UOR Observables
- IRI
https://uor.foundation/observable/- Prefix
observable:- Space
- bridge
- Comment
- Observable quantities and metrics computed by the UOR kernel. Includes ring-metric, Hamming-metric, curvature, holonomy, and catastrophe-theoretic observables.
Imports
https://uor.foundation/op/https://uor.foundation/schema/https://uor.foundation/partition/https://uor.foundation/type/
Classes
| Name | IRI | Subclass Of | Disjoint With | Comment |
|---|---|---|---|---|
| Observable | https://uor.foundation/observable/Observable | http://www.w3.org/2002/07/owl#Thing | A measurable quantity in the UOR Framework. All observables are kernel-computed and user-consumed. | |
| StratumObservable | https://uor.foundation/observable/StratumObservable | https://uor.foundation/observable/Observable | An observable measuring stratum-level properties: position within the ring's layer structure. | |
| MetricObservable | https://uor.foundation/observable/MetricObservable | https://uor.foundation/observable/Observable | An observable measuring geometric distance between ring elements under a specific metric. | |
| PathObservable | https://uor.foundation/observable/PathObservable | https://uor.foundation/observable/Observable | An observable measuring properties of paths through the ring: path length, total variation, winding number. | |
| ReductionObservable | https://uor.foundation/observable/ReductionObservable | https://uor.foundation/observable/Observable | An observable measuring reduction properties: the length and count of operation sequences. | |
| CatastropheObservable | https://uor.foundation/observable/CatastropheObservable | https://uor.foundation/observable/Observable | An observable measuring catastrophe-theoretic properties: thresholds at which qualitative changes occur in the partition. | |
| CurvatureObservable | https://uor.foundation/observable/CurvatureObservable | https://uor.foundation/observable/Observable | An observable measuring the curvature of the UOR geometry: the gap between ring-isometry and Hamming-isometry for a given transform. | |
| HolonomyObservable | https://uor.foundation/observable/HolonomyObservable | https://uor.foundation/observable/Observable | An observable measuring holonomy: the accumulated transformation when traversing a closed path in the ring. | |
| RingMetric | https://uor.foundation/observable/RingMetric | https://uor.foundation/observable/MetricObservable | Distance between two ring elements under the ring metric: d_R(x, y) = |x - y| mod 2^n. | |
| HammingMetric | https://uor.foundation/observable/HammingMetric | https://uor.foundation/observable/MetricObservable | Distance between two ring elements under the Hamming metric: the number of bit positions where they differ. | |
| IncompatibilityMetric | https://uor.foundation/observable/IncompatibilityMetric | https://uor.foundation/observable/MetricObservable | The metric incompatibility between two ring elements: the divergence between their ring-metric and Hamming-metric distances, measuring geometric curvature. | |
| StratumValue | https://uor.foundation/observable/StratumValue | https://uor.foundation/observable/StratumObservable | The stratum index of a ring element. | |
| StratumDelta | https://uor.foundation/observable/StratumDelta | https://uor.foundation/observable/StratumObservable | The difference in stratum between two ring elements. | |
| StratumTrajectory | https://uor.foundation/observable/StratumTrajectory | https://uor.foundation/observable/StratumObservable | The sequence of strata traversed by a path through the ring. | |
| PathLength | https://uor.foundation/observable/PathLength | https://uor.foundation/observable/PathObservable | The length of a path through the ring, measured in operation steps. | |
| TotalVariation | https://uor.foundation/observable/TotalVariation | https://uor.foundation/observable/PathObservable | The total variation of a path: the sum of metric distances between consecutive elements. | |
| WindingNumber | https://uor.foundation/observable/WindingNumber | https://uor.foundation/observable/PathObservable | The winding number of a closed path: the number of times the path wraps around the ring. | |
| ReductionLength | https://uor.foundation/observable/ReductionLength | https://uor.foundation/observable/ReductionObservable | The number of operation applications in a reduction sequence. | |
| ReductionCount | https://uor.foundation/observable/ReductionCount | https://uor.foundation/observable/ReductionObservable | The number of distinct reduction sequences in a computation. | |
| CatastropheThreshold | https://uor.foundation/observable/CatastropheThreshold | https://uor.foundation/observable/CatastropheObservable | A critical value at which a qualitative change occurs in the partition structure. | |
| CatastropheCount | https://uor.foundation/observable/CatastropheCount | https://uor.foundation/observable/CatastropheObservable | The number of catastrophe events (qualitative partition changes) in a computation. | |
| Commutator | https://uor.foundation/observable/Commutator | https://uor.foundation/observable/CurvatureObservable | The commutator [f, g](x) = f(g(x)) - g(f(x)) of two operations, measuring their non-commutativity. | |
| CurvatureFlux | https://uor.foundation/observable/CurvatureFlux | https://uor.foundation/observable/CurvatureObservable | The integrated curvature over a region of type space: the total metric incompatibility accumulated. | |
| Monodromy | https://uor.foundation/observable/Monodromy | https://uor.foundation/observable/HolonomyObservable | The monodromy of a closed path: the net transformation accumulated when traversing a loop in the type space. | |
| ParallelTransport | https://uor.foundation/observable/ParallelTransport | https://uor.foundation/observable/HolonomyObservable | The parallel transport of a vector along a path: the canonical lift of the path to the tangent bundle of the ring. | |
| DihedralElement | https://uor.foundation/observable/DihedralElement | https://uor.foundation/observable/HolonomyObservable | An element of the dihedral group D_{2^n} acting on the type space. Each dihedral element induces an isometry of 𝒯_n. | |
| MeasurementUnit | https://uor.foundation/observable/MeasurementUnit | http://www.w3.org/2002/07/owl#Thing | A unit of measurement for observable quantities. Each MeasurementUnit individual names a specific unit (bits, ring steps, dimensionless) replacing the string-valued observable:unit property. | |
| Jacobian | https://uor.foundation/observable/Jacobian | https://uor.foundation/observable/CurvatureObservable | Site-by-site curvature decomposition. J_k measures the discrete derivative of the incompatibility metric at site position k: J_k = |d_R(x, succ(x)) - d_H(x, succ(x))| restricted to position k. | |
| TopologicalObservable | https://uor.foundation/observable/TopologicalObservable | https://uor.foundation/observable/Observable | An observable measuring a topological invariant of the resolution space. Topological observables are invariant under continuous deformations of the constraint configuration. | |
| BettiNumber | https://uor.foundation/observable/BettiNumber | https://uor.foundation/observable/TopologicalObservable | The rank of a homology group of the constraint nerve. β_k = rank(H_k(N(C))) counts the k-dimensional holes in the constraint configuration. | |
| SpectralGap | https://uor.foundation/observable/SpectralGap | https://uor.foundation/observable/TopologicalObservable | The smallest positive eigenvalue of the constraint nerve Laplacian. Controls the convergence rate of iterative resolution: larger gap = faster convergence. | |
| ThermoObservable | https://uor.foundation/observable/ThermoObservable | https://uor.foundation/observable/Observable | An observable measuring thermodynamic properties of the resolution process: residual entropy, Landauer cost, and reduction distribution statistics. | |
| ResidualEntropy | https://uor.foundation/observable/ResidualEntropy | https://uor.foundation/observable/ThermoObservable | S_residual: the residual Shannon entropy of the site distribution after partial resolution. Computed as S = (Σ κ_k − χ(N(C))) × ln 2 (IT_7b). Unit: Nats. | |
| LandauerCost | https://uor.foundation/observable/LandauerCost | https://uor.foundation/observable/ThermoObservable | The minimum thermodynamic cost (in units of k_B T ln 2) of erasing one bit of site uncertainty. The UOR ring operates at β* = ln 2 — the Landauer temperature. | |
| ReductionEntropy | https://uor.foundation/observable/ReductionEntropy | https://uor.foundation/observable/ThermoObservable | The Shannon entropy of the reduction distribution P(j) = 2^{−j}. At the Landauer temperature, this equals ln 2 per reduction step — each step erases exactly one bit of site uncertainty. | |
| SynthesisSignature | https://uor.foundation/observable/SynthesisSignature | http://www.w3.org/2002/07/owl#Thing | A named topological signature: a pair (realised Euler characteristic, realised Betti profile). Linked from TypeSynthesisResult. Allows comparison between the goal signature and the actually achieved signature. | |
| SpectralSequencePage | https://uor.foundation/observable/SpectralSequencePage | http://www.w3.org/2002/07/owl#Thing | A single page E_r of the quantum level spectral sequence. Carries the page index r and the differential d_r. The sequence converges when all differentials vanish — typically by E_3 for simple constraint configurations. | |
| LiftObstructionClass | https://uor.foundation/observable/LiftObstructionClass | http://www.w3.org/2002/07/owl#Thing | The cohomology class in H^2(N(C(T))) representing the LiftObstruction for a specific WittLift. The class is zero iff the obstruction is trivial. When non-zero, it indexes the specific site pair at Q_{n+1} that cannot be closed by the lifted constraint set alone. | |
| MonodromyClass | https://uor.foundation/observable/MonodromyClass | http://www.w3.org/2002/07/owl#Thing | A classification of a type's holonomy: the subgroup of D_{2^n} generated by all Monodromy observables computed over closed paths in the type's constraint nerve. Trivial iff every closed constraint path returns to its starting site assignment without net dihedral transformation. | |
| HolonomyGroup | https://uor.foundation/observable/HolonomyGroup | http://www.w3.org/2002/07/owl#Thing | The holonomy group of a ConstrainedType: the group of all Monodromy elements achievable by closed paths in the constraint nerve. Always a subgroup of D_{2^n}. Trivial iff the type has trivial monodromy everywhere; equals D_{2^n} iff paths involving both neg and bnot involutions are present. | |
| ClosedConstraintPath | https://uor.foundation/observable/ClosedConstraintPath | http://www.w3.org/2002/07/owl#Thing | A sequence of constraint applications forming a closed loop in the constraint nerve — beginning and ending at the same site assignment. The Monodromy of the loop is the net DihedralElement accumulated when traversing it. | |
| PhaseBoundaryType | https://uor.foundation/observable/PhaseBoundaryType | http://www.w3.org/2002/07/owl#Thing | A classification of phase boundary in the catastrophe diagram: period boundary (g divides 2^n − 1) or power-of-two boundary (g = 2^k). | |
| AchievabilityStatus | https://uor.foundation/observable/AchievabilityStatus | http://www.w3.org/2002/07/owl#Thing | The achievability classification of a topological signature in the morphospace. Either Achievable or Forbidden (witnessed by ImpossibilityWitness). | |
| HomotopyGroup | https://uor.foundation/observable/HomotopyGroup | http://www.w3.org/2002/07/owl#Thing | The k-th homotopy group πk(N(C), v) of the constraint nerve based at vertex v. | |
| HigherMonodromy | https://uor.foundation/observable/HigherMonodromy | http://www.w3.org/2002/07/owl#Thing | The image of πk(N(C)) → Aut(sitek) for k > 1. Generalises the MN_6 monodromy homomorphism. | |
| WhiteheadProduct | https://uor.foundation/observable/WhiteheadProduct | http://www.w3.org/2002/07/owl#Thing | The Whitehead product [α, β] ∈ πp+q−1 for α ∈ πp, β ∈ πq. | |
| StratificationRecord | https://uor.foundation/observable/StratificationRecord | http://www.w3.org/2002/07/owl#Thing | A record of the holonomy stratification of the moduli space at a given quantum level: the list of HolonomyStrata, their codimensions, and their relationship to the MorphospaceBoundary. | |
| BaseMetric | https://uor.foundation/observable/BaseMetric | https://uor.foundation/observable/Observable | Superclass for the six universal measurements. Every computation on the ring produces these six quantities: d_Δ, σ, J_k, β_k, χ, r. | |
| GroundingObservable | https://uor.foundation/observable/GroundingObservable | https://uor.foundation/observable/Observable | The grounding metric σ = pinned sites / total sites. Ranges from 0 (no sites pinned) to 1 (fully grounded). | |
| EulerCharacteristicObservable | https://uor.foundation/observable/EulerCharacteristicObservable | https://uor.foundation/observable/Observable | The Euler characteristic χ = Σ(−1)^k β_k of the constraint nerve. An integer-valued topological invariant. |
Properties
| Name | Kind | Functional | Domain | Range | Comment |
|---|---|---|---|---|---|
| value | Datatype | true | https://uor.foundation/observable/Observable | http://www.w3.org/2001/XMLSchema#decimal | The numeric value of an observable measurement. |
| source | Object | true | https://uor.foundation/observable/Observable | http://www.w3.org/2002/07/owl#Thing | The source object of this measurement (datum, partition, or path start point). |
| target | Object | true | https://uor.foundation/observable/Observable | http://www.w3.org/2002/07/owl#Thing | The target object of this measurement (for metrics and path-end measurements). |
| hasUnit | Object | true | https://uor.foundation/observable/Observable | https://uor.foundation/observable/MeasurementUnit | The measurement unit of this observable. Replaces the string-valued observable:unit property with a typed reference to a MeasurementUnit individual. |
| sitePosition | Datatype | true | https://uor.foundation/observable/Jacobian | http://www.w3.org/2001/XMLSchema#nonNegativeInteger | The site position k at which this Jacobian entry is measured. |
| derivativeValue | Datatype | true | https://uor.foundation/observable/Jacobian | http://www.w3.org/2001/XMLSchema#decimal | The discrete derivative value at this site position. |
| dimension | Datatype | true | https://uor.foundation/observable/TopologicalObservable | http://www.w3.org/2001/XMLSchema#nonNegativeInteger | The dimension k of the topological observable (e.g., the degree of the Betti number or the dimension of the spectral gap). |
| realisedEuler | Datatype | true | https://uor.foundation/observable/SynthesisSignature | http://www.w3.org/2001/XMLSchema#integer | The Euler characteristic actually achieved by this synthesis signature. |
| realisedBetti | Datatype | false | https://uor.foundation/observable/SynthesisSignature | http://www.w3.org/2001/XMLSchema#nonNegativeInteger | Non-functional. Realised Betti number values, one assertion per homological degree. |
| pageIndex | Datatype | true | https://uor.foundation/observable/SpectralSequencePage | http://www.w3.org/2001/XMLSchema#nonNegativeInteger | The page r of this spectral sequence page. r=1 is the initial page; convergence is declared when all d_r are zero. |
| differentialIsZero | Datatype | true | https://uor.foundation/observable/SpectralSequencePage | http://www.w3.org/2001/XMLSchema#boolean | True iff d_r = 0 on this page — no further corrections to the lifted homology. |
| convergedAt | Datatype | true | https://uor.foundation/observable/SpectralSequencePage | http://www.w3.org/2001/XMLSchema#nonNegativeInteger | The page index r at which the spectral sequence converged (all subsequent differentials zero). |
| obstructionClass | Object | true | https://uor.foundation/observable/LiftObstructionClass | https://uor.foundation/cohomology/CohomologyGroup | The cohomology class in H^2(N(C(T))) representing this obstruction. |
| monodromyLoop | Object | true | https://uor.foundation/observable/Monodromy | https://uor.foundation/observable/ClosedConstraintPath | The closed path that generates this monodromy value. |
| monodromyElement | Object | true | https://uor.foundation/observable/Monodromy | https://uor.foundation/observable/DihedralElement | The dihedral element g in D_{2^n} accumulated when traversing the loop. The monodromy is trivial iff this element is the group identity. |
| isTrivialMonodromy | Datatype | true | https://uor.foundation/observable/Monodromy | http://www.w3.org/2001/XMLSchema#boolean | True iff the monodromyElement is the identity in D_{2^n}. |
| holonomyGroup | Object | false | https://uor.foundation/observable/HolonomyGroup | https://uor.foundation/observable/DihedralElement | Non-functional. The generators of the holonomy group: one DihedralElement per generating monodromy. |
| holonomyGroupOrder | Datatype | true | https://uor.foundation/observable/HolonomyGroup | http://www.w3.org/2001/XMLSchema#positiveInteger | The order of the holonomy group as a subgroup of D_{2^n}. For a FlatType: 1. For full dihedral holonomy: 2^{n+1}. |
| pathLength | Datatype | true | https://uor.foundation/observable/ClosedConstraintPath | http://www.w3.org/2001/XMLSchema#nonNegativeInteger | The number of constraint application steps in this closed path. |
| pathConstraints | Object | false | https://uor.foundation/observable/ClosedConstraintPath | https://uor.foundation/type/Constraint | Non-functional. The ordered sequence of constraints traversed. One assertion per step. |
| dihedralElementValue | Object | false | https://uor.foundation/observable/DihedralElement | https://uor.foundation/op/Operation | Non-functional. One assertion per generator in the normal form of the element — the sequence of neg and/or bnot operations that realises this dihedral element when composed. |
| isIdentityElement | Datatype | true | https://uor.foundation/observable/DihedralElement | http://www.w3.org/2001/XMLSchema#boolean | True iff this element is the group identity (the trivial monodromy value). |
| elementOrder | Datatype | true | https://uor.foundation/observable/DihedralElement | http://www.w3.org/2001/XMLSchema#positiveInteger | The order of this element in D_{2^n}: the smallest k >= 1 such that g^k = id. For neg and bnot: order 2. |
| hardnessEstimate | Datatype | true | https://uor.foundation/observable/ThermoObservable | http://www.w3.org/2001/XMLSchema#decimal | An estimated computational hardness for a ThermoObservable, connecting thermodynamic cost to complexity (TH_9 realisation). |
| phaseN | Datatype | true | https://uor.foundation/observable/CatastropheObservable | http://www.w3.org/2001/XMLSchema#positiveInteger | The ring dimension coordinate n in the (n, g) catastrophe phase diagram (PD_1 n-coordinate). |
| phaseG | Datatype | true | https://uor.foundation/observable/CatastropheObservable | http://www.w3.org/2001/XMLSchema#positiveInteger | The group-order coordinate g in the (n, g) catastrophe phase diagram (PD_1 g-coordinate). |
| onResonanceLine | Datatype | true | https://uor.foundation/observable/CatastropheObservable | http://www.w3.org/2001/XMLSchema#boolean | True when g divides 2^n − 1, placing this observable on a resonance line in the phase diagram (PD_4). |
| phaseBoundaryType | Object | true | https://uor.foundation/observable/CatastropheObservable | https://uor.foundation/observable/PhaseBoundaryType | The type of phase boundary at this point in the catastrophe diagram: PeriodBoundary or PowerOfTwoBoundary (PD_2). |
| achievabilityStatus | Object | true | https://uor.foundation/observable/SynthesisSignature | https://uor.foundation/observable/AchievabilityStatus | The achievability classification of this observable's topological signature in the morphospace. |
| isAchievable | Datatype | true | https://uor.foundation/observable/SynthesisSignature | http://www.w3.org/2001/XMLSchema#boolean | Whether this signature has been empirically verified as achievable at some quantum level. |
| isForbidden | Datatype | true | https://uor.foundation/observable/SynthesisSignature | http://www.w3.org/2001/XMLSchema#boolean | Whether this signature has been formally proven impossible by an ImpossibilityWitness. |
| achievabilityWitness | Object | true | https://uor.foundation/observable/SynthesisSignature | https://uor.foundation/proof/Proof | The proof individual (ImpossibilityWitness or AxiomaticDerivation) that grounds this signature's achievability classification. |
| rotationExponent | Datatype | true | https://uor.foundation/observable/DihedralElement | http://www.w3.org/2001/XMLSchema#nonNegativeInteger | The rotation exponent k ∈ \[0, 2^n) of this dihedral element in the standard representation r^k s^s. Together with reflectionBit, uniquely identifies the element within D_\{2^n\}. |
| reflectionBit | Datatype | true | https://uor.foundation/observable/DihedralElement | http://www.w3.org/2001/XMLSchema#boolean | The reflection flag s ∈ \{0,1\} of this dihedral element. False = pure rotation (r^k), true = reflection (r^k·s). D_7 defines composition: (r^a s^p)(r^b s^q) = r^(a + (-1)^p b) s^(p XOR q). |
| homotopyDimension | Datatype | true | https://uor.foundation/observable/HomotopyGroup | http://www.w3.org/2001/XMLSchema#nonNegativeInteger | The dimension k of this homotopy group πk. |
| homotopyRank | Datatype | true | https://uor.foundation/observable/HomotopyGroup | http://www.w3.org/2001/XMLSchema#nonNegativeInteger | The rank of this homotopy group (number of free generators). |
| homotopyBasepoint | Object | true | https://uor.foundation/observable/HomotopyGroup | https://uor.foundation/type/Constraint | The basepoint vertex v at which this homotopy group is computed. |
| higherMonodromyDimension | Datatype | true | https://uor.foundation/observable/HigherMonodromy | http://www.w3.org/2001/XMLSchema#nonNegativeInteger | The dimension k > 1 at which this higher monodromy acts. |
| whiteheadTrivial | Datatype | true | https://uor.foundation/observable/WhiteheadProduct | http://www.w3.org/2001/XMLSchema#boolean | True iff this Whitehead product is trivial (zero in πp+q−1). |
| postnikovTruncation | Object | true | https://uor.foundation/observable/SpectralSequencePage | https://uor.foundation/homology/PostnikovTruncation | The Postnikov truncation associated with this spectral sequence page. |
| stratificationLevel | Object | true | https://uor.foundation/observable/StratificationRecord | https://uor.foundation/schema/WittLevel | The quantum level at which this stratification is computed. |
| stratificationStratum | Object | false | https://uor.foundation/observable/StratificationRecord | https://uor.foundation/type/HolonomyStratum | A HolonomyStratum in this stratification record. |
| metricDomain | Datatype | true | https://uor.foundation/observable/BaseMetric | http://www.w3.org/2001/XMLSchema#string | The mathematical domain of this base metric. |
| metricRange | Datatype | true | https://uor.foundation/observable/BaseMetric | http://www.w3.org/2001/XMLSchema#string | The mathematical range (codomain) of this base metric. |
| metricComposition | Object | true | https://uor.foundation/observable/BaseMetric | https://uor.foundation/schema/TermExpression | How this metric composes with others in the measurement tower. |
| referencesClass | Object | true | https://uor.foundation/observable/BaseMetric | https://uor.foundation/observable/Observable | The existing observable class that this base metric references. |
| referencesIdentity | Object | true | https://uor.foundation/observable/BaseMetric | https://uor.foundation/op/Identity | The existing identity that defines this base metric. |
| saturationNumerator | Datatype | true | https://uor.foundation/observable/GroundingObservable | http://www.w3.org/2001/XMLSchema#nonNegativeInteger | The count of pinned sites (numerator of σ). |
| saturationDenominator | Datatype | true | https://uor.foundation/observable/GroundingObservable | http://www.w3.org/2001/XMLSchema#positiveInteger | The total site count (denominator of σ). |
| alternatingSum | Object | true | https://uor.foundation/observable/EulerCharacteristicObservable | https://uor.foundation/schema/TermExpression | The alternating sum formula for Euler characteristic. |
| metricUnit | Object | true | https://uor.foundation/observable/BaseMetric | https://uor.foundation/observable/MeasurementUnit | The unit of measurement for this base metric. |
| metricPrecision | Datatype | true | https://uor.foundation/observable/BaseMetric | http://www.w3.org/2001/XMLSchema#nonNegativeInteger | The precision or resolution of this base metric. |
| metricMonotonicity | Object | true | https://uor.foundation/observable/BaseMetric | https://uor.foundation/schema/TermExpression | Monotonicity property of this metric (e.g., non-decreasing). |
| metricDecomposition | Object | true | https://uor.foundation/observable/BaseMetric | https://uor.foundation/schema/TermExpression | The decomposition rule for this metric into sub-metrics. |
| metricTowerPosition | Datatype | true | https://uor.foundation/observable/BaseMetric | http://www.w3.org/2001/XMLSchema#nonNegativeInteger | The position of this metric in the metric tower. |
| metricComputationCost | Object | true | https://uor.foundation/observable/BaseMetric | https://uor.foundation/schema/TermExpression | The computational cost of evaluating this metric. |
| metricBound | Object | true | https://uor.foundation/observable/BaseMetric | https://uor.foundation/schema/TermExpression | Upper or lower bound on the metric value. |
Named Individuals
| Name | Type | Properties | Comment |
|---|---|---|---|
| Bits | https://uor.foundation/observable/MeasurementUnit | — | Information-theoretic unit: the measurement is in bits (e.g., Hamming weight, entropy). |
| RingSteps | https://uor.foundation/observable/MeasurementUnit | — | Ring-arithmetic unit: the measurement is in ring distance steps (|x - y| mod 2^n). |
| Dimensionless | https://uor.foundation/observable/MeasurementUnit | — | Dimensionless unit: the measurement is a pure number (e.g., winding number, Betti number, spectral gap). |
| Nats | https://uor.foundation/observable/MeasurementUnit | — | Natural information unit: entropy measured in nats (using natural logarithm). S_residual is expressed in nats when computed as (Σ κ_k − χ) × ln 2. |
| PeriodBoundary | https://uor.foundation/observable/PhaseBoundaryType | — | A phase boundary where g divides 2^n − 1, meaning g is a period of the multiplicative structure of R_n. |
| PowerOfTwoBoundary | https://uor.foundation/observable/PhaseBoundaryType | — | A phase boundary where g = 2^k, meaning g aligns with the binary stratification of R_n. |
| Achievable | https://uor.foundation/observable/AchievabilityStatus | — | The signature has been verified as achievable at some quantum level by an AxiomaticDerivation proof. |
| Forbidden | https://uor.foundation/observable/AchievabilityStatus | — | The signature has been formally proven impossible by an ImpossibilityWitness deriving from MS_1, MS_2, or other impossibility theorems. |
| d_delta_metric | https://uor.foundation/observable/BaseMetric |
| d_Δ: the incompatibility metric |d_R − d_H| per site pair. |
| sigma_metric | https://uor.foundation/observable/BaseMetric |
| σ: the grounding metric, pinned sites / total sites. |
| jacobian_metric | https://uor.foundation/observable/BaseMetric |
| J_k: per-site curvature, ∂_R f_k. |
| betti_metric | https://uor.foundation/observable/BaseMetric |
| β_k: per-dimension Betti number of the constraint nerve. |
| euler_metric | https://uor.foundation/observable/BaseMetric |
| χ: Euler characteristic, Σ(−1)^k β_k. |
| residual_metric | https://uor.foundation/observable/BaseMetric |
| r: count of free (unpinned) sites, the residual entropy. |