A constitutive source artifact with explicit theorem, gate, and gap boundaries. UD Constitutive Receipt-Index Event Operator v0.1 Status: PASS_SCOPED / EXACT_CONDITIONAL CONSTITUTIVE OPERATOR DERIVED Physical promotion: 0 1. Source state The current finite source chain already supplies: exact frame alignment from a source-complete oriented C1C_1 transport receipt; a receipt-complete Hamming-budget law μΘ,n=indFn; \mu_{\Theta,n}=\operatorname{ind}F_n; pre-event pair-integrity receipts Tp,npairT_{p,n}^{\mathrm pair}; pre-event quadratic burden weights Wp,nB2W_{p,n}^{B2}; an integrity-constrained target-shell optimizer. The prior signed-support polarity lane remains conditional on one irreducible aligned-versus-anti-aligned sign axiom. It is therefore not used as the primary constitutive source when a receipt-index source packet is available. 2. Continuation-map cancellation theorem Let Fn:ℒn−→ℒn+ F_n:\mathcal L_n^-\to\mathcal L_n^+ be any finite-dimensional continuation map between the already-declared viable pre-event and receipt-constructed viable output-obligation spaces. Write dn−:=dimℒn−,dn+:=dimℒn+,rn:=rankFn. d_n^-:=\dim\mathcal L_n^-, \qquad d_n^+:=\dim\mathcal L_n^+, \qquad r_n:=\operatorname{rank}F_n. Rank-nullity gives dimkerFn=dn−−rn, \dim\ker F_n=d_n^--r_n, and dimcokerFn=dn+−rn. \dim\operatorname{coker}F_n=d_n^+-r_n. Therefore indFn=dimkerFn−dimcokerFn=dn−−dn+. \boxed{ \operatorname{ind}F_n = \dim\ker F_n-\dim\operatorname{coker}F_n = d_n^- - d_n^+. } Consequence The net Hamming budget does not require predecessor matching or the explicit continuation map once the two finite obligation spaces are independently and target-blindly constructed: μΘ,n=dn−−dn+. \boxed{ \mu_{\Theta,n} = d_n^- - d_n^+. } The continuation map remains necessary for: separate NsinkN_{\mathrm sink} and NaddN_{\mathrm add}; gross lineage turnover; predecessor identity; source versus renewal provenance; overlap/merger provenance. Thus: net event polarity/multiplicity≠full lineage provenance. \boxed{ \text{net event polarity/multiplicity} \neq \text{full lineage provenance}. } This is a source-gate reduction, not an endpoint reconstruction. 3. Minimal net-budget source packet For net event polarity/multiplicity, the minimal admissible source packet is: the frame-aligned current Θ\Theta state θ‾n−; \bar\theta_n^-; a target-hidden, source-complete construction of the finite viable input-obligation space ℒn−; \mathcal L_n^-; a target-hidden, source-complete construction of the finite viable output-obligation space ℒn+; \mathcal L_n^+; pair-integrity receipts Tp,npair; T_{p,n}^{\mathrm pair}; pre-event quadratic burden histories Wp,nB2. W_{p,n}^{B2}. No observed endpoint θn+\theta_n^+ is admissible. No terminal residual is admissible. No signed-support alignment axiom is required in this receipt-complete lane. 4. Net polarity and multiplicity theorem Define μΘ,n=dn−−dn+. \boxed{ \mu_{\Theta,n} = d_n^- - d_n^+. } Then the target Hamming count is kn+=wH(θ‾n−)+μΘ,n. \boxed{ k_n^+ = w_H(\bar\theta_n^-) + \mu_{\Theta,n}. } A source packet is carrier-admissible only when 0≤kn+≤3. \boxed{ 0\le k_n^+\le3. } Since Dcl(θ)=6−2wH(θ), D_{\mathrm cl}(\theta)=6-2w_H(\theta), the predicted closure-capacity change is ΔDcl,n=−2μΘ,n. \boxed{ \Delta D_{{\mathrm cl},n} = -2\mu_{\Theta,n}. } Hence: μΘ,n>0\mu_{\Theta,n}>0: net closure; μΘ,n<0\mu_{\Theta,n}<0: net reopening; μΘ,n=0\mu_{\Theta,n}=0: fixed-Hamming exchange or hold. This polarity follows from the receipt-index orientation already present in the source law; no signed-support polarity axiom is used. 5. Exact three-pair address selector For pair-integrity receipts Tp∈{0,1}T_p\in\{0,1\}, define 𝒜k(T)={θ∈Θ:wH(θ)=k,(1−θp)(1−Tp)=0∀p}. \mathcal A_k(T) = \left\{ \theta\in\Theta: w_H(\theta)=k,\ (1-\theta_p)(1-T_p)=0\ \forall p \right\}. Let W=(WA,WB,WC) W=(W_A,W_B,W_C) be the source-derived quadratic burden weights. Define Θopt=arg minθ∈𝒜k(T)∑pθpWp. \boxed{ \Theta_{\mathrm opt} = \operatorname*{arg\,min}_{\theta\in\mathcal A_k(T)} \sum_p\theta_pW_p. } For the three-pair carrier this has the following closed form. k=0k=0 The only candidate is 000. 000. It is admissible iff TA=TB=TC=1. T_A=T_B=T_C=1. k=3k=3 The only candidate is 111. 111. It is always integrity-admissible under the declared open-pair integrity rule. k=1k=1 Exactly one pair is closed. A candidate closed pair pp is admissible iff the other two pairs pass integrity. Among admissible candidates choose the unique minimum WpW_p. k=2k=2 Exactly one pair is open. The open pair must pass integrity. Since minimizing the closed burden sum is equivalent to maximizing the burden left open, choose the unique maximum WpW_p among integrity-passing open-pair candidates. If the admissible set is empty, return CONSTITUTIVE_INFEASIBLE. If the optimum is tied, return HOLD_UNRESOLVED_SYMMETRY. No arbitrary tie-break is admitted. 6. Constitutive Receipt-Index Event Operator When the exact relative-frame gate has supplied rnr_n, define θ‾n−=Pπℳ(rn)θn−. \bar\theta_n^- = P_{\pi_{\mathcal M}(r_n)}\theta_n^-. The source-complete constitutive operator is the partial map ℰnRI:(θ‾n−,ℒn−,ℒn+,Tn,WnB2)↦θn+ \boxed{ \mathcal E_n^{\mathrm RI}: \left( \bar\theta_n^-, \mathcal L_n^-, \mathcal L_n^+, T_n, W_n^{B2} \right) \longmapsto \theta_n^+ } with the sequence d−−d+→μΘ→k+→𝒜k+(T)→Θopt. d^- - d^+ \to \mu_\Theta \to k^+ \to \mathcal A_{k^+}(T) \to \Theta_{\mathrm opt}. If the optimizer is a singleton, θn+ \theta_n^+ is predicted without endpoint access. Then define cΘ,n=(1−θ‾n−)⊙θn+, c_{\Theta,n} = (1-\bar\theta_n^-)\odot\theta_n^+, oΘ,n=θ‾n−⊙(1−θn+), 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}. These obey the exact identities |cΘ,n|−|oΘ,n|=μΘ,n, \boxed{ |c_{\Theta,n}|-|o_{\Theta,n}| = \mu_{\Theta,n}, } and |cΘ,n|+|oΘ,n|=wH(an). \boxed{ |c_{\Theta,n}|+|o_{\Theta,n}| = w_H(a_n). } Therefore the operator predicts both: net multiplicity μΘ\mu_\Theta; gross pair-toggle multiplicity wH(an)w_H(a_n); as well as the directed close/reopen addresses when the optimizer is unique. Finally, UnΘ=X(an)ΓΘ(πℳ(rn)) \boxed{ U_n^\Theta = X(a_n)\Gamma_\Theta(\pi_{\mathcal M}(r_n)) } follows exactly from the already-locked affine receipt representation. 7. Source hierarchy repair The source-completeness requirements split into two levels. NET-BUDGET SOURCE GATE Required: independent viable input-obligation space; independent viable output-obligation space; target-hidden construction of both spaces. Not required: predecessor bijection; explicit continuation matrix. This gate is sufficient for μΘ. \mu_\Theta. LINEAGE-PROVENANCE SOURCE GATE Additionally required: predecessor relations; continuation map; source/sink/renewal type; duplicate/merger equivalence; overlap provenance. This stronger gate is required for gross lineage interpretation. Therefore the earlier rule requiring the full continuation map before any Hamming-budget output was stricter than mathematically necessary. 8. Signed-support lane disposition The one-seat sign candidate μΘsign=−sgnGΣS± \mu_\Theta^{\mathrm sign} = -\operatorname{sgn}G_\Sigma^{S\pm} remains useful only as an independent consistency diagnostic when a receipt-index packet exists. The equality μΘRI=μΘsign \boxed{ \mu_\Theta^{\mathrm RI} = \mu_\Theta^{\mathrm sign} } is a falsifiable comparison. It is not imposed. A disagreement does not permit the sign lane to overwrite the receipt-index source. 9. Remaining constitutive gap This result closes the constitutive operator on the source-complete receipt domain. It does not derive the source receipts themselves from the closed K4K_4/Hodge carrier. The remaining upstream question is now: What native pre-event mechanism constructs ℒn− and ℒn+? \boxed{ \text{What native pre-event mechanism constructs } \mathcal L_n^- \text{ and } \mathcal L_n^+ \text{?} } That is narrower than the previous statement that event polarity/multiplicity as a whole remained unspecified. 10. Claim status continuation-map cancellation: EXACT_INTERNAL; net receipt-index budget: EXACT_CONDITIONAL SOURCE LAW; polarity/capacity relation: EXACT_CONDITIONAL; three-pair address selector: EXACT_CONDITIONAL; full ℰnRI\mathcal E_n^{\mathrm RI}: EXACT_CONDITIONAL / COMPUTED CONSTITUTIVE OPERATOR; signed-support lane: SECONDARY CONDITIONAL DIAGNOSTIC; primitive source of obligation-space receipts: OPEN; physical promotion: 0.