1
Overview
▶
1.1
What VR is
1.2
The three architectural layers
1.3
Dependency structure
1.4
Axiom profile across the cycle
1.5
What to read first
1.6
Position relative to existing programmes
1.7
Provenance and reproducibility
2
VR Formal System
▶
2.1
Position
2.2
Primitives
2.3
The named principles A3, A4 (theorem and recursor)
2.4
Leibnizian equality
2.5
Von Neumann ordinals
2.6
Arithmetic
2.7
Peano correspondence
2.8
Acyclicity of membership
2.9
Main theorem — VR \(\simeq \) PA
2.10
Axiom profile
2.11
References
3
VR-Numbers
▶
3.1
Position
3.2
Integers \(\mathbb {Z}_{\mathrm{VR}}\)
3.3
Rationals \(\mathbb {Q}_{\mathrm{VR}}\)
3.4
Reals \(\mathbb {R}_{\mathrm{VR}}\)
3.5
Complex \(\mathbb {C}_{\mathrm{VR}}\)
3.6
Axiom profile boundary
3.7
The § VI.5 list of non-primitive objects
3.8
Methodological observations
3.9
Axiom profile
3.10
References
4
VR-Sets
▶
4.1
Position
4.2
Foundation (Part II)
4.3
ZFC axioms as closure theorems (Part III)
4.4
Modes (Part IV)
▶
4.4.1
ZFA boundary: Quine atom and AFA refuted in PSet
4.5
Conjectures (Part IV)
4.6
VR numbers bridge (Part V)
4.7
Operational consequences
4.8
Axiom profile analysis
4.9
Methodological observations (Part X)
▶
Group A: Quotient-base structure
Group B: Five structural boundaries
Group C: Axiom-minimal patterns
Group D: Methodological convergences
4.10
Axiom profile
4.11
References
5
VR-Sets-ZFA
▶
5.1
Position
5.2
Architecture (Part II)
5.3
AFA as theorem (Part III)
5.4
Embedding \(\mathrm{OSet} \hookrightarrow \mathrm{OSetZFA}\) (Part IV)
5.5
Demonstrations (Part V)
5.6
ZF\(^-\) axioms on \(\mathrm{OSetZFA}\) (Part VI)
5.7
Conjecture IV.2 closed
5.8
Methodological observations
5.9
Position relative to existing work (Part VII)
5.10
Axiom profile
5.11
References
6
VR-Forms
▶
6.1
Position
6.2
Formal language (Part II)
6.3
Realisability (Part II § II.7)
6.4
Transit (Part III)
▶
6.4.1
The conservativity boundary (Theorem III.1)
6.5
The ZFC reading and the bridge theorems (Part V)
6.6
Examples and mixed formulas (Parts V–VII)
6.7
Substrate (Stage 6)
6.8
Conservativity: from boundary to formalised result (§ IX.2)
6.9
Methodological observations (Part IX)
▶
§ IX.1 Foundation-level properties
§ IX.3 Structural patterns
§ IX.4 Cross-cycle integration
6.10
Axiom profile
6.11
References
7
VR-Audit
▶
7.1
Position
7.2
Operational structures (Stages 1–4)
7.3
Main theorem
7.4
The transit pattern in action
▶
Step 1: Riesz invoked as black box
Step 2: Explicit extension
Step 3: Operationality via orthogonal projection
Step 4: Norm equality
7.5
Why Path B, not Path A
7.6
Non-vacuity instance
7.7
Open programme
7.8
Position relative to neighbouring frameworks
7.9
Methodological observations
7.10
Axiom profile
7.11
References
8
VR-Apparatus
▶
8.1
Position
8.2
Architecture
8.3
Mode A (Part III)
8.4
Mode B and Factorisable (Part IV)
8.5
Apparatus instances and morphisms (Part V)
8.6
Compositional algebra and twelve findings
8.7
Position relative to existing frameworks (Part VII)
8.8
Honest scope limitations
8.9
Axiom profile
8.10
References
9
VR-Algebra
▶
9.1
Position
9.2
Operational typeclasses (Parts II–IV)
9.3
Mode A closure theorems
9.4
Concrete instances
9.5
Mode B audit: image of operational subgroup
▶
Structural contrast with VR-Audit Mode B
9.6
Structural exceptions in field-inverse reasoning
9.7
Recognition discipline: bidirectional signature
9.8
Twenty findings catalogued (A0–A19)
9.9
Position relative to the Apparatus framework
9.10
Axiom profile
9.11
References
10
VR-Topology
▶
10.1
Position
10.2
FormalTopology structure
10.3
Operational layer
10.4
Product of formal topologies
10.5
Compact witnesses and
OperationalCompact
10.6
Binary Tychonoff
10.7
Bridge to
Order.Frame
10.8
Honest scope: deferred abstract instance
10.9
Architectural amendments (T_FINDINGS)
10.10
Position relative to existing work
10.11
Axiom profile
10.12
References
11
VR-Transit
▶
11.1
Position
11.2
Conservativity (I)
11.3
The four-source decomposition of cost
11.4
The witness library (II)
11.5
Axiom-attribution audit
11.6
Findings
11.7
Honest scope
11.8
Position relative to existing work
11.9
References
12
Brouwer via Sperner (mathlib-bound)
▶
12.1
Position
12.2
The architecture (stage pipeline)
12.3
Load-bearing node I: the dimension recursion
12.4
Load-bearing node II: a vacuous abstraction, caught
12.5
Load-bearing node III: the differential witness (machine-checked)
12.6
The general theorem — compact convex sets
12.7
Honest scope
12.8
References
13
Operational continuum and number spectrum (exploratory)
▶
13.1
Position
13.2
The number spectrum
13.3
Boundary I: the Markov line (decidability of zero)
13.4
Boundary II: content versus packaging (machine-checked)
13.5
Two rationals, by necessity
13.6
Doing, not being (meta)
13.7
The three classical pillars, operationally
13.8
Honest scope
14
Operational set universe (Brouwer path)
▶
14.1
Position
14.2
The carrier: sets as revealing functionalities
14.3
Operations
14.4
Foundation is a predicate, not a choice
14.5
Cardinality, the diagonal, and choice — the corrected core
14.6
Honest scope
15
Operational sets under ZTL verdicts (VR Part II)
▶
15.1
Position
15.2
Choice sequences as the lazy register (step (б))
15.3
The survival ledger (step (в))
Dependency graph
VR Cycle Blueprint
Vitaly Reznik
1
Overview
1.1
What VR is
1.2
The three architectural layers
1.3
Dependency structure
1.4
Axiom profile across the cycle
1.5
What to read first
1.6
Position relative to existing programmes
1.7
Provenance and reproducibility
2
VR Formal System
2.1
Position
2.2
Primitives
2.3
The named principles A3, A4 (theorem and recursor)
2.4
Leibnizian equality
2.5
Von Neumann ordinals
2.6
Arithmetic
2.7
Peano correspondence
2.8
Acyclicity of membership
2.9
Main theorem — VR \(\simeq \) PA
2.10
Axiom profile
2.11
References
3
VR-Numbers
3.1
Position
3.2
Integers \(\mathbb {Z}_{\mathrm{VR}}\)
3.3
Rationals \(\mathbb {Q}_{\mathrm{VR}}\)
3.4
Reals \(\mathbb {R}_{\mathrm{VR}}\)
3.5
Complex \(\mathbb {C}_{\mathrm{VR}}\)
3.6
Axiom profile boundary
3.7
The § VI.5 list of non-primitive objects
3.8
Methodological observations
3.9
Axiom profile
3.10
References
4
VR-Sets
4.1
Position
4.2
Foundation (Part II)
4.3
ZFC axioms as closure theorems (Part III)
4.4
Modes (Part IV)
4.4.1
ZFA boundary: Quine atom and AFA refuted in PSet
4.5
Conjectures (Part IV)
4.6
VR numbers bridge (Part V)
4.7
Operational consequences
4.8
Axiom profile analysis
4.9
Methodological observations (Part X)
Group A: Quotient-base structure
Group B: Five structural boundaries
Group C: Axiom-minimal patterns
Group D: Methodological convergences
4.10
Axiom profile
4.11
References
5
VR-Sets-ZFA
5.1
Position
5.2
Architecture (Part II)
5.3
AFA as theorem (Part III)
5.4
Embedding \(\mathrm{OSet} \hookrightarrow \mathrm{OSetZFA}\) (Part IV)
5.5
Demonstrations (Part V)
5.6
ZF\(^-\) axioms on \(\mathrm{OSetZFA}\) (Part VI)
5.7
Conjecture IV.2 closed
5.8
Methodological observations
5.9
Position relative to existing work (Part VII)
5.10
Axiom profile
5.11
References
6
VR-Forms
6.1
Position
6.2
Formal language (Part II)
6.3
Realisability (Part II § II.7)
6.4
Transit (Part III)
6.4.1
The conservativity boundary (Theorem III.1)
6.5
The ZFC reading and the bridge theorems (Part V)
6.6
Examples and mixed formulas (Parts V–VII)
6.7
Substrate (Stage 6)
6.8
Conservativity: from boundary to formalised result (§ IX.2)
6.9
Methodological observations (Part IX)
§ IX.1 Foundation-level properties
§ IX.3 Structural patterns
§ IX.4 Cross-cycle integration
6.10
Axiom profile
6.11
References
7
VR-Audit
7.1
Position
7.2
Operational structures (Stages 1–4)
7.3
Main theorem
7.4
The transit pattern in action
Step 1: Riesz invoked as black box
Step 2: Explicit extension
Step 3: Operationality via orthogonal projection
Step 4: Norm equality
7.5
Why Path B, not Path A
7.6
Non-vacuity instance
7.7
Open programme
7.8
Position relative to neighbouring frameworks
7.9
Methodological observations
7.10
Axiom profile
7.11
References
8
VR-Apparatus
8.1
Position
8.2
Architecture
8.3
Mode A (Part III)
8.4
Mode B and Factorisable (Part IV)
8.5
Apparatus instances and morphisms (Part V)
8.6
Compositional algebra and twelve findings
8.7
Position relative to existing frameworks (Part VII)
8.8
Honest scope limitations
8.9
Axiom profile
8.10
References
9
VR-Algebra
9.1
Position
9.2
Operational typeclasses (Parts II–IV)
9.3
Mode A closure theorems
9.4
Concrete instances
9.5
Mode B audit: image of operational subgroup
Structural contrast with VR-Audit Mode B
9.6
Structural exceptions in field-inverse reasoning
9.7
Recognition discipline: bidirectional signature
9.8
Twenty findings catalogued (A0–A19)
9.9
Position relative to the Apparatus framework
9.10
Axiom profile
9.11
References
10
VR-Topology
10.1
Position
10.2
FormalTopology structure
10.3
Operational layer
10.4
Product of formal topologies
10.5
Compact witnesses and
OperationalCompact
10.6
Binary Tychonoff
10.7
Bridge to
Order.Frame
10.8
Honest scope: deferred abstract instance
10.9
Architectural amendments (T_FINDINGS)
10.10
Position relative to existing work
10.11
Axiom profile
10.12
References
11
VR-Transit
11.1
Position
11.2
Conservativity (I)
11.3
The four-source decomposition of cost
11.4
The witness library (II)
11.5
Axiom-attribution audit
11.6
Findings
11.7
Honest scope
11.8
Position relative to existing work
11.9
References
12
Brouwer via Sperner (mathlib-bound)
12.1
Position
12.2
The architecture (stage pipeline)
12.3
Load-bearing node I: the dimension recursion
12.4
Load-bearing node II: a vacuous abstraction, caught
12.5
Load-bearing node III: the differential witness (machine-checked)
12.6
The general theorem — compact convex sets
12.7
Honest scope
12.8
References
13
Operational continuum and number spectrum (exploratory)
13.1
Position
13.2
The number spectrum
13.3
Boundary I: the Markov line (decidability of zero)
13.4
Boundary II: content versus packaging (machine-checked)
13.5
Two rationals, by necessity
13.6
Doing, not being (meta)
13.7
The three classical pillars, operationally
13.8
Honest scope
14
Operational set universe (Brouwer path)
14.1
Position
14.2
The carrier: sets as revealing functionalities
14.3
Operations
14.4
Foundation is a predicate, not a choice
14.5
Cardinality, the diagonal, and choice — the corrected core
14.6
Honest scope
15
Operational sets under ZTL verdicts (VR Part II)
15.1
Position
15.2
Choice sequences as the lazy register (step (б))
15.3
The survival ledger (step (в))