Subjective Numbers

Abstract

A subjective number is a value together with the perspective that holds it, written a(s). A judgment relating two subjective numbers is itself made from a perspective, written a(s) ⊑_p b(t): from the standpoint of p, the value a held at s stands in the judged relation to the value b held at t. The position that holds a value and the standpoint that judges a relation are separate indices. This article gives the semantics of such judgments and proves what the separation buys.

Each evaluating perspective p starts from a base kernel of judgments. Its closed relation is the least fixpoint of an operator that adds reflexivity, transitivity, substitution of equal values, and adoption of other perspectives' judgments through a guard CPA_p(q,u) that selects by the position u at which an adopted judgment lands. The closed relation at each perspective is a preorder (Theorem 3.2), the derivation rules are sound and complete for it (Theorem 3.3), and the guard is equivalent to importing every judgment of q that lands at a permitted position and then closing (Theorem 3.6). Derivations form a category, histories form a category, and provenance is a strict functor between them (Theorems 4.1 to 4.3).

The central results concern collapse. A valuation assigns one value to each position, and it is coherent at p when it respects every closed judgment of p. Identifying two positions is a kernel entry, and the Licensing Theorem (Theorem 5.2) says exactly what it does: it keeps the coherent valuations that already agree on the two positions and discards the rest. An identification is therefore licensed, forbidden, or open according to whether all, none, or some coherent valuations agree, and a perspective that identifies positions becomes incoherent exactly when its identifications are jointly forbidden (Theorem 5.3). The Classical Collapse Theorem (Theorem 5.4) shows that with every guard open and every position identified the whole system is one ordinary preorder on the value space. Ordinary mathematics is the collapsed case. The extension is proper (Theorem 5.7): there are finite systems, the Penrose staircase and the liar sentence among them, in which a perspective holding every judgment is coherent and the collapse is forbidden.

The impossible objects then receive an arithmetic. On a perspective graph whose channels shift values by fixed amounts, an identification is licensed exactly when the walk sum between the identified positions vanishes, so the verdict depends on the value group: the closed four-flight staircase is forbidden over the integers and licensed modulo four (Corollary 8.4). The obstructions to collapse form the group H¹(Γ;V)≅V^b₁, and they add: two impossible objects can sum to a possible one, and the liar added to itself is the truth-teller (Theorem 9.1, Corollary 9.2). Over a field K the readings of subjective numbers from the positions of the unfolding form a ring in which every reading is a ring homomorphism, the classical numbers are the constants, and the twisted numbers form the group algebra of the character group of the loops (Theorem 10.3). The liar is the unit σ of that ring with σ²=1, reading true from one position and false from the other, sent to −σ by re-reading after the loop, and descending to no single position (Theorem 10.5). The liar stands to the classical values as √−1 stands to the reals, and the parity theorem is its descent criterion.

The article also proves that closed sections are analytic and universally measurable under standard Borel hypotheses, which yields a pseudometric between perspectives (Theorems 6.1 and 6.2), and gives termination and complexity bounds for finite universes and a conservativity theorem for conjunctive fusion of perspectives.

Subjective Numbers in Plain Terms

A number in ordinary mathematics belongs to no one: seven is seven whoever counts it. A subjective number is a number together with where it is held. Clock times already work this way: nine o'clock in London and nine o'clock in New York both read nine, and they are not the same moment. Writing 9(London) and 9(New York) keeps that difference on the page. The framework does the same for any value: 7(a) is seven as held at position a.

Relating two of them is done from somewhere, by a stated rule. London relates its nine o'clock to New York's four o'clock by "subtract five hours"; that rule is London's judgment, and New York may hold a different one. Each perspective keeps the rules it holds outright, adds what follows from them by chaining, and may borrow another perspective's rules under a permission that names which destinations it trusts that perspective about. Everything derived keeps a record of where it came from.

The addresses matter on a loop. Travel east and set your watch forward an hour at each time zone. After going all the way around, your watch agrees with the clocks at home and your calendar is a day ahead. Every step was right; only the loop is wrong, by exactly what the trip added. The Penrose staircase is the same thing drawn: four flights that each go up, joined into a loop. Nothing is wrong with any flight. What cannot be done is to declare the top of the last flight to be the bottom of the first, because four is not zero. Count heights modulo four, as a watch counts hours modulo twelve, and the staircase is a possible object. The date line does this for the calendar: it is where the impossible declaration is refused by a jump.

Declaring that two positions hold one value is itself a judgment, and it can be checked in advance. If every consistent way of filling in the values already agrees at the two positions, the declaration changes nothing. If none does, it destroys consistency. If some do and some do not, it is a choice. Ordinary mathematics is what is left when every such declaration is made and every permission is open: one clock for the whole world. Section 5 says exactly when that is safe.

The liar sentence is the smallest loop where it is not. "This sentence is false" occupies two positions, the one that speaks and the one spoken about, and says that they carry opposite values. A Möbius strip is the picture: at every point it has two sides, and one trip around brings you back on the other side, so "which side" has a local answer everywhere and no global one. The liar's value σ is that kind of object. It reads true from one position and false from the other, one trip around the loop turns σ into −σ, a second turns it back, and σ times itself is one. From any single position it adds and multiplies like an ordinary number. The classical demand is that the strip have one side.

Subjective numbers extend ordinary numbers the way complex numbers do: by admitting objects that solve equations with no ordinary solution, and by recording in a conjugation, here the trip around the loop, exactly what was admitted. The article proves that ordinary mathematics is the collapsed case, says when collapse is safe, gives the objects that cannot be collapsed their arithmetic, and traces every contradiction to the declaration that caused it.

Notation Guide

1. Introduction

1.1 Values with a Standpoint

Every judgment in mathematics is made from somewhere, and ordinary mathematics is built so that it does not matter where. A relation on a set holds or fails regardless of who asks, and a value is the same value wherever it occurs. That convention is a choice. It amounts to identifying every standpoint with every other, and every occurrence of a value with every other occurrence of it. This article withholds the identification, studies the structure that remains, and proves that the ordinary case is recovered exactly when the identification is granted.

A subje ctive number is written a(s), with value a ∈ V and intrinsic perspective s ∈ P. For each evaluating perspective p, a base kernel R^∘_p records the judgments p holds outright, and the closed relation R^⋆_p is generated from the kernels by a least fixpoint closure. A closed judgment a(s) ⊑_p b(t) carries three perspective indices: the positions s and t that hold the two values, and the standpoint p that validates the relation between them.

Kernels can be sparse and asymmetric. Closure supplies reflexivity, transitivity inside a fixed evaluating perspective, substitution of equal values, and adoption of other perspectives' judgments under the guard CPA_p(q,u). Preference aggregation, distributed knowledge, and multi-agent coordination all involve claims that one party accepts and another does not, and the framework lets each party keep a coherent view without forcing every judgment into one global relation.

Three questions organize the article. What structure do indexed judgments have? Each closed relation is a preorder, derivations form a category, and provenance is a functor (Sections 3 and 4). When can the indices be dropped? Exactly when every coherent valuation already agrees across the identified positions (Section 5). What is the system with the indices dropped? One ordinary preorder on the value space, generated by the union of the kernels (Theorem 5.4). What arithmetic do the objects that cannot be collapsed obey? They are units in a ring of readings, where the liar squares to one and reads differently from its two positions, and their obstructions to collapse form a group that adds (Sections 8 to 10). The Penrose staircase and the liar paradox are the two smallest systems in which dropping the indices is forbidden, and the second is treated in full in its own article.

1.2 Reading a Judgment

The judgment a(s) ⊑_p b(t) reads: from the standpoint of p, the value a held at s stands in the judged relation to the value b held at t. The content of the relation is whatever the kernel of p makes it. In Example A1 it is an order, an equality, and a proportion in turn. The framework fixes only the structural laws: reflexivity, transitivity inside a fixed evaluating perspective, substitution of equal values, and guarded adoption.

Section 5 uses one specific reading, determination: a at s fixes b at t. Under that reading a kernel is a system of constraints on the values held at positions, a valuation is a candidate assignment of one value to each position, and coherence is the assignment respecting every constraint. The liar paradox is analyzed under this reading.

1.3 Positive Judgments, Absence, and Rejection Logs

The system is positive. A judgment is present in a base kernel, derivable after closure, or absent from the closed relation. Absence is a different thing from falsity. Some examples also record explicit rejections in an optional negative log N^∘_p. These rejections are annotations. They do not enter the closure operator.

Saying that a(s) ⊑_p b(t) is not derivable means that the positive system has no proof of it. Saying that p rejects the judgment means that N^∘_p records a negative attitude. The two notions are kept apart throughout.

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2. Formal Data

2.1 Basic Notation

• P is a nonempty set of perspectives.
• V is a value space equipped with equality =.
• The set of subjective numbers is X=P×V. The pair (s,a) is written a(s).
• For each p ∈ P, the base kernel is R^∘_p⊆X×X, equivalently R^∘_p:P×P×V×V→{0,1}.
• The closed relation is R^⋆_p⊆X×X.
• The closed judgment a(s) ⊑_p b(t) means (a(s),b(t))∈R^⋆_p.

2.2 Value Substitution

Kernels respect equality on V. If a=a' and b=b' in V, then R^∘_p(s,t,a,b)=R^∘_p(s,t,a',b'). When equality on V is literal identity, this condition holds automatically. If V is presented with an equivalence relation, the same rule says that the kernels are invariant under replacement of representatives. One may instead quotient V first and then form X=P×V from equivalence classes.

2.3 The Adoption Guard

For each evaluating perspective p, the adoption guard is a Boolean predicate

The value CPA_p(q,u)=1 means that p accepts the judgments of q that land at position u. Formally, p may compose any judgment of its own that ends at b(t) with a judgment of q that starts there and lands at u:

Every subjective number is related to itself at every perspective, so the judgment of p that ends at b(t) can be the reflexive one, and then the rule delivers b(t) ⊑_p c(u) itself. The guard is therefore a permission to import, and it selects by where the imported judgment lands. Theorem 3.6 makes this exact. Two judgments of q that land at different positions can be treated differently by p, and the guard is directional: CPA_p(q,u) and CPA_q(p,u) are independent.

2.4 Measurable Structure

The discrepancy construction of Section 6 requires additional measurable data. For that section, assume that P and V are standard Borel spaces, that ν is a complete probability measure on P, and that μ_V is a complete probability measure on V. The product measure on V×V is μ_V×V.

Let the global base set and global guard set be

For the measurable results, require B and C to be Borel. Then each base section A^∘_p;s,t={(a,b):R^∘_p(s,t,a,b)=1} is Borel.

3. Two-Tier Semantics: Closure and Derivability

3.1 Global Atomic Predicates

For each r∈P, let E_r(s,t,a,b) be an atomic predicate. A base axiom asserts E_r(s,t,a,b) whenever R^∘_r(s,t,a,b)=1.

3.2 Derivation Rules

1. Base: infer E_p(s,t,a,b) from R^∘_p(s,t,a,b)=1.
2. Refl: infer E_p(s,s,a,a).
3. Subst: from E_p(s,t,a,b) and a=a', b=b', infer E_p(s,t,a',b').
4. Trans: from E_p(s,t,a,b) and E_p(t,u,b,c), infer E_p(s,u,a,c).
5. CPA: from E_p(s,t,a,b), E_q(t,u,b,c), and CPA_p(q,u)=1, infer E_p(s,u,a,c).

The five rules are a Datalog program in the sense of Ceri, Gottlob, and Tanca (1989): finitely many Horn rules over the atoms E_r, the kernels, and the guards. The least model of that program is the closure defined next.

3.3 Least Fixpoint Closure

Let D=X×X and let L=∏_r∈P𝒫(D) be the complete lattice of perspective-indexed relations ordered by componentwise inclusion, in the standard sense of Davey and Priestley (2002). For S∈L, define a one-step operator T:S↦T(S). For each p, T_p(S) is the union of:

1. R^∘_p.
2. The diagonal {(a(s),a(s)):s∈P,a∈V}.
3. All value-substitution variants of members of S_p.
4. All (a(s),c(u)) such that for some b(t), (a(s),b(t))∈S_p and (b(t),c(u))∈S_p.
5. All (a(s),c(u)) such that for some q∈P and b(t), (a(s),b(t))∈S_p, (b(t),c(u))∈S_q, and CPA_p(q,u)=1.

Define S⁰=⊥, the family of empty relations, and Sⁿ⁺¹=T(Sⁿ). The closed relation is

3.4 Local Preorders

For each fixed evaluating perspective p, the relation ⊑_p is reflexive and transitive after closure. Symmetry is not imposed.

3.5 Soundness and Completeness

3.6 Positive Consistency of the Core Language

The core language derives positive judgments only. It has no rule that converts absence into negation and no rule that turns a rejection log into a positive judgment.

Section 5 introduces a different notion, incoherence: a perspective is incoherent when no assignment of values to positions respects all of its closed judgments. Incoherence is a property of the closed relation under the determination reading. It is never a derived negation, so Theorem 3.4 and the results of Section 5 are compatible.

3.7 The Import Form of the Guard

For perspectives p and q, the imports of p from q are the closed judgments of q whose landing position the guard permits:

The generators of p are G_p=R^∘_p∪Imp_p.

The theorem fixes what a guard restricts. It does not restrict where an adopted judgment starts, because the left premise of the CPA rule can always be reflexive. It restricts where the adopted judgment lands. A perspective that opens CPA_p(q,u) takes on everything q concludes about position u, and nothing q concludes about other positions.

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4. Categorical Provenance

Category theory enters through derivational provenance. The objects are subjective numbers. The morphisms are typed proof terms witnessing closed judgments. Raw values such as 0 or true are never treated as identities; the identity at an object is the reflexive proof, and the identity history is the empty word.

4.1 Proof Terms

Proof terms record whether a judgment came from a base kernel, reflexivity, substitution, transitivity, or guarded adoption. A term concluding E_p(s,t,a,b) is a derivation from a(s) to b(t) at perspective p. Write src(π) and tgt(π) for the source and target subjective numbers of a proof term π.

The proof-term grammar is generated by the following constructors:

The constructor base_p(s,t,a,b) records the kernel entry R^∘_p(s,t,a,b)=1. The constructor refl_p(x) records identity at the subjective number x. The constructor subst records the value identifications used. The constructor trans(π,ρ) records same-perspective composition. The constructor cpa_p,q,u(π,ρ) records guarded composition of a p-derivation π with a q-derivation ρ landing at intrinsic perspective u; it is well typed only when CPA_p(q,u)=1.

Proof terms are taken modulo the congruence generated by the unit and associativity equations for trans: trans(refl_p(x),π)=π, trans(π,refl_p(y))=π, and trans(trans(π,ρ),σ)=trans(π,trans(ρ,σ)). The congruence identifies no two distinct base entries, substitution events, or adoption events, so provenance labels survive the quotient.

4.2 Derivation Category

For each p, the category Deriv_p has objects X. A morphism x→y is a congruence class of proof terms deriving E_p(x,y). The identity at x is the class of refl_p(x). Composition is induced by the Trans rule:

4.3 History Category

Let Σ be the alphabet of provenance events: one letter ⟨base,p,s,t,a,b⟩ for each possible kernel entry, one letter ⟨subst,a=a',b=b'⟩ for each substitution, and one letter ⟨adopt,p,q,u⟩ for each possible adoption. A typed history from x to y is a triple (x,h,y) with h∈Σ^*. The history category Hist has objects X and morphisms (x,h,y):x→y. The identity at x is (x,ε,x). Composition is concatenation:

4.4 Strict Provenance Functor

The history of a proof term is defined by recursion on the constructors:

The functor H_p:Deriv_p→Hist is the identity on objects and sends the class of π:x→y to (x,h(π),y). An adoption event names the importing perspective, the perspective adopted from, and the landing position, and the history of the adopted derivation follows it, so every base entry used anywhere in a derivation appears in the history together with the chain of adoptions through which it arrived.

5. Collapse and Licensing

This section fixes the determination reading of judgments and asks when the perspective indices can be dropped. Throughout the section, equality on V is literal identity, so the substitution clause is vacuous.

5.1 Valuations and Coherence

Fix an evaluating perspective p. Let G_p=R^∘_p∪Imp_p be its generators (Theorem 3.6) and let Pos_p be the set of intrinsic perspectives that occur as the source or the target of a generator. A valuation for p is a function v:Pos_p→V. It respects a judgment a(s) ⊑_p b(t) with s,t∈Pos_p when v(s)=a implies v(t)=b. It is coherent at p when it respects every closed judgment of p between positions of Pos_p. The set of coherent valuations is Coh_p, and p is coherent when Coh_p is nonempty.

A closed judgment of p that involves a position outside Pos_p is reflexive, because by Theorem 3.6 every closed judgment is a chain of generators and no generator touches such a position. So nothing is lost by restricting valuations to Pos_p.

5.2 Identification and the Licensing Theorem

For positions s and t, the identification of s and t is the set of identity edges

For a set S of unordered pairs of positions, I(S) is the union of the corresponding identifications, and C_P=I(all pairs) is the total identification. To collapse S at p is to add I(S) to the kernel of p; the resulting perspective is written p^S.

Adding entries to the kernel of p can change the closures of perspectives that adopt from p, and through them the imports of p itself. The results below assume stable imports: Imp_p is the same before and after the collapse. This holds when CPA_p is identically 0, and in general whenever no perspective that p adopts from adopts, directly or through other perspectives, from p.

Let p be coherent and let every pair in S consist of positions of p. The collapse of S is licensed at p if every coherent valuation of p agrees on each pair in S, forbidden if no coherent valuation agrees on every pair in S, and open otherwise. It is determining if all coherent valuations of p^S assign the same value to each identified pair. For a single pair the three cases read: all coherent valuations agree, none does, some do.

5.3 The Collapse Theorem

A perspective can hold every judgment of a dependency and still have no coherent valuation once it identifies positions. The next theorem locates that failure, gives it a syntactic witness in the Boolean case, and traces it to its source.

Part (2) is the criterion for two-literal satisfiability, applied to a closure rather than to a formula. It says that when every constraint is an equation between two positions, an evaluator is incoherent exactly when its closure forbids both values at some position. Part (3) is the general form of a fact the companion article proves for the liar: the contradiction never comes from the content alone, and every derivation of it can be opened to show the identification responsible. Part (4) is the general form of transmission: an identification travels between perspectives only through a guard, and the record of the derivation names the route.

5.4 The Classical Collapse Theorem

Ordinary mathematics uses one standpoint and treats every occurrence of a value as the same value. In the present terms that is a system in which every guard is open and every position is identified. The following theorem says what such a system is.

Theorem 5.4 and Corollary 5.6 say two different things about the same operation. The theorem says that collapse always produces a relation: whatever the kernels are, identifying every position and opening every guard yields a preorder on the values. The corollary says when that relation still has a coherent reading. A collapse can produce a perfectly good preorder whose constraints no assignment of values satisfies. That is what a paradox is in this framework.

5.5 The Extension Is Proper

In both systems the collapsed relation exists. For the staircase it is the usual order of the integers, generated by a≤a+1. For the two-position system it is the preorder in which 0 and 1 are equivalent. Neither relation is defective as a relation. What fails is coherence: no assignment of values to positions satisfies the constraints once the positions are one. The framework therefore contains ordinary relational mathematics as the collapsed case and contains, in addition, systems whose collapse has no coherent reading. Those are the impossible objects, and the liar is the smallest of them. Sections 8 to 10 give them an arithmetic.

6. Measurable Discrepancies and a Pseudometric

For each p,s,t∈P, define the closed section

6.1 Analyticity and Universal Measurability

The facts about analytic sets used below are standard (Kechris 1995): in a standard Borel space, Borel sets are analytic; analytic sets are closed under countable unions, countable intersections, Borel preimages, and projections along standard Borel spaces; sections of analytic sets are analytic; and, by Lusin's theorem (Kechris 1995, Theorem 21.10), analytic sets are universally measurable.

6.2 Localized Discrepancy

This is the measure of the symmetric difference between the closed sections for p and q at intrinsic perspectives s,t:

6.3 Aggregated Pseudometric

The integral is taken in the completed product measure. The global closed relation R^⋆ is analytic by Theorem 6.1. The fiber-measure theorem for analytic sets (Bertsekas and Shreve 1978, Proposition 7.48) states that if A⊆X×Y is analytic in a product of standard Borel spaces and μ is a probability measure on Y, then the function x↦μ(A_x) is upper semianalytic and therefore universally measurable.

Apply this theorem to the closed-section families for p and q and to their intersection. Since

the function (s,t)↦δ(p,q;s,t) is universally measurable and bounded by 1. Hence the integral defining Δ is well defined in the completed product measure.

The construction follows the pattern of the metrics between labelled Markov processes of Desharnais, Gupta, Jagadeesan, and Panangaden (2004), where the distance between two systems is a measure of the disagreement between their observable behaviors. Here the observable behavior of a perspective is its closed section, and the disagreement between two perspectives is the measure of the set of judgments on which they differ.

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7. Finite-Universe Termination and Complexity Bounds

Suppose |P|=m and |V|=n. Then |X|=mn. For each evaluating perspective there are at most |X|²=m²n² possible positive atoms, and across all evaluating perspectives there are at most

possible atoms.

The bound is crude. The closure is a Datalog program, and semi-naive evaluation (Ceri, Gottlob, and Tanca 1989) processes each derived atom once rather than rescanning the whole relation at every stage; for a fixed program the running time is polynomial in the size of the data, with the exponent fixed by the number of variables in the rules.

8. Transport Systems on Perspective Graphs

The kernels of Sections 2 to 5 are arbitrary sets of judgments. This section and the next two study the kernels in which every judgment says the same thing about every value: crossing a channel between two positions shifts the value by a fixed amount. Such kernels are the ones in which the value operation and the perspective structure constrain each other, and they are the home of the impossible objects.

8.1 Perspective Graphs and Transports

A perspective graph is a finite connected directed multigraph Γ=(P,E,src,tgt). Its vertices are positions, its edges are channels along which a value held at one position is read at another. A walk is a sequence of edges each traversed forward or backward, and a walk is closed when it ends where it started. Let (V,+) be an abelian group. A transport system on Γ over V is a function d:E→V. Its kernel, held by an evaluating perspective e, is the uniform family

Crossing e forward adds d_e; crossing it backward subtracts it. The walk sum d(w) of a walk is the signed sum of the transports along it, and the circulation of a closed walk is its walk sum. The liar over V=ℤ/2 is the transport system with one edge from Ptr to Tgt and d=1: the four judgments of K_d are exactly the liar edges of the companion article. A staircase of four ascending flights is a path of four edges with d=1 over V=ℤ.

8.2 The Circulation Criterion

The criterion is Kirchhoff's voltage law for a network whose edge voltages are the transports (Kirchhoff 1847; Biggs 1993): a potential exists exactly when every loop has zero net voltage. An evaluator whose guards let it see only a spanning tree is always coherent, whatever the transports are; incoherence needs a closed loop in view.

8.3 Licensing on Transport Systems

So on a transport system the third status of Section 5 disappears: the content settles every identification one way or the other. The status depends on the value group as much as on the graph.

The second part contains the parity theorem of the companion article as the case k=2: a cycle of liars is coherent exactly when its length is even. The first part is the sense in which the impossible staircase is a fact about the integers. Its four flights are consistent with one another, the drawing is honest at every corner, and the demand that the top of the fourth flight be the bottom of the first fails because 4≠0. Where heights are counted modulo four the same drawing is a possible object.

9. Impossibility Classes and Their Arithmetic

Two transport systems that differ only by relabeling the values held at positions have the same sections up to that relabeling and the same licensing verdicts. Factoring out the relabelings leaves an invariant that measures exactly how far a system is from having a section, and the invariants of a fixed graph form a group.

9.1 Gauge and Class

For c:P→V, the coboundary δc:E→V is δc(e)=c(tgt e)−c(src e). Replacing every value v(x) by v(x)+c(x) carries the sections of d to the sections of d+δc, so d and d+δc are the same system in different coordinates. The impossibility class of d is

and b₁(Γ)=|E|−|P|+1 is the cycle rank of Γ, the number of edges outside a spanning tree.

The quotient H¹(Γ;V) is the first cohomology of the graph with coefficients in V (Hatcher 2002). Penrose (1992) uses the same group, with the multiplicative group of positive reals as coefficients, to measure the impossibility of the figures that bear his name, and Abramsky, Barbosa, Kishida, Lal, and Mansfield (2015) use cohomological obstructions to witness the contextuality of liar cycles. Theorem 9.1 places both in the licensing theory: the class of a system is the obstruction to a section, and Theorem 8.3 says which identifications the obstruction forbids.

9.2 Adding Impossibilities

Adding two systems edgewise composes what they say. The sum of the liar with itself is the sentence that applies the liar's transport twice, "this sentence is the negation of the negation of itself," and that sentence is the truth-teller. A staircase that rises by four around a loop added to one that descends by four is a staircase that is level, and it has a height function.

9.3 Quotients

Identifying positions is passing to a quotient graph. For a partition π of P, the quotient Γ/π has the blocks as positions and the same edges, and a transport system on Γ is a transport system on Γ/π with the same values.

For the unfolding of the liar, the path r₀→r₁→r₂→… with d=1 over ℤ/2, a partition is licensed exactly when every block contains positions of one parity, which is the coarsest-quotient theorem of the companion article. For the staircase path with d=1 over ℤ, no partition that joins two distinct positions is licensed, and over ℤ/4 exactly the partitions whose blocks are spaced by multiples of four are.

10. The Reading Ring and the Subjective Units

Sections 8 and 9 decide which identifications a transport system allows. This section gives the objects that survive when an identification is refused an arithmetic. A subjective number is taken to be a single object whose reading depends on the position from which it is read, and the impossible objects appear as units of that arithmetic: elements that read differently from different positions and cannot be given one value, in the way that a square root of −1 cannot be given a real value.

10.1 The Unfolding and the Deck Group

Let Γ be a connected perspective graph with a base position x₀. The unfolding Γ̃ of Γ is its universal cover: its positions are the walks from x₀ with no immediate backtracking, and each edge of Γ lifts to an edge between the walks it extends. The unfolding is a tree, and the deck group G of closed reduced walks at x₀ acts freely on the positions of Γ̃ by concatenation, with the positions of Γ as the orbits. The group G is free on b₁(Γ) generators, one for each edge outside a spanning tree (Serre 1980; Hatcher 2002). For the loop graph, one position with one edge, the unfolding is the line of integers and the deck group is ℤ acting by translation; this is the two-sided form of the unfolding of the liar in the companion article, which uses the ray that starts at the outermost occurrence. Write P̃ for the positions of Γ̃ and γx for the action.

10.2 Readings

Fix a field K. A subjective number over Γ with values in K is a function f:P̃→K; the value f(x) is the reading of f at the unfolded position x. The reading ring is

with pointwise addition and multiplication. The deck group acts by (γ·f)(x)=f(γ⁻¹x): the number γ·f is f read again after the loop γ has been traversed. The notation a(s) of the earlier sections is the reading: a subjective number of Sections 2 to 5 is a single reading, and a valuation is a family of readings, one per position.

Part (1) is local classicality in its arithmetic form: from any single position, subjective numbers add and multiply as ordinary numbers. What a position cannot see is how its readings relate to the readings elsewhere, and that relation is what the deck group records.

10.3 Twisted Numbers

Let Ĝ=Hom(G,K^×) be the group of characters of the deck group. For χ∈Ĝ, the χ-twisted subjective numbers are

A twisted number changes by a fixed factor when it is re-read after a loop, and the factor depends only on the loop. The positional numbers are S₁.

For the loop graph, P has one element, G=ℤ, a character is the choice of the unit χ(1)∈K^×, and the twisted ring is the group algebra K[K^×]: one copy of K for each possible factor by which a re-reading around the loop can multiply a number.

10.4 Descent

A subgroup H≤G gives an intermediate quotient Γ̃/H, whose positions are the H-orbits; it lies between the unfolding and Γ=Γ̃/G, and passing to it identifies each unfolded position with its images under H. A subjective number descends to Γ̃/H when it is a function of the H-orbit.

Descent is the licensing question asked of a single subjective number instead of a kernel: the identification of the unfolded positions along H is licensed for f exactly when f is H-invariant, and Proposition 10.9 below shows that the two questions agree where both apply.

10.5 The Liar Unit

Let Γ be the loop graph, so that Γ̃ is the line of integers, and let K have characteristic different from 2. Let χ be the character with χ(1)=−1 and let σ=u_χ with D={0}: the subjective number with readings σ(n)=(−1)ⁿ. Encode true as 1 and false as −1, so that negation is multiplication by −1 and the liar's transport is the character χ.

Read against the companion article, the theorem says what the liar's value is. Its two coherent valuations are σ and −σ: true at the pointer and false at the target, or the reverse. Neither is classical, and by (3) neither descends to a single position, which is the Observer Theorem. The two-position representation is the coarsest quotient on which the liar's value is a function of position, which is the coarsest-quotient theorem. The classical demand for one value of the liar is the demand that σ be a constant, and σ is not. What the demand overlooks is that σ is an ordinary element of an ordinary ring: it adds, multiplies, and squares to 1. The liar stands to the classical values as √−1 stands to the reals. There is no real number whose square is −1 and no classical value whose negation is itself, and in each case the missing object exists one ring up, with conjugation as the record of what was refused: complex conjugation sends i to −i, and re-reading after the loop sends σ to −σ. When K=ℝ, the ring ℝ[σ] with σ²=1 is the ring of split-complex numbers (Harkin and Harkin 2004).

Over the rationals or the reals there is no three-valued liar unit, because there is no primitive cube root of unity; over the complex numbers, or over ℤ/7, there is. Whether a many-valued liar exists as a number is decided by the value field, and where it exists its cycles of coherence are decided by divisibility. This is the counterpart, inside the reading ring, of Corollary 8.4.

10.6 The Staircase Unit

Not every subjective number is twisted. On the loop graph let n denote the height function of the infinite spiral staircase, the subjective number with reading n at position n. Re-reading after the loop shifts it: γ·n=n−1.

The two units record the two ways a re-reading can change a number: by a factor, as for the liar, or by a shift, as for the staircase. In the theory of difference equations these are the semisimple and unipotent solutions of the equations y(x+1)=λy(x) and y(x+1)=y(x)+1 over the constants K, and the group generated by the re-readings that fix the classical numbers is their difference Galois group (van der Put and Singer 1997). The classical numbers are the constants of the shift, and every non-classical subjective number solves an equation that has no classical solution.

10.7 Impossible Times Impossible

A staircase that doubles a quantity around a loop, multiplied by one that halves it, is level. The liar multiplied by itself is the truth-teller. Two impossible objects can compose to a possible one, and the rule that says when is the group law of the character group.

10.8 Sections Are Twisted Numbers

So the licensing theory of Sections 5, 8, and 9 and the descent theory of the reading ring are one theory seen from two sides. A transport system is a rule for re-reading; its sections are the subjective numbers that obey the rule; an identification is licensed for the system exactly when it is licensed for each of them.

Universe 00110000

11. Asymmetric Models and Perspective Incompleteness

Let P={p,q} and V={0,1}. Define a base kernel by setting R^∘_p(p,q,1,1)=1 and leaving the reverse judgment absent from R^∘_q. After closure, perspective p validates 1(p) ⊑_p1(q), while q does not validate 1(q) ⊑_q1(p).

This is a finite model of an asymmetric subjective preorder. The closed relations remain reflexive and transitive inside each fixed perspective.

11.1 Perspective Incompleteness

A perspective can fail to derive a judgment from its own kernel while deriving it after an adoption step. This is perspective incompleteness. The missing judgment is a judgment whose derivation runs through another perspective, and the history of the derivation records the route.

Universe 00110000

12. Selective Fusion of Perspectives

Fusion creates a perspective p⊕q from two existing perspectives. Either p⊕q∈P is already specified, or it is adjoined to P together with its base kernel and adoption guard. The fused kernel and guard are built pointwise from those of p and q by Boolean operations f and g:

The fused perspective is isolated when the other perspectives are unaffected by its addition: their kernels are unchanged and mention no subjective number at intrinsic perspective p⊕q, and no perspective other than p⊕q adopts from it, that is, CPA_r(p⊕q,u)=0 for all r≠p⊕q and all u. When f and g are monotone, fusion respects inclusion of base information and guards.

Disjunctive fusion, f=g=∨, behaves in the opposite way: it can validate chains that neither perspective validates alone, because a left edge from p can compose with a right edge from q. It models a collective perspective and it is expansive. Example A4 uses it.

Universe 00110000

13. Neighbors of the Framework

Each of the following structures shares a component with subjective numbers. The comparison states, for each, what it has and what the present framework adds.

13.1 Preorder and Modal Semantics

For a fixed evaluating perspective p, the closed relation ⊑_p is a preorder on X, so the single-perspective fragment has the shape of an S4 frame. A modal translation that sends a(s) ⊑_p b(t) to a formula such as □_p(a=b) loses the target perspective and the adoption structure. A faithful translation must keep the indexed relation itself, through primitive predicates E_p(s,t,a,b) and guard predicates CPA_p(q,u), as in Section 3.

13.2 Epistemic Logic and Trust

Multi-agent epistemic logic (Fagin, Halpern, Moses, and Vardi 1995) indexes knowledge by agent: K_pφ says that p knows φ. A judgment a(s) ⊑_p b(t) can be written as K_p applied to a relation between two values, but in the epistemic setting the values carry no index of their own. Here each value carries its position, so a judgment is a relation between two objects, and objects can be chained, adopted, and traced. Trust logics (Liau 2003) have an operator by which one agent trusts another on a topic. The guard CPA_p(q,u) has the same arity, with the landing position as the topic, and Theorem 3.6 says exactly what trusting q about u amounts to: taking on every judgment of q that lands at u.

13.3 Provenance and Justification

Provenance semirings (Green, Karvounarakis, and Tannen 2007) annotate database tuples with polynomials that record which source tuples produced them and how. Justification logic (Artemov 2001) annotates formulas with terms that record the evidence for them. The provenance functor of Section 4 belongs to this family. Its histories record kernel entries and adoption events in the order they were used, and Theorem 5.3 uses them to attribute an incoherence to the identification that produced it and to the perspective it came from.

13.4 Fixpoints and Datalog

The closure of Section 3 is the least fixpoint of a monotone operator (Tarski 1955; Kleene 1952) and, equivalently, the least model of a Datalog program with five rules (Ceri, Gottlob, and Tanca 1989). Theorem 3.6 is a program transformation: the guarded composition rule can be replaced by an import rule and the transitivity rule without changing the least model.

13.5 Descriptive Set Theory and Process Metrics

Section 6 uses the closure properties of analytic sets and Lusin's theorem (Kechris 1995) and the measurability of fiber measures of analytic sets (Bertsekas and Shreve 1978). The pseudometric follows the pattern of the metrics for labelled Markov processes of Desharnais, Gupta, Jagadeesan, and Panangaden (2004).

13.6 Many-Valued Logics

Many-valued logics vary the value space. Subjective numbers vary the evaluating perspective, the intrinsic perspectives of the values compared, and the guarded rules for adopting judgments. The two can be combined, since nothing in Sections 3 to 5 requires V to have two elements; only Theorem 5.3 (2) does.

13.7 Indexed and Enriched Structures

The framework is an indexed family of preorders coupled by adoption rules. Judgments live over indices, as in fibrational and enriched settings, but the formal system is elementary: a family of relations, a monotone closure operator, and explicit proof terms.

13.8 Nearest Relatives

Four bodies of work put an observer or a subject next to a value or a truth. Benioff (2018) lets the value of a number depend on the observer's location, so that a numerical quantity is a field over observers; Khots and Khots (2007) relativize arithmetic to an observer's finite precision. Jøsang (2016) attaches opinions with degrees of belief and uncertainty to the sources that hold them. Lasersohn (2017) and MacFarlane (2014) make the truth of a sentence relative to a judge or an assessor. In each, one party is indexed: the observer whose value it is, the source whose opinion it is, the judge whose truth it is. In none does a value carry its own perspective as an index that a second, evaluating perspective relates to another such value under a guard. That double index is what subjective numbers add, and the results of Section 5 use both indices at once.

13.9 What the Combination Adds

None of the neighbors carries all three of the following at once: an intrinsic index on each value, a separate evaluating index on each judgment, and a guard on adoption indexed by the position at which an adopted judgment lands. The Licensing Theorem, the Collapse Theorem, and the Classical Collapse Theorem depend on the combination. The first two need positions that can be identified or kept apart inside one evaluator; the third needs guards that can be opened all at once and positions that can be identified all at once, so that the ordinary case appears as a limit of the indexed one. The reading ring of Section 10 needs the same double index: a reading is a value at an unfolded position, and re-reading is the action of a loop on the family of readings.

13.10 Covering Spaces, Group Algebras, and Difference Equations

The reading ring is the ring of functions on the universal cover of a graph, its twisted part is the group algebra of the character group of the fundamental group, and the liar unit generates the split-complex numbers. Each of these is classical mathematics (Hatcher 2002; Lang 2002; Harkin and Harkin 2004), and the shift on the unfolding of the loop graph makes the reading ring a difference ring whose constants are the classical numbers (van der Put and Singer 1997). Four bodies of work come closest to Sections 8 to 10. Penrose (1992) measures the impossibility of a figure by a cohomology class, and Ghrist and Cooperband (2025) and Ghrist and Ghrist (2026) treat the Penrose staircase and its relatives with torsors, potentials, double covers, and holonomy; Theorem 9.1 is the same obstruction, and Corollary 8.4 adds its dependence on the value group and its reading as a licensing verdict. Spencer-Brown (1969) and Kauffman (1987, 2002) give the liar an imaginary value modeled on √−1, reading the sentence as the oscillation true, false, true, false and its value as a solution of x=−1/x; Mar and Grim (1991) study the same oscillation as a dynamical system. The liar unit of Theorem 10.5 differs from the imaginary value in three ways: it satisfies σ²=1 and generates the split-complex numbers rather than the complex numbers; it is derived from the unfolding and its deck group rather than assigned; and it comes with the theorem that descending it and licensing the collapse are the same question (Proposition 10.9), so that the liar and the staircase are units of one ring with one descent criterion.

14. Conclusion

Subjective numbers are values with a standpoint, and judgments about them are made from a standpoint. The two-tier semantics separates the judgments a perspective holds outright from the judgments it derives. Closure is a least fixpoint, each closed relation is a preorder, adoption between perspectives is import selected by landing position, and every derivation carries a history that names its sources.

Identifying two positions is a judgment, and the Licensing Theorem says what it does: it keeps the coherent valuations that already agree on the two positions and removes the rest. A collapse is safe exactly when nothing is removed, and it is a paradox exactly when everything is. With every guard open and every position identified, the framework is ordinary relational mathematics, a single preorder on the values. Ordinary mathematics is the collapsed case, and the Penrose staircase and the liar sentence are the smallest systems in which the collapse is forbidden.

The objects that cannot be collapsed have an arithmetic. On a perspective graph with fixed transports the obstructions to collapse form the group H¹(Γ;V), they add, and whether a given identification is forbidden depends on the value group as much as on the graph. In the ring of readings every position does ordinary arithmetic, the classical numbers are the readings that agree everywhere, and the liar is the unit σ with σ²=1 whose two readings are the two values the classical evaluator tried to merge. Subjective numbers extend the classical values as the complex numbers extend the reals: by admitting the solutions of equations that have none, and by recording in a conjugation, here the re-reading around a loop, exactly what was admitted.

The measure-theoretic construction yields a pseudometric between perspectives: two perspectives that differ only on null sets have zero discrepancy, and the distance between two perspectives is the measure of the judgments on which they disagree.

14.1 Further Questions

• Non-well-founded provenance for infinite dependency chains, in the setting of Aczel (1988).
• Faster closure algorithms for restricted classes of guard graphs.
• Licensing and collapse for value spaces with more than two elements, where the syntactic witness of Theorem 5.3 (2) is not available.
• Transport systems with non-abelian transports, where the impossibility classes form a set of holonomy representations rather than a group.
• The reading ring for kernels that are not transport systems, where sections are cut out by inequalities or conjunctive constraints rather than by shifts.

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Appendix: Mathematical Examples for Subjective Numbers

The examples below illustrate the framework with explicit kernels. In each, R^∘ denotes base information and R^⋆ the closure generated by the rules of Section 3. The displayed consequences are selected consequences. Once reflexivity, transitivity, substitution, and guarded adoption are applied, R^⋆ may contain further judgments generated by longer chains.

Example A1: A Model with Explicit Relation Functions

Take three perspectives P = {p, q, r} and values in ℝ. Each perspective judges from its own position, and the content of its judgments differs.

For perspective p, define R^∘_p as follows:

• R^∘_p(p, p, a, a) = 1 for all a ∈ ℝ.
• R^∘_p(p, q, a, b) = 1 if and only if a = b.
• R^∘_p(p, r, a, b) = 1 if and only if a ≤ b.

For perspective q, define R^∘_q as follows:

• R^∘_q(q, q, a, a) = 1 for all a ∈ ℝ.
• R^∘_q(q, p, a, b) = 1 if and only if a ≥ b.
• R^∘_q(q, r, a, b) = 1 if and only if a = 2b.

For perspective r, define R^∘_r as follows:

• R^∘_r(r, r, a, a) = 1 for all a ∈ ℝ.
• R^∘_r(r, p, a, b) = 1 if and only if a = b + 1.
• R^∘_r(r, q, a, b) = 1 if and only if a = b/2.

Reflexivity is present in each kernel for same-perspective comparisons and is added by closure for every subjective number. Asymmetry is visible: R^∘_p(p,r,3,5)=1 since 3≤5, while R^∘_r(r,p,5,3) is absent because 5≠3+1. Transitivity holds in the closed relations by Theorem 3.2.

Selected judgments from perspective p:

• 5(p) ⊑_p 5(q), by value equality with q.
• 3(p) ⊑_p 7(r), since 3 ≤ 7.
• 7(p) ⊑_p 3(r) is absent from the kernel because 7 ≰ 3.

From perspective q:

• 8(q) ⊑_q 4(p), since 8 ≥ 4.
• 10(q) ⊑_q 5(r), since 10 = 2 × 5.
• 3(q) ⊑_q 7(p) is absent from the kernel because 3 ≱ 7.

From perspective r:

• 6(r) ⊑_r 5(p), since 6 = 5 + 1.
• 4(r) ⊑_r 8(q), since 4 = 8/2.
• 3(r) ⊑_r 9(p) is absent from the kernel because 3 ≠ 9 + 1.

The kernels here record comparisons, and the determination reading of Section 5 does not apply to them: 3(p) ⊑_p 5(r) and 3(p) ⊑_p 7(r) both hold, and they are comparisons, not constraints that fix the value at r.

Example A2: Computing in the Reading Ring

Take the loop graph, so that the unfolding is the line of integers, and take K=ℚ. The liar unit σ has readings 1,−1,1,−1,… at positions 0,1,2,3,…. Some computations, each done reading by reading:

• σ² reads 1 everywhere: the truth-teller, a classical number.
• 1+σ reads 2,0,2,0,…: twice the pointer projection (1+σ)/2, which reads 1 at even positions and 0 at odd ones.
• 3+2σ reads 5,1,5,1,…. Its image in ℚ×ℚ under (ev₀,ev₁) is (5,1), and re-reading after the loop gives 3−2σ, whose image is (1,5): the swap.
• (3+2σ)(3−2σ)=9−4σ²=5: a subjective number times its re-reading is classical, as (a+bi)(a−bi) is real.

Now take K=ℤ/7, in which 2 is a primitive cube root of unity. The character χ(1)=2 gives the unit u with readings 1,2,4,1,2,4,…: a three-valued liar, with u³=1, which descends to the quotients with 3,6,9,… positions and to no others (Corollary 10.6). Over ℚ no such unit exists.

Finally the staircase. Over V=ℤ the path r₀→r₁→r₂→r₃→r₄ with d=1 has the section v(r_i)=i, and the identification of r₄ with r₀ is forbidden because the walk sum is 4. Over V=ℤ/4 the same path has the section 0,1,2,3,0, and the identification is licensed: the closed staircase exists, and walking around it four steps up returns to the starting height.

Example A3: Path-Dependent Derivability

Different guarded paths make different judgments available to an evaluating perspective. Take P = {α, β, γ} with the following base information:

1. For perspective α:
   • R^∘_α(α,β,5,5)=1.
   • R^∘_α(α,α,5,3) is absent from the kernel.
   • No direct base relation with γ is specified.
2. For perspective β:
   • R^∘_β(β,γ,5,5)=1.
   • No direct base relation with α is specified.
3. For perspective γ:
   • R^∘_γ(γ,α,5,3)=1.
   • No direct base relation with β is specified.

Guards:

• CPA_α(β,γ)=1.
• CPA_α(γ,α)=1.

The judgment 5(α) ⊑_α3(α) is not in R^∘_α. This is absence from the kernel, and the closure of the kernel alone does not contain it either. Through the chain α → β → γ → α:

1. Base: 5(α) ⊑_α5(β).
2. Base at β: 5(β) ⊑_β5(γ).
3. By CPA_α(β,γ), perspective α imports the second edge and obtains 5(α) ⊑_α5(γ).
4. Base at γ: 5(γ) ⊑_γ3(α).
5. By CPA_α(γ,α), perspective α imports the fourth edge and obtains 5(α) ⊑_α3(α).

So 5(α) ⊑_α3(α) belongs to R^⋆_α through adoption, and its history names β and γ as the perspectives adopted from. Under the determination reading this judgment forbids the value 5 at position α: no coherent valuation of α has v(α)=5.

Example A4: Perspective Incompleteness

Certain judgments are inaccessible from individual kernels and become available through fusion. Take P = {α, β, γ} and the subjective numbers 7(α), 4(β), and 10(γ), with base information:

1. For perspective α:
   • R^∘_α(α,β,7,4)=1.
   • R^∘_α(α,γ,7,10) is absent.
2. For perspective β:
   • R^∘_β(β,γ,4,10)=1.
   • R^∘_β(β,α,4,7) is absent.
3. For perspective γ:
   • R^∘_γ(γ,β,10,4) is absent.
   • The optional rejection log records N^∘_γ(γ,α,10,7)=1. This rejection is an annotation and does not enter the closure operator.

Adjoin the fused perspective α⊕β with the disjunctive kernel

From the fused perspective:

• 7(α) ⊑_α⊕β4(β), inherited from α.
• 4(β) ⊑_α⊕β10(γ), inherited from β.
• By transitivity, 7(α) ⊑_α⊕β10(γ).

The relation between 7(α) and 10(γ) is present in neither individual kernel and becomes accessible through fusion. The rejection log at γ records a conflicting attitude, and because it is not part of the closure it creates no derived judgment.

Example A5: The Penrose Staircase

The Penrose staircase (Penrose and Penrose 1958) is a drawing of four flights of stairs joined in a loop so that each flight rises from the one before it and the last returns to the first. Each corner is drawn consistently; the whole has no consistent height function. The following system is its analogue.

1. Perspectives as corners. Five positions p₁,…,p₅, the bottom of the first flight, the three corners, and the top of the fourth flight, together with an observer o.
2. Values as heights. V=ℤ, so that h(p_i) is the height held at corner i.
3. Transports of ascent. Each flight raises the height by one: the transport system with d=1 on each edge p_i→p_i+1, held by the corner p_i, whose kernel holds a(p_i) ⊑ (a+1)(p_i+1) for every height a and i=1,…,4.
4. The drawing. The picture declares that p₅ is p₁: the top of the fourth flight is the bottom of the first. That declaration is the identification of the two positions.

An observer o with the guards CPA_o(p_i,p_i+1)=1 for the four transitions imports all four edges and holds the whole path. Under the determination reading o is coherent: the valuation v(p_i)=i−1, which reads 4 at the top of the fourth flight, respects every edge. What o cannot do over the integers is treat the top of the fourth flight as the bottom of the first. By Corollary 8.4 that identification is licensed exactly when 4=0 in the value group: forbidden over ℤ, licensed over ℤ/4. The drawing is coherent at every corner and has no global height, and the system is coherent at every perspective and has no coherent collapse over the integers.