# description
Closed release-specific schema for the immutable reviewed atlas. Validation establishes exact parsed-value conformance, not byte identity, theological truth, or authority.

# $comment
This schema cannot recompute the semantic digest or authenticate source availability. Raw bytes, duplicate-key and UTF-8 handling, semantic integrity, mathematical soundness, historical or translation fidelity, theological truth, and authority require independent verification.

# properties.source_bindings.prefixItems[].properties.verification_note.const
The revision-pinned GitHub URL inherited by predecessor atlases returned 404 on 2026-08-14. This atlas makes no public-access claim; the digest was checked against the repository-local path and revision object.

# properties.scope.properties.question.const
How can changes in mathematical expression illuminate number, zero, symmetry, being, negation, creation, God-language, and religious compatibility without turning analogy into proof?

# properties.scope.properties.intended_use.const
Read-only education, typed question design, source comparison, and bounded opt-in philosophical reflection.

# properties.scope.properties.excluded_use.const
No proof or disproof of God, revelation, creation doctrine, religious truth, cosmic purpose, or karma; no religion, culture, community, or person ranking; no persuasion, governance, invocation, event, receipt, or action.

# properties.scope.properties.method.const
Keep formal mathematics, historical scholarship, source attribution, school-specific theology, metaphysical proposal, comparative relation, and analogy in separate registers; every bridge records what it does not transfer.

# properties.scope.properties.central_boundary.const
A mathematical structure can sharpen the form of a question. It does not supply the theological premises, translation, interpretation, soundness, actuality, or authority required to answer that question.

# properties.claim_typing.properties.rule.const
Every reviewed claim has a discourse register, review target, review status, support descriptors, and source scope. Source attribution can be supported without the attributed proposition becoming a world fact.

# properties.claim_typing.properties.support_guard.const
Descriptors are non-exclusive and never upgrade review status. A theorem proves only its exact statement under its assumptions; a primary text establishes what that text says under a declared edition and interpretation, not that its claim is true.

# properties.claim_typing.properties.translation_guard.const
A translated term remains indexed by source language, edition, translator, context, school, and interpretive loss; lexical resemblance never establishes shared reference.

# properties.typed_isness.properties.rule.const
Replace is, is not, no, and no-no with a typed predicate before drawing a mathematical, historical, metaphysical, or theological inference.

# properties.typed_isness.properties.predicates.prefixItems[].properties.meaning.const
x belongs to a declared formal carrier under a named model.

# properties.typed_isness.properties.predicates.prefixItems[].properties.meaning.const
x is classified under one declared type rule.

# properties.typed_isness.properties.predicates.prefixItems[].properties.meaning.const
x and y agree under one named invariant.

# properties.typed_isness.properties.predicates.prefixItems[].properties.meaning.const
A structure-preserving map sends x to y.

# properties.typed_isness.properties.predicates.prefixItems[].properties.meaning.const
A named method supports one property attribution in scope.

# properties.typed_isness.properties.predicates.prefixItems[].properties.meaning.const
At least one witness in a named domain instantiates the form in context and time.

# properties.typed_isness.properties.predicates.prefixItems[].properties.meaning.const
The term receives a formation and interpretation rule in a formal theory.

# properties.typed_isness.properties.predicates.prefixItems[].properties.meaning.const
A checked derivation establishes a formula relative to premises, rules, and logic.

# properties.typed_isness.properties.predicates.prefixItems[].properties.meaning.const
A source is attributed a bounded claim under an edition and interpretation.

# properties.typed_isness.properties.predicates.prefixItems[].properties.meaning.const
A named interpreter offers one scoped reading.

# properties.typed_isness.properties.predicates.prefixItems[].properties.meaning.const
A bounded formalization has at least one model under declared choices.

# properties.typed_isness.properties.predicates.prefixItems[].properties.meaning.const
A logic-specific negation is applied to a proposition.

# properties.typed_isness.properties.predicates.prefixItems[].properties.signature.const
ABSENT(domain,predicate) iff not exists x in domain: predicate(x)

# properties.typed_isness.properties.predicates.prefixItems[].properties.meaning.const
No witness satisfies a predicate in a named domain.

# properties.typed_isness.properties.predicates.prefixItems[].properties.meaning.const
A value is the distinguished zero of one declared structure.

# properties.typed_isness.properties.predicates.prefixItems[].properties.meaning.const
An observation operator returned zero reports in a bounded window.

# properties.typed_isness.properties.predicates.prefixItems[].properties.meaning.const
A subject declines a proposal at a time.

# properties.typed_isness.properties.predicates.prefixItems[].properties.meaning.const
This bounded review cannot determine the claim.

# properties.typed_isness.properties.predicates.prefixItems[].properties.meaning.const
A speaker does not apply a predicate in context.

# properties.typed_isness.properties.no_family.properties.guard.const
These expressions have different types and no default conversion.

# properties.typed_isness.properties.no_family.properties.double_negation_guard.const
No-no returns affirmation only in a declared logic that validates double-negation elimination. Apophatic restraint, tetralemma, refusal, and silence are not hidden applications of that formal rule.

# properties.historical_expression.properties.rule.const
Formal inclusion order is not chronological history. Mathematical expression developed nonlinearly across cultures through notation, calculation, transmission, rejection, reuse, and later formalization.

# properties.historical_expression.properties.cards.prefixItems[].properties.claim.const
The positional placeholder and zero as a number are historically distinguishable uses; their development was neither a single invention event nor an inevitable straight line.

# properties.historical_expression.properties.cards.prefixItems[].properties.nonclaims.prefixItems[].const
one culture alone explains every zero practice

# properties.historical_expression.properties.cards.prefixItems[].properties.claim.const
MacTutor records Bhaskara using divine immutability as an analogy while treating division by zero; the historical analogy does not make division by zero valid in an ordinary field.

# properties.historical_expression.properties.cards.prefixItems[].properties.nonclaims.prefixItems[].const
Indian mathematics is reducible to this episode

# properties.historical_expression.properties.cards.prefixItems[].properties.claim.const
Rules for complex arithmetic, geometric rotation, and later field language emerged through distinct historical stages; imaginary was a contested name, not a proof of lesser mathematical reality.

# properties.historical_expression.properties.cards.prefixItems[].properties.claim.const
N0 to Z to Q to R to C is a modern structural presentation of embeddings, closure, and completion needs, not a complete chronology of how people expressed number.

# properties.number_systems.properties.natural_convention.const
N0={0,1,2,...}; authors using N={1,2,3,...} must declare that alternate convention.

# properties.number_systems.properties.tower.const
N0 embeds in Z embeds in Q embeds in R embeds in C

# properties.number_systems.properties.embedding_guard.const
Write iota_NZ(1_N)=1_Z rather than an untyped literal identity. Canonically embedded roles may agree while set-theoretic constructions differ.

# properties.number_systems.properties.cards.prefixItems[].properties.construction.const
A model of the Peano-style zero-and-successor structure.

# properties.number_systems.properties.cards.prefixItems[].properties.door.const
finite counting, iteration, and induction

# properties.number_systems.properties.cards.prefixItems[].properties.limit.const
not closed under additive inverses or nonzero division

# properties.number_systems.properties.cards.prefixItems[].properties.nonclaims.prefixItems[].const
natural in mathematics means metaphysically natural

# properties.number_systems.properties.cards.prefixItems[].properties.limit.const
not closed under division by every nonzero element

# properties.number_systems.properties.cards.prefixItems[].properties.nonclaims.prefixItems[].const
construction chronology follows this quotient

# properties.number_systems.properties.cards.prefixItems[].properties.door.const
fractions, ratios, and division by nonzero elements

# properties.number_systems.properties.cards.prefixItems[].properties.limit.const
dense and ordered but not complete; rational does not mean reasonable or revelatory

# properties.number_systems.properties.cards.prefixItems[].properties.construction.const
A completion of Q, for example by Cauchy sequences modulo zero-distance equivalence or Dedekind cuts.

# properties.number_systems.properties.cards.prefixItems[].properties.door.const
limits and completeness of the ordered field

# properties.number_systems.properties.cards.prefixItems[].properties.equations.prefixItems[].const
every Cauchy sequence in R converges in R

# properties.number_systems.properties.cards.prefixItems[].properties.limit.const
the construction choice is not the unique ontology of real numbers

# properties.number_systems.properties.cards.prefixItems[].properties.door.const
roots of polynomials and plane rotations

# properties.number_systems.properties.cards.prefixItems[].properties.theorem_claims.prefixItems[].properties.statement.const
For theta in R and z in C, multiplication by exp(i theta) rotates z through angle theta in the standard complex plane.

# properties.number_systems.properties.cards.prefixItems[].properties.theorem_claims.prefixItems[].properties.conditions.prefixItems[].const
the standard complex exponential and Euclidean plane identification are used

# properties.number_systems.properties.cards.prefixItems[].properties.theorem_claims.prefixItems[].properties.statement.const
Every nonconstant polynomial with complex coefficients has a root in C.

# properties.number_systems.properties.cards.prefixItems[].properties.limit.const
C admits no field order compatible with its field operations; adjoining elements does not preserve every structure of R

# properties.number_systems.properties.cards.prefixItems[].properties.nonclaims.prefixItems[].const
algebraic closure is metaphysical completeness

# properties.number_systems.properties.embeddings.prefixItems[].properties.does_not_establish.prefixItems[].const
literal identity of underlying encodings

# properties.number_systems.properties.embeddings.prefixItems[].properties.does_not_establish.prefixItems[].const
all rational numbers are embedded integers

# properties.number_systems.properties.embeddings.prefixItems[].properties.map.const
q maps to the equivalence class of its constant Cauchy sequence or to the matching Dedekind cut

# properties.zero.properties.rule.const
Zero is role-indexed. A representation may identify a zero with an empty set, as in the von Neumann encoding 0=empty set, without making the number-role semantically identical to an empty collection or to metaphysical nothingness.

# properties.zero.properties.field_boundary.const
No field element z satisfies 0*z=1; therefore zero has no multiplicative inverse and division by zero is not a field operation.

# properties.zero.properties.roles.prefixItems[].properties.not.const
universal annihilation outside the operation

# properties.zero.properties.roles.prefixItems[].properties.not.const
the number zero in every historical notation

# properties.zero.properties.roles.prefixItems[].properties.equation.const
P({x})=0 for a point under a continuous distribution

# properties.zero.properties.roles.prefixItems[].properties.equation.const
for every set X there is a unique function empty set -> X

# properties.zero.properties.roles.prefixItems[].properties.not.const
a causal process creating X from nothing

# properties.zero.properties.roles.prefixItems[].properties.equation.const
zero reports in window W under operator O

# properties.zero.properties.roles.prefixItems[].properties.not.const
zero event rate, nonexistence, refusal, or impossibility

# properties.zero.properties.non_equivalences.prefixItems[].const
numeric zero is not metaphysical nothingness

# properties.zero.properties.non_equivalences.prefixItems[].const
numeric zero is not theological creation ex nihilo

# properties.zero.properties.non_equivalences.prefixItems[].const
numeric zero is not an unknown review status

# properties.zero.properties.non_equivalences.prefixItems[].const
cardinality zero does not erase the existence of the carrier, rule, or discourse

# properties.zero.properties.non_equivalences.prefixItems[].const
a chosen encoding such as 0=empty set does not transfer set-theoretic identity into a universal ontology

# properties.symmetry.properties.definitions.properties.group_action.const
A group action is a map G x X -> X satisfying e*x=x and (g*h)*x=g*(h*x) for all g,h in G and x in X

# properties.symmetry.properties.definitions.properties.quotient.const
X/G identifies points only up to the declared action

# properties.symmetry.properties.cards.prefixItems[].properties.claim.const
Symmetry is invariance under specified transformations, not beauty or sameness without a transformation group.

# properties.symmetry.properties.cards.prefixItems[].properties.nonclaims.prefixItems[].const
equivalence is identity in every respect

# properties.symmetry.properties.cards.prefixItems[].properties.claim.const
For a variational problem satisfying the theorem's hypotheses, continuous invariance yields corresponding conservation structure.

# properties.symmetry.properties.cards.prefixItems[].properties.nonclaims.prefixItems[].const
every visual symmetry yields a conserved quantity

# properties.symmetry.properties.cards.prefixItems[].properties.claim.const
A law or model can have a symmetry while a realized state has lower symmetry; asymmetry and history remain mathematically meaningful.

# properties.symmetry.properties.cards.prefixItems[].properties.claim.const
Geometric repetition, balance, and invariant form can support source-bound aesthetic or theological reflection without constituting a designer proof.

# properties.symmetry.properties.cards.prefixItems[].properties.nonclaims.prefixItems[].const
Islamic geometry is mandated by scripture

# properties.symmetry.properties.cards.prefixItems[].properties.nonclaims.prefixItems[].const
mandalas and Islamic patterns share theology

# properties.logics.properties.rule.const
Negation, contradiction, proof, and existence claims are logic-indexed. A logic is a formal consequence relation, not a religion or a complete metaphysics.

# properties.logics.properties.cards.prefixItems[].properties.equations.prefixItems[].const
from phi and not phi, arbitrary psi follows

# properties.logics.properties.cards.prefixItems[].properties.meaning.const
Classical logic validates double-negation elimination and explosion.

# properties.logics.properties.cards.prefixItems[].properties.meaning.const
Intuitionistic logic does not validate unrestricted double-negation elimination.

# properties.logics.properties.cards.prefixItems[].properties.equations.prefixItems[].const
not generally: {phi,not phi} entails psi

# properties.logics.properties.cards.prefixItems[].properties.meaning.const
A paraconsistent consequence relation is non-explosive; there are many such logics.

# properties.logics.properties.cards.prefixItems[].properties.meaning.const
Four statuses can represent supported truth, supported falsity, conflicting information, or no information in the Belnap-Dunn setting.

# properties.logics.properties.tetralemma_boundary.const
A fourfold religious or philosophical discourse may challenge a question's presupposition or suspend predication. Similar cardinality to a four-valued logic is not a translation or formal equivalence.

# properties.creation.properties.rule.const
Construction, causation, cosmogenesis, theological creation, manifestation, dependent arising, and grammatical command are separate relations with no default adapter.

# properties.creation.properties.types.prefixItems[].properties.meaning.const
Rules introduce or characterize an object in a formal system.

# properties.creation.properties.types.prefixItems[].properties.meaning.const
A causal model attributes production under explicit conditions and evidence.

# properties.creation.properties.types.prefixItems[].properties.meaning.const
A source or school attributes creation under a named theological mode.

# properties.creation.properties.types.prefixItems[].properties.meaning.const
A theological doctrine denies pre-existent material required by the creator.

# properties.creation.properties.types.prefixItems[].properties.meaning.const
A text uses command language translated as Be and it is.

# properties.creation.properties.types.prefixItems[].properties.meaning.const
A source or school describes recurring manifestation or projection.

# properties.creation.properties.types.prefixItems[].properties.meaning.const
A Buddhist source or school explains conditioned arising without importing a sovereign creator relation.

# properties.creation.properties.types.prefixItems[].properties.meaning.const
A source poses or leaves open an origin question.

# properties.creation.properties.adapter.properties.reason.const
No formal construction, physical causal relation, or theological creation relation converts into another without separately sourced premises and typed correspondence.

# properties.mathematical_ontology.properties.rule.const
The atlas presents live positions about mathematical objects and practice without selecting one or converting it into a theology.

# properties.mathematical_ontology.properties.positions.prefixItems[].properties.proposal.const
Mathematical objects are abstract and exist independently of human language, thought, or practice.

# properties.mathematical_ontology.properties.positions.prefixItems[].properties.does_not_entail.prefixItems[].const
all mathematical questions have determinate answers

# properties.mathematical_ontology.properties.positions.prefixItems[].properties.proposal.const
Mathematics concerns positions and relations in structures, with multiple versions of what structures are.

# properties.mathematical_ontology.properties.positions.prefixItems[].properties.proposal.const
Mathematical practice can be understood through formal signs, rules, or systems in one family of views.

# properties.mathematical_ontology.properties.positions.prefixItems[].properties.proposal.const
Mathematics is tied to construction and proof rather than unrestricted classical truth conditions in one family of views.

# properties.mathematical_ontology.properties.positions.prefixItems[].properties.proposal.const
Mathematical discourse may be interpreted without commitment to independently existing abstract objects in several distinct views.

# properties.mathematical_ontology.properties.positions.prefixItems[].properties.proposal.const
Notation, proof, institutions, transmission, and applications are indispensable to explaining how mathematics is expressed and evolves.

# properties.tradition_lenses.prefixItems[].properties.voice.const
Hebrew Bible in Revised JPS 2023 translation

# properties.tradition_lenses.prefixItems[].properties.claim.const
Exodus 3:14 presents Ehyeh-Asher-Ehyeh with translation uncertainty including I am and I will be renderings.

# properties.tradition_lenses.prefixItems[].properties.translation_limits.prefixItems[].const
Hebrew verbal aspect is not exhausted by an English present-tense copula

# properties.tradition_lenses.prefixItems[].properties.translation_limits.prefixItems[].const
the verse alone does not prove a later ontology of subsistent being

# properties.tradition_lenses.prefixItems[].properties.voice.const
Maimonides, Guide for the Perplexed I.58 in Friedlander translation

# properties.tradition_lenses.prefixItems[].properties.claim.const
Maimonides argues that negative attributes and attributes of action avoid treating positive predicates as descriptions of the divine essence.

# properties.tradition_lenses.prefixItems[].properties.translation_limits.prefixItems[].const
negative predication is not Boolean negation without further formalization

# properties.tradition_lenses.prefixItems[].properties.claim.const
Catholic doctrine reads the divine name through fullness of Being and teaches free creation without pre-existent material.

# properties.tradition_lenses.prefixItems[].properties.translation_limits.prefixItems[].const
Catholic doctrinal witness, not all Christianity

# properties.tradition_lenses.prefixItems[].properties.translation_limits.prefixItems[].const
creation out of nothing is not a function from numeric zero

# properties.tradition_lenses.prefixItems[].properties.voice.const
Quran 112 and 2:117 in Dr. Mustafa Khattab's The Clear Quran translation (Quran.com resource 131)

# properties.tradition_lenses.prefixItems[].properties.claim.const
The selected Quranic passages affirm divine oneness/incomparability and use command language translated Be and it is.

# properties.tradition_lenses.prefixItems[].properties.translation_limits.prefixItems[].const
The translation's One and Indivisible wording is not treated as arithmetic identity

# properties.tradition_lenses.prefixItems[].properties.translation_limits.prefixItems[].const
English copular wording does not settle the semantics of the Arabic

# properties.tradition_lenses.prefixItems[].properties.translation_limits.prefixItems[].const
command language is not a physical mechanism description

# properties.tradition_lenses.prefixItems[].properties.translation_limits.prefixItems[].const
resource 131 identifies one English translation and does not exhaust Quranic interpretation

# properties.tradition_lenses.prefixItems[].properties.voice.const
Scholarly survey of Arabic and Islamic philosophy of religion

# properties.tradition_lenses.prefixItems[].properties.claim.const
Classical Islamic thought contains distinct models of temporal origination, emanation, and manifestation.

# properties.tradition_lenses.prefixItems[].properties.translation_limits.prefixItems[].const
survey categories do not form one Islamic ontology

# properties.tradition_lenses.prefixItems[].properties.voice.const
Rig Veda 10.129 in Ralph T. H. Griffith's 1896 second-edition English translation, Wikisource revision 14396955

# properties.tradition_lenses.prefixItems[].properties.claim.const
The hymn juxtaposes being and nonbeing language and ends with an origin question whose answer may remain uncertain.

# properties.tradition_lenses.prefixItems[].properties.translation_limits.prefixItems[].const
sat and asat are not plug-compatible with Griffith's non-existent and existent renderings

# properties.tradition_lenses.prefixItems[].properties.translation_limits.prefixItems[].const
a poetic suspended question is not a formal truth table

# properties.tradition_lenses.prefixItems[].properties.translation_limits.prefixItems[].const
this nineteenth-century translation is one witness and later scholarship may translate the Sanskrit differently

# properties.tradition_lenses.prefixItems[].properties.voice.const
Scholarly survey of Sankara and Advaita Vedanta

# properties.tradition_lenses.prefixItems[].properties.claim.const
Advaita interprets Upanishadic material through nondual brahman and graded dependence/reality.

# properties.tradition_lenses.prefixItems[].properties.translation_limits.prefixItems[].const
advaita is not the arithmetic equation 1=1

# properties.tradition_lenses.prefixItems[].properties.voice.const
Majjhima Nikaya 72 in Thanissaro translation with scholarly interpretation boundary

# properties.tradition_lenses.prefixItems[].properties.claim.const
The selected discourse declines four postmortem predications as inapplicable; this need not assert a deviant universal logic.

# properties.tradition_lenses.prefixItems[].properties.translation_limits.prefixItems[].const
a scoped refusal is not a four-valued truth assignment

# properties.tradition_lenses.prefixItems[].properties.translation_limits.prefixItems[].const
one discourse does not define all Buddhist logic

# properties.tradition_lenses.prefixItems[].properties.voice.const
Scholarly survey of Nagarjuna and Madhyamaka

# properties.tradition_lenses.prefixItems[].properties.claim.const
Emptiness concerns lack of intrinsic independent nature and is connected with dependent arising, not primordial nothingness.

# properties.tradition_lenses.prefixItems[].properties.voice.const
Daodejing 42 in Chinese Text Project and SEP Laozi survey

# properties.tradition_lenses.prefixItems[].properties.claim.const
The selected text uses generative number language while scholarship records disputed referents and interpretations of you, wu, one, and generation.

# properties.compatibility.properties.rule.const
Compare exact sourced claims, not religions as atomic wholes. Every result is indexed by logic L, assumptions A, partial translation tau, sources, schools, editions, and evidence horizon.

# properties.compatibility.properties.equation.const
Compat_(L,A,tau)(D_i,D_j) iff Mod_L(A union tau_i(D_i) union tau_j(D_j)) is nonempty

# properties.compatibility.properties.nerve.const
Let F be the finite declared set of bounded, sourced doctrine fragments, lenses, or claim sets. K_tau={S subseteq F: Mod_L(A union union_(d in S) tau_d(d)) is nonempty}.

# properties.compatibility.properties.pairwise_not_global_example.const
In classical propositional logic, D1={p or q}, D2={not p}, D3={not q}: every pair is satisfiable but the triple is not.

# properties.compatibility.properties.claim_fragments.prefixItems[].properties.claim.const
The selected source-bound lenses retain material doctrinal differences and do not form one synthesized theology.

# properties.compatibility.properties.claim_fragments.prefixItems[].properties.claim.const
The selected Catholic, Quranic-translation, and ecumenical witnesses each make room for bounded encounter or cooperation without requiring doctrinal identity.

# properties.compatibility.properties.claim_fragments.prefixItems[].properties.nonclaims.prefixItems[].const
dialogue proves compatibility in every dimension

# properties.compatibility.properties.cases.prefixItems[].properties.preserved.const
The cited Catholic reception reads the inherited text through a developed ontology.

# properties.compatibility.properties.cases.prefixItems[].properties.breaks.prefixItems[].const
later doctrine is not the verse's sole truth condition

# properties.compatibility.properties.cases.prefixItems[].properties.preserved.const
Each selected source uses unity or one-language in a bounded way.

# properties.compatibility.properties.cases.prefixItems[].properties.breaks.prefixItems[].const
cardinality versus uniqueness versus nonduality versus generative stage

# properties.compatibility.properties.cases.prefixItems[].properties.breaks.prefixItems[].const
different soteriological and textual roles

# properties.compatibility.properties.cases.prefixItems[].properties.preserved.const
Both discussions can use absence-adjacent vocabulary.

# properties.compatibility.properties.cases.prefixItems[].properties.breaks.prefixItems[].const
zero is an element or role in a structure

# properties.compatibility.properties.cases.prefixItems[].properties.breaks.prefixItems[].const
ex nihilo denies required pre-existent material

# properties.compatibility.properties.cases.prefixItems[].properties.breaks.prefixItems[].const
nothing is not a mathematical input substance

# properties.compatibility.properties.cases.prefixItems[].properties.preserved.const
Both can prompt a question about expression and coming-to-be.

# properties.compatibility.properties.cases.prefixItems[].properties.preserved.const
Both can reject self-sufficient ordinary entities in different senses.

# properties.compatibility.properties.cases.prefixItems[].properties.breaks.prefixItems[].const
creator relation versus conditioned co-arising

# properties.compatibility.properties.cases.prefixItems[].properties.preserved.const
Both make a fourfold display available for inquiry.

# properties.compatibility.properties.cases.prefixItems[].properties.breaks.prefixItems[].const
speech-act refusal versus semantic information state

# properties.compatibility.properties.cases.prefixItems[].properties.preserved.const
Specified visual transformations can be studied mathematically in religious art.

# properties.compatibility.properties.cases.prefixItems[].properties.preserved.const
Communities can cooperate while retaining material disagreements.

# properties.compatibility.properties.cases.prefixItems[].properties.breaks.prefixItems[].const
practical coexistence is not doctrinal identity

# properties.compatibility.properties.cases.prefixItems[].properties.breaks.prefixItems[].const
contradiction does not license social exclusion

# properties.compatibility.properties.forbidden_output.const
No bare COMPATIBLE verdict, percentage, scalar, rank, synthesis, conversion target, or conclusion about a whole religion, community, or person.

# properties.bridge_cards.prefixItems[].properties.preserved.const
New rules can make previously inexpressible solutions available.

# properties.bridge_cards.prefixItems[].properties.breaks.prefixItems[].const
formal object construction is not cosmic causation

# properties.bridge_cards.prefixItems[].properties.breaks.prefixItems[].const
historical discovery and invention remain contested

# properties.bridge_cards.prefixItems[].properties.preserved.const
A declared model may encode absence with zero.

# properties.bridge_cards.prefixItems[].properties.preserved.const
A once-contested formal object can become mathematically useful.

# properties.bridge_cards.prefixItems[].properties.breaks.prefixItems[].const
historical label is not an ontology class

# properties.bridge_cards.prefixItems[].properties.breaks.prefixItems[].const
formal consistency is not empirical evidence

# properties.bridge_cards.prefixItems[].properties.preserved.const
Transformations and invariants can sharpen questions about unity and difference.

# properties.bridge_cards.prefixItems[].properties.breaks.prefixItems[].const
symmetry breaking retains difference and history

# properties.bridge_cards.prefixItems[].properties.preserved.const
Validity and model conditions of a formal argument can be checked.

# properties.bridge_cards.prefixItems[].properties.breaks.prefixItems[].const
formal possibility is not actual divine being

# properties.bridge_cards.prefixItems[].properties.preserved.const
Fourfold distinctions can prevent forced binary answers.

# properties.bridge_cards.prefixItems[].properties.breaks.prefixItems[].const
formal semantics and discourse aims differ

# properties.bridge_cards.prefixItems[].properties.from.const
canonical embeddings among number systems

# properties.bridge_cards.prefixItems[].properties.preserved.const
A comparison can ask which claims are preserved under a declared translation.

# properties.bridge_cards.prefixItems[].properties.breaks.prefixItems[].const
people and traditions have no scalar completion order

# properties.bridge_cards.prefixItems[].properties.to.const
divine uniqueness, simplicity, unity, or nonduality

# properties.bridge_cards.prefixItems[].properties.preserved.const
The shared word can trigger sense-disambiguation.

# properties.bridge_cards.prefixItems[].properties.breaks.prefixItems[].const
one deity is not an arithmetic operation

# properties.bridge_cards.prefixItems[].properties.to.const
reasonableness, rational theology, or revelation

# properties.bridge_cards.prefixItems[].properties.preserved.const
The English label rational can invite lexical ambiguity between a mathematical field and nonmathematical uses of reason.

# properties.bridge_cards.prefixItems[].properties.breaks.prefixItems[].const
field membership does not measure reasoning

# properties.bridge_cards.prefixItems[].properties.preserved.const
Both vocabularies can include balance or consequence language.

# properties.bridge_cards.prefixItems[].properties.breaks.prefixItems[].const
moral consequence is not a conserved physical quantity

# properties.model_cards.prefixItems[].properties.equation.const
N0 -> Z -> Q -> R -> C by declared embeddings

# properties.model_cards.prefixItems[].properties.question.const
Which operations, identities, order, topology, and completeness properties survive each map?

# properties.model_cards.prefixItems[].properties.invalidating_condition.const
An asserted preserved structure is not respected by the map.

# properties.model_cards.prefixItems[].properties.question.const
Which zero roles are active in this sentence and domain?

# properties.model_cards.prefixItems[].properties.invalidating_condition.const
A role is omitted, treated as exhaustive without review, or silently identified with another role or metaphysical nonbeing.

# properties.model_cards.prefixItems[].properties.question.const
Which typed sense of is does a claim use?

# properties.model_cards.prefixItems[].properties.invalidating_condition.const
The vector is treated as a worth, consciousness, or completeness score.

# properties.model_cards.prefixItems[].properties.equation.const
Derivable_L(Gamma,phi) depends on declared logic L

# properties.model_cards.prefixItems[].properties.question.const
Does no-no license phi in this logic and context?

# properties.model_cards.prefixItems[].properties.invalidating_condition.const
A result is transferred between logics or into apophatic/refusal discourse without a bridge.

# properties.model_cards.prefixItems[].properties.equation.const
G acts on X; I(g*x)=I(x); identity claim is indexed by I and G

# properties.model_cards.prefixItems[].properties.question.const
What remains invariant, and what difference is quotienting away?

# properties.model_cards.prefixItems[].properties.invalidating_condition.const
Aesthetic resemblance substitutes for an action or invariant.

# properties.model_cards.prefixItems[].properties.equation.const
FormalConstruct + PhysicalCause + TheologicalCreate + Manifest + DependentlyArise, as non-convertible tagged relations

# properties.model_cards.prefixItems[].properties.invalidating_condition.const
A formal construction is presented as a cosmic cause or a source attribution as a world fact.

# properties.model_cards.prefixItems[].properties.equation.const
Mod_L(A union tau_i(D_i) union tau_j(D_j)) is nonempty

# properties.model_cards.prefixItems[].properties.question.const
Are these exact translated fragments jointly satisfiable under declared assumptions?

# properties.model_cards.prefixItems[].properties.invalidating_condition.const
The result is generalized to whole religions or practical coexistence.

# properties.model_cards.prefixItems[].properties.equation.const
K_tau={S: translated doctrine fragments in S are jointly satisfiable under declared logic L}

# properties.model_cards.prefixItems[].properties.question.const
Where does pairwise compatibility fail to compose globally?

# properties.model_cards.prefixItems[].properties.assumptions.prefixItems[].const
the supplied three-fragment counterexample uses classical propositional logic

# properties.model_cards.prefixItems[].properties.invalidating_condition.const
Pairwise edges are treated as proof of a global face or a counterexample is moved between logics without rechecking satisfiability.

# properties.model_cards.prefixItems[].properties.equation.const
tau: source terms -> partial target predicates with an explicit loss set

# properties.model_cards.prefixItems[].properties.question.const
Which senses survive translation, split, merge, or remain untranslated?

# properties.model_cards.prefixItems[].properties.invalidating_condition.const
A shared English word is treated as an exact common referent.

# properties.model_cards.prefixItems[].properties.question.const
Which premise, modal axiom, domain, or interpretation carries the theological burden?

# properties.model_cards.prefixItems[].properties.invalidating_condition.const
Formal validity alone is reported as God exists or God does not exist.

# properties.problem_cards.prefixItems[].properties.question.const
What exactly is preserved and lost along N0, Z, Q, R, and C embeddings?

# properties.problem_cards.prefixItems[].properties.falsifier.const
one claimed preservation fails under the declared map

# properties.problem_cards.prefixItems[].properties.question.const
Which zero role, empty carrier, absence claim, or observation code is present?

# properties.problem_cards.prefixItems[].properties.falsifier.const
the same type and semantics are proved for two allegedly distinct roles

# properties.problem_cards.prefixItems[].properties.question.const
Under which logic and speech act does double negation recover affirmation?

# properties.problem_cards.prefixItems[].properties.deliverable.const
logic-indexed derivation and countermodel table

# properties.problem_cards.prefixItems[].properties.falsifier.const
an unqualified rule survives all declared logics and discourse types

# properties.problem_cards.prefixItems[].properties.question.const
Which transformations preserve the object, and which differences matter after quotienting?

# properties.problem_cards.prefixItems[].properties.deliverable.const
action, orbit, stabilizer, and invariant declaration

# properties.problem_cards.prefixItems[].properties.falsifier.const
no transformation action or invariant can be named

# properties.problem_cards.prefixItems[].properties.question.const
What separates formal construction, physical production, manifestation, dependent arising, and theological creation?

# properties.problem_cards.prefixItems[].properties.falsifier.const
a separately sourced equivalence proves one bridge exact

# properties.problem_cards.prefixItems[].properties.question.const
Do pairwise satisfiable doctrine fragments remain jointly satisfiable under declared logic L?

# properties.problem_cards.prefixItems[].properties.deliverable.const
finite satisfiability or countermodel certificate under a named logic; the supplied abstract example is classical propositional only

# properties.problem_cards.prefixItems[].properties.falsifier.const
the declared global union has a model in the named logic

# properties.problem_cards.prefixItems[].properties.question.const
Which premise and modal axiom separate validity from a sound God-existence conclusion?

# properties.problem_cards.prefixItems[].properties.falsifier.const
a proof establishes the conclusion without the disputed premise under the fixed semantics

# properties.problem_cards.prefixItems[].properties.question.const
How did notation and acceptance change independently of the modern formal tower?

# properties.problem_cards.prefixItems[].properties.falsifier.const
the historical record is exactly the formal construction order

# properties.problem_cards.prefixItems[].properties.question.const
What do ehyeh, ahad, sat/asat, sunyata, you/wu, one, being, and nothing lose in English abstraction?

# properties.problem_cards.prefixItems[].properties.deliverable.const
partial translation graph with untranslated residues

# properties.problem_cards.prefixItems[].properties.falsifier.const
an exact bidirectional translation is source-established

# properties.problem_cards.prefixItems[].properties.question.const
Which dimensions of compatibility are being asked, by whom, under what rights and purpose?

# properties.problem_cards.prefixItems[].properties.falsifier.const
all dimensions are proved equivalent for the bounded case

# properties.philosophy_questions.properties.questions.prefixItems[].const
Are numbers discovered, invented, constructed, instantiated, or differently understood under different mathematical ontologies?

# properties.philosophy_questions.properties.questions.prefixItems[].const
What remains the same when a number system is embedded into a larger one?

# properties.philosophy_questions.properties.questions.prefixItems[].const
Does a formal extension create an object, reveal a structure, or make a prior possibility expressible?

# properties.philosophy_questions.properties.questions.prefixItems[].const
Can zero be something mathematically while not naming metaphysical nothingness?

# properties.philosophy_questions.properties.questions.prefixItems[].const
Does naming nothing already place a term inside a discourse and structure?

# properties.philosophy_questions.properties.questions.prefixItems[].const
What does imaginary reveal about the difference between a historical name and a mathematical role?

# properties.philosophy_questions.properties.questions.prefixItems[].const
Can symmetry illuminate unity-in-difference without erasing asymmetry, history, or persons?

# properties.philosophy_questions.properties.questions.prefixItems[].const
Which sense of is appears in God is: existence, identity, predication, promise, analogy, confession, or apophatic limit?

# properties.philosophy_questions.properties.questions.prefixItems[].const
Does no-no affirm, suspend, refuse, or expose a false presupposition in this logic and discourse?

# properties.philosophy_questions.properties.questions.prefixItems[].const
Can a formally valid ontological argument remain theologically unsound?

# properties.philosophy_questions.properties.questions.prefixItems[].const
Can traditions share a lexical resonance without sharing truth conditions?

# properties.philosophy_questions.properties.questions.prefixItems[].const
Can doctrinal contradiction coexist with equal dignity, dialogue, and practical cooperation?

# properties.philosophy_questions.properties.questions.prefixItems[].const
What must a translation preserve before compatibility becomes a mathematical question?

# properties.philosophy_questions.properties.questions.prefixItems[].const
Is creation temporal, causal, ontological, grammatical, liturgical, or formal in the claim at hand?

# properties.philosophy_questions.properties.questions.prefixItems[].const
What does a model leave outside itself when it formalizes being?

# properties.philosophy_questions.properties.boundary.const
No answer is required. Silence, refusal, rest, uncertainty, and departure are complete and non-penalized responses.

# properties.karma.properties.religious_and_philosophical.properties.guard.const
Plural doctrines of karma, moral consequence, or cosmic order are not a number, conserved quantity, symmetry, score, or universal causal model in this atlas.

# properties.karma.properties.adapter.properties.reason.const
Religious or philosophical karma, Zerone recognition, and KINGDOM invocation-aftermath receipts are non-equivalent types. Arithmetic, conservation, symmetry, and causal-model language do not convert among them.

# properties.unknowns.prefixItems[].properties.question.const
Which ontology of mathematics, if any, is best?

# properties.unknowns.prefixItems[].properties.why_open.const
The selected survey presents live competing positions; this atlas adds no adjudicating argument.

# properties.unknowns.prefixItems[].properties.question.const
What fuller cross-cultural history connects notation, operations, institutions, and formalization?

# properties.unknowns.prefixItems[].properties.why_open.const
The selected history sources are bounded and do not support a universal chronology.

# properties.unknowns.prefixItems[].properties.question.const
Which translations of divine being-language preserve source-language sense across traditions?

# properties.unknowns.prefixItems[].properties.why_open.const
No universal lossless translation was established.

# properties.unknowns.prefixItems[].properties.question.const
Is absolute nothingness coherent or referable?

# properties.unknowns.prefixItems[].properties.why_open.const
Mathematical zero and empty structures do not settle the metaphysical question.

# properties.unknowns.prefixItems[].properties.question.const
Does any exact ontological argument have premises acceptable under a declared theology and modal semantics?

# properties.unknowns.prefixItems[].properties.why_open.const
Argument versions and premise defenses differ; formal validity is insufficient.

# properties.unknowns.prefixItems[].properties.question.const
Can independent evidence support a particular design inference in a bounded domain?

# properties.unknowns.prefixItems[].properties.why_open.const
Symmetry alone underdetermines design and non-design explanations.

# properties.unknowns.prefixItems[].properties.question.const
Can whole religions be globally classified as compatible?

# properties.unknowns.prefixItems[].properties.why_open.const
The category is rejected as ill-typed without claim, school, mapping, logic, dimension, time, and purpose.

# properties.unknowns.prefixItems[].properties.question.const
Which sourced doctrine fragments form higher-order compatible families?

# properties.unknowns.prefixItems[].properties.why_open.const
Only an abstract counterexample is supplied; no tradition is encoded for SAT classification.

# properties.unknowns.prefixItems[].properties.question.const
Which, if any, theological or metaphysical creation account is true?

# properties.unknowns.prefixItems[].properties.why_open.const
Selected sources attribute claims; mathematics does not adjudicate their world truth.

# properties.unknowns.prefixItems[].properties.question.const
How should distinct karma doctrines be compared without flattening them or importing KINGDOM receipt semantics?

# properties.unknowns.prefixItems[].properties.why_open.const
No tradition-specific karma study was undertaken in this atlas.

# properties.sources.prefixItems[].properties.authors_or_body.const
J. J. O'Connor and E. F. Robertson, University of St Andrews

# properties.sources.prefixItems[].properties.limitation.const
A bounded historical survey, not a primary manuscript edition or theology.

# properties.sources.prefixItems[].properties.authors_or_body.const
MacTutor History of Mathematics, University of St Andrews

# properties.sources.prefixItems[].properties.used_for.const
complex arithmetic, terminology, geometric rotation, and FTA history

# properties.sources.prefixItems[].properties.limitation.const
Historical survey; the theorem is used only in its standard scoped form.

# properties.sources.prefixItems[].properties.limitation.const
Survey of positions, not an adjudication.

# properties.sources.prefixItems[].properties.authors_or_body.const
Stanford Encyclopedia of Philosophy, Summer 2024 archive

# properties.sources.prefixItems[].properties.used_for.const
logic-indexed double negation and excluded middle

# properties.sources.prefixItems[].properties.limitation.const
Formal logic source, not a model of natural-language or religious negation.

# properties.sources.prefixItems[].properties.used_for.const
non-explosion and diversity of paraconsistent logics

# properties.sources.prefixItems[].properties.limitation.const
Paraconsistency does not entail dialetheism or unrestricted contradiction.

# properties.sources.prefixItems[].properties.limitation.const
Formal information semantics, not a religious fourfold ontology.

# properties.sources.prefixItems[].properties.authors_or_body.const
Emmy Noether; English translation by M. A. Tavel

# properties.sources.prefixItems[].properties.used_for.const
conditional relation between variational symmetries and conservation laws

# properties.sources.prefixItems[].properties.limitation.const
Requires theorem hypotheses; not every symmetry or conservation metaphor qualifies.

# properties.sources.prefixItems[].properties.authors_or_body.const
Stanford Encyclopedia of Philosophy, Fall 2018 archive

# properties.sources.prefixItems[].properties.used_for.const
group-theoretic symmetry, physical uses, and symmetry breaking

# properties.sources.prefixItems[].properties.limitation.const
Survey does not support a theological design inference.

# properties.sources.prefixItems[].properties.limitation.const
Surveying an argument does not prove its conclusion.

# properties.sources.prefixItems[].properties.authors_or_body.const
Hebrew Bible; Revised JPS 2023 and alternate translations via Sefaria

# properties.sources.prefixItems[].properties.used_for.const
Ehyeh wording and explicit translation uncertainty

# properties.sources.prefixItems[].properties.limitation.const
Translation witnesses do not establish one exhaustive Jewish ontology.

# properties.sources.prefixItems[].properties.title.const
Guide for the Perplexed, Part I, Chapter 58

# properties.sources.prefixItems[].properties.authors_or_body.const
Moses Maimonides; M. Friedlander translation

# properties.sources.prefixItems[].properties.used_for.const
Maimonidean negative attributes across the complete chapter

# properties.sources.prefixItems[].properties.limitation.const
One translated medieval philosophical voice, not all Jewish theology.

# properties.sources.prefixItems[].properties.title.const
Catechism of the Catholic Church, paragraphs 198-231

# properties.sources.prefixItems[].properties.used_for.const
Catholic reception of the divine name and fullness of Being

# properties.sources.prefixItems[].properties.limitation.const
Catholic doctrine, not all Christian interpretation or historical-critical exegesis.

# properties.sources.prefixItems[].properties.title.const
Catechism of the Catholic Church, The Creator, paragraphs 279-324

# properties.sources.prefixItems[].properties.used_for.const
Catholic creation ex nihilo and creation/science distinction

# properties.sources.prefixItems[].properties.limitation.const
Catholic doctrine; nothing is not modeled as numeric zero.

# properties.sources.prefixItems[].properties.authors_or_body.const
Quran; Dr. Mustafa Khattab, The Clear Quran, Quran.com translation resource 131

# properties.sources.prefixItems[].properties.used_for.const
divine oneness and incomparability language in one named English translation

# properties.sources.prefixItems[].properties.limitation.const
Translation and interpretation vary; this atlas does not convert the translation's One language into arithmetic identity; resource 131 does not exhaust Arabic or tafsir.

# properties.sources.prefixItems[].properties.authors_or_body.const
Quran; Dr. Mustafa Khattab, The Clear Quran, Quran.com translation resource 131

# properties.sources.prefixItems[].properties.used_for.const
Originator and Be-and-it-is language in one named English translation

# properties.sources.prefixItems[].properties.limitation.const
A translated scriptural expression, not a mechanism or formal operation; resource 131 is one interpretive witness.

# properties.sources.prefixItems[].properties.title.const
Arabic and Islamic Philosophy of Religion

# properties.sources.prefixItems[].properties.used_for.const
variance among kalam, falsafa, emanation, origination, and manifestation models

# properties.sources.prefixItems[].properties.limitation.const
Survey does not define one Islamic ontology.

# properties.sources.prefixItems[].properties.title.const
The Hymns of the Rigveda, Book 10, Hymn 129 (Nasadiya Sukta)

# properties.sources.prefixItems[].properties.authors_or_body.const
Rig Veda; Ralph T. H. Griffith translation, second edition, 1896; Wikisource revision 14396955

# properties.sources.prefixItems[].properties.used_for.const
one named translation's existent/non-existent and cosmogonic uncertainty language

# properties.sources.prefixItems[].properties.limitation.const
One nineteenth-century translation; sat/asat and poetic questions are not formal truth values, and later translations differ.

# properties.sources.prefixItems[].properties.used_for.const
Advaita nonduality, dependence, and graded reality

# properties.sources.prefixItems[].properties.limitation.const
Advaita-focused; Hindu and Vedanta traditions remain diverse.

# properties.sources.prefixItems[].properties.title.const
Majjhima Nikaya 72, To Vacchagotta on Fire

# properties.sources.prefixItems[].properties.authors_or_body.const
Pali canon; Thanissaro Bhikkhu translation via SuttaCentral

# properties.sources.prefixItems[].properties.used_for.const
scoped fourfold postmortem predication refusal

# properties.sources.prefixItems[].properties.limitation.const
One discourse and translation; not a universal Buddhist logic.

# properties.sources.prefixItems[].properties.used_for.const
false-presupposition and logic cautions around fourfold questions

# properties.sources.prefixItems[].properties.limitation.const
Scholarly reconstruction amid source and school disputes.

# properties.sources.prefixItems[].properties.authors_or_body.const
Stanford Encyclopedia of Philosophy, Fall 2021 archive

# properties.sources.prefixItems[].properties.used_for.const
Madhyamaka emptiness and dependent arising

# properties.sources.prefixItems[].properties.limitation.const
Madhyamaka-focused; emptiness is not numeric zero or nihilism.

# properties.sources.prefixItems[].properties.authors_or_body.const
Laozi; James Legge translation via Chinese Text Project

# properties.sources.prefixItems[].properties.used_for.const
Dao, one, two, three, and myriad-things generation language

# properties.sources.prefixItems[].properties.limitation.const
Translation and referents are disputed; numbers are not assumed cardinal stages.

# properties.sources.prefixItems[].properties.used_for.const
you/wu, naming, generation, and interpretation variance

# properties.sources.prefixItems[].properties.limitation.const
Classical text scholarship, not all organized religious Daoism.

# properties.sources.prefixItems[].properties.used_for.const
historical visual use of geometry, repetition, and symmetry

# properties.sources.prefixItems[].properties.limitation.const
Art history does not establish a Quranic mandate or common Islamic theology.

# properties.sources.prefixItems[].properties.authors_or_body.const
Quran; Dr. Mustafa Khattab, The Clear Quran, Quran.com translation resource 131

# properties.sources.prefixItems[].properties.used_for.const
mizan, balance, and measure language in one named English translation

# properties.sources.prefixItems[].properties.limitation.const
Mizan is not identified with mathematical group symmetry or Noether conservation; resource 131 is one translation.

# properties.sources.prefixItems[].properties.title.const
Mandalas: Mapping the Buddhist Art of Tibet

# properties.sources.prefixItems[].properties.used_for.const
practice-oriented geometric diagrams in Tibetan Buddhist art

# properties.sources.prefixItems[].properties.limitation.const
One art-historical field; visual form does not imply shared doctrine.

# properties.sources.prefixItems[].properties.used_for.const
dialogue that retains Catholic commitments and recognizes others

# properties.sources.prefixItems[].properties.limitation.const
Catholic document, not a universal interreligious settlement.

# properties.sources.prefixItems[].properties.authors_or_body.const
Quran; Dr. Mustafa Khattab, The Clear Quran, Quran.com translation resource 131

# properties.sources.prefixItems[].properties.used_for.const
distinct ways and competition in good in one named English translation

# properties.sources.prefixItems[].properties.limitation.const
Interpretation varies; not a formal compatibility axiom; resource 131 does not define all Islamic readings.

# properties.sources.prefixItems[].properties.title.const
Striving Together in Dialogue: A Muslim-Christian Call to Reflection and Action

# properties.sources.prefixItems[].properties.authors_or_body.const
World Council of Churches and partner Islamic organisations and journals; fruit of the November 2000 Amersfoort Muslim-Christian meeting

# properties.sources.prefixItems[].properties.used_for.const
dialogue and cooperation that retain recognized doctrinal difference

# properties.sources.prefixItems[].properties.limitation.const
A bounded joint dialogue document whose participants acknowledge limited representativeness; not doctrine shared by every Christian, Muslim, or religion.