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.

This is a bridge-space namespace in the Resolve stage of the PRISM pipeline. It provides the resolution infrastructure — queries, partitions, observables, proofs, derivations, and traces that transform inputs into certified results.

Learn more: Pipeline Overview · Observables & Measurement

Class hierarchy

Imports

https://uor.foundation/op/
https://uor.foundation/schema/
https://uor.foundation/partition/
https://uor.foundation/type/

Classes

Name	Subclass Of	Comment
`Observable`	`Thing`	A measurable quantity in the UOR Framework. All observables are kernel-computed and user-consumed.
`StratumObservable`	`Observable`	An observable measuring stratum-level properties: position within the ring's layer structure.
`MetricObservable`	`Observable`	An observable measuring geometric distance between ring elements under a specific metric.
`PathObservable`	`Observable`	An observable measuring properties of paths through the ring: path length, total variation, winding number.
`ReductionObservable`	`Observable`	An observable measuring reduction properties: the length and count of operation sequences.
`CatastropheObservable`	`Observable`	An observable measuring catastrophe-theoretic properties: thresholds at which qualitative changes occur in the partition.
`CurvatureObservable`	`Observable`	An observable measuring the curvature of the UOR geometry: the gap between ring-isometry and Hamming-isometry for a given transform.
`HolonomyObservable`	`Observable`	An observable measuring holonomy: the accumulated transformation when traversing a closed path in the ring.
`RingMetric`	`MetricObservable`	Distance between two ring elements under the ring metric: d_R(x, y) = \|x - y\| mod 2^n.
`HammingMetric`	`MetricObservable`	Distance between two ring elements under the Hamming metric: the number of bit positions where they differ.
`IncompatibilityMetric`	`MetricObservable`	The metric incompatibility between two ring elements: the divergence between their ring-metric and Hamming-metric distances, measuring geometric curvature.
`StratumValue`	`StratumObservable`	The stratum index of a ring element.
`StratumDelta`	`StratumObservable`	The difference in stratum between two ring elements.
`StratumTrajectory`	`StratumObservable`	The sequence of strata traversed by a path through the ring.
`PathLength`	`PathObservable`	The length of a path through the ring, measured in operation steps.
`TotalVariation`	`PathObservable`	The total variation of a path: the sum of metric distances between consecutive elements.
`WindingNumber`	`PathObservable`	The winding number of a closed path: the number of times the path wraps around the ring.
`ReductionLength`	`ReductionObservable`	The number of operation applications in a reduction sequence.
`ReductionCount`	`ReductionObservable`	The number of distinct reduction sequences in a computation.
`CatastropheThreshold`	`CatastropheObservable`	A critical value at which a qualitative change occurs in the partition structure.
`CatastropheCount`	`CatastropheObservable`	The number of catastrophe events (qualitative partition changes) in a computation.
`Commutator`	`CurvatureObservable`	The commutator [f, g](x) = f(g(x)) - g(f(x)) of two operations, measuring their non-commutativity.
`CurvatureFlux`	`CurvatureObservable`	The integrated curvature over a region of type space: the total metric incompatibility accumulated.
`Monodromy`	`HolonomyObservable`	The monodromy of a closed path: the net transformation accumulated when traversing a loop in the type space.
`ParallelTransport`	`HolonomyObservable`	The parallel transport of a vector along a path: the canonical lift of the path to the tangent bundle of the ring.
`DihedralElement`	`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`	`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`	`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`	`Observable`	An observable measuring a topological invariant of the resolution space. Topological observables are invariant under continuous deformations of the constraint configuration.
`BettiNumber`	`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`	`TopologicalObservable`	The smallest positive eigenvalue of the constraint nerve Laplacian. Controls the convergence rate of iterative resolution: larger gap = faster convergence.
`ThermoObservable`	`Observable`	An observable measuring thermodynamic properties of the resolution process: residual entropy, Landauer cost, and reduction distribution statistics.
`ResidualEntropy`	`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`	`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`	`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`	`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`	`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`	`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`	`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`	`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`	`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`	`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`	`Thing`	The achievability classification of a topological signature in the morphospace. Either Achievable or Forbidden (witnessed by ImpossibilityWitness).
`HomotopyGroup`	`Thing`	The k-th homotopy group πk(N(C), v) of the constraint nerve based at vertex v.
`HigherMonodromy`	`Thing`	The image of πk(N(C)) → Aut(sitek) for k > 1. Generalises the MN_6 monodromy homomorphism.
`WhiteheadProduct`	`Thing`	The Whitehead product [α, β] ∈ πp+q−1 for α ∈ πp, β ∈ πq.
`StratificationRecord`	`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`	`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`	`Observable`	The grounding metric σ = pinned sites / total sites. Ranges from 0 (no sites pinned) to 1 (fully grounded).
`EulerCharacteristicObservable`	`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	`Observable`	`decimal`	The numeric value of an observable measurement.
`source`	Object	true	`Observable`	`Thing`	The source object of this measurement (datum, partition, or path start point).
`target`	Object	true	`Observable`	`Thing`	The target object of this measurement (for metrics and path-end measurements).
`hasUnit`	Object	true	`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	`Jacobian`	`nonNegativeInteger`	The site position k at which this Jacobian entry is measured.
`derivativeValue`	Datatype	true	`Jacobian`	`decimal`	The discrete derivative value at this site position.
`dimension`	Datatype	true	`TopologicalObservable`	`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	`SynthesisSignature`	`integer`	The Euler characteristic actually achieved by this synthesis signature.
`realisedBetti`	Datatype	false	`SynthesisSignature`	`nonNegativeInteger`	Non-functional. Realised Betti number values, one assertion per homological degree.
`pageIndex`	Datatype	true	`SpectralSequencePage`	`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	`SpectralSequencePage`	`boolean`	True iff d_r = 0 on this page — no further corrections to the lifted homology.
`convergedAt`	Datatype	true	`SpectralSequencePage`	`nonNegativeInteger`	The page index r at which the spectral sequence converged (all subsequent differentials zero).
`obstructionClass`	Object	true	`LiftObstructionClass`	`CohomologyGroup`	The cohomology class in H^2(N(C(T))) representing this obstruction.
`monodromyLoop`	Object	true	`Monodromy`	`ClosedConstraintPath`	The closed path that generates this monodromy value.
`monodromyElement`	Object	true	`Monodromy`	`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	`Monodromy`	`boolean`	True iff the monodromyElement is the identity in D_{2^n}.
`holonomyGroup`	Object	false	`HolonomyGroup`	`DihedralElement`	Non-functional. The generators of the holonomy group: one DihedralElement per generating monodromy.
`holonomyGroupOrder`	Datatype	true	`HolonomyGroup`	`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	`ClosedConstraintPath`	`nonNegativeInteger`	The number of constraint application steps in this closed path.
`pathConstraints`	Object	false	`ClosedConstraintPath`	`Constraint`	Non-functional. The ordered sequence of constraints traversed. One assertion per step.
`dihedralElementValue`	Object	false	`DihedralElement`	`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	`DihedralElement`	`boolean`	True iff this element is the group identity (the trivial monodromy value).
`elementOrder`	Datatype	true	`DihedralElement`	`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	`ThermoObservable`	`decimal`	An estimated computational hardness for a ThermoObservable, connecting thermodynamic cost to complexity (TH_9 realisation).
`phaseN`	Datatype	true	`CatastropheObservable`	`positiveInteger`	The ring dimension coordinate n in the (n, g) catastrophe phase diagram (PD_1 n-coordinate).
`phaseG`	Datatype	true	`CatastropheObservable`	`positiveInteger`	The group-order coordinate g in the (n, g) catastrophe phase diagram (PD_1 g-coordinate).
`onResonanceLine`	Datatype	true	`CatastropheObservable`	`boolean`	True when g divides 2^n − 1, placing this observable on a resonance line in the phase diagram (PD_4).
`phaseBoundaryType`	Object	true	`CatastropheObservable`	`PhaseBoundaryType`	The type of phase boundary at this point in the catastrophe diagram: PeriodBoundary or PowerOfTwoBoundary (PD_2).
`achievabilityStatus`	Object	true	`SynthesisSignature`	`AchievabilityStatus`	The achievability classification of this observable's topological signature in the morphospace.
`isAchievable`	Datatype	true	`SynthesisSignature`	`boolean`	Whether this signature has been empirically verified as achievable at some quantum level.
`isForbidden`	Datatype	true	`SynthesisSignature`	`boolean`	Whether this signature has been formally proven impossible by an ImpossibilityWitness.
`achievabilityWitness`	Object	true	`SynthesisSignature`	`Proof`	The proof individual (ImpossibilityWitness or AxiomaticDerivation) that grounds this signature's achievability classification.
`rotationExponent`	Datatype	true	`DihedralElement`	`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	`DihedralElement`	`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	`HomotopyGroup`	`nonNegativeInteger`	The dimension k of this homotopy group πk.
`homotopyRank`	Datatype	true	`HomotopyGroup`	`nonNegativeInteger`	The rank of this homotopy group (number of free generators).
`homotopyBasepoint`	Object	true	`HomotopyGroup`	`Constraint`	The basepoint vertex v at which this homotopy group is computed.
`higherMonodromyDimension`	Datatype	true	`HigherMonodromy`	`nonNegativeInteger`	The dimension k > 1 at which this higher monodromy acts.
`whiteheadTrivial`	Datatype	true	`WhiteheadProduct`	`boolean`	True iff this Whitehead product is trivial (zero in πp+q−1).
`postnikovTruncation`	Object	true	`SpectralSequencePage`	`PostnikovTruncation`	The Postnikov truncation associated with this spectral sequence page.
`stratificationLevel`	Object	true	`StratificationRecord`	`WittLevel`	The quantum level at which this stratification is computed.
`stratificationStratum`	Object	false	`StratificationRecord`	`HolonomyStratum`	A HolonomyStratum in this stratification record.
`metricDomain`	Datatype	true	`BaseMetric`	`string`	The mathematical domain of this base metric.
`metricRange`	Datatype	true	`BaseMetric`	`string`	The mathematical range (codomain) of this base metric.
`metricComposition`	Object	true	`BaseMetric`	`TermExpression`	How this metric composes with others in the measurement tower.
`referencesClass`	Object	true	`BaseMetric`	`Observable`	The existing observable class that this base metric references.
`referencesIdentity`	Object	true	`BaseMetric`	`Identity`	The existing identity that defines this base metric.
`saturationNumerator`	Datatype	true	`GroundingObservable`	`nonNegativeInteger`	The count of pinned sites (numerator of σ).
`saturationDenominator`	Datatype	true	`GroundingObservable`	`positiveInteger`	The total site count (denominator of σ).
`alternatingSum`	Object	true	`EulerCharacteristicObservable`	`TermExpression`	The alternating sum formula for Euler characteristic.
`metricUnit`	Object	true	`BaseMetric`	`MeasurementUnit`	The unit of measurement for this base metric.
`metricPrecision`	Datatype	true	`BaseMetric`	`nonNegativeInteger`	The precision or resolution of this base metric.
`metricMonotonicity`	Object	true	`BaseMetric`	`TermExpression`	Monotonicity property of this metric (e.g., non-decreasing).
`metricDecomposition`	Object	true	`BaseMetric`	`TermExpression`	The decomposition rule for this metric into sub-metrics.
`metricTowerPosition`	Datatype	true	`BaseMetric`	`nonNegativeInteger`	The position of this metric in the metric tower.
`metricComputationCost`	Object	true	`BaseMetric`	`TermExpression`	The computational cost of evaluating this metric.
`metricBound`	Object	true	`BaseMetric`	`TermExpression`	Upper or lower bound on the metric value.

Named Individuals

Name	Type	Comment
`Bits`	`MeasurementUnit`	Information-theoretic unit: the measurement is in bits (e.g., Hamming weight, entropy).
`RingSteps`	`MeasurementUnit`	Ring-arithmetic unit: the measurement is in ring distance steps (\|x - y\| mod 2^n).
`Dimensionless`	`MeasurementUnit`	Dimensionless unit: the measurement is a pure number (e.g., winding number, Betti number, spectral gap).
`Nats`	`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`	`PhaseBoundaryType`	A phase boundary where g divides 2^n − 1, meaning g is a period of the multiplicative structure of R_n.
`PowerOfTwoBoundary`	`PhaseBoundaryType`	A phase boundary where g = 2^k, meaning g aligns with the binary stratification of R_n.
`Achievable`	`AchievabilityStatus`	The signature has been verified as achievable at some quantum level by an AxiomaticDerivation proof.
`Forbidden`	`AchievabilityStatus`	The signature has been formally proven impossible by an ImpossibilityWitness deriving from MS_1, MS_2, or other impossibility theorems.
`d_delta_metric`	`BaseMetric`	d_Δ: the incompatibility metric \|d_R − d_H\| per site pair.
`metricDomain`: pair of ring elements `metricRange`: non-negative integer `referencesClass`: `IncompatibilityMetric`
`sigma_metric`	`BaseMetric`	σ: the grounding metric, pinned sites / total sites.
`metricDomain`: computation state `metricRange`: decimal in [0, 1] `referencesIdentity`: `GS_2`
`jacobian_metric`	`BaseMetric`	J_k: per-site curvature, ∂_R f_k.
`metricDomain`: computation state × site index `metricRange`: decimal `referencesClass`: `Jacobian` `referencesIdentity`: `DC_6`
`betti_metric`	`BaseMetric`	β_k: per-dimension Betti number of the constraint nerve.
`metricDomain`: simplicial complex × dimension `metricRange`: non-negative integer `referencesClass`: `BettiNumber`
`euler_metric`	`BaseMetric`	χ: Euler characteristic, Σ(−1)^k β_k.
`metricDomain`: simplicial complex `metricRange`: integer `referencesIdentity`: `IT_2`
`residual_metric`	`BaseMetric`	r: count of free (unpinned) sites, the residual entropy.
`metricDomain`: computation state `metricRange`: non-negative integer `referencesClass`: `ResidualEntropy`

UOR Observables

Imports

Classes

Properties

Named Individuals

Related Concepts