A=0. } Equivalently, det(λI−𝒜)=λ(λ2+4)7. \det(\lambda I-\mathcal A)=\lambda(\lambda^2+4)^7. Thus the single-cell carrier contains one harmonic zero mode and fourteen nonzero directions with frequency magnitude 22. The resulting recurrence parameter is π\pi; it is an internal flow parameter, not physical time. The chain carrier decomposes orthogonally as 𝒞=H0⊕(C0⊥⊕imB1T)⊕(imB2⊕imB2T)⊕(imB3⊕C3), \mathcal C = H_0 \oplus (C_0^\perp\oplus\operatorname{im}B_1^T) \oplus (\operatorname{im}B_2\oplus\operatorname{im}B_2^T) \oplus (\operatorname{im}B_3\oplus C_3), with dimensions 1,6,6,2. 1,\;6,\;6,\;2. On C1C_1, Πcut=14B1TB1,Πcyc=14B2B2T,Πcut+Πcyc=IC1. \Pi_{\mathrm cut}=\frac14B_1^TB_1, \qquad \Pi_{\mathrm cyc}=\frac14B_2B_2^T, \qquad \Pi_{\mathrm cut}+\Pi_{\mathrm cyc}=I_{C_1}. Because H1=0H_1=0, kerB1=imB2. \ker B_1=\operatorname{im}B_2. For a normalized pure edge-cycle z∈kerB1z\in\ker B_1, c1(s)=cos(2s)z,c2(s)=12sin(2s)B2Tz,c0(s)=c3(s)=0. c_1(s)=\cos(2s)z, \qquad c_2(s)=\frac12\sin(2s)B_2^Tz, \qquad c_0(s)=c_3(s)=0. 3.3 Hasse currents and control-volume balance For incident adjacent-degree simplices σ⊂η\sigma\subset\eta, define Jση=2Bσηcσcη. \boxed{ J_{\sigma\eta} = 2B_{\sigma\eta}c_\sigma c_\eta. } For any finite Hasse control volume ℛ\mathcal R, Sμ(ℛ)=∑σ∈ℛcσ2. S_\mu(\mathcal R)=\sum_{\sigma\in\mathcal R}c_\sigma^2. Theorem 3.1 (Exact support control-volume balance). ddsSμ(ℛ)=Jin(ℛ)−Jout(ℛ), \boxed{ \frac{d}{ds}S_\mu(\mathcal R) = J_{\mathrm in}(\mathcal R)-J_{\mathrm out}(\mathcal R), } where only Hasse incidences crossing the cut ℛ|𝒦\ℛ\mathcal R\,|\,\mathcal K\setminus\mathcal R contribute. Internal incidence-current contributions cancel pairwise. Taking ℛ=𝒦\mathcal R=\mathcal K recovers global support conservation. On a glued tetrahedral complex, shared simplices occur once in the global chain carrier. Equal-top-star ownership wστeq=1n3(σ) w_{\sigma\tau}^{\mathrm eq}=\frac1{n_3(\sigma)} partitions each unique-simplex support exactly: ∑τ⊃σwστeq=1. \sum_{\tau\supset\sigma}w_{\sigma\tau}^{\mathrm eq}=1. Hence ∑τSμ,τeq=Sμglobal. \sum_\tau S_{\mu,\tau}^{\mathrm eq}=S_\mu^{\mathrm global}. These gluing identities are exact finite bookkeeping statements. 4. Branch and receipt dynamics 4.1 Exact Hamming branch structure On the basis {|θ⟩:θ∈Θ}\{|\theta\rangle:\theta\in\Theta\}, define WΘ|θ⟩=wH(θ)|θ⟩. W_\Theta|\theta\rangle=w_H(\theta)|\theta\rangle. For w∈{0,1,2,3}w\in\{0,1,2,3\}, define PwΘ=∏m=0m≠w3WΘ−mIw−m. P_w^\Theta = \prod_{\substack{m=0\\m\neq w}}^3 \frac{W_\Theta-mI}{w-m}. Then PwΘPw′Θ=δww′PwΘ,∑wPwΘ=IΘ, P_w^\Theta P_{w'}^\Theta=\delta_{ww'}P_w^\Theta, \qquad \sum_wP_w^\Theta=I_\Theta, and rank(P0Θ,P1Θ,P2Θ,P3Θ)=(1,3,3,1). \boxed{ \operatorname{rank} (P_0^\Theta,P_1^\Theta,P_2^\Theta,P_3^\Theta) = (1,3,3,1). } The induced S4S_4 action permutes the three matching coordinates and preserves Hamming weight: [ρΘ(g),WΘ]=0. [\rho_\Theta(g),W_\Theta]=0. The complement involution obeys 𝒞WΘ𝒞−1=3I−WΘ, \mathcal C W_\Theta\mathcal C^{-1}=3I-W_\Theta, so the weight-one and weight-two sectors are exactly paired. Shell multiplicity therefore does not select one over the other. If an operator UeΘU_e^\Theta is supplied, branch preservation is exact iff [UeΘ,WΘ]=0. \boxed{ [U_e^\Theta,W_\Theta]=0. } 4.2 Receipt-conditioned affine transport For exposure endpoints x−=(g−,θ−),x+=(g+,θ+), x^-=(g^-,\theta^-), \qquad x^+=(g^+,\theta^+), define the relative-frame receipt re=g+(g−)−1 r_e=g^+(g^-)^{-1} and the parity-transition receipt ae=θ+⊕Pπ(re)θ−. a_e=\theta^+\oplus P_{\pi(r_e)}\theta^-. The receipt-conditioned visible operator is UeΘ,aff=X(ae)ΓΘ(π(re)). \boxed{ U_e^{\Theta,\mathrm aff} = X(a_e)\Gamma_\Theta(\pi(r_e)). } It is unitary. If r=wH(ae), r=w_H(a_e), then the branch-transition ranks satisfy rank(Pw′ΘUeΘ,affPwΘ)=(rs)(3−rw−s),s=w+r−w′2, \operatorname{rank} \left( P_{w'}^\Theta U_e^{\Theta,\mathrm aff} P_w^\Theta \right) = \binom rs \binom{3-r}{w-s}, \qquad s=\frac{w+r-w'}2, for admissible integral ss, and vanish otherwise. Moreover, 18∥[UeΘ,aff,WΘ]∥F2=wH(ae). \boxed{ \frac18 \|[U_e^{\Theta,\mathrm aff},W_\Theta]\|_F^2 = w_H(a_e). } Thus branch mixing can be nonzero while the receipt-conditioned transport remains unitary. 4.3 Exact relative-frame extraction Let the ordered pre-event transport on the oriented edge carrier be 𝒯1,npre. \mathcal T_{1,n}^{\mathrm pre}. Since B1B1T=4I4−J4, B_1B_1^T=4I_4-J_4, define P̂0,n=14(B1𝒯1,npreB1T+J4). \boxed{ \widehat P_{0,n} = \frac14 \left( B_1\mathcal T_{1,n}^{\mathrm pre}B_1^T+J_4 \right). } If P̂0,n\widehat P_{0,n} is exactly a four-vertex permutation matrix and B1𝒯1,npre=P̂0,nB1, \boxed{ B_1\mathcal T_{1,n}^{\mathrm pre} = \widehat P_{0,n}B_1, } then there is a unique rn∈S4r_n\in S_4 satisfying P0(rn)=P̂0,n. P_0(r_n)=\widehat P_{0,n}. Let P1(rn)P_1(r_n) be the induced signed edge action and define Kn=P1(rn)−1𝒯1,npre. K_n=P_1(r_n)^{-1}\mathcal T_{1,n}^{\mathrm pre}. Then B1Kn=B1. \boxed{ B_1K_n=B_1. } Thus the residual transport is boundary silent. If 𝒯1,npre\mathcal T_{1,n}^{\mathrm pre} is orthogonal, the residual fixes the cut space and acts only on the cycle space. 4.4 Receipt-index constitutive event operator For receipt-complete events the constitutive frontier can be narrowed further. Let ℒn− \mathcal L_n^- be the finite viable pre-event obligation space and ℒn+ \mathcal L_n^+ the finite viable output-obligation space constructed from target-hidden source receipts. For any pre-specified continuation map Fn:ℒn−→ℒn+, F_n:\mathcal L_n^-\to\mathcal L_n^+, finite rank-nullity gives indFn=dimℒn−−dimℒn+. \operatorname{ind}F_n = \dim\mathcal L_n^- - \dim\mathcal L_n^+. Hence the net Hamming budget is μΘ,n=dimℒn−−dimℒn+. \boxed{ \mu_{\Theta,n} = \dim\mathcal L_n^- - \dim\mathcal L_n^+. } The explicit predecessor/continuation map is therefore unnecessary for the net event budget once the two obligation spaces are independently constructed. It remains necessary for separate sink/addition counts and lineage provenance. After exact frame alignment, kn+=wH(θ‾n−)+μΘ,n. k_n^+ = w_H(\bar\theta_n^-)+\mu_{\Theta,n}. Carrier admissibility requires 0≤kn+≤3. 0\le k_n^+\le3. The integrity-constrained endpoint set is 𝒜kn+(T)={θ∈Θ:wH(θ)=kn+,(1−θp)(1−Tp)=0}. \mathcal A_{k_n^+}(T) = \left\{ \theta\in\Theta: w_H(\theta)=k_n^+,\ (1-\theta_p)(1-T_p)=0 \right\}. Using only the pre-event quadratic burden histories, Θopt,n=arg minθ∈𝒜kn+(T)∑pθpWp,nB2. \Theta_{{\mathrm opt},n} = \operatorname*{arg\,min}_{\theta\in\mathcal A_{k_n^+}(T)} \sum_p\theta_pW_{p,n}^{B2}. A singleton optimizer predicts θn+\theta_n^+ without endpoint access. Empty admissibility is a source inconsistency; an exact optimum tie returns hold. Define cΘ,n=(1−θ‾n−)⊙θn+,oΘ,n=θ‾n−⊙(1−θn+), c_{\Theta,n} = (1-\bar\theta_n^-)\odot\theta_n^+, \qquad o_{\Theta,n} = \bar\theta_n^-\odot(1-\theta_n^+), and an=cΘ,n⊕oΘ,n. a_n=c_{\Theta,n}\oplus o_{\Theta,n}. Then |cΘ,n|−|oΘ,n|=μΘ,n, \boxed{ |c_{\Theta,n}|-|o_{\Theta,n}| = \mu_{\Theta,n}, } while |cΘ,n|+|oΘ,n|=wH(an). \boxed{ |c_{\Theta,n}|+|o_{\Theta,n}| = w_H(a_n). } Thus a source-complete receipt packet and unique burden optimum determine net polarity, net multiplicity, gross pair-toggle multiplicity, and directed addresses. The earlier signed-support sign law is retained only as an independent consistency diagnostic on this domain. The remaining upstream gap is no longer the event operator itself. It is the native construction of the pre-event obligation spaces ℒn±\mathcal L_n^\pm from more primitive packet dynamics. 4.5 Primitive-facing obligation-space candidate The receipt-index event operator still requires a source construction for ℒn±\mathcal L_n^\pm. The current exact support/route structure narrows the possible constructor. For each edge, ΛeS=2|c1,e|(|c0,i|+|c0,j|) \Lambda_e^S = 2|c_{1,e}| (|c_{0,i}|+|c_{0,j}|) is the pre-gate latent support activity. For each opposite-edge pair define Λppair=∑e∈MpΛeS, \Lambda_p^{\mathrm pair} = \sum_{e\in M_p}\Lambda_e^S, and Tppair=∏e∈MpTint,ehard. T_p^{\mathrm pair} = \prod_{e\in M_p} T_{{\mathrm int},e}^{\mathrm hard}. Gated route flow cannot serve as the source because a closed pair has zero gated throughput even when its pre-gate latent activity is nonzero. A reopening-aware source must therefore precede the closure gate. The minimal source-bound binary candidate is ωplat=Tppair𝟏[Λppair>0]. \boxed{ \omega_p^{\mathrm lat} = T_p^{\mathrm pair} \mathbf1[ \Lambda_p^{\mathrm pair}>0 ]. } This object is exactly S4S_4-covariant and uses no fitted tolerance. The constitutive step is the POC-1 hypothesis: a viable latent pair-support lane is an obligation for that pair to be open at the next completed local event. At a completed event boundary this gives the saturation receipt 1−θτΘ=ωτΘlat. 1-\theta_{\tau_\Theta} = \omega_{\tau_\Theta}^{\mathrm lat}. At the first subsequent exact change in ωlat\omega^{\mathrm lat}, define ℒ−=span{ep:ωp,τΘlat=1}, \mathcal L^-= \operatorname{span}\{e_p:\omega_{p,\tau_\Theta}^{\mathrm lat}=1\}, and ℒ+=span{ep:ωp,n⋆lat=1}. \mathcal L^+= \operatorname{span}\{e_p:\omega_{p,n_\star}^{\mathrm lat}=1\}. Then μΘ=dimℒ−−dimℒ+. \mu_\Theta = \dim\mathcal L^--\dim\mathcal L^+. The pair-addressed minimal-capacity endpoint is θPOC1+=𝟏−ωn⋆lat. \boxed{ \theta_{\mathrm POC1}^+ = \mathbf1-\omega_{n_\star}^{\mathrm lat}. } Conditional on POC-1, the net and gross multiplicity identities of Sec. 4.4 follow exactly and the burden optimizer becomes an independent diagnostic rather than the address source. The mathematical carrier does not prove the POC-1 interpretation: a lawful source/sink mechanism could extinguish or create obligations beyond instantaneous latent activity. POC-1 is therefore a preregistered constitutive candidate, not an exact primitive theorem. Its held-out falsifier is exact endpoint disagreement after previous-event saturation and source-authenticated pre-event acquisition. 4.6 Exact viable-lane source and sink receipts The source/sink ambiguity in Sec. 4.5 can be separated from the POC-1 response hypothesis. For edge e={i,j}e=\{i,j\}, define the nonnegative polynomial latent-support witness Re=c1,e2(c0,i2+c0,j2). R_e=c_{1,e}^2(c_{0,i}^2+c_{0,j}^2). Then Re>0⇔ΛeS>0. R_e>0 \iff \Lambda_e^S>0. For opposite pair pp, Rp=∑e∈MpRe,Lp=𝟏[Rp>0], R_p=\sum_{e\in M_p}R_e, \qquad L_p=\mathbf1[R_p>0], and retain the hard pair integrity Tp=∏e∈MpTint,ehard. T_p=\prod_{e\in M_p}T_{{\mathrm int},e}^{\mathrm hard}. The viable latent-lane state is ωp=TpLp. \boxed{ \omega_p=T_pL_p. } For two ordered source states define Ap=(1−ωp−)ωp+ \boxed{ A_p=(1-\omega_p^-)\omega_p^+ } and Sp=ωp−(1−ωp+). \boxed{ S_p=\omega_p^-(1-\omega_p^+). } These are exact pair-lane addition and sink receipts, independent of any Θ\Theta endpoint law. They satisfy ωp+−ωp−=Ap−Sp. \omega_p^+-\omega_p^-=A_p-S_p. Since pair labels are fixed, the canonical continuation map Fω:span{ep:ωp−=1}→span{ep:ωp+=1} F_\omega: \operatorname{span}\{e_p:\omega_p^-=1\} \to \operatorname{span}\{e_p:\omega_p^+=1\} keeps persistent pair basis elements and annihilates disappearing ones. Hence dimkerFω=∑pSp,dimcokerFω=∑pAp, \boxed{ \dim\ker F_\omega=\sum_pS_p, \qquad \dim\operatorname{coker}F_\omega=\sum_pA_p, } and indFω=Nsink−Nadd=|ω−|−|ω+|. \boxed{ \operatorname{ind}F_\omega = N_{\mathrm sink}-N_{\mathrm add} = |\omega^-|-|\omega^+|. } The pair-lane receipts also admit an exact six-way primitive classification: latent-support onset, integrity restoration, joint onset/restoration, latent-support extinction, integrity loss, and joint extinction/loss. This closes source/sink accounting on the viable latent-pair carrier without calling the lanes “obligations.” POC-1 is consequently reduced to the response statement cΘ=S,oΘ=A, \boxed{ c_{\Theta}=S, \qquad o_{\Theta}=A, } or, under previous-event saturation, θ+=1−ω+. \theta^+=1-\omega^+. The source/sink receipts themselves do not imply this response. The same pre-event receipt set admits both the POC-1 endpoint and a hold endpoint unless a constitutive response law is supplied. Thus the remaining open problem is now the response coupling from exact viable-lane source/sink receipts to Θ\Theta rather than the construction of source/sink receipts themselves. 4.7 Maximal-transport response no-go The exact viable-lane source/sink receipts do not by themselves determine the closure response. One natural candidate is maximal admissible transport. At pair level define Vpair(θ∣ω)=∑p(1−θp)ωp. V_{\mathrm pair}(\theta\mid\omega) = \sum_p(1-\theta_p)\omega_p. Every maximizer opens all viable lanes, ωp≤1−θp, \omega_p\le1-\theta_p, but leaves θp\theta_p free whenever ωp=0\omega_p=0. Consequently, on the previous-saturation domain, viable-lane appearance forces reopening under maximal pair transport: Ap=1⇒oΘ,p=1. A_p=1\Rightarrow o_{\Theta,p}=1. Viable-lane disappearance does not force closure: Sp=1⇏cΘ,p=1. S_p=1\not\Rightarrow c_{\Theta,p}=1. The actual edge-level admitted latent activity is Vedge(θ)=∑e(1−θp(e))TeΛeS=∑p(1−θp)Gp, V_{\mathrm edge}(\theta) = \sum_e(1-\theta_{p(e)})T_e\Lambda_e^S = \sum_p(1-\theta_p)G_p, where Gp=∑e∈MpTeΛeS. G_p=\sum_{e\in M_p}T_e\Lambda_e^S. Every maximizer opens a pair whenever Gp>0G_p>0. This creates an exact partial-integrity divergence. If one edge of a pair loses integrity while another integrity-valid edge retains latent support, then ωp=0butGp>0. \omega_p=0 \quad\text{but}\quad G_p>0. POC-1 proposes closure, whereas edge-level maximal transport requires the pair remain open. Support-continuity identities do not resolve this choice because they are valid after either hard gate mask is supplied; they govern support flow conditional on the gate and do not generate the gate state. The POC-1 relation 1−θp=ωp 1-\theta_p=\omega_p is exactly equivalent to: ωp≤1−θp \omega_p\le1-\theta_p (all viable lanes open) plus 1−θp≤ωp 1-\theta_p\le\omega_p (no idle open pair). The first follows from maximal viable-pair transport. The second is an additional closure-economy principle. A lexicographic rule—maximize pair transport, then maximize closure count—uniquely recovers POC-1, but the second objective is not derived from the current Hasse/support-continuity structure. Thus the remaining constitutive gap is sharpened again: the missing theorem is specifically a justification, or falsification, of the no-idle-open-pair/closure-economy response principle. 4.8 Native-structure audit of closure economy The remaining no-idle-open-pair principle can be tested against symmetry, idempotence, recurrence and gate-history composition. Let q=1−θ q=1-\theta be the closure-open response vector, typed separately from the viable-lane receipt ω\omega. Under the induced S3S_3 permutation symmetry on the three matching classes there are exactly 64 equivariant Boolean response maps ω↦q\omega\mapsto q. Requiring viable completeness ω≤q \omega\le q leaves eight. Under a formal coordinate-copy idempotence test all eight remain idempotent. Requiring monotonicity leaves four memoryless symmetric response laws: R0(ω)=ω, R_0(\omega)=\omega, R2(ω)={ω,wH(ω)≤1,111,wH(ω)≥2, R_2(\omega)= \begin{cases} \omega,&w_H(\omega)\le1,\\ 111,&w_H(\omega)\ge2, \end{cases} R1(ω)={000,wH(ω)=0,111,wH(ω)≥1, R_1(\omega)= \begin{cases} 000,&w_H(\omega)=0,\\ 111,&w_H(\omega)\ge1, \end{cases} and R⊤(ω)=111. R_\top(\omega)=111. All four are source-recurrence compatible because they are single-valued functions of the current receipt state. POC-1 is distinguished only by being the pointwise least transport-complete response: RPOC1(ω)=arg minq≥ωwH(q)=ω. \boxed{ R_{\mathrm POC1}(\omega) = \operatorname*{arg\,min}_{q\ge\omega}w_H(q) = \omega. } Thus “minimal open state,” “maximal closure,” and “no idle open pair” are equivalent formulations of the same constitutive selection and do not independently derive one another. Other natural minimality notions do not support POC-1. Minimizing gate changes across a sink transition leaves the now-unneeded pair open, while minimizing wH(θ)w_H(\theta) favors the fully closure-open reference state. Gate-history composition also remains non-unique: both close-on-sink and hold-on-sink local update laws are symmetric, deterministic, idempotent under repeated updates, and commute on distinct pair events. Zero gate-history holonomy would select close-on-sink on an elementary source/sink loop, but loop return is not a native UD theorem; local admissibility may compose to a nontrivial residual. Hence the current native structure does not derive closure economy. It characterizes POC-1 as the least complete response and isolates the missing principle to the selection of that least element. 4.9 Native functional audit of closure economy The least-response characterization raises the possibility that an already-derived burden, action, or gate functional might independently select that least element. The current source does not provide such a functional. The exact number of closure-open edges is Nopen(q)=2wH(q). N_{\mathrm open}(q)=2w_H(q). Minimizing NopenN_{\mathrm open} subject to viable completeness q≥ωq\ge\omega uniquely yields q=ω. q=\omega. However, no current theorem declares NopenN_{\mathrm open} to be a quantity minimized by the response dynamics. That minimization is exactly the closure-economy principle restated as a scalar objective. The existing quadratic burden response score has the opposite declared direction: ℛB(q;W)=14∑pqpWpB2, \mathcal R_B(q;W)=\frac14\sum_pq_pW_p^{B2}, and is maximized in the fixed-Hamming event law. Without an independently fixed Hamming shell, generic positive WpW_p therefore favor opening all pairs rather than the least complete response. Reversing the optimization sign would constitute a new response law. Likewise the burden-mixing relaxation power is −dEBds=14∑eqp(e)Te[(𝖡Tb)e]2, -\frac{dE_B}{ds} = \frac14 \sum_e q_{p(e)}T_e[(\mathsf B^Tb)_e]^2, so steepest burden relaxation favors every gate carrying a nonzero integrity-admitted gradient. The canonical Hodge action 𝔍ν=π∥Pνx∥2 \mathfrak J_\nu=\pi\|P_\nu x\|^2 does not contain qq in its current definition and therefore cannot order the eight gate responses at fixed chain state. The lifted loop action and event-contraction receipt are accumulated/post-event objects, while QTQ_T is an integrity filter rather than an idle-open penalty. Thus no existing independently directed native scalar currently selects POC-1. A future cost closes the gap only if both its form and its optimization direction are derived upstream rather than chosen because their optimizer is q=ωq=\omega. 4.10 Structured-zero, restore, and saturation audit The remaining least-response direction is not supplied by the current restore vocabulary. Three typed Boolean zeros must be distinguished: the viable-lane zero 0Ω0_\Omega, the gate-open zero 0Q0_Q, and the Θ\Theta zero 0Θ0_\Theta. Since q=1−θq=1-\theta, 0Θ↦111Q, 0_\Theta\mapsto111_Q, and the current Θ\Theta reference state is fully closure-open. Thus restore to the canonical Θ\Theta reference does not produce POC-1. The breathing/source-interface restore step is presently procedural: evaluate–act–record–restore probe cycles return to a frozen pre-event source state. It does not define an event gate projection. A history-preserving gate restore has the exact minimal-change form q+=q−∨ω+, q^+=q^-\vee\omega^+, which is transport-complete and idempotent but leaves obsolete gates open on sink transitions. By contrast, “restore to the least current source-complete gate support” gives q+=ω+, q^+=\omega^+, but this is exactly the closure-economy principle expressed as restore semantics. The completed-event saturation receipt likewise cannot derive the response because it is itself defined by q=ω q=\omega as the POC-1 constitutive hypothesis. A useful conditional uniqueness result remains. If the response is pair-local, qp=g(ωp), q_p=g(\omega_p), transport completeness gives g(1)=1g(1)=1. If one can independently derive the typed zero-source-silence bridge R(0Ω)=0Q, R(0_\Omega)=0_Q, then g(0)=0g(0)=0 and POC-1 follows uniquely. Current sources do not provide that cross-type zero bridge, and numerical equality of the bit triples is not sufficient typed provenance. The response gap is therefore reduced to whether structured-zero dynamics can independently prove that zero viable lanes imply zero closure-open lanes. 4.11 Support-projector and annihilator response no-go The existing Hodge projectors do not supply the remaining zero-source-to-zero-gate bridge. On the filled K4K_4 carrier, P2=I−PH0=−𝒜2/4. P_2=I-P_{H_0}=-\mathcal A^2/4. If P2c=0P_2c=0, then cc is purely harmonic and its C1C_1 component vanishes. Hence every latent pair-support receipt vanishes and ω=000. \omega=000. But all eight gate states produce the same zero route throughput because every factor c1,ec_{1,e} is zero. The support annihilator therefore makes the gate state unidentifiable rather than selecting q=000q=000. Pairwise, Pp(1)c1=0P_p^{(1)}c_1=0 implies zero latent support on pair pp, but both qp=0q_p=0 and qp=1q_p=1 yield identical pair throughput. The zero branch-contact operator is also non-invertible to closure because it may result from closure, failed integrity, absent edge support, or absent endpoint support. The ray-plane projector Q[x]=xxT+(Jx)(Jx)TxTx Q_{[x]} = \frac{xx^T+(Jx)(Jx)^T}{x^Tx} is defined only for x≠0x\neq0. Scale invariance makes a canonical continuous extension to x=0x=0 impossible: approaching zero along distinct rays retains distinct projectors. Thus support-projector annihilation supplies a source-zero certificate but not a gate-zero response. The remaining candidate “annihilator silence” rule, ωp=0⇒qp=0, \omega_p=0\Rightarrow q_p=0, is equivalent to the already isolated zero-source-silence/closure-economy principle and is not independent evidence. 5. Retriangulation and finite covariance constraints 5.1 Oriented Pachner pair On vertices 0,1,2,3,40,1,2,3,4, define K−=[0123]−[0124], K_-=[0123]-[0124], and K+=−[1234]+[0234]−[0134]. K_+=-[1234]+[0234]-[0134]. Then K−−K+=∂[01234], \boxed{ K_--K_+=\partial[01234], } so ∂K−=∂K+. \boxed{ \partial K_-=\partial K_+. } A separate exact no-go shows that no fixed binary vertex labeling makes all five participating tetrahedra simultaneously satisfy the declared 2:22{:}2 condition. Topological admissibility therefore does not imply fixed-label micro-admissibility. 5.2 Finite A/B/C relay Use A={0123,0124}, A=\{0123,0124\}, B={0123,0124,0134}, B=\{0123,0124,0134\}, C={0134,0234,1234}. C=\{0134,0234,1234\}. Their exact ff-vectors are f(A)=(5,9,7,2), f(A)=(5,9,7,2), and f(B)=f(C)=(5,10,9,3). f(B)=f(C)=(5,10,9,3). The involution p=(03)(14) p=(0\ 3)(1\ 4) with 22 fixed induces a signed-permutation simplicial isomorphism B≅C. B\cong C. 5.3 Exact spectra and no common recurrence The global unique-simplex Hodge–Dirac spectra are Spec(𝒜A)=0(1)∪(±i3)(4)∪(±i5)(7), \boxed{ \operatorname{Spec}(\mathcal A_A) = 0^{(1)} \cup (\pm i\sqrt3)^{(4)} \cup (\pm i\sqrt5)^{(7)}, } and Spec(𝒜B)=Spec(𝒜C)=0(1)∪(±i2)(2)∪(±i5)(11). \boxed{ \operatorname{Spec}(\mathcal A_B) = \operatorname{Spec}(\mathcal A_C) = 0^{(1)} \cup (\pm i\sqrt2)^{(2)} \cup (\pm i\sqrt5)^{(11)}. } Equivalently, χA(λ)=λ(λ2+3)4(λ2+5)7, \chi_A(\lambda) = \lambda(\lambda^2+3)^4(\lambda^2+5)^7, χB(λ)=χC(λ)=λ(λ2+2)2(λ2+5)11. \chi_B(\lambda) = \chi_C(\lambda) = \lambda(\lambda^2+2)^2(\lambda^2+5)^{11}. Since 53∉ℚ,52∉ℚ, \sqrt{\frac53}\notin\mathbb Q, \qquad \sqrt{\frac52}\notin\mathbb Q, there is no finite T>0T>0 satisfying eT𝒜=I e^{T\mathcal A}=I for the full flow on any of the current A/B/C complexes. The current A/B/C relay admits no finite nonzero global Hodge–Dirac recurrence. \boxed{ \text{The current A/B/C relay admits no finite nonzero global Hodge–Dirac recurrence.} } Thus the exact single-cell recurrence cannot be promoted to a universal clock on the glued carriers. Detailed intertwiners, inclusion residuals, protected/vulnerable sector decomposition, and canonical internal action-transfer results are preserved in Supplement S4 and the source manuscript. 6. Projective Support Receipt 6.1 Ray nonidentifiability before PSR The previously available event, route, lineage, Hasse-current, and total-support receipts do not identify one chain ray. In particular, the parity-degree rescaling Gλ(c0,c1,c2,c3)=(λc0,λ−1c1,λc2,λ−1c3),λ>0, G_\lambda(c_0,c_1,c_2,c_3) = (\lambda c_0,\lambda^{-1}c_1,\lambda c_2,\lambda^{-1}c_3), \qquad \lambda>0, leaves adjacent-degree bilinear Hasse currents invariant. An additional exact witness preserves SμS_\mu while changing the ray. Therefore {event/route/lineage receipts+all Hasse currents+Sμ}⇏[c]. \boxed{ \{\text{event/route/lineage receipts} +\text{all Hasse currents}+S_\mu\} \not\Rightarrow[c]. } This is a scoped identifiability no-go. 6.2 Sufficient receipt and reconstruction theorem For c≠0c\neq0, define ρσ=cσ2Sμ(c). \rho_\sigma=\frac{c_\sigma^2}{S_\mu(c)}. On the Hasse graph restricted to nonzero-support simplices, choose a deterministic spanning forest FF. For each forest incidence record χστ=sgn(cσcτ)=sgn(BστJστ), \chi_{\sigma\tau} = \operatorname{sgn}(c_\sigma c_\tau) = \operatorname{sgn}(B_{\sigma\tau}J_{\sigma\tau}), and record one coefficient sign ϵC\epsilon_C at a root of each connected support component. Define PSR(c)=({ρσ},{χστ}F,{ϵC}). \boxed{ \operatorname{PSR}(c) = \left( \{\rho_\sigma\}, \{\chi_{\sigma\tau}\}_{F}, \{\epsilon_C\} \right). } Propagate signs through the forest and set ĉσ=sσρσ. \widehat c_\sigma=s_\sigma\sqrt{\rho_\sigma}. Then ℛ(PSR(c))=cSμ(c). \boxed{ \mathcal R(\operatorname{PSR}(c)) = \frac{c}{\sqrt{S_\mu(c)}}. } Thus the PSR determines the positive projective chain ray while leaving absolute support normalization separate. 6.3 Pre-event acquisition algebra At a frozen pre-event state, acquire qσ=cσ2 q_\sigma=c_\sigma^2 for each unique simplex and Jστ=2Bστcσcτ J_{\sigma\tau}=2B_{\sigma\tau}c_\sigma c_\tau on the forest incidences. The exact consistency identity is Jστ2=4qσqτ. \boxed{ J_{\sigma\tau}^2=4q_\sigma q_\tau. } One common sign flip per connected support component remains invisible to stocks and bilinear currents. For a deterministic root rr, a matched calibrated probe gives qr+=(cr+α)2,qr−=(cr−α)2, q_r^+=(c_r+\alpha)^2, \qquad q_r^-=(c_r-\alpha)^2, so qr+−qr−=4αcr. \boxed{ q_r^+-q_r^-=4\alpha c_r. } The probe amplitude is a measurement control, not an internal coefficient. 6.4 Forest gauge and A/B/C event covariance Let TT denote the canonical A-to-B inclusion and PP the signed B-to-C simplicial permutation. At the event map, TTT=IA,PTP=IB,𝒜CP=P𝒜B. T^TT=I_A, \qquad P^TP=I_B, \qquad \mathcal A_CP=P\mathcal A_B. The PSR reconstruction commutes with these event maps: ℛB(PSR(TcA))=TℛA(PSR(cA)), \boxed{ \mathcal R_B(\operatorname{PSR}(Tc_A)) = T\mathcal R_A(\operatorname{PSR}(c_A)), } and ℛC(PSR(PcB))=PℛB(PSR(cB)). \boxed{ \mathcal R_C(\operatorname{PSR}(Pc_B)) = P\mathcal R_B(\operatorname{PSR}(c_B)). } A mapped forest and a freshly recomputed canonical forest may use different incidence edges yet reconstruct the same ray. Forest choice is therefore a receipt-serialization gauge rather than a new state variable. These are event-level statements. They do not erase the known dynamical residual of the dimension-changing A-to-B move. 6.5 Authentication and interval uncertainty Receipt consistency does not establish source authenticity: PSR schema integrity⇏identity with a separately intended external state. \boxed{ \text{PSR schema integrity} \not\Rightarrow \text{identity with a separately intended external state}. } A measurement-facing receipt therefore separates source authentication from mathematical feasibility. Cryptographic authentication establishes packet integrity and trusted-key possession; it does not establish physical truth. For strictly positive measured support intervals, write xσ=lnqσ. x_\sigma=\ln q_\sigma. The exact current law gives the interval constraint 2ln(Jστmin2)≤xσ+xτ≤2ln(Jστmax2). 2\ln\!\left(\frac{J_{\sigma\tau}^{\min}}2\right) \le x_\sigma+x_\tau \le 2\ln\!\left(\frac{J_{\sigma\tau}^{\max}}2\right). Together with the logged support intervals, magnitude compatibility becomes a linear feasibility problem in the transformed measured fields. If a required support or sign interval crosses zero, the result is indeterminate rather than repaired by a fitted tolerance. 7. Faithful observable realization and held-out test 7.1 Intrinsic held-out design The fully supported filled tetrahedron has 15 simplex coordinates and exactly 28 Hasse incidences: 12C0↔C1+12C1↔C2+4C2↔C3=28. 12_{C_0\leftrightarrow C_1} + 12_{C_1\leftrightarrow C_2} + 4_{C_2\leftrightarrow C_3} = 28. A spanning tree on the 15 supported simplex vertices contains 15−1=14 15-1=14 incidences. The remaining 14 incidences are not required for reconstruction. Hence 28=14reconstruction+14heldout. \boxed{ 28 = 14_{\mathrm reconstruction} + 14_{\mathrm heldout}. } The training PSR defines a frozen feasible set ℱPSR\mathcal F_{\mathrm PSR}. For each held-out Hasse incidence, Iστpred=rangec∈ℱPSR(2Bστcσcτ). \boxed{ I_{\sigma\tau}^{\mathrm pred} = \operatorname{range}_{c\in\mathcal F_{\mathrm PSR}} \left( 2B_{\sigma\tau}c_\sigma c_\tau \right). } Held-out data are released only after these predictions are frozen. The held-out channels may not repair the reconstruction receipt. 7.2 Coherent photonic representation A coherent multi-mode interferometer provides a faithful representation of the PSR observables. Assign one optical mode to each simplex coordinate and define Pscience=∑τ|Eτ|2. P_{\mathrm science}=\sum_\tau |E_\tau|^2. Then ρσ=|Eσ|2Pscience. \boxed{ \rho_\sigma = \frac{|E_\sigma|^2}{P_{\mathrm science}}. } For one Hasse-adjacent pair, a calibrated 50:50 analyzer yields P+=|Eσ+Eτ|22,P−=|Eσ−Eτ|22, P_+ = \frac{|E_\sigma+E_\tau|^2}{2}, \qquad P_- = \frac{|E_\sigma-E_\tau|^2}{2}, and therefore P+−P−=2Re(Eσ*Eτ). P_+-P_- = 2\operatorname{Re}(E_\sigma^\ast E_\tau). On the admitted common real quadrature, Jστopt=BστP+−P−Pscience=2Bστĉσĉτ. \boxed{ J_{\sigma\tau}^{\mathrm opt} = B_{\sigma\tau} \frac{P_+-P_-}{P_{\mathrm science}} = 2B_{\sigma\tau}\widehat c_\sigma\widehat c_\tau. } A phase-locked auxiliary provides the matched root-sign probe. Under the publication correspondence scale adopted here, this optical map is CRL-1: faithful representation. It establishes an explicit observable algebra map; it does not identify the optical processor as the physical UD substrate. 7.3 Experimental status A public-corpus admissibility audit was performed against the frozen POC-1 source fields. The strongest surveyed external datasets included a physical K4K_4 quantum-walk experiment, raw voltage/current switching datasets, experimental simplicial-oscillator time series, photonic processor reproducibility packages, and repeated-syndrome QEC corpora. None simultaneously exposes the native four vertex-chain channels, six independent edge-chain channels, hard edge-integrity receipts, prior three-pair Θ\Theta state, ordered pre-event history, and a separately hidden next Θ\Theta endpoint. Accordingly, no currently surveyed public corpus is source-complete for a native POC-1 score. \boxed{ \text{no currently surveyed public corpus is source-complete for a native POC-1 score}. } No empirical POC-1 verdict is reported. Constructing a new proxy map after corpus selection is prohibited. The next admissible empirical route is a purpose-built independent acquisition or a future public corpus whose correspondence map is frozen before target exposure. A 20-mode implementation has been preregistered with 1–15=science,16=reference,17=root auxiliary,18–20=calibration/guard. 1\text{--}15=\text{science}, \qquad 16=\text{reference}, \qquad 17=\text{root auxiliary}, \qquad 18\text{--}20=\text{calibration/guard}. The deterministic 14 reconstruction and 14 held-out incidences are frozen. Device/laboratory qualification, authenticated calibration intake, and zero-touch held-out release tooling are maintained as supplementary reproducibility artifacts. No real held-out optical result exists at this manuscript state. Thus: CRL-1 mathematical representation: specified, \boxed{ \text{CRL-1 mathematical representation: specified}, } CRL-2 operational same-observable test: pending, \boxed{ \text{CRL-2 operational same-observable test: pending}, } CRL-3 constitutive physical-carrier identification: open. \boxed{ \text{CRL-3 constitutive physical-carrier identification: open}. } 7.4 Purpose-built native POC-1 constitutive test Because no surveyed public corpus exposes the complete native POC-1 source tuple, a purpose-built independent event experiment has been preregistered. The experimental carrier provides four independently acquired C0C_0 channels, six independently acquired C1C_1 edge channels, six hard integrity receipts, three opposite-edge-pair endpoint latches, and blinded endpoint custody. The scientific event engine must be authored independently of POC-1 and committed before prediction. At the first exact change in the source-derived latent obligation receipt, the predictor freezes θ̂+=𝟏−ωlat. \widehat\theta^+ = \mathbf1-\omega^{\mathrm lat}. Only then is the endpoint released. The primary deterministic verdict uses exact endpoint equality. One admitted source-authenticated mismatch falsifies POC-1 on the tested bench domain. A POC-1-programmed event engine is permitted only as an apparatus control and carries no scientific evidentiary weight. A successful independent run would establish at most CRL-2 operational same-observable constitutive transfer on the purpose-built bench. CRL-3 physical-carrier identification would remain open. A laboratory implementation packet now freezes the native channel map, source-calibration fields, timing/custody barriers, DAQ/source-stream interface, and independent event-engine handoff contract. Hardware-specific sample rate, electrical range, timing tolerance, and latency remain laboratory-sourced rather than UD constants. No real bench run has yet occurred. 7.5 Pretarget multi-model response sidecar Before any external laboratory endpoint data exist, five surviving deterministic response laws have been frozen as a blinded sidecar: R0,R2,R1,R⊤,MTA. R_0,\quad R_2,\quad R_1,\quad R_\top,\quad \mathrm{MTA}. Using q=1−θq=1-\theta, their definitions are R0(ω)=ω, R_0(\omega)=\omega, R2(ω)={ω,wH(ω)≤1,111,wH(ω)≥2, R_2(\omega)= \begin{cases} \omega,&w_H(\omega)\le1,\\ 111,&w_H(\omega)\ge2, \end{cases} R1(ω)={000,wH(ω)=0,111,wH(ω)≥1,R⊤(ω)=111, R_1(\omega)= \begin{cases} 000,&w_H(\omega)=0,\\ 111,&w_H(\omega)\ge1, \end{cases} \qquad R_\top(\omega)=111, and qpMTA=𝟏[Gp>0],Gp=∑e∈MpTeΛeS. q_p^{\mathrm MTA}=\mathbf1[G_p>0], \qquad G_p=\sum_{e\in M_p}T_e\Lambda_e^S. The sidecar freezes all resolved predictions from the same source stream and trigger as the primary POC-1 test before endpoint release. Exact endpoint equality is the only model-survival criterion. Two event classes have special discriminating value. Clean zero-source events ω=000,G=000 \omega=000,\quad G=000 test zero-source silence against the fully-open response. Partial-integrity events ωp=0,Gp>0 \omega_p=0,\quad G_p>0 directly distinguish POC-1 from MTA on that pair. The sidecar does not alter the already-preregistered POC-1 primary verdict and has not yet been distributed as a laboratory-protocol amendment. 7.6 Response-law internal frontier checkpoint A consolidated audit of the current response-law derivations now separates exact source accounting from the unresolved response selector. Exact viable-lane source/sink receipts, maximal-transport constraints, response-map classification, native functional audits, restore/zero audits, and projector-annihilator audits do not uniquely derive the POC-1 response without an additional selection principle. Several formulations are mathematically equivalent under transport completeness, including least complete response, no idle open pair, minimum open-gate cardinality, maximum closure count, and least-current-support restore. They are therefore treated as one unresolved constitutive family rather than independent arguments. The current internal response frontier is frozen against further reformulations of that family. Re-entry requires a genuinely new typed gate dynamics, zero bridge, gate-holonomy theorem, independently directed pre-event functional, new source receipt, or new blinded empirical evidence. The externally distributed five-model registry R0,R2,R1,R⊤,MTA R_0,\ R_2,\ R_1,\ R_\top,\ \mathrm{MTA} remains the active independent discriminator while laboratory feasibility and blinded data are pending. 8. Discussion 8.1 Novelty relative to standard machinery The paper does not claim novelty for the existence of simplicial boundary operators, Hodge/Dirac-style finite generators, controlled-unitary constructions, or Kraus dilation identities. The intended new contribution is the specific finite tetrahedral packet architecture and the receipt structure derived on it. The strongest distinct results are the PSR identifiability no-go, sufficient projective reconstruction theorem, forest-gauge result, event covariance through the current A/B/C relay, authenticated interval-feasibility construction, and the intrinsic 28=14+1428=14+14 held-out design. The source-complete relative-frame extractor and the finite retriangulation spectral constraints are additional framework-specific results. 8.2 What is not claimed Nothing in this paper identifies Sμ S_\mu with physical energy, 𝒜 \mathcal A with a quantum Hamiltonian, the finite tetrahedral carrier with physical spacetime, the Θ\Theta fiber with a Standard Model gauge or matter degree of freedom, or any internal finite ratio with a measured physical coupling. The optical representation is a test of the receipt/observable algebra in a calibrated medium. Even a successful held-out optical test would establish operational implementability of the mapped observables in that device, not a constitutive theory of nature. 8.3 Highest open problem The highest upstream gap is constitutive event selection. The framework currently contains exact transport structure, exact receipt reconstruction once source data exist, and a falsifiable observable layer. It does not yet derive from the closed finite carrier alone a unique source law that decides the general event polarity and multiplicity. This gap is upstream of most proposed physical interpretations. Progress on the constitutive source law therefore has greater explanatory value than adding additional analog hardware platforms or numerical correspondences. 9. Conclusion The finite tetrahedral carrier supports a replayable chain complex, exact skew Hodge–Dirac support transport, finite branch structure, retriangulation constraints, and an exact projective receipt for the chain ray. The PSR identifies precisely which support and sign information is sufficient, proves which earlier receipt summaries are insufficient, and yields an intrinsic held-out test on the same observable algebra. These results define a finite mathematical and experimental-methods program. They do not yet constitute a physical theory of spacetime, matter, gauge fields, or measured constants. The next scientific step is therefore twofold but ordered: execute the already-preregistered observable realization, and independently advance the constitutive event-source law that determines which receipt is produced. Appendix A. Explicit tetrahedral boundary matrices In the ordered bases (01,02,03,12,13,23),(012,013,023,123), (01,02,03,12,13,23), \qquad (012,013,023,123), the canonical matrices are B1=(−1−1−1000100−1−1001010−1001011), B_1= \begin{pmatrix} -1&-1&-1&0&0&0\\ 1&0&0&-1&-1&0\\ 0&1&0&1&0&-1\\ 0&0&1&0&1&1 \end{pmatrix}, B2=(1100−10100−1−101001010−10011), B_2= \begin{pmatrix} 1&1&0&0\\ -1&0&1&0\\ 0&-1&-1&0\\ 1&0&0&1\\ 0&1&0&-1\\ 0&0&1&1 \end{pmatrix}, and B3=(−11−11). B_3= \begin{pmatrix} -1\\ 1\\ -1\\ 1 \end{pmatrix}. Direct multiplication gives B1B2=0,B2B3=0. B_1B_2=0, \qquad B_2B_3=0. Appendix B. Retriangulation spectra and preserved relay details The exact characteristic polynomials used in Section 5 are χA(λ)=λ(λ2+3)4(λ2+5)7, \chi_A(\lambda) = \lambda(\lambda^2+3)^4(\lambda^2+5)^7, χB(λ)=χC(λ)=λ(λ2+2)2(λ2+5)11. \chi_B(\lambda) = \chi_C(\lambda) = \lambda(\lambda^2+2)^2(\lambda^2+5)^{11}. The full source manuscript additionally derives: - the exact B/C signed-permutation chain isomorphism; - the dimension-changing A-to-B inclusion residual; - protected and vulnerable spectral subspaces; - exact action-budget transfer through the current relay; - the lifted loop-history action functional; - the no-go for one universal scalar conversion from endpoint loop defect to accumulated action. Those statements are preserved unchanged in Supplement S4 and in the full v1.9 source manuscript. They are not required for the central PSR theorem chain. Appendix C. PSR proof and uncertainty details For positive support on one forest edge (σ,τ)(\sigma,\tau), Jστ2=4qσqτ. J_{\sigma\tau}^2=4q_\sigma q_\tau. A resolved relative sign and one root sign per support component determine every coefficient sign on the forest. The normalized coefficient magnitudes are ρσ\sqrt{\rho_\sigma}, so reconstruction follows by sign propagation. With certified current interval 0<Jmin≤|Jστ|≤Jmax, 0<J_{\min}\le |J_{\sigma\tau}|\le J_{\max}, and xσ=lnqσx_\sigma=\ln q_\sigma, 2ln(Jmin/2)≤xσ+xτ≤2ln(Jmax/2). 2\ln(J_{\min}/2) \le x_\sigma+x_\tau \le 2\ln(J_{\max}/2). Together with the logged stock intervals, the magnitude feasibility problem is linear. Root-probe constraints remain exact quadratic preimages. Zero-crossing support or sign intervals are reported as indeterminate. The complete derivations, authenticated receipt schema, robustness cases, and experimental custody protocol are retained in the supplementary research package. Publication claim boundary This v2.0 manuscript is an editorial refactor of the governed v1.9 scientific source. No theorem, no-go, claim status, ledger value, or provenance ruling is changed by this refactor.