GRM BASE-t RECONSTRUCTION OF INTER-UNIVERSAL TEICHMÜLLER THEORY
GRM BASE-t RECONSTRUCTION OF INTER-UNIVERSAL TEICHMÜLLER THEORY
Part I — Reconstruction Protocol
1. Why Rebuild IUT
1.1 Reconstruction versus exposition
1.2 Source-native structure versus retrospective interpretation
1.3 REPRESENTATION ≠ SOURCE
1.4 Why theorem names cannot seed mechanisms
1.5 Why “abc”, “universe”, “HoTT path”, and “non-liftability” are initially erased
1.6 Reconstruction success criterion
2. GRM Base-t Execution Law
2.1 SOURCE_STATE(IUT)
2.2 Target-blind normalization TBLIND
2.3 Forward dependency hypergraph \(FB\)
2.4 Reverse necessity graph \(RB\)
2.5 \(GEN^\star\): source-native generation
2.6 \(TR^\star\): licensed transport
2.7 \(CLOSE^\star\): reconstruction/gluing closure
2.8 Carrier antichain \(C^\star\)
2.9 Minimal cuts \(CUT^\star\)
2.10 Residue, counterkernel, successor, liftback
3. Anti-Leakage Tests
3.1 Target-vocabulary erasure
3.2 Requirement-free reconstruction test
3.3 Carrier-neutrality test
3.4 No common-ring assumption
3.5 No generic-transport collapse
3.6 No downstream inequality imported upstream
Part II — IUT Source State
4. Initial Θ-Data
4.1 Elliptic/arithmetic starting package
4.2 Prime \(\ell\) and valuation-indexed data
4.3 Covering data
4.4 Arithmetic and tempered fundamental groups
4.5 Frobenioid data
4.6 Theta/q-data
4.7 Local/global organization
4.8 Provenance and ownership
5. Source-Generated Arithmetic Structures
5.1 Punctured elliptic curves
5.2 Prime-strips
5.3 Multiplicative monoids
5.4 Log-shells
5.5 Galois actions
5.6 Reconstruction interfaces
5.7 Local/global Frobenioid structure
6. Hodge-Theater Formation
6.1 Hodge theater as execution regime, not “universe”
6.2 Theater-local ring/scheme structure
6.3 Theater-local arithmetic fundamental groups
6.4 Theater-local multiplicative structures
6.5 Labeling and provenance
6.6 Formation of mutually nonidentified copies
6.7 Log-theta-lattice organization
Part III — Native Carrier Geometry
7. The Carrier Antichain \(C^\star_{\mathrm{IUT}}\)
7.1 Full Hodge-theater carrier
7.2 Frobenioid carrier
7.3 Prime-strip carrier
7.4 Abstract-monoid carrier
7.5 Arithmetic-fundamental-group carrier
7.6 Log-shell carrier
7.7 Kummer carrier
7.8 Multiradial carrier
7.9 Why no single carrier dominates
8. Forgetting and Reconstruction
8.1 Full arithmetic structure
8.2 Forgetting ring/scheme reconstruction history
8.3 Abstract multiplicative reducts
8.4 What survives forgetting
8.5 What becomes unavailable
8.6 Mono-anabelian reconstruction
8.7 Reconstruction is not transport
9. Nonidentification as a Structural Operation
9.1 Isomorphic abstract object versus identical reconstructed object
9.2 Multiple copies without common ambient identity
9.3 Labels as anti-collapse devices
9.4 Basepoint nonidentification
9.5 Group-isomorphism indeterminacy
9.6 Ring nonidentification
9.7 Why premature identification destroys usable distinctions
Part IV — Link-Specific Transport
10. Transport as a Role, Not a Single Map
10.1 Architectural TRANSPORT
10.2 IUT-specific transport instances
10.3 Preservation profile
10.4 Destruction profile
10.5 Domain/codomain typing
10.6 Composition constraints
10.7 Noncommuting transport diagrams
11. Θ-Links
11.1 Horizontal role in the log-theta lattice
11.2 Stripped Frobenius-like interfaces
11.3 Theta-value assignment
11.4 \(j^2\)-indexed structure
11.5 Selected Galois compatibility
11.6 \(\widehat{\mathbf Z}^{\times}\)-indeterminacy
11.7 Structures preserved
11.8 Ring/scheme structure not carried
12. Log-Links
12.1 Vertical role
12.2 \(p\)-adic logarithmic relation
12.3 Multiplicative/additive interaction
12.4 Abstract topological-group isomorphisms
12.5 Basepoint indeterminacy
12.6 Failure of ring-homomorphic identification
12.7 Structures preserved and destroyed
13. Kummer Transport and Detachment
13.1 Kummer-theoretic data
13.2 Detachment from original arithmetic realization
13.3 Rigidity requirements
13.4 Conjugate synchronization
13.5 Indeterminacy control
13.6 Reattachment/reconstruction obligations
14. Multiradial Comparison
14.1 Why multiradiality is not ordinary transport
14.2 Simultaneous Frobenius-like components
14.3 Alien arithmetic-holomorphic structure
14.4 Distinguished theta families
14.5 Synchronization across components
14.6 Multiradial compatibility certificate
Part V — TRANSPORT → DEBT → BUDGET
15. The IUT Transport–Debt Law
15.1 \(T_i:C_i\rightsquigarrow C_j\)
15.2 \(\mathrm{Pres}(T_i)\)
15.3 \(\mathrm{Lost}(T_i)\)
15.4 Transport-generated reconstruction obligation
15.5 Typed rather than scalar loss
16. Debt Ledger
16.1 \(D_{\mathrm{ring}}\): unavailable ring structure
16.2 \(D_{\mathrm{basepoint}}\)
16.3 \(D_{\mathrm{group}}\)
16.4 \(D_{\widehat{\mathbf Z}^{\times}}\)
16.5 \(D_{\mathrm{reconstruction}}\)
16.6 \(D_{\mathrm{Kummer}}\)
16.7 \(D_{\mathrm{conjugacy}}\)
16.8 \(D_{\mathrm{multiradial}}\)
16.9 \(D_{\mathrm{local/global}}\)
17. Budget Formation
17.1 Why debt cannot merely remain qualitative
17.2 Indeterminacy budget
17.3 Exponentiation budget
17.4 Reconstruction budget
17.5 Exceptional/local budget
17.6 Log-volume distortion budget
17.7 Cumulative versus noncomposable budgets
17.8 Budget exhaustion criterion
18. Typed Residues
18.1 \(R_{\mathrm{ring}}\)
18.2 \(R_{\mathrm{basepoint}}\)
18.3 \(R_{\mathrm{group}}\)
18.4 \(R_{\widehat{\mathbf Z}^{\times}}\)
18.5 \(R_{\mathrm{reconstruction}}\)
18.6 \(R_{\mathrm{log-volume}}\)
18.7 Residue ownership
18.8 Residue transport across carrier changes
Part VI — The Native Cut
19. Common-Ring Counterkernel
19.1 Candidate common ambient ring
19.2 Cross-log identification gained
19.3 Θ-separation destroyed
19.4 Incompatible simultaneous requirements
19.5 Minimal cut localization
20. Successor Architecture
20.1 Reject common-ring collapse
20.2 Preserve separate theaters
20.3 Work through reduct carriers
20.4 Link-specific comparison
20.5 Independent reconstruction
20.6 Explicit indeterminacy
20.7 Multiradial successor capability
Part VII — Reconstruction and Global Closure
21. Mono-Anabelian Reconstruction
21.1 Group-theoretic input
21.2 Reconstruction targets
21.3 Local arithmetic recovery
21.4 Reconstruction uniqueness/indeterminacy
21.5 Compatibility with link histories
22. Local Reconstruction Packets
22.1 Local packet definition
22.2 Ownership
22.3 Preservation/loss ancestry
22.4 Local consistency
22.5 Failure residues
23. Global Tensor/Packet: Formation and Faithful Localization
23.1 Hodge-theater and capsule inputs
23.2 Labeled global fields
23.3 Module tensor packet \(M_A\)
23.4 Induced ring structure
23.5 Unique decomposition into number fields
23.6 Factor embeddings \(\iota_\alpha\)
23.7 Log-link-mediated localization
23.8 Injectivity and packet-as-image identification
23.9 Nonarchimedean integral compatibility
23.10 Archimedean integral compatibility
23.11 Log-shifted localization
23.12 GRM transport grade
23.13 Residual debt: holomorphic/mono-analytic discrepancy
23.14 Boundary: Prop. 3.3 closure versus genuine diagrammatic GLOBAL_CLOSE
24. GRM Diagrammatic Gluing Audit
24.1 Patch family
24.2 Overlap hypergraph
24.3 Higher-order intersections
24.4 Gluing residues
24.5 Compatible cocones
24.6 Noncommuting lift chains
24.7 Global closure test
Part VIII — Multiradial Output and Quantitative Control
25. From Reconstruction to Multiradiality
25.1 Reconstructed arithmetic data
25.2 Theta/Kummer synchronization
25.3 Multiradial representation
25.4 Residual indeterminacy orbit
25.5 Budget accounting
26. Log-Volume Layer
26.1 Arithmetic log-volume as readout
26.2 Inputs licensed for comparison
26.3 Contributions from indeterminacy
26.4 Distortion accumulation
26.5 Budget inequality
26.6 Failure conditions
26.7 Reconstruction-versus-readout distinction
27. Closure Certificate
27.1 Every transport typed
27.2 Every loss assigned to debt
27.3 Every debt assigned a budget or discharged
27.4 Every residue owned
27.5 Every reconstruction ancestry-complete
27.6 Every local packet globally compatible
27.7 No illicit ring reidentification
27.8 No hidden target import
Part IX — Liftback and External Consequences
28. Liftback Discipline
28.1 Required transport grade
28.2 Reverse reconstruction plan
28.3 Loss continuity
28.4 Source recovery
28.5 Failure of reverse transport
28.6 Exact versus observational liftback
29. Arithmetic Readout
29.1 Log-volume comparison
29.2 Arithmetic degree/height consequences
29.3 Diophantine inequality layer
29.4 Separation of IUT mechanism from later consequences
29.5 No downstream result promoted into source ontology
Part X — Reconstruction Audit
30. Factorization Tests
30.1 Alternative carrier factorizations
30.2 Equivalent reconstruction routes
30.3 Representation-independent cuts
30.4 Carrier-stable residues
30.5 Genuine versus representation-induced obstruction
31. Adversarial Failure Tests
31.1 Collapse all theaters into one ring
31.2 Treat Θ-link as ring map
31.3 Treat log-link as ring map
31.4 Identify abstractly isomorphic groups canonically
31.5 Ignore \(\widehat{\mathbf Z}^{\times}\)-indeterminacy
31.6 Replace reconstruction with transport
31.7 Infer global closure from pairwise compatibility
31.8 Spend budget without ownership
31.9 Hide residue by representation change
32. Final GRM Reconstruction Statement
32.1 Source-state recovered
32.2 \(GEN^\star\) recovered
32.3 Carrier antichain recovered
32.4 Transport species recovered
32.5 TRANSPORT → DEBT → BUDGET recovered
32.6 Native counterkernel recovered
32.7 Reconstruction/gluing architecture recovered
32.8 Quantitative readout recovered
32.9 Remaining source gaps
32.10 Reconstruction certificate versus theorem-validity certificate
Appendix A — IUT → GRM Dictionary
Initial Θ-data → SOURCE_STATE
Hodge-theater formation → \(GEN^\star\)
Hodge theaters/reducts → \(C^\star\)
Θ/log/Kummer links → typed \(TR^\star_i\)
lost/unavailable structure → DEBT_i
controlled indeterminacy → BUDGET_i
surviving incompatibility → \(\rho_i\)
common-ring incompatibility → \(CUT^\star/\)counterkernel
mono-anabelian recovery → reconstruction
multiradial packet → comparison carrier
local/global packet compatibility → GLUE/CLOSE^\star
log-volume consequence → readout/liftback.
Appendix B — HoTT Only After Reconstruction
B.1 What a dependent-type model would need to preserve
B.2 Why ordinary identity paths may be insufficient
B.3 Heterogeneous link species
B.4 Displayed/directed alternatives
B.5 HoTT fidelity test against the reconstructed IUT dependency graph
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