2026 TSCT/GRM Post-Ramanujan Developments, Structural Descent, and Mathematical Discovery

 

2026 TSCT/GRM

Post-Ramanujan Developments, Structural Descent, and Mathematical Discovery

Table of Contents

1. RAMANUJAN’S FRONTIER: THE STRUCTURE HE COULD SEE BUT NOT CLOSE
   1.1 Ramanujan’s final $q$-series investigations
   1.2 Mock theta functions as anomalous theta-like objects
   1.3 Why ordinary theta functions were an insufficient carrier
   1.4 Holomorphicity as both strength and obstruction
   1.5 Singular behavior near roots of unity
   1.6 Transformation behavior without an adequate transformation theory
   1.7 Identity production versus structural explanation
   1.8 Ramanujan’s frontier as persistent mathematical residue
   1.9 What information survived the failure of the classical theta basin
   1.10 The post-Ramanujan problem: identify the missing carrier

2. FROM THETA FUNCTIONS TO INDEFINITE THETA GEOMETRY
   2.1 Definite quadratic forms and classical theta modularity
   2.2 What changes when the quadratic form becomes indefinite
   2.3 Loss of automatic convergence
   2.4 Restriction to cones and proper subsets of the lattice
   2.5 Convergence recovered at the cost of modularity
   2.6 Sign kernels as discontinuous geometric selectors
   2.7 Chamber structure and wall structure
   2.8 Positive and negative directions as competing geometric sectors
   2.9 The first structural tradeoff:

   convergence
   ↓
   cone restriction
   ↓
   loss of modularity

   2.10 Why the defect is more informative than the original function

3. ZWEGERS: THE FIRST SUCCESSFUL ASCENT
   3.1 Signature $(1,n-1)$ indefinite theta series
   3.2 Mock theta functions inside a larger modular structure
   3.3 Error functions as smoothing operators
   3.4 Holomorphic mock part and nonholomorphic correction
   3.5 Modular completion
   3.6 Shadow as exposed transformation residue
   3.7 Harmonic Maass interpretation
   3.8 Appell–Lerch realization
   3.9 Meromorphic Jacobi realization
   3.10 Multiple carriers for one deeper structure
   3.11 Zwegers’ three constructions as a first carrier-comparison experiment
   3.12 Why the completion, rather than the mock function alone, is the natural reconstructed object

4. VIGNÉRAS: THE DIFFERENTIAL CARRIER UNDER THE COMPLETION
   4.1 Theta series with general kernels
   4.2 The Vignéras differential equation
   4.3 Differential admissibility and modular transformation
   4.4 Integrability and convergence conditions
   4.5 Kernel deformation as the source of modular recovery
   4.6 Separating:

   kernel geometry
   ↓
   differential constraint
   ↓
   theta summation
   ↓
   modular transformation

   4.7 Why error functions are solutions, not necessarily primitives
   4.8 The inverse problem: what data force a Vignéras-admissible kernel?

Nazaroglu emphasizes that generalized error functions work because they satisfy the Vignéras equation, while convergence remains a separate nontrivial obligation.

5. DOUBLE ERROR FUNCTIONS AND THE FIRST HIGHER-SIGNATURE FRACTURE
   5.1 Signature $(2,n-2)$
   5.2 Why ordinary error-function completion no longer carries enough structure
   5.3 Double error functions
   5.4 Multiple positive directions
   5.5 Coupled wall smoothing
   5.6 Higher-dimensional chamber geometry
   5.7 Recursive shadow behavior
   5.8 Evidence that depth is structural rather than merely notational

6. NAZAROGLU’S $r$-TUPLE ERROR FUNCTIONS
   6.1 From double error functions to $r$-tuple error functions
   6.2 Definition of $M_r$
   6.3 Definition of $E_r$
   6.4 Gaussian convolution of sign data
   6.5 Permutation invariance
   6.6 Scale invariance of defining directions
   6.7 Orthogonal covariance
   6.8 Factorization under orthogonal decomposition
   6.9 Complementary error functions and exponentially suppressed sectors
   6.10 Smooth $E_r$ versus discontinuous asymptotic sign kernel
   6.11 Boosted error functions for arbitrary nondegenerate bilinear forms

7. THE CRITICAL DISCOVERY: $r$-TUPLE FUNCTIONS HAVE A DESCENT LAW
   7.1 Directional derivatives of $E_r$
   7.2 Directional derivatives of $M_r$
   7.3 Rank reduction under differentiation
   7.4 Shadow of $E_r$ expressed through $E_{r-1}$
   7.5 Shadow of $M_r$ expressed through $M_{r-1}$
   7.6 Recursive descent:

   $E_r$
   ↓
   $E_{r-1}$
   ↓
   $E_{r-2}$
   ↓
   …
   ↓
   $E_1$

   7.7 Rank as an exposed descent coordinate
   7.8 Shadow ancestry as a candidate definition of depth
   7.9 Why this is stronger than “higher signature requires a higher error function”

Nazaroglu proves that the shadow of $E_r$ and $M_r$ is assembled from lower-rank $E_{r-1}$ and $M_{r-1}$ terms.

8. THE SUBSET LATTICE HIDDEN INSIDE THE ANALYTIC COMPLETION
   8.1 Decomposition over subsets $S\subseteq[r]$
   8.2 Boolean incidence structure
   8.3 Full-dimensional chamber
   8.4 Codimension-one faces
   8.5 Codimension-two intersections
   8.6 Higher intersections
   8.7 Terminal strata
   8.8 Analytic rank versus combinatorial codimension
   8.9 Complementary $M$-functions attached to lower-dimensional strata
   8.10 Cancellation of discontinuities across incidence relations
   8.11 The subset lattice as latent structural skeleton
   8.12 Error functions as analytic realization of an incidence complex

9. HIGHER-DEPTH INDEFINITE THETA SERIES
   9.1 Signature $(r,n-r)$ lattices
   9.2 The holomorphic sign-difference kernel
   9.3 $2r$ defining vectors
   9.4 Positive subspace conditions
   9.5 Gram determinant conditions
   9.6 Cofactor conditions
   9.7 Negative-definite control of the support
   9.8 Absolute convergence
   9.9 Replacement of sign kernels by $E_r$-kernels
   9.10 Modular completion
   9.11 Vector-valued Jacobi transformation law
   9.12 Separating convergence proof from modularity proof

Theorem 4.1 makes this separation explicit: the sign-restricted holomorphic series is first shown convergent under geometric hypotheses, and the $E_r$-completed kernel then yields the Jacobi transformation law.

10. TSCT REINTERPRETATION: THE COMPLETION AS A DESCENT BASIN
    10.1 Semantic object: indefinite-theta completion
    10.2 Admissible descent operators
    10.3 Shadow descent
    10.4 Rank descent
    10.5 Wall restriction
    10.6 Face restriction
    10.7 Representation ablation
    10.8 Asymptotic-sign ablation
    10.9 Orthogonal decomposition
    10.10 Factorization descent
    10.11 Determining what survives all admissible descents
    10.12 Basin identity from stable transformation response
    10.13 Basin fracture where modular ancestry changes
    10.14 Stragglers as unreconstructed boundary data
    10.15 Bottom as minimal replay-stable modular structure

This matches TSCT’s supplied discovery architecture in which descent, fill, middle erasure, straggler audit and bottom formation are explicit stages rather than descriptive metaphors.

11. THE SHADOW COMPLEX
    11.1 Definition of a candidate structural carrier
    11.2 Nodes: residue objects at each depth
    11.3 Edges: shadow/descent maps
    11.4 Incidence labels
    11.5 Gaussian weights
    11.6 Wall orientations
    11.7 Asymptotic sign data
    11.8 Transformation weight
    11.9 Multiplier data
    11.10 Growth data
    11.11 Principal parts
    11.12 Homogeneous modular ambiguity
    11.13 Replay data
    11.14 Boundary debt
    11.15 Candidate depth invariant

12. FROM SIGNATURE DEPTH TO RESIDUE DEPTH
    12.1 Why $r$ is not automatically the fundamental notion of depth
    12.2 Geometric rank
    12.3 Analytic rank
    12.4 Shadow rank
    12.5 Residue depth
    12.6 Cancellation depth
    12.7 Effective depth after theta summation
    12.8 Degenerate depth
    12.9 Carrier-independent depth
    12.10 Proposed invariant:

    depth = length of nontrivial irreducible shadow ancestry

    12.11 When signature bounds depth
    12.12 When signature and depth separate
    12.13 Conditions under which deepest residues cancel
    12.14 Native depth versus presentation depth

13. GRM ASCENT: RECONSTRUCTING THE COMPLETION FROM ITS RESIDUES
    13.1 Terminal demand: modularly transforming completed object
    13.2 Forward build from the holomorphic source
    13.3 Reverse build from modular covariance
    13.4 Earliest common carrier
    13.5 First noninvertible arrow
    13.6 Shadow data as reconstruction obligation
    13.7 Asymptotic behavior as boundary data
    13.8 Principal part as uniqueness data
    13.9 Growth conditions
    13.10 Homogeneous modular ambiguity
    13.11 Minimal correction
    13.12 Recursive ascent from lower rank
    13.13 Reconstruction of $E_1$
    13.14 Reconstruction of $E_2$
    13.15 Reconstruction of general $E_r$
    13.16 Determine whether generalized error functions are forced or merely convenient

GRM’s supplied architecture defines the reverse build from the demanded terminal state, forward build from earned source transformations, their earliest comparison carrier, and the first noninvertible arrow as the point of concentrated work.

14. A RECONSTRUCTION THEOREM PROGRAM
    14.1 Input: piecewise polynomial or sign-type kernel
    14.2 Compute distributional modular residue
    14.3 Stratify residue by wall support
    14.4 Descend through wall intersections
    14.5 Record lower-rank shadow objects
    14.6 Establish termination
    14.7 Supply asymptotic chamber data
    14.8 Solve minimal Vignéras-compatible correction
    14.9 Establish convergence
    14.10 Establish modular covariance
    14.11 Establish uniqueness modulo homogeneous solutions
    14.12 Lift back to the original holomorphic object
    14.13 Cold replay under representation change
    14.14 Candidate theorem schema for finite modular-residue ancestry

15. CARRIER ABLATION: INDEFINITE THETA IS NOT ALLOWED TO DEFINE THE THEORY
    15.1 Remove the lattice presentation
    15.2 Remove the sign-kernel presentation
    15.3 Remove the generalized-error-function presentation
    15.4 Preserve only transformation consequences
    15.5 Ask whether shadow ancestry survives
    15.6 If yes: identify carrier-independent structure
    15.7 If no: localize what indefinite-theta geometry contributes irreducibly
    15.8 Representation-dependence court
    15.9 Source-versus-readout court
    15.10 Reconstruction under alternate carriers

16. APPELL–LERCH ASCENT
    16.1 Zwegers’ Appell–Lerch carrier
    16.2 Generalized Appell functions
    16.3 Elliptic transformation structure
    16.4 Pole geometry
    16.5 Mock contribution
    16.6 Completion contribution
    16.7 Extract shadow ancestry
    16.8 Compare with indefinite-theta shadow complex
    16.9 Identify common bottom
    16.10 GRM reconstruction of Appell realization from common structural data

17. MEROMORPHIC JACOBI ASCENT
    17.1 Meromorphic Jacobi forms as a second alternate carrier
    17.2 Polar part
    17.3 Finite part
    17.4 Mock component
    17.5 Completion
    17.6 Pole data versus wall data
    17.7 Descent to common shadow ancestry
    17.8 Carrier-specific versus carrier-invariant information
    17.9 Reconstruction of Jacobi realization from the common bottom

Nazaroglu explicitly identifies Appell–Lerch sums and meromorphic Jacobi forms as Zwegers’ other closely related constructions and points toward corresponding higher-$r$ generalizations.

18. THE FIRST MAJOR 2026 DISCOVERY COURT: THREE-CARRIER AGREEMENT
    18.1 Construct one object in indefinite-theta form
    18.2 Construct its Appell-type realization
    18.3 Construct its meromorphic-Jacobi realization
    18.4 Descend each independently
    18.5 Erase names and conventional presentations
    18.6 Compare shadow DAGs
    18.7 Compare weights and multipliers
    18.8 Compare residue incidence
    18.9 Compare asymptotic data
    18.10 Compare principal parts
    18.11 Compare reconstruction ambiguity
    18.12 Earn common structural identity only after replay

19. NULL VECTORS: THE EXPLICIT FRONTIER LEFT OPEN BY NAZAROGLU
    19.1 Why positive-definite span assumptions matter
    19.2 Null directions
    19.3 Degeneration of projectors
    19.4 Degeneration of dual bases
    19.5 Coalescing walls
    19.6 Failure of existing convergence estimates
    19.7 Change in asymptotic sectors
    19.8 Null-limit shadow behavior
    19.9 First failed reconstruction map
    19.10 Candidate successor kernel
    19.11 Convergence after null degeneration
    19.12 Modular covariance after null degeneration

20. LINEAR DEPENDENCE AS A BASIN FRACTURE
    20.1 Independent wall directions
    20.2 Dependent wall directions
    20.3 Collapse of Boolean subset geometry
    20.4 Quotient incidence structures
    20.5 Redundant shadow descendants
    20.6 Minimal generating wall family
    20.7 Native arity after dependency
    20.8 Reconstruction from a non-free incidence complex
    20.9 New completion candidates
    20.10 Replay against independent approximants

Nazaroglu names null vectors and linear dependencies as important extensions required to enlarge the range of indefinite-theta applications.

21. FROM BOOLEAN SHADOW COMPLEXES TO GENERAL INCIDENCE COMPLEXES
    21.1 Boolean arrangement as the nondegenerate case
    21.2 Matroidal dependence
    21.3 Oriented matroids
    21.4 Hyperplane arrangements
    21.5 Cone complexes
    21.6 Face posets
    21.7 Degenerate incidence
    21.8 Weighted incidence
    21.9 Shadow differential on incidence objects
    21.10 Candidate homological interpretation
    21.11 Which portions are mathematical evidence versus 2026 conjectural extension

22. A POSSIBLE HOMOLOGICAL THEORY OF MODULAR DEFECT
    22.1 Modular defect as boundary data
    22.2 Shadow as first boundary operator
    22.3 Higher shadows as iterated boundaries
    22.4 Boundary-of-boundary constraints
    22.5 Cancellation of adjacent residues
    22.6 Exact versus nonexact modular defects
    22.7 Cohomological obstruction candidate
    22.8 Completion as filling
    22.9 Mock modular form as boundary value of a completed object
    22.10 Conditions required before this analogy can become a theorem

23. TSCT OPERATOR BASIS FOR MOCK-MODULAR DISCOVERY
    23.1 Shadow descent
    23.2 Wall restriction
    23.3 Rank reduction
    23.4 Orthogonal factorization
    23.5 Cone deformation
    23.6 Null degeneration
    23.7 Dependency collapse
    23.8 Representation swap
    23.9 Carrier swap
    23.10 Asymptotic deformation
    23.11 Principal-part ablation
    23.12 Multiplier ablation
    23.13 Replay under all admitted operators
    23.14 Selecting a nonredundant descent generating family

24. NATIVE ARITY IN HIGHER-DEPTH MOCK STRUCTURE
    24.1 Why $r$ directions do not automatically establish $r$-ary necessity
    24.2 Remove one direction
    24.3 Attempt reconstruction from the others
    24.4 Pairwise reconstruction tests
    24.5 Higher-order irreducibility
    24.6 Dependent versus genuinely joint wall structure
    24.7 Earned $r$-arity
    24.8 Relation between native arity and mock depth

25. NONCOMMUTING DESCENTS
    25.1 Shadow before degeneration
    25.2 Degeneration before shadow
    25.3 Carrier swap before rank reduction
    25.4 Rank reduction before carrier swap
    25.5 Detecting order-sensitive structure
    25.6 Commutators of descent operations
    25.7 Higher-order commutators
    25.8 New invariants from descent order
    25.9 Candidate source of previously unseen higher-depth phenomena

26. STRAGGLERS AND UNDISCHARGED MODULAR RESIDUE
    26.1 Residues that survive standard completion
    26.2 Residues supported on degenerate strata
    26.3 Residues that survive carrier change
    26.4 Noise versus genuine structure
    26.5 Grain refinement
    26.6 Basin fracture
    26.7 Successor requirement
    26.8 New primitive candidate
    26.9 Frontier payload rather than premature theorem

27. BOTTOMS IN THE POST-RAMANUJAN BASIN
    27.1 Ordinary modular form as one possible bottom
    27.2 Unary theta shadow bottom
    27.3 Finite incidence bottom
    27.4 Residue-free Vignéras kernel
    27.5 Multiple bottoms under different carriers
    27.6 Why same bottom does not imply same upstream object
    27.7 Bottom prediction
    27.8 Bottom replay
    27.9 When descent should stop

28. AUTOMATED DISCOVERY OF COMPLETIONS
    28.1 Input a holomorphic $q$-series
    28.2 Detect transformation mismatch
    28.3 Estimate weight and multiplier
    28.4 Search for wall/cone representation
    28.5 Generate candidate shadow
    28.6 Descend candidate shadow
    28.7 Infer depth
    28.8 Reverse-build modular requirements
    28.9 Solve correction equation
    28.10 Test convergence
    28.11 Test modular transformation
    28.12 Perform representation swap
    28.13 Cold replay
    28.14 Reject self-validating discoveries

29. SEARCHING FOR UNKNOWN MOCK STRUCTURES
    29.1 Search by residue ancestry rather than known special-function family
    29.2 Enumerate low-depth shadow complexes
    29.3 Realize them in multiple carriers
    29.4 Search for unoccupied incidence patterns
    29.5 Search for admissible but unrealized shadow DAGs
    29.6 Search for unexpected cancellation depth
    29.7 Search for higher native arity
    29.8 Search null and dependent-wall basins
    29.9 Search beyond lattice realizations
    29.10 Demand external mathematical consequences before promotion

30. FROM CLASSIFICATION OF FUNCTIONS TO CLASSIFICATION OF DESCENT ANCESTRIES
    30.1 Traditional classification by formula
    30.2 Classification by weight
    30.3 Classification by multiplier
    30.4 Classification by lattice signature
    30.5 Classification by mock depth
    30.6 Classification by shadow DAG
    30.7 Classification by incidence type
    30.8 Classification by reconstruction ambiguity
    30.9 Classification by admissible carrier set
    30.10 Candidate new equivalence relation:

    two objects are structurally equivalent when their replay-stable descent ancestries agree at declared grain

31. RAMANUJAN REVISITED THROUGH TSCT
    31.1 Start from a Ramanujan mock theta function
    31.2 Remove its historical name
    31.3 Remove its particular $q$-series notation
    31.4 Determine transformation defect
    31.5 Extract shadow
    31.6 Descend shadow
    31.7 Locate bottom
    31.8 Compare with other mock theta functions
    31.9 Determine common basins
    31.10 Determine genuine fractures between Ramanujan’s examples

32. RAMANUJAN REVISITED THROUGH GRM
    32.1 Begin from the demanded modular completion
    32.2 Reverse-build its necessary transformation data
    32.3 Forward-build from Ramanujan’s holomorphic series
    32.4 Locate earliest comparison carrier
    32.5 Identify the first noninvertible arrow
    32.6 Recover shadow
    32.7 Recover nonholomorphic correction
    32.8 Recover the completed object
    32.9 Determine which parts Ramanujan could have inferred inside his available ontology
    32.10 Determine exactly where a successor mathematical language became necessary

33. ZWEGERS REVISITED
    33.1 What Zwegers actually added
    33.2 Which structures were representation-specific
    33.3 Which structures survived subsequent higher-signature generalization
    33.4 Shadow as persistent invariant
    33.5 Error function as replaceable realization
    33.6 Completion as reconstruction
    33.7 The transition from one positive direction to $r$ positive directions
    33.8 What remains invariant across that transition

34. NAZAROGLU REVISITED
    34.1 $r$-tuple error functions as analytic constructors
    34.2 Rank-lowering derivatives as descent operators
    34.3 Subset decompositions as incidence data
    34.4 Vignéras equation as reconstruction constraint
    34.5 Convergence conditions as source-side obligations
    34.6 Null directions as an explicit frontier
    34.7 Linear dependence as an explicit frontier
    34.8 Higher-$r$ Appell/Jacobi correspondence as an explicit frontier

35. THE 2026 SYNTHESIS
    35.1 Ramanujan discovered anomalous objects
    35.2 Zwegers discovered their larger modular carrier
    35.3 Vignéras supplied the differential admissibility mechanism
    35.4 Higher-signature work generalized the geometric carrier
    35.5 Nazaroglu exposed a recursive $r$-tuple hierarchy
    35.6 TSCT extracts the invariant descent architecture
    35.7 GRM reconstructs admissible carriers from that architecture
    35.8 The object of study shifts:

    special function
    ↓
    completion
    ↓
    shadow
    ↓
    shadow ancestry
    ↓
    carrier-independent structural complex

36. THE PRIMARY 2026 DISCOVERY CONJECTURE
    

    Chapter 36 — Automorphic Obstruction Complexes 

    This revision replaces the earlier “mock modularity as finite modular-residue ancestry” formulation. The evidence now supports a broader research target: mock, false, quantum, Jacobi, Eisenstein, Eichler-integral and related post-Ramanujan structures repeatedly involve modular obstruction, completion, recursive depth, differential operators and carrier change. False-theta theory explicitly computes an obstruction to modularity; higher-depth quantum modular forms encode errors of modularity; higher-depth mock forms admit recursive defect structure; and independent carriers such as indefinite theta and coupled Eisenstein series can realize higher depth.

    1. 36.1 The Primary 2026 Conjecture: Automorphic Families as Realizations of Reduced Obstruction Complexes

       • 36.1.1 From named function classes to structural invariants

       • 36.1.2 Erasing historical labels without erasing mathematical consequences

       • 36.1.3 Why “mock modular form” is too carrier-specific to serve as the primitive

       • 36.1.4 Why “shadow” is too narrow to serve as the universal primitive

       • 36.1.5 Why lattice signature cannot serve as the universal classifier

       • 36.1.6 Why formal higher depth cannot serve as the full classifier

       • 36.1.7 Candidate thesis:

         concrete automorphic object
         ↓
         transformation-closure defect
         ↓
         operator-generated obstruction architecture
         ↓
         reduced carrier-independent structure

       • 36.1.8 Scope: conjectural structural unification, not established equivalence theorem

    2. 36.2 Transformation-Closure Defect as the Candidate Primitive

       • 36.2.1 Modular transformation as a closure condition

       • 36.2.2 Candidate defect:

         $$\Delta_f(\gamma) = f|_k\gamma-f$$

       • 36.2.3 Why the defect must be interpreted relative to an action

       • 36.2.4 Weight as structural context

       • 36.2.5 Multiplier as structural context

       • 36.2.6 Domain of transformation

       • 36.2.7 Boundary locus

       • 36.2.8 Allowed correction/completion class

       • 36.2.9 Distinguishing zero defect from removable defect

       • 36.2.10 Distinguishing local closure from global closure

    3. 36.3 The Defect Packet

       • 36.3.1 Minimal candidate data structure

         $$\mathfrak D(f) = \langle \Delta_f, \Gamma, k, \mu, B, \mathcal C \rangle$$

       • 36.3.2 $\Delta_f$: transformation failure

       • 36.3.3 $\Gamma$: acting group or semigroup

       • 36.3.4 $k$: weight data

       • 36.3.5 $\mu$: multiplier/representation data

       • 36.3.6 $B$: closure geometry

       • 36.3.7 $\mathcal C$: admissible completion class

       • 36.3.8 Which components are indispensable under ablation

       • 36.3.9 When two defect packets should count as equivalent

    4. 36.4 Shadow as One First-Order Defect Readout

       • 36.4.1 Classical depth-one mock situation

       • 36.4.2 Shadow extracted from a completed object

       • 36.4.3 $\bar\partial$-type readout

       • 36.4.4 $\xi$-operator readout

       • 36.4.5 Why shadow is consequential but not ontologically universal

       • 36.4.6 Shadow versus full transformation defect

       • 36.4.7 Shadow versus boundary obstruction

       • 36.4.8 Shadow versus false-theta obstruction

       • 36.4.9 Shadow as one coordinate of $\mathfrak D(f)$, not the whole object

    5. 36.5 Higher Depth as Recursive Defect Architecture

       • 36.5.1 Recursive depth in higher-depth mock modularity

       • 36.5.2 Depth-$d$ completion descending to depth-$(d-1)$ completion data

       • 36.5.3 Terminal depth zero as ordinary modular data

       • 36.5.4 Branching versus linear descent

       • 36.5.5 Tensor/product-valued descendants

       • 36.5.6 Coupled descendants

       • 36.5.7 Why a DAG is more faithful than a chain

       • 36.5.8 Residue ancestry as a special case of obstruction ancestry

    Bringmann–Nazaroglu define higher depth recursively so that derivatives of depth-$d$ completions land in combinations involving depth-$(d-1)$ completions and modular forms. They also note that products of depth-one mock forms produce trivial depth-two examples.

    6. 36.6 From Residue Chain to Obstruction DAG

       • 36.6.1 Nodes as defect objects

       • 36.6.2 Directed edges as descent/exposure relations

       • 36.6.3 Branch multiplicity

       • 36.6.4 Coupling coefficients

       • 36.6.5 Signs and orientations

       • 36.6.6 Weight shifts

       • 36.6.7 Multiplier changes

       • 36.6.8 Terminal modular nodes

       • 36.6.9 Boundary nodes

       • 36.6.10 Degenerate or unresolved nodes

       • 36.6.11 Ancestry equivalence under carrier change

    7. 36.7 Coupling Topology as Structural Data

       • 36.7.1 Why a list of descendants is insufficient

       • 36.7.2 Coupled Eisenstein depth two

       • 36.7.3 Antisymmetric and signed coupling patterns

       • 36.7.4 Coupling order

       • 36.7.5 Coupling multiplicity

       • 36.7.6 Coupling as a possible primitive invariant

       • 36.7.7 Distinguishing two depth-two objects with different coupling topology

       • 36.7.8 Coupling topology under representation ablation

    8. 36.8 Formal Depth versus Primitive Depth

       • 36.8.1 Formal recursive depth

       • 36.8.2 Trivial product depth

       • 36.8.3 Genuinely coupled depth

       • 36.8.4 Decomposable residue sector

       • 36.8.5 Candidate quotient:

         $$[\rho]_d \in \mathcal O_d/\mathrm{Dec}_d$$

       • 36.8.6 Primitive depth

         $$d_{\mathrm{prim}}(f) = \max\{d:[\rho]_d\neq0\}$$

       • 36.8.7 Cancellation depth

       • 36.8.8 Effective depth after summation

       • 36.8.9 Depth under carrier change

       • 36.8.10 Conditions required before primitive depth becomes a theorem

    9. 36.9 Signature as Generator, Not Definition

       • 36.9.1 Signature $(r,n-r)$

       • 36.9.2 $r$-tuple error-function construction

       • 36.9.3 Rank-lowering shadows

       • 36.9.4 Signature as capacity for recursive depth

       • 36.9.5 Why null directions can disrupt rank/depth identification

       • 36.9.6 Why dependent directions can collapse effective depth

       • 36.9.7 Why cancellation can lower realized depth

       • 36.9.8 Why Eisenstein coupling defeats signature sovereignty

       • 36.9.9 Signature as one constructor of an obstruction DAG

    10. 36.10 Carrier as Realization Rather Than Ontology

        • 36.10.1 Indefinite theta carrier

        • 36.10.2 Appell–Lerch carrier

        • 36.10.3 Meromorphic Jacobi carrier

        • 36.10.4 Eichler-integral carrier

        • 36.10.5 Harmonic Maass carrier

        • 36.10.6 Eisenstein-series carrier

        • 36.10.7 Poincaré-series carrier

        • 36.10.8 False-theta carrier

        • 36.10.9 Quantum-modular carrier

        • 36.10.10 Carrier ablation as a structural test

    The corpus contains explicit carrier plurality: higher-depth quantum/false-theta companions can be represented both by double Eichler integrals and by nonholomorphic theta series with double-error-function coefficients. Coupled Eisenstein series also provide an independent construction of depth-two mock modularity beyond the indefinite-theta route.

    11. 36.11 False Modularity as an Adversarial Court

        • 36.11.1 False theta functions and modular obstruction

        • 36.11.2 Modular completion of false theta functions

        • 36.11.3 Higher-rank false theta

        • 36.11.4 Higher-depth false modularity

        • 36.11.5 Structural parallels with mock modularity

        • 36.11.6 Structural differences from mock modularity

        • 36.11.7 Why equal depth must not imply equal basin

        • 36.11.8 What data must separate mock and false obstruction complexes

    Higher-depth false modular forms were explicitly developed as a structure parallel to higher-depth mock modular forms.

    12. 36.12 Quantum Modularity and Closure on a Different Locus

        • 36.12.1 Upper-half-plane closure versus boundary closure

        • 36.12.2 Errors of modularity

        • 36.12.3 Rational boundary data

        • 36.12.4 Higher Mordell integrals

        • 36.12.5 Higher-depth quantum modularity

        • 36.12.6 Why closure locus must enter the classifier

        • 36.12.7 Preventing false identification with mock modularity

        • 36.12.8 Boundary geometry as structural information

    13. 36.13 Cohomological Reinterpretation

        • 36.13.1 Transformation defect as cocycle-like data

        • 36.13.2 Coboundary equations

        • 36.13.3 Completion as solving a defect equation

        • 36.13.4 Classical $1$-cohomology completion

        • 36.13.5 Extended cohomology

        • 36.13.6 Failure of classical cocycle closure by products of simpler maps

        • 36.13.7 Higher-depth ancestry as multiplicative cocycle failure

        • 36.13.8 Iterated integrals as structured higher defect

        • 36.13.9 When an ordinary complex is insufficient

        • 36.13.10 Need for multiplicative/branched obstruction structure

    Bringmann–Diamantis–Raum developed a completion technique for $1$-cohomology parallel to mock modular completion, while Bringmann–Diamantis later introduced an extended cohomology whose maps fail classical cocycle conditions by products of simpler maps.

    14. 36.14 The Extended Cohomological Obstruction Complex

        • 36.14.1 Candidate object:

          $$\mathfrak O(f) = \langle V,E,\delta,\mathcal D,\mathcal P,B,K \rangle$$

        • 36.14.2 $V$: obstruction objects

        • 36.14.3 $E$: ancestry/incidence

        • 36.14.4 $\delta$: transformation/coboundary structure

        • 36.14.5 $\mathcal D$: operator family

        • 36.14.6 $\mathcal P$: product/coupling structure

        • 36.14.7 $B$: boundary and closure geometry

        • 36.14.8 $K$: homogeneous reconstruction kernel

        • 36.14.9 Optional weight and multiplier decorations

        • 36.14.10 Finite versus infinite complexes

        • 36.14.11 Criteria for structural equivalence

    15. 36.15 Operator Ecology

        • 36.15.1 Why no evidence currently justifies one universal scalar operator

        • 36.15.2 Antiholomorphic derivative

        • 36.15.3 Shadow operator

        • 36.15.4 $\xi$-operator

        • 36.15.5 Bol operator

        • 36.15.6 Maass raising operator

        • 36.15.7 Flipping operator

        • 36.15.8 Jacobi differential operators

        • 36.15.9 Operator composition

        • 36.15.10 Operator commutators

        • 36.15.11 Operator equivalence under reconstruction

        • 36.15.12 Selecting a minimal nonredundant operator family

    Bringmann’s corpus explicitly treats shadow and Bol operators as controlling different canonical pieces of harmonic Maass forms and introduces a flipping operator exchanging those roles. Maass raising operators also appear constructively, realizing new objects from Poincaré-series sources.

    16. 36.16 Closure Geometry

        • 36.16.1 Modular closure on $\mathbb H$

        • 36.16.2 Jacobi closure

        • 36.16.3 Harmonic Maass closure

        • 36.16.4 Lower-half-plane companions

        • 36.16.5 Quantum/rational-boundary closure

        • 36.16.6 Local versus global closure

        • 36.16.7 Singular closure

        • 36.16.8 Mixed closure regimes

        • 36.16.9 Why closure geometry cannot be quotiented away prematurely

    17. 36.17 Completion as the Inverse Defect Problem

        • 36.17.1 Given defect $\rho$, solve:

          $$\delta F=\rho$$

        • 36.17.2 Existence of a completion

        • 36.17.3 Nonuniqueness

        • 36.17.4 Homogeneous ambiguity:

          $$\delta H=0$$

        • 36.17.5 Why $F+H$ has the same defect

        • 36.17.6 Principal part as reconstruction data

        • 36.17.7 Growth as reconstruction data

        • 36.17.8 Asymptotics as reconstruction data

        • 36.17.9 Boundary values as reconstruction data

        • 36.17.10 Minimal admissible completion

    18. 36.18 GRM Ascent from the Obstruction Complex

        • 36.18.1 Terminal demand

        • 36.18.2 Reverse-build the required closure law

        • 36.18.3 Forward-build from observed defect

        • 36.18.4 Earliest common carrier

        • 36.18.5 First noninvertible reconstruction step

        • 36.18.6 Solve deepest obstruction first

        • 36.18.7 Restore higher coupling levels

        • 36.18.8 Apply boundary constraints

        • 36.18.9 Quotient homogeneous ambiguity

        • 36.18.10 Lift back into candidate analytic carriers

        • 36.18.11 Replay against the original source

    19. 36.19 TSCT Descent to the Reduced Obstruction Complex

        • 36.19.1 Object audit

        • 36.19.2 Representation ablation

        • 36.19.3 Carrier swap

        • 36.19.4 Defect extraction

        • 36.19.5 Operator-family execution

        • 36.19.6 Coupling decomposition

        • 36.19.7 Remove decomposable ancestry

        • 36.19.8 Preserve irreducible obstruction

        • 36.19.9 Detect basin fractures

        • 36.19.10 Resolve stragglers

        • 36.19.11 Determine bottom

        • 36.19.12 Produce reduced obstruction object

    20. 36.20 The Reduced Complex

        • 36.20.1 Define decomposable sector

        • 36.20.2 Remove representation artifacts

        • 36.20.3 Remove redundant operators

        • 36.20.4 Collapse reconstructible descendants

        • 36.20.5 Preserve nontrivial coupling topology

        • 36.20.6 Preserve closure geometry

        • 36.20.7 Preserve reconstruction kernel

        • 36.20.8 Candidate notation:

          $$\mathfrak O_{\mathrm{red}}(f)$$

        • 36.20.9 Minimality criterion

        • 36.20.10 Replay-stability criterion

    21. 36.21 Carrier-Independent Equivalence

        • 36.21.1 Candidate criterion:

          $$\mathfrak O_{\mathrm{red}}(f) \cong \mathfrak O_{\mathrm{red}}(g)$$

        • 36.21.2 Preservation of operator labels

        • 36.21.3 Preservation of coupling

        • 36.21.4 Preservation of weights and multipliers

        • 36.21.5 Preservation of closure locus

        • 36.21.6 Preservation of reconstruction kernel

        • 36.21.7 Carrier-specific decorations that may be erased

        • 36.21.8 Consequential data that may not be erased

        • 36.21.9 Distinguishing isomorphism from superficial similarity

    22. 36.22 Cross-Carrier Court I: Indefinite Theta versus Eisenstein Coupling

        • 36.22.1 Select a depth-two object with both realizations

        • 36.22.2 Freeze object and consequence set

        • 36.22.3 Descend indefinite-theta realization

        • 36.22.4 Descend Eisenstein realization independently

        • 36.22.5 Erase carrier-specific notation

        • 36.22.6 Compare reduced obstruction DAGs

        • 36.22.7 Compare coupling topology

        • 36.22.8 Compare closure data

        • 36.22.9 Compare reconstruction kernels

        • 36.22.10 Carrier independence survives only if the reduced complexes agree

    23. 36.23 Cross-Carrier Court II: Eichler Integral versus Double-Error Theta

        • 36.23.1 Higher-depth quantum/false-theta companion

        • 36.23.2 Double Eichler-integral realization

        • 36.23.3 Nonholomorphic theta realization

        • 36.23.4 Double-error-function coefficients

        • 36.23.5 Independent descent

        • 36.23.6 Compare obstruction complexes

        • 36.23.7 Identify shared invariant

        • 36.23.8 Localize carrier-specific residue

    24. 36.24 Cross-Theory Court: Mock versus False versus Quantum

        • 36.24.1 Same depth is deliberately allowed

        • 36.24.2 Same operator count is deliberately allowed

        • 36.24.3 Test whether reduced complexes remain distinguishable

        • 36.24.4 Mock closure signature

        • 36.24.5 False-modular closure signature

        • 36.24.6 Quantum boundary signature

        • 36.24.7 If all collapse, the classifier is too coarse

        • 36.24.8 Identify the missing structural coordinate

    25. 36.25 Common Source, Multiple Realizations

        • 36.25.1 Partition-source algebras

        • 36.25.2 Theta realization

        • 36.25.3 Quasi-Jacobi realization

        • 36.25.4 Appell–Lerch realization

        • 36.25.5 False-theta realization

        • 36.25.6 Determine whether common source algebra maps to common obstruction bottom

        • 36.25.7 Source structure versus analytic realization

    Bringmann–van Ittersum–Kaszian exhibit algebras of theta-like functions on partitions whose generating functions lead to theta, quasi-Jacobi, Appell–Lerch and false-theta objects.

    26. 36.26 Native Arity of Obstruction Couplings

        • 36.26.1 Unary obstruction

        • 36.26.2 Binary coupling

        • 36.26.3 Higher coupling

        • 36.26.4 Remove one branch

        • 36.26.5 Attempt reconstruction

        • 36.26.6 Pairwise sufficiency test

        • 36.26.7 Genuine higher-arity obstruction

        • 36.26.8 Relation between native arity and primitive depth

    27. 36.27 Noncommuting Operators

        • 36.27.1 Apply defect extraction before raising

        • 36.27.2 Apply raising before defect extraction

        • 36.27.3 Apply carrier swap before completion

        • 36.27.4 Apply completion before carrier swap

        • 36.27.5 Detect order-sensitive consequences

        • 36.27.6 Operator commutator as structural invariant

        • 36.27.7 Higher commutators

        • 36.27.8 Potential source of new depth phenomena

    28. 36.28 Obstruction Bottoms

        • 36.28.1 What counts as terminal modular data?

        • 36.28.2 Ordinary modular bottom

        • 36.28.3 Cocycle bottom

        • 36.28.4 Boundary bottom

        • 36.28.5 Multiple admissible bottoms

        • 36.28.6 Bottom dependence on closure geometry

        • 36.28.7 Bottom equivalence across carriers

        • 36.28.8 Conditions for descent termination

    29. 36.29 Discovery Criterion

        • 36.29.1 Representation change alone is not discovery

        • 36.29.2 New named special function alone is not discovery

        • 36.29.3 Increased formal depth alone is not discovery

        • 36.29.4 Candidate discovery event:

          existing obstruction complex
          ↓
          persistent irreducible straggler
          ↓
          all licensed reconstructions fail
          ↓
          new operator/coupling/closure component required

        • 36.29.5 Successor structure must improve replay

        • 36.29.6 No ontology expansion without consequential necessity

    30. 36.30 Reconstruction Criterion

        • 36.30.1 Supply only reduced obstruction data

        • 36.30.2 Supply minimum boundary packet

        • 36.30.3 Hide known carrier

        • 36.30.4 GRM reconstructs admissible completion

        • 36.30.5 Compare against known object

        • 36.30.6 Characterize homogeneous ambiguity

        • 36.30.7 Failure localizes missing structural data

    31. 36.31 The Blinded Post-Ramanujan Discovery Experiment

        • 36.31.1 Remove names Ramanujan, Zwegers, Maass, Appell, Jacobi

        • 36.31.2 Remove known completion formulas

        • 36.31.3 Preserve only source $q$-series and transformation evidence

        • 36.31.4 Extract defect

        • 36.31.5 Descend obstruction

        • 36.31.6 Generate candidate completion mechanism

        • 36.31.7 Test modular consequences

        • 36.31.8 Name structure only after reconstruction

        • 36.31.9 Distinguish rediscovery from target leakage

    32. 36.32 Falsification Conditions

        • 36.32.1 Same mathematical object yields incompatible reduced complexes under two valid carriers

        • 36.32.2 Distinct theories collapse to the same complex despite different consequential behavior

        • 36.32.3 Reconstruction from the proposed complex repeatedly fails

        • 36.32.4 Required carrier information cannot be eliminated

        • 36.32.5 Primitive depth fails to distinguish trivial from nontrivial higher depth

        • 36.32.6 Closure geometry proves insufficient

        • 36.32.7 Operator family is representation-selected rather than source-earned

        • 36.32.8 Infinite ancestry defeats finite-complex assumption

        • 36.32.9 Ad hoc repairs proliferate without predictive gain

    33. 36.33 Evidence Status

        • 36.33.1 ESTABLISHED: modular and nonmodular phenomena admit multiple completion mechanisms

        • 36.33.2 ESTABLISHED: false theta functions possess computable modular obstructions

        • 36.33.3 ESTABLISHED: higher-depth mock modularity is recursively defined and permits trivial product depth

        • 36.33.4 ESTABLISHED: higher-depth false modular theory parallels higher-depth mock theory

        • 36.33.5 ESTABLISHED: higher-depth quantum modular structures possess explicit modularity errors

        • 36.33.6 ESTABLISHED: cohomological completion and extended cocycle failure structures exist

        • 36.33.7 ESTABLISHED: independent Eisenstein coupling produces depth-two mock modularity

        • 36.33.8 DIRECT INFERENCE: these structures motivate a common obstruction architecture

        • 36.33.9 UNRESOLVED: existence of a universal reduced obstruction complex

        • 36.33.10 UNRESOLVED: completeness of $\mathfrak O_{\mathrm{red}}$ as a classifier

        • 36.33.11 UNRESOLVED: uniqueness of GRM reconstruction from reduced obstruction data

    34. 36.34 Final 2026 Conjecture

        • 36.34.1 Weak form:

          A broad class of post-Ramanujan automorphic phenomena admits useful organization by transformation defects, completions and recursive obstruction data.

        • 36.34.2 Strong form:

          After carrier ablation, these objects possess replay-stable reduced obstruction complexes.

        • 36.34.3 Strongest form:

          Mock, false, quantum, Jacobi, theta, Eisenstein and Eichler-integral realizations are reconstructible manifestations of a finite family of extended cohomological obstruction architectures.

        • 36.34.4 Required theorem:

          $$\mathfrak O_{\mathrm{red}}(f) \cong \mathfrak O_{\mathrm{red}}(g)$$

          if and only if $f$ and $g$ are structurally equivalent at the declared automorphic grain.

        • 36.34.5 Required reconstruction theorem:

          reduced obstruction complex
          ↑
          minimal boundary data
          ↑
          admissible completion modulo controlled homogeneous kernel

        • 36.34.6 Required falsification:

          cross-carrier agreement
          +
          cross-theory separation
          +
          successful blinded reconstruction

        • 36.34.7 Status: 2026 TSCT/GRM discovery conjecture — not established theorem 

37. THE PRIMARY 2026 RECONSTRUCTION CONJECTURE
    37.1 Input:

    shadow complex
    asymptotic chamber data
    weight
    multiplier
    growth
    principal part

    37.2 Output: modular completion
    37.3 Uniqueness modulo homogeneous modular solutions
    37.4 Recovery of generalized error functions when the source is an indefinite theta cone
    37.5 Recovery of Appell structure under elliptic/pole constraints
    37.6 Recovery of meromorphic Jacobi structure under Jacobi constraints
    37.7 Failure conditions
    37.8 Counterexamples that would destroy the conjecture

38. FALSIFICATION PROGRAM
    38.1 Find two objects with identical proposed shadow complexes but inequivalent consequential behavior
    38.2 Find depth not detected by recursive shadow descent
    38.3 Find a completion not reconstructible from residue plus boundary data
    38.4 Find carrier information that survives every proposed ablation
    38.5 Find native arity invisible to wall incidence
    38.6 Find null degenerations requiring fundamentally different primitives
    38.7 Find divergent shadow ancestry
    38.8 Find infinite-depth objects outside the finite architecture
    38.9 Reject the 2026 model if these failures cannot be repaired without ontology inflation

39. EXTERNAL MATHEMATICAL COURTS
    39.1 Known Zwegers examples
    39.2 Known signature $(2,n-2)$ examples
    39.3 Nazaroglu $r=3$ cases
    39.4 Nazaroglu nonfactorizable $r=4$ example
    39.5 Generalized Appell examples
    39.6 Meromorphic Jacobi examples
    39.7 Null-limit examples
    39.8 Linearly dependent wall examples
    39.9 Representation-swapped replay
    39.10 Held-out examples excluded during construction

40. DISCOVERY STATUS DISCIPLINE
    40.1 Established theorem
    40.2 Established construction
    40.3 Direct structural inference
    40.4 TSCT hypothesis
    40.5 GRM reconstruction hypothesis
    40.6 Computational evidence
    40.7 External replay
    40.8 Counterexample
    40.9 Open frontier
    40.10 No promotion from elegance or internal coherence alone

41. A GENERAL THEORY OF MATHEMATICAL FRONTIERS
    41.1 Ramanujan frontier
    41.2 Persistent defect
    41.3 Exhaustion of current representational language
    41.4 Successor carrier
    41.5 Descent to invariant residue
    41.6 Reconstruction into a larger theory
    41.7 Replay into the original object
    41.8 Discovery as earned ontology expansion

42. POST-RAMANUJAN DISCOVERY BEYOND MOCK MODULAR FORMS
    42.1 Search for the same architecture in automorphic forms
    42.2 Period integrals
    42.3 Eichler integrals
    42.4 Quantum modular forms
    42.5 False theta functions
    42.6 Partial theta functions
    42.7 Indefinite theta lifts
    42.8 Higher-depth automorphic objects
    42.9 Determine which share the same residue architecture
    42.10 Prevent premature identification across distinct basins

43. TSCT/GRM AS A BIDIRECTIONAL DISCOVERY MACHINE
    43.1 TSCT:

    concrete object
    ↓
    descent
    ↓
    invariant structure

    43.2 GRM:

    invariant structure
    ↑
    necessity
    ↑
    reconstructible realization

    43.3 Round-trip court
    43.4 Descent failure
    43.5 Ascent failure
    43.6 Mismatch as new discovery residue
    43.7 Successor generation
    43.8 Replay

44. FINAL SYNTHESIS: FROM RAMANUJAN’S MOCKNESS TO STRUCTURAL DISCOVERY
    44.1 Ramanujan found the fracture
    44.2 Zwegers found one completion mechanism
    44.3 Higher-signature work exposed recursive completion structure
    44.4 Nazaroglu made rank descent explicit
    44.5 TSCT turns rank descent into a general structural discovery operator
    44.6 GRM turns shadow ancestry into a reconstruction problem
    44.7 The next target is no longer another named special function
    44.8 The next target is the invariant architecture generating whole families of completions
    44.9 Post-Ramanujan mathematics as a descent/ascent system
    44.10 2026 frontier:

    classify modular residue ancestries
    ↓
    reconstruct their admissible carriers
    ↓
    discover previously unknown modular structures