Fixed-integrity crossing determines a rank sign. O09 action ordering requires canonical pre-event maps and normalization. Governed release package · 0.1Latent Crossing and O09 SelectionFixed-integrity crossing determines a rank sign. O09 action ordering requires canonical pre-event maps and normalization.StatusEXACT CONDITIONAL / SOURCE GATEPhysical promotion0Package files7Hash manifestVerified · 6 filesDownload immutable release ZIP ↓Package compositionpackage root7Show all 7 preserved pathsUD_LATENT_CROSSING_O09_SELECTOR_HASH_MANIFEST_v0.1.jsonUD_LATENT_CROSSING_O09_SELECTOR_VERIFICATION_v0.1.jsonUD_LATENT_SUPPORT_CROSSING_TO_LANE_RANK_SIGN_AUDIT_v0.1.mdUD_MANUSCRIPT_MILESTONE_ADDENDUM_LATENT_CROSSING_O09_SELECTOR_v0.1.mdUD_O09_SOURCE_SELECTING_PAIRING_ACTION_AUDIT_v0.1.mdUD_PROVENANCE_LEDGER_ADDENDUM_LATENT_CROSSING_O09_SELECTOR_v0.1.mdverify_ud_latent_crossing_o09_selector_v0_1.pyUD_LATENT_SUPPORT_CROSSING_TO_LANE_RANK_SIGN_AUDIT_v0.1.mdUD Latent-Support Crossing to Lane-Rank Sign Audit v0.1 Date: 2026-09-14 Status: RESTRICTED_THEOREM_PASS / GLOBAL_OP03_SIGN_BRIDGE_NO_GO Physical promotion: 0 0. Question Test whether the exact current objects Q216:ṘeQ216:\dot R_e, Q217:LpQ217:L_p, Q218:Ap,SpQ218:A_p,S_p, Q219Q219 source/sink subtype partition, Q220:FωQ220:F_\omega, Q221:indFωQ221:\operatorname{ind}F_\omega, force the sign of Nadd−Nsink N_{\mathrm add}-N_{\mathrm sink} at first admissible crossings without adding a constitutive response axiom. 1. Exact viable-lane algebra For pair pp, Rp=∑e∈MpRe≥0,Lp=𝟏[Rp>0],ωp=TpLp. R_p=\sum_{e\in M_p}R_e\ge0, \qquad L_p=\mathbf1[R_p>0], \qquad \omega_p=T_pL_p. For two ordered source states, Ap=(1−ωp−)ωp+,Sp=ωp−(1−ωp+). A_p=(1-\omega_p^-)\omega_p^+, \qquad S_p=\omega_p^-(1-\omega_p^+). Therefore Ap−Sp=ωp+−ωp−. \boxed{ A_p-S_p=\omega_p^+-\omega_p^-. } Summing over pairs, Nadd−Nsink=|ω+|−|ω−|. \boxed{ N_{\mathrm add}-N_{\mathrm sink} = |\omega^+|-|\omega^-|. } This is exact source/sink mathematics and does not use a Θ\Theta-endpoint law. 2. Restricted pure-latent crossing theorem Assume one pair changes and hard integrity is unchanged and open: Tp−=Tp+=1. T_p^-=T_p^+=1. Then ωp±=Lp±. \omega_p^\pm=L_p^\pm. Hence Nadd−Nsink=Lp+−Lp−. \boxed{ N_{\mathrm add}-N_{\mathrm sink} = L_p^+-L_p^-. } Because Lp=𝟏[Rp>0]L_p=\mathbf1[R_p>0]: Onset Rp−=0,Rp+>0 R_p^-=0,\qquad R_p^+>0 gives ΔRp>0,Nadd−Nsink=+1. \Delta R_p>0,\qquad N_{\mathrm add}-N_{\mathrm sink}=+1. Extinction Rp−>0,Rp+=0 R_p^->0,\qquad R_p^+=0 gives ΔRp<0,Nadd−Nsink=−1. \Delta R_p<0,\qquad N_{\mathrm add}-N_{\mathrm sink}=-1. Therefore sgn(Nadd−Nsink)=sgn(Rp+−Rp−) \boxed{ \operatorname{sgn}(N_{\mathrm add}-N_{\mathrm sink}) = \operatorname{sgn}(R_p^+-R_p^-) } for one-pair, fixed-integrity, pure latent-support crossings. AUTO_LOCK ONE_PAIR_FIXED_INTEGRITY_LATENT_CROSSING_FIXES_LANE_RANK_SIGN_THEOREM_v0.1 This is a finite-difference theorem. It is not yet the OP03 signed-support theorem. 3. Instantaneous-rate no-go at the exact zero RpR_p is a nonnegative polynomial witness. Under smooth Hodge evolution it is differentiable. At an interior point s*s_* satisfying Rp(s*)=0, R_p(s_*)=0, nonnegativity makes s*s_* a local minimum. Therefore Ṙp(s*)=0. \boxed{ \dot R_p(s_*)=0. } Thus the instantaneous Q216Q216 rate at the exact zero cannot by itself distinguish onset from extinction. A one-sided or finite-interval history is required. AUTO_LOCK NO-GO INSTANTANEOUS_Q216_AT_EXACT_LATENT_ZERO_CANNOT_SELECT_CROSSING_POLARITY_v0.1 4. Integrity transitions break a pure-latent sign theorem Because ωp=TpLp, \omega_p=T_pL_p, lane rank can change with LpL_p unchanged. Example: (L−,T−)=(1,0),(L+,T+)=(1,1) (L^-,T^-)=(1,0), \qquad (L^+,T^+)=(1,1) gives ω−=0,ω+=1, \omega^- =0,\qquad \omega^+=1, so Nadd−Nsink=+1 N_{\mathrm add}-N_{\mathrm sink}=+1 despite ΔL=0. \Delta L=0. Similarly, (1,1)→(1,0) (1,1)\to(1,0) produces a sink with ΔL=0\Delta L=0. Exhaustive one-pair binary audit: all 24=162^4=16 (L−,T−,L+,T+)(L^-,T^-,L^+,T^+) transitions checked; 66 are rank-changing; 22 rank-changing cases have ΔL=0\Delta L=0; fixed-integrity T=1T=1 latent crossings: 2/22/2 satisfy the theorem. Factorization: 16=24,6=2⋅3,2 prime. 16=2^4,\qquad 6=2\cdot3,\qquad 2\text{ prime}. AUTO_LOCK NO-GO LATENT_OCCUPANCY_CHANGE_ALONE_DOES_NOT_DETERMINE_VIABLE_LANE_RANK_CHANGE_WHEN_INTEGRITY_CAN_CHANGE_v0.1 5. GΣS±G_\Sigma^{S\pm} is not Ṙ\dot R The current signed support source uses λpS±=2c1TPp(1)(B1Tc0−B2c2). \lambda_p^{S\pm} = 2c_1^TP_p^{(1)} (B_1^Tc_0-B_2c_2). On the exact Hodge flow, ċ1=B1Tc0−B2c2, \dot c_1=B_1^Tc_0-B_2c_2, so λpS±=dds∥Pp(1)c1∥2. \boxed{ \lambda_p^{S\pm} = \frac{d}{ds}\|P_p^{(1)}c_1\|^2. } By contrast, Rp=∑e={i,j}∈Mpc1,e2(c0,i2+c0,j2) R_p = \sum_{e=\{i,j\}\in M_p} c_{1,e}^2(c_{0,i}^2+c_{0,j}^2) contains vertex-amplitude factors and therefore has a different derivative. Exact counterexample Use the canonical K4 orientation and set c0=(−2,−2,−2,−1), c_0=(-2,-2,-2,-1), c1=(−2,−2,−2,−2,−2,−2),c2=0. c_1=(-2,-2,-2,-2,-2,-2), \qquad c_2=0. The Hodge equations give ċ0=(−6,−2,2,6), \dot c_0=(-6,-2,2,6), ċ1=(0,0,1,0,1,1). \dot c_1=(0,0,1,0,1,1). For every opposite-edge pair p=A,B,Cp=A,B,C, λpS±=−4, \lambda_p^{S\pm}=-4, while Rp=52,Ṙp=+28. R_p=52, \qquad \dot R_p=+28. Therefore λpS±<0whileṘp>0 \boxed{ \lambda_p^{S\pm}<0 \quad\text{while}\quad \dot R_p>0 } on all three pair classes simultaneously. Summed over the full edge carrier, GΣ-rate=−12,Ṙtotal=84. G_\Sigma\text{-rate}=-12, \qquad \dot R_{\mathrm total}=84. Hence GΣS± and latent-witness change do not have a universal common sign. \boxed{ G_\Sigma^{S\pm} \text{ and latent-witness change do not have a universal common sign.} } AUTO_LOCK NO-GO SIGNED_EDGE_SUPPORT_FLOW_IS_NOT_LATENT_WITNESS_RATE_THEOREM_v0.1 6. Verdict The requested frontier partially closes: pure latent crossing + fixed integrity⇒lane-rank sign. \boxed{ \text{pure latent crossing + fixed integrity} \Rightarrow \text{lane-rank sign}. } But Q216−Q221⇏sgn(Nadd−Nsink)=sgnGΣS± \boxed{ Q216\!-\!Q221 \not\Rightarrow \operatorname{sgn}(N_{\mathrm add}-N_{\mathrm sink}) = \operatorname{sgn}G_\Sigma^{S\pm} } under current general grammar. The remaining OP03 bridge is still a constitutive relation between two differently typed source objects, not a hidden consequence of the latent witness. Physical promotion remains 0. UD_MANUSCRIPT_MILESTONE_ADDENDUM_LATENT_CROSSING_O09_SELECTOR_v0.1.mdUD Manuscript Milestone Addendum — Latent Crossing and O09 Source Selector v0.1 Date: 2026-09-14 Physical promotion: 0 1. Latent crossing result For a single pair with fixed passing integrity, Tp−=Tp+=1, T_p^-=T_p^+=1, the viable-lane source/sink algebra yields Nadd−Nsink=Lp+−Lp−. N_{\mathrm add}-N_{\mathrm sink} = L_p^+-L_p^-. Because Lp=𝟏[Rp>0], L_p=\mathbf1[R_p>0], a pure latent onset has positive rank flow and a pure extinction has negative rank flow: sgn(Nadd−Nsink)=sgn(Rp+−Rp−). \boxed{ \operatorname{sgn}(N_{\mathrm add}-N_{\mathrm sink}) = \operatorname{sgn}(R_p^+-R_p^-). } This theorem does not extend to arbitrary integrity transitions. Moreover, the instantaneous Ṙp\dot R_p vanishes at an exact smooth zero of the nonnegative witness, so finite history is required. The OP03 signed support object GΣS±G_\Sigma^{S\pm} is not the same derivative. An exact Hodge-flow witness has λpS±=−4\lambda_p^{S\pm}=-4 and Ṙp=+28\dot R_p=+28 on each pair simultaneously. Therefore the OP03 support-flow/rank-flow sign bridge remains open. 2. O09 source-selection result The current O09 action operates on lifted loop/Hodge history. It cannot compare burden-gradient, boundary, defect, and transport source branches unless each branch has a canonical pre-event source-to-Hodge map with fixed normalization. Without a fixed normalization, branch action ordering can be reversed by rescaling the source injection. Current Q250/Q252 no-gos additionally block reusing response-blind or post-event action as a pre-event selector. The lawful interim object is a typed source-action packet, not a scalar weighted sum. 3. Governance AUTO_LOCK ONE_PAIR_FIXED_INTEGRITY_LATENT_CROSSING_FIXES_LANE_RANK_SIGN_THEOREM_v0.1 AUTO_LOCK NO-GO INSTANTANEOUS_Q216_AT_EXACT_LATENT_ZERO_CANNOT_SELECT_CROSSING_POLARITY_v0.1 AUTO_LOCK NO-GO SIGNED_EDGE_SUPPORT_FLOW_IS_NOT_LATENT_WITNESS_RATE_THEOREM_v0.1 AUTO_LOCK O09_SOURCE_SELECTION_REQUIRES_CANONICAL_PRE_EVENT_SOURCE_TO_HODGE_INTERTWINERS_THEOREM_v0.1 AUTO_LOCK NO-GO UNNORMALIZED_SOURCE_TO_HODGE_MAPS_CAN_REVERSE_PAIRING_SOURCE_ACTION_RANKING_NO_GO_v0.1 AUTO_FREEZE ARBITRARY_WEIGHTED_SUM_OF_TYPED_PAIRING_SOURCE_ACTIONS_USED_AS_SELECTOR_v0.1 OPEN OP03 support-flow ↔\leftrightarrow rank-flow sign bridge; canonical source-to-Hodge maps for pairing source branches; common source-action comparison normalization; physical bridge. Physical promotion remains 0. UD_O09_SOURCE_SELECTING_PAIRING_ACTION_AUDIT_v0.1.mdUD O09 Source-Selecting Pairing Action Audit v0.1 Date: 2026-09-14 Status: INTERFACE_THEOREM_PASS / CURRENT_SELECTOR_NO_GO Physical promotion: 0 0. Objective Test whether the current mainline lifted loop-history action ℐloop→Hodge \mathcal I_{\mathrm loop\to Hodge} can be extended into a source-selecting action that discriminates burden-gradient, defect, boundary, and transport A2A_2-compatible pairing sources without violating existing action and causal-order no-gos. 1. Current O09 action object The current mainline action map is ℐlift=12∑ω>0aωΔϕ̃ω. \boxed{ \mathcal I_{\mathrm lift} = \frac12 \sum_{\omega>0} a_\omega\, \Delta\widetilde\phi_\omega. } For a closed history, ℐclosed=π∑ω>0nωaω. \boxed{ \mathcal I_{\mathrm closed} = \pi\sum_{\omega>0}n_\omega a_\omega. } O09 is temporally additive and covariant under the declared orthogonal Hodge intertwiners. It is a loop/Hodge history object, not a generic source-branch scalar. 2. Necessary source-selection interface Let pairing-source branch α\alpha have native typed source carrier Xα. X_\alpha. To compare branch α\alpha using O09, one first needs a pre-event map Jα:Xα→ℋO09 \boxed{ J_\alpha: X_\alpha \longrightarrow \mathcal H_{\mathrm O09} } into the exact lifted Hodge-history carrier consumed by O09. A lawful selector therefore requires all of: typed source map JαJ_\alpha; pre-event causality — JαXαJ_\alpha X_\alpha exists before the event being selected; canonical normalization — no free branch scale; carrier covariance under the branch and Hodge symmetries; common comparison type across branches; derived optimization direction — minimization/maximization cannot be chosen post hoc. Only after those gates pass may one define ℐα=ℐO09(JαXα) \mathcal I_\alpha = \mathcal I_{\mathrm O09}(J_\alpha X_\alpha) and compare branches. AUTO_LOCK O09_SOURCE_SELECTION_REQUIRES_CANONICAL_PRE_EVENT_SOURCE_TO_HODGE_INTERTWINERS_THEOREM_v0.1 3. Why current O09 cannot yet select the source Current source branches are typed on different carriers: burden-gradient pairing source: b→WpBG→A2⊗ℝ; b\to W_p^{BG}\to A_2\otimes\mathbb R; signed transport/support: C1C_1/Hasse-flow receipts; radial boundary anisotropy: retained mm-operator sector; defect/obstruction candidates: their own precursor carriers. No current theorem supplies canonical maps JBG,Jsupport,J∂,Jdefect J_{BG},\quad J_{\mathrm support},\quad J_{\partial},\quad J_{\mathrm defect} from all of these carriers into O09’s lifted Hodge-history input with fixed relative normalization. Therefore O09 has no current common branch-action comparison domain. 4. Normalization nonidentifiability theorem Suppose one introduces a linear source-to-Hodge map by hand. For branch α\alpha, Jα↦cαJα,cα>0. J_\alpha\mapsto c_\alpha J_\alpha, \qquad c_\alpha>0. Since O09 is homogeneous in the injected mode amplitudes aωa_\omega at fixed lifted phase history, ℐα↦cαℐα. \mathcal I_\alpha \mapsto c_\alpha\mathcal I_\alpha. Thus branch ordering is not invariant under unsourced branch normalization. Exact witness: (ℐ1,ℐ2)=(1,2) (\mathcal I_1,\mathcal I_2)=(1,2) gives 1<2. 1<2. Choose equally admissible unnormalized source-map scales (c1,c2)=(3,1). (c_1,c_2)=(3,1). Then (c1ℐ1,c2ℐ2)=(3,2), (c_1\mathcal I_1,c_2\mathcal I_2) = (3,2), and the ranking reverses: 3>2. 3>2. Therefore an action ranking without canonical cross-branch normalization is nonidentifiable. \boxed{ \text{an action ranking without canonical cross-branch normalization is nonidentifiable.} } AUTO_LOCK NO-GO UNNORMALIZED_SOURCE_TO_HODGE_MAPS_CAN_REVERSE_PAIRING_SOURCE_ACTION_RANKING_NO_GO_v0.1 5. Existing action no-gos remain active The current response audit already locks: Jν=π∥Pνx∥2 J_\nu=\pi\|P_\nu x\|^2 has no q/Hq/H response argument. Therefore current canonical Hodge action is response-blind. The same audit locks lifted loop action and event contraction as source-late for pre-event response selection. Thus adding a gate mask or using a post-event action difference to choose the source would create a new source law and violate causal ordering. AUTO_LOCK NO-GO CURRENT_O09_ACTION_CANNOT_BE_RETROFITTED_AS_PRE_EVENT_PAIRING_SOURCE_SELECTOR_v0.1 6. Lawful development path: typed action bundle A lawful extension can preserve provenance without falsely scalarizing the branches. Define a source-action packet ℑsrc=(ℐO09,ℐBG,ℐ∂,ℐdefect,ℐtransport), \boxed{ \mathfrak I_{\mathrm src} = \left( \mathcal I_{\mathrm O09}, \mathcal I_{BG}, \mathcal I_{\partial}, \mathcal I_{\mathrm defect}, \mathcal I_{\mathrm transport} \right), } where each component remains undefined until its own native source-to-action map is independently derived. This is a typed direct-product ledger, not an additive scalar action. Do not write ℐtotal=ℐO09+cBGℐBG+c∂ℐ∂+⋯ \mathcal I_{\mathrm total} = \mathcal I_{\mathrm O09} + c_{BG}\mathcal I_{BG} + c_\partial\mathcal I_\partial+\cdots unless every coefficient and unit map is source-derived. AUTO_LOCK PAIRING_SOURCE_ACTION_PACKET_MUST_PRESERVE_BRANCH_PROVENANCE_UNTIL_COMMON_ACTION_MAP_IS_DERIVED_v0.1 AUTO_FREEZE ARBITRARY_WEIGHTED_SUM_OF_TYPED_PAIRING_SOURCE_ACTIONS_USED_AS_SELECTOR_v0.1 7. Current verdict The attempted source-selecting action does not close as a unique scalar law. What closes is the exact interface theorem: source selection by O09⇒canonical pre-event source-to-Hodge maps + common normalization. \boxed{ \text{source selection by O09} \Rightarrow \text{canonical pre-event source-to-Hodge maps + common normalization}. } Those maps are currently absent. Therefore O09 source-selecting pairing action=OPEN. \boxed{ \text{O09 source-selecting pairing action}=\text{OPEN}. } The burden-gradient branch remains the current executable source by source completeness, not because O09 minimizes or selects it. Physical promotion remains 0. UD_PROVENANCE_LEDGER_ADDENDUM_LATENT_CROSSING_O09_SELECTOR_v0.1.mdUD Provenance Ledger Addendum — Latent Crossing / O09 Selector v0.1 Date: 2026-09-14 Physical promotion: 0 ID Object Exact form Claim bin Status Next gate LC01 one-pair pure latent crossing sign fixed T=1T=1: sgn(Nadd−Nsink)=sgnΔRp\operatorname{sgn}(N_{\mathrm add}-N_{\mathrm sink})=\operatorname{sgn}\Delta R_p EXACT_CONDITIONAL AUTO_LOCK extend only with new integrity/source theorem LC02 instantaneous Q216Q216 zero-crossing polarity Rp=0⇒Ṙp=0R_p=0\Rightarrow\dot R_p=0 for smooth nonnegative witness LOCKED_NO_GO AUTO_LOCK use one-sided/finite history LC03 integrity-only rank changes ΔL=0\Delta L=0 can still give A−S=±1A-S=\pm1 LOCKED_NO_GO AUTO_LOCK preserve Q219 typing LC04 OP03 signed support vs latent witness rate no universal sign identity; exact opposite-sign Hodge witness LOCKED_NO_GO AUTO_LOCK independent sign bridge AS01 O09 source-selector interface canonical pre-event Jα:Xα→ℋO09J_\alpha:X_\alpha\to\mathcal H_{\mathrm O09} required EXACT_INTERNAL AUTO_LOCK derive branch maps AS02 source-map normalization arbitrary scales can reverse action ranking LOCKED_NO_GO AUTO_LOCK canonical normalization AS03 O09 as pre-event selector current action response-blind/source-late LOCKED_NO_GO AUTO_LOCK derive new pre-event source-action law AS04 typed source-action packet preserve branch provenance; no scalar sum yet LOCKED_SOURCE_STRUCTURE AUTO_LOCK common action map and selector