Research

Research program overview

Registration Theory is organized into several interconnected branches. Registration Geometry remains the most developed. The new Emergent Mathematics branch investigates how counting and algebra could arise from registration, distinct from the existing Registration Mathematics toolbox.

Research map →Current frontier →

Branches

The current frontier is the bridge from exact and model-specific registration results to primitive closed dynamics, localized persistent bodies, effective metric geometry, energy-momentum, and the controlled recovery of classical and relativistic dynamics.

Registration Geometry

Primary active branch

The most developed branch. Current work includes ACRF-4, rank-three relational geometry, system-relative dimensionality, retained-history geometry, maintained-distinction time, Lorentzian history structure, and conditional routes toward effective metric and gravitational dynamics.

Registration Dynamics

Active branch

Contains explicit dynamical models, including conservative ledger dynamics, Lyapunov-controlled evolution, interaction-triggered symmetry breaking, persistent records, finite-memory registration updates, and cross-host consequence dynamics. The current frontier includes primitive closed dynamics and recovery of effective classical motion.

Registration Information

Developing branch

Focuses on consequence sufficiency, distinguishability, representation completeness, closed-event consequence accounting, and the relation between physical records and operational information. Formal results now exist, while the branch remains less mature than Geometry and Dynamics.

Registration Memory

Active branch

Covers retained registration, path dependence, fading-memory history functionals, retained relational geometry, and the role of prior registration history in later dynamics. Several exact finite-model results are now established.

Emergent Mathematics

Developing branch

A distinct foundational branch asking whether mathematical identity, counting, compositional algebra, and eventually geometry can arise from primitive distinction, retained consequence, and recursive daughter registration. The inaugural October 2026 paper proves conditional counting and quotient-algebra results and identifies persistence and host-accessibility obstructions; a universal derivation from Trinity remains open.

Inaugural paper (DOI) ↗

Registration Mathematics

Active branch

Provides the formal machinery underlying the program: invariant persistence, persistent lineages, quotient and symmetry structures, rank results, dilation theorems, zero-set and generator theorems, representation-independence results, and mathematical audits of ACRF-4.

Research threads

Cross-cutting lines of work across the Registration Theory program. Status is a coarse label for how settled a thread is, not a completion percentage — several threads have exact results only within a specific model or under stated assumptions.

Established
Active
Conditional
Computational
Open

Registration Geometry

Active

The five-paper completion spine (registration tensor → coordinate-free geometry → Lorentzian signature → physical metric candidate → field-to-metric map) plus the canonical ACRF-4 1 ⊕ 3 decomposition.

Persistent Registration and Memory

Conditional

Exact model-level results for retained relational history (an exponentially weighted, fading-memory functional) and for when a changing organization counts as one persistent lineage rather than unrelated states.

Host Structure

Computational

Pre-geometric host resolution and reciprocal retained-boundary interaction, currently documented as a computational research note rather than a closed-form theorem.

Conjugate / Brokerage Structure

Conditional

An abstract brokerage-generator theorem yields hyperbolic (1,1) structure from conjugate-channel assumptions; a candidate relativistic brokerage law connects it to the energy–momentum relation. Both replace the withdrawn direct physical-sector identification.

Consequence Propagation

Computational

Exact ACRF-4 boundary consequence now supports a computational consequence-network construction: effective edges are constituted by an intervention-defined receiver change rather than an independently assigned bond probability. In the declared reference ensemble, the non-gauge mean channel and retained shared boundary strongly shift the resulting finite-type operator. The present operator still propagates pre-interaction receiver types; the next target is a Cross-Registration Successor Kernel that propagates the actual receiver successor state.

Metric Reconstruction

Conditional

Phenomenological, one-way effective-metric bridges exist on a supplied Lorentzian background; a completed, general theorem reconstructing a physical metric from registration structure remains open.

Dimensionality

Open

ACRF-4 contains an exact three-dimensional relative registration sector, but the physical identification and stabilization of macroscopic 3+1 geometry across hosts is stated by the source as a remaining theorem obligation.

Continuum Limits

Active

Discrete-to-continuum work on ACRF-4's tangent geometry, including an exact four-channel zero-set theorem and exact discrete-to-continuum transport for retained history.

Physical Spacetime Reconstruction

Open

ACRF-4 now has exact node-block nonautonomy and strict retained-edge densification, but its complete positive K-support graph does not derive sparse primitive-neighbor locality. The forward-generated boundary-growth and inter-host transfer structure required for physical spacetime remains unresolved.

Bilateral Registration

Conditional

The foundational bilateral principle remains proposed, but canonical finite ACRF-4 now supplies an exact global model realization: one quotient carrier is jointly consequence-sufficient, participant exchange acts involutively, the generic quotient is a fibered participant-relative pair, and the global invariant pair reconstructs the complete quotient. The next target is a dynamically selected bridge from the complete bilateral carrier to reduced compatibility coordinates.

For the underlying claims and their exact status, see Status, or the full research map.

Open problems

The program’s current working list of unresolved obligations, each linked to the threads and sources it bears on. None are marked solved.

Open Problem

Resolution existence

That a system-relative resolution map exists and is well-defined for a given host.

Open Problem

Effective rank and dimension agreement

That independently registering systems resolve compatible effective dimension.

Related threads: Dimensionality, Registration Geometry

Sources: System-Relative Dimensionality

Open Problem

Cross-registration consistency

That different hosts' resolutions of a shared event remain mutually consistent.

Related threads: Host Structure

Open Problem

Macroscopic stabilization

That persistent, localized, classical-body-like records can be derived rather than conceptually proposed.

Related threads: Physical Spacetime Reconstruction

Open Problem

Temporal asymmetry / Lorentzian signature

A non-circular derivation of a single time-facing sign, not assumed as input.

Open Problem

Metric and continuum reconstruction

A completed, general theorem reconstructing a physical metric from registration structure.

Related threads: Metric Reconstruction, Continuum Limits

Sources: Registration Geometry I: Stable Records and a Memory-Sensitive Effective Metric on a Supplied Lorentzian Background, Registration Geometry V: Induced Einstein Dynamics from a Retained Registration Spectrum

Open Problem

Observer diversity / measurable residuals

Testable, host-dependent residuals that distinguish this program from a fixed-background theory.

Related threads: Host Structure, Dimensionality

Open Problem

Registration Closure–Boundary theorem

Canonical ACRF-4 now proves that persistent internal registration can coexist with identically zero receiver-directed boundary consequence on an exact successor-invariant sector. The remaining open task is to derive the general physical theorem linking invariant registration closure to zero net outward space-facing production and bounded consequential support.

Sources: Invariant Registration Closure, Compact Closure and Noncompact Conjugate Escape

Open Problem

Particle–Field Conjugacy theorem

Derive localized/persistent and field-facing maps from one complete registration and prove their reciprocal Trinity-conjugate relation rather than postulating the physical identification.

Sources: Particle–Field Trinity Conjugacy

Open Problem

Multi-horizon boundary-field theorem

Partially resolved in canonical ACRF-4: the complete instantaneous zero-boundary-field set is not forward invariant, but an exact internally active successor-invariant zero-field sector exists and contains a nonzero persistent fixed registration. The remaining task is to characterize the full family and stability of persistent zero-field sectors.

Related threads: Host Structure, Consequence Propagation

Sources: Particle–Field Trinity Conjugacy, Compact Closure and Noncompact Conjugate Escape

Open Problem

Relative Charge-Orientation theorem

Derive a gauge-independent relational orientation functional with a composition or conservation law and establish its effect on receiver consequence. The complex-orientation no-go theorem now shows that the local sign J versus -J is not itself gauge-independent on the current bilateral quotient, so charge cannot be identified with that sign without additional structure.

Sources: Particle–Field Trinity Conjugacy, Canonical Complex Structure Without Canonical Orientation

Open Problem

Mass Coarse-Graining theorem

Derive a physically calibrated mass functional from persistent localized multi-registration organization and recover inertial and energetic mass observables in an appropriate effective regime.

Sources: Particle–Field Trinity Conjugacy

Open Problem

Closure-Loss / Radiative-Propagation theorem

Derive conditions under which loss or reorganization of persistent localized closure produces a sustained outward field-conjugate consequence mode with controlled propagation. The compact-closure/noncompact-escape result supplies an algebraic precursor but does not yet establish this physical bridge.

Sources: Particle–Field Trinity Conjugacy, Invariant Registration Closure, Compact Closure and Noncompact Conjugate Escape

Open Problem

Cross-Registration Successor Kernel

Replace consequence-incidence propagation by a receiver-successor law: derive or control a kernel T_ij^k(delta) for the probability that a consequential source/receiver interaction sends the receiver into successor type k, and determine whether the resulting dynamic operator closes on a natural RT quotient.

Related threads: Consequence Propagation, Host Structure

Sources: Cross-Registration Consequence Networks

Open Problem

Bilateral carrier to compatibility-coordinate bridge

Derive dynamically or physically selected maps p=f_S(rho_{S|C}) and q=f_C(rho_{C|S}) from the complete bilateral registration carrier, rather than choosing scalar coordinates for algebraic convenience, before applying the separate hyperbolic compatibility geometry.

Sources: Bilateral Participatory Registration in ACRF-4

Open Problem

Common Closed-Sector Theorem

Derive from primitive Registration Theory dynamics the physically selected closed quotient on which conjugate exchange and closure-orientation parity act as the required anticommuting involutions.

Sources: Closure-Induced Complex Structure

Open Problem

Physical Phase-Succession Bridge

Show that intrinsically individuated physical registration succession is generated by or faithfully represented by the compact phase action induced by the closure-generated complex structure.

Sources: Closure-Induced Complex Structure, Canonical Local Complex Structure in ACRF-4, Canonical Complex Structure Without Canonical Orientation

Open Problem

Transverse Stability of the Persistent Closed Registration

Compute the transverse Jacobian and spectrum of the nonzero persistent zero-field fixed registration to determine whether exact persistent closure is dynamically attracting, repelling, or saddle-like.

Related threads: Host Structure, Consequence Propagation

Sources: Compact Closure and Noncompact Conjugate Escape

Open Problem

Global Finite-Amplitude Closed-Sector Complex Structure

Determine whether the canonically derived local tangent parity/exchange structure extends to a globally defined finite-amplitude closed-sector complex structure over the persistent ACRF-4 closure manifold without inserting the exchange map by hand.

Sources: Canonical Local Complex Structure in ACRF-4, Canonical Complex Structure Without Canonical Orientation, Compact Closure and Noncompact Conjugate Escape