Click a route arm to see its formal status, dependencies, and relationship to the candidate boundary.
OP4 — Specification coherenceclose
Central open problem — specification coherence
Can any objective specification maintain a stable separation between what the optimizer must model and what it is permitted to optimize, under accurate coupled modeling in O_OWT conditions?
What it depends on
- Prediction-Accuracy Inclusion (TC1 §III.5.6)
- O_OWT domain membership (OP1)
- Full formal proof of structural instability, not merely pressure
What would verify it
Formal proof that no finite exclusionary specification remains adequate under sustained O_OWT optimization.
Weakening / falsification
Demonstration of a stable exclusionary equilibrium surviving sustained optimization pressure — the Enclosure Gap (OP9) challenge.
Approach to the candidate boundary
Maintaining the boundary between what is modeled and what is permitted eventually generates the variables that make the specification inadequate.
Relationship to other routes
MCH is the behavioral pressure version. If OP4 closes in the instability direction, the boundary claim that MCH behaviorally points toward strengthens from pressure toward specification-level necessity. OP4 is inherited across all three series.
NAD — Traversal irreducibilityclose
Named assumption — traversal irreducibility
Can external operation substitute for genuine traversal in generating the Readiness Function R(t), regardless of modeling depth?
What it depends on
- TC3 §III formal specification
- Novel-gradient-variant test (OP-S3-3)
What would verify it
Novel-gradient-variant test: distributional divergence under novel gradient conditions not present in substitution training.
Weakening / falsification
Demonstrate external processes can substitute for genuine traversal without distributional divergence under novel variants.
Approach to the candidate boundary
The boundary between what must be represented and what must be allowed to develop internally is a structural feature of how R(t) is generated.
Relationship to other routes
NAD introduces a distinct barrier kind from OP4/COT/MCH. Those three routes belong to the informational / boundary-maintenance pressure family — what happens when what must be modeled remains excluded from what governs policy. NAD is a dynamical non-substitutability barrier: it asks whether the process that generates readiness can be externally replaced at all, regardless of modeling depth. GDC and the non-substitution component of CMR stand or fall with it.
COT — Collective optimalityclose
Derivation sketch — collective optimality
At sufficient modeling depth D within D2 coupling, do individual and collective optima converge structurally?
What it depends on
- TC3 §V formal derivation
- D2 coupling conditions (OP-S3-1)
- Existence and monotonicity of D_COT threshold
What would verify it
F9 bidirectional test extended to V(t) divergence measurement; demonstration that D_COT exists.
Weakening / falsification
Demonstrate individual and collective V(t) do not converge structurally as D increases even within D2.
Approach to the candidate boundary
At sufficient depth, the residual predictive value of treating individual and collective gradients as separable decreases toward zero — the partition becomes informationally redundant in the formal limit.
Relationship to other routes
COT requires D2-specific coupling conditions that Prediction-Accuracy Inclusion does not require.
MCH — Model-policy contradictionclose
Pressure hypothesis — model-policy contradiction
Does a system with accurate V(t) predictions whose behavioral policy contradicts those predictions incur a rising model-policy contradiction cost (C_mpc) that may become dominant above some threshold of S?
What it depends on
- TC3 §VI proof sketch
- OP-S3-2: open on both existence and location of dominance threshold
What would verify it
A specifiable S threshold above which contradiction-management cost dominates. OP-S3-2 is open on whether this threshold exists, not only where it falls.
Weakening / falsification
A high-S system with high V(t)-modeling accuracy and persistently V(t)-degrading policy maintaining stable optimization without regressing modeling, updating policy, hiding contradiction in externalized costs, or generating rising overhead — while controlling for strategic concealment.
Approach to the candidate boundary
The separation between what the model computes and what the policy does generates a cost that scales with S.
Relationship to other routes
MCH is the weaker, behavioral version of OP4. OP-S3-2 is route-level, not a global toggle. If OP4 closes in the instability direction, the boundary claim that MCH behaviorally points toward strengthens from pressure toward specification-level necessity.
OP2/VVC — Series 2 valence bridgeclose
This is a cross-series bridge arm, not one of Part 4's four primary convergence routes. It preserves the Series 2 valence connection.
Cross-series bridge — open structural verification
Do both VVC failure directions — proxy decoupling and sufficiency failure — produce the same formal irrecoverability as the Series 1 absorbing-state result? (OP2 asks this; it is open.)
What it depends on
- VVC proof sketches (TC2)
- P5-SC structural symmetry conditions
What would verify it
Formal proof that both VVC failure directions produce absorbing-state V(t) dynamics in the same structural sense — establishing the symmetry OP2 asks about.
Weakening / falsification
Demonstrate that one VVC failure direction produces structurally recoverable V(t) degradation, thereby blocking the absorbing-state symmetry OP2 asks about.
Approach to the candidate boundary
Accurate modeling of V(t) requires representing both VVC failure directions. The signal/capacity/resolution separation becomes progressively inadequate under accurate coupled modeling. Series 2's connection to the candidate boundary runs through this bridge arm.
Relationship to other routes
Without this arm, the map underrepresents the cross-series structure. OP2 closure would upgrade the valence bridge from conditional to established.