Open Theorem And Observer-Conditioned Roadmap

Status snapshot: June 2026 proof-spine and observer-conditioned research route.

This page concentrates two routes that are otherwise spread across the preprint, Appendix A, Appendix C, Appendix H, the review protocol, and the public replication package.

The Open Theorem Roadmap is CCT's formal proof spine: bounded theorem targets, verifier repairs, proof obligations, counterexample searches, and method artifacts. Observer-conditioned physics is the long-horizon theory track, called Layer 3 in ontology-specific docs: calibration transport, observer/compiler structure, boundary/interface behavior, effective adjacency, OP0 specificity filters, and stability-depth questions.

OP labels are roadmap handles inside that spine. They name open theorem/problem lanes, while a theorem note, verifier, scaffold, schema, route surface, or review packet from that lane carries its own earned status. This lets a lane remain active while a bounded piece of it becomes inspectable and reviewable.

The observer-conditioned roadmap is the broader long-horizon theory route. Some observer-conditioned questions become OP lanes, while others first appear as calibration rows, ledgers, state/coherence objects, effective-adjacency capsules, mission-architecture templates, or Tau-X resource questions before they become theorem targets.

Both routes are active parts of the same staged program. They generate formal objects, simulations, ledgers, null routes, and review gates before any stronger interpretation is promoted.

Current Implemented Method Artifacts

Several theorem-roadmap and observer-conditioned targets have crossed into public-safe method artifacts:

Track Current public artifact What it establishes
BT3/BT5 command attribution / OP3 Repaired verifier route plus OP3 route-classifier and proof draft for memoryless and history-conditioned command attribution. Raw actuator-output influence no longer substitutes for controller-attributable influence; hidden predictors, shared clocks, state sufficiency, joint capacity, denominator scope, and task-functional promotion now have explicit route guards.
Scalar OP4/BT4 Declared-envelope verifier with square-root toy case plus gamma, log/sublinear, saturating, threshold, denominator, and hidden-resource diagnostics. Resource-envelope claims have a broader synthetic discriminator than the original toy case.
Vector OP4 Multi-resource simulator, scalarized route cases, finite resource-front verifier, manifest freeze, and post-review confirmation. Accepted finite/discrete V1 resource-front language is available for pure-row, gate-passing strategies with positive resource costs and declared policies; broader continuous, mixture, hardware, and architecture interpretations stay on later routes.
BT6 Basin/path-measure verifier, finite-state discriminator, finite-sample terminal/coarse KL interval diagnostics, and finite-support KL concentration/perturbation proof draft. Basin movement is routed through declared support, kernel/path ledgers, incumbent routes, denominator checks, interval diagnostics, and narrow accepted V1 finite-support KL language.
OP2 Logged-dependence estimator, randomized holdout route surface, finite-sample holdout-delta interval diagnostics, and exact conditional tau-grid randomization-inference proof draft. Observation-to-control claims are separated from observation quality, command-effect sublemma, hidden-channel, denominator, incumbent, holdout-integrity gates, and narrow accepted V1 tau-grid language.
BT7b Passive aperture/operator-norm verifier, proof-review stub, and finite-dimensional theorem-status review. Exact-linear-response amplitude/operator-norm language is available in the finite-dimensional V1 scope; boundary classes inherit hidden-gain, incident-power, resonance, near-field, nonlinear, and denominator routes.
Scalar multiwell anti-uniqueness / OP0a Scalar theorem note, synthetic route companion, and post-specialist formal-review record. Hierarchy-like basin counts and local curvature matches become specificity-boundary objects: expressivity is visible, and selection requires stronger filters.
QFT-data specificity-filter scaffold / OP0b Public-safe Phi(C,[x_*]) -> QFTData schema, route examples, route-semantics checker, equivalence-audit fixtures, and QFT / representation scaffold-review clearance. Standard-Model-facing specificity is routed through source-object completeness, equivalence invariance, field-status discipline, null/incumbent closure, compression/holdout checks, and review gates.
Regime-local RFH metrology envelope / OP1 Accepted algebraic finite-window envelope, manifest examples, finite-window uncertainty sidecar, Bridge V1 Fisher-certificate sufficient-condition note, proof draft, formal metrology/probability review, and Bridge V1 certificate verifier. Measurement-scaling claims are routed by declared resource class, estimator policy, back-action/disturbance envelope, hidden-resource handling, finite-window uncertainty discipline, and machine-auditable Bridge V1 route fixtures in the narrow V1a/V1b scope.
Calibration holonomy Public-safe closed-loop retuning capsule. Calibration transport is expressed as loop rows with ordinary drift, hysteresis, estimator-offset, repeatability, and incumbent routes.
Effective adjacency and Tau-X ledgers Effective-neighborhood, effective-adjacency object-family, feedback-cycle timing, environmental-handle, state/coherence payload, support-decomposition, H2B passive-boundary support, and H2B-C2 broad-group support-reducer route artifacts. Space-and-motion intuition is translated into reachability, propagation, reconstruction, correction, timing, environmental, mission-architecture, support-accounting, and resource-ledger rows, with C3 evidence-bearing witness work as the next Tau-X route object.

These are method-validation, branch-narrowing, proof-review, and Tau-X architecture / resource-ledger artifacts. Their job is to make assumptions, routes, and failure modes inspectable.

Active Formal Queue

The open queue remains live:

Item Current role Next burden
Scalar multiwell anti-uniqueness / OP0a Scalar theorem object for hierarchy expressivity and anti-uniqueness. Move from post-specialist scalar theorem-readiness into accepted scalar theorem text; keep high-dimensional R^m language behind relative Morse / handle-construction lemmas.
QFT-data specificity-filter scaffold / OP0b Public-safe scaffold for Phi(C,[x_*]) -> QFTData review. Build future non-synthetic target-data routes only after the equivalence relation, complexity denominator, anomaly/locality/unitarity/Lorentz/RG/topology payloads, and target-leakage policy are frozen.
OP1 Regime-local RFH theorem under physical constraints. Use accepted algebraic finite-window language in the structural R=A+D scope; Bridge V1a Fisher-certificate and V1b direct all-pair R replacement routes now have public-safe certificate fixtures; statistical coverage, QFI, nuisance, and dependence bridges stay separate.
OP2 Observation-to-control bridge. Use narrow exact conditional tau-grid V1 language in the randomized holdout scope; develop command-effect sublemma paths, sensing/compute ledgers, normalized-ratio routes, and architecture-specific corollaries separately.
OP3 Forbidden designs and attribution limits. Route-classifier and formal-review-ready proof-route artifacts are ready for command-attribution review; next objects are theorem handoff, route-schema hardening, and history-conditioned capacity/hidden-channel proof obligations.
OP4 Vector no-free-lunch resource accounting. Use accepted finite/discrete resource-front V1 language in the declared pure-row scope; keep continuous fronts, mixtures, stronger Pareto/resource-theory language, hardware, and architecture interpretation separate.
BT6 Basin/path-measure extension. Use narrow finite-support terminal/coarse KL concentration/perturbation V1 language; develop dependent-sample, continuous-time, diffusion, capacity-selection, denominator-ratio, and branch/bench interpretation corollaries separately.
BT7b Passive boundary/operator-norm theorem. Use finite-dimensional exact-linear-response amplitude/operator-norm language; develop the Born-linearized scalar Helmholtz/Green-function theorem and broader boundary taxonomy next.
Calibration transport Observer-conditioned effective-law bridge. Formal transport object, repeatability routes, perturbation controls, and calibration/metrology review.
Boundary grammar Passive/active/resonant/nonlinear/hidden-gain taxonomy. Build after BT7b so boundary claims inherit a theorem-ready norm and denominator policy.

Observer-Conditioned Physics Route

Observer-conditioned physics is the theory track that asks how finite observers, instruments, controllers, calibration regimes, and environments condition what becomes stable, legible, and steerable.

In first-touch docs, this is usually called observer-conditioned physics. In ontology-specific docs, it is called Layer 3. The difference is mainly audience and genre: the public review route uses operational language; the ontology route carries the deeper interpretation.

The current public route is:

  1. generate the formal candidate or discriminator;
  2. build a public-safe simulator, theorem stub, ledger, or route surface;
  3. run incumbent, null, denominator, support, and hidden-channel checks;
  4. seek specialist review for the formal or method object;
  5. promote the claim class that the artifact has earned.

This keeps the long-horizon theory active while assigning each artifact to the claim class it has earned.

Recent quantum-information work on holographic codes and quantum "magic" is useful for the observer-conditioned roadmap. Here, quantum "magic" means a technical non-Clifford / non-stabilizer resource, not a CCT keyword.

The relevant lesson is resource accounting. In those code models, entanglement alone can make geometry-like connectivity too rigid, while non-stabilizer resources help distinguish fixed-background code behavior from state-dependent geometry-like response. CCT reads this as a related theory signal: effective-geometry claims should track reconstruction resources, approximation class, recovery error, encoding complexity, and hidden target insertion, not just information volume.

That signal touches four CCT routes:

  • Observer-conditioned / Layer 3: reconstruction class and code resources can matter for what geometry-like structure is recoverable.
  • OP0b specificity filters: derived structure must be separated from target structure inserted through a powerful code, equivalence choice, projection, or recovery policy.
  • Vector OP4 resource accounting: non-energy resource axes such as non-stabilizer cost, recovery error, approximation burden, and encoding complexity may need explicit ledgers.
  • Tau-X: state/coherence and effective-adjacency questions can borrow this resource vocabulary, while mission primitives and capability claims remain on the evidence-bearing route.

Hardware And Bench Connection

The theorem and observer-conditioned tracks are not separate from hardware. They stage hardware exposure by specifying what a physical test is being asked to decide.

The path is:

  • formal targets define assumptions and denominators;
  • public verifiers and synthetic capsules make route behavior inspectable;
  • simulations identify operating regions and fragile assumptions;
  • preregistration locks controls, baselines, nulls, and decision labels;
  • benches test selected regimes against real instruments, materials, drift, noise, and replication;
  • results feed back into the theorem queue and observer-conditioned track.

Hardware therefore appears as the physical exposure layer of the roadmap, not as an afterthought.