Bilateral Registration
Can one interaction-generated registration be represented as a common carrier with two participant-relative manifestations, and how does that carrier connect to later compatibility geometry?
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.
Established
- The bilateral consequence-factorization theorem is exact conditional mathematics.
- Canonical finite ACRF-4 has one exact common quotient carrier for both host successor records.
- The matched receiver-isolation intervention is exactly covariant under participant exchange.
- The generic quotient decomposes as a participant-relative fiber product, and the invariant construction extends this bilateral organization globally across symmetry-restoration strata.
Active work
- Derive physically selected scalar coordinates p=f_S(rho_{S|C}) and q=f_C(rho_{C|S}) from the complete bilateral carrier.
- Keep the identification with primitive Trinity conjugation open until a separate foundational derivation is supplied.
Open problems
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.
Related threads: Bilateral Registration, Conjugate / Brokerage Structure
Relevant claims
Bilateral Participatory Registration Principle
A primitive interaction-generated registration is proposed to be one common relational object with two simultaneous participant-relative manifestations, rho_{A|B} and rho_{B|A}, exchanged involutively rather than related by an internal ancestry arrow.
Scope: Candidate foundational refinement of Registration Theory introduced in the bilateral ACRF-4 paper; it supplies the architecture tested by the exact conditional and finite-model results.
- Not derived from the existing canonical RT axioms.
- Does not identify participant exchange with Trinity conjugation.
- Does not imply equal downstream consequences for asymmetric participants.
Last reviewed 2026-09-08
Bilateral Consequence Factorization Theorem
For a declared bilateral representation, joint consequence sufficiency is equivalent to constancy of both participant consequence signatures on representation fibers and to factorization of both consequences through the same common representation.
Scope: Exact representation-theoretic result conditional on the declared bilateral event domain, representation, and consequence signatures.
- Does not by itself prove that a proposed representation is physically primitive.
- Physical completeness still requires that consequential retained-boundary information not be omitted.
Last reviewed 2026-09-08
Global Bilateral Quotient Reconstruction in Canonical ACRF-4
Canonical finite ACRF-4 admits two globally defined participant-relative records that are exchanged involutively and jointly reconstruct the complete orbit-separating quotient; the same common carrier remains upstream of both matched-intervention consequences.
Scope: Exact within canonical finite ACRF-4, including symmetry-restoration strata through Q=(E,UU^T,VV^T,c^2,cUV^T).
- The global invariant representation is algebraically redundant and does not make the stratified quotient a globally smooth product.
- The ACRF participant-relative records are not identified with primitive Trinity registrations.
- No physical metric, Lorentz structure, force, charge, or universal speed follows from this theorem.
Source
Last reviewed 2026-09-08
Relevant publications
Bilateral Participatory Registration in Canonical ACRF-4: Exact Consequence Factorization, Participant-Exchange Covariance, and a Global Two-Sided Quotient Carrier
Develops the Bilateral Participatory Registration Principle and proves an exact canonical ACRF-4 realization with one common quotient carrier, participant-exchange covariance, generic fibered participant-relative decomposition, and a global invariant two-sided reconstruction.
- Registration Theory
- ACRF-4
- bilateral participatory registration
- participant-relative registration
- consequence factorization
- participant exchange
- exchange covariance
- matched intervention
- quotient dynamics
- permutation symmetry
- invariant quotient
- orbit separation
- stratified quotient
- relational ontology
- pre-geometric physics
- Trinity
Exact Finite-Amplitude Permutation-Quotient Cross-Host Consequence Dynamics in Canonical ACRF-4: Orbit Completeness, Stratified Host-Boundary State, and a Nine-Dimensional Generic Quotient
Proves an exact finite-amplitude S2 x S2 permutation quotient for canonical ACRF-4: the quotient separates internal-label orbits, supports exact nonlinear descent for every finite horizon, and is sufficient for permutation-invariant host consequences. The generic orbit quotient has dimension nine, with an interpretable 3+3+3 host-host-boundary chart and a stratified global quotient.
- Registration Theory
- ACRF-4
- permutation quotient
- invariant theory
- adaptive closed registration
- retained boundary
- cross-host consequence
- finite-amplitude dynamics
- orbit completeness
- stratified quotient
- consequence sufficiency
- relational alignment
Structural Trinity Conjecture in Registration Theory
Current physical-interpretation version of the Structural Trinity Conjecture; supersedes the v1.0 formulation below.