Foundations — Registration Theory

Foundations

The more formal counterpart to Theory: primitive ontology, definitions, exact finite-model structure, and the program's current open proof obligations.

Primitive ontology

Axiom

Registration Theory takes persistent physical registration as the primitive physical organization from which stable objects, operational time, spatial relation, information, and effective geometry may emerge. Registration and Trinity are treated as two descriptions of the same primitive organization: space-facing, time-facing, and energy-facing allocations are mutually related aspects of that organization, and maximally space-facing propagation defines a universal local limit.

Ordinary spacetime geometry is derived, not assumed at the outset. Effective geometry must be reconstructed from relational registration structure rather than assumed at the primitive level; persistent objects are stabilized registration records, and the ledger is persistent physical organization rather than abstract information.

Sources: Canonical Foundations of Registration Theory (v1.0)

Definitions

Definition

Registration tensor: a covariant object defined on registration-state space — not assumed a priori on a spacetime manifold — representing physically available relational organization.

Host / resolution map: a system-specific map that selects, weights, and integrates registration structure into an effective, system-relative representation. Different hosts may resolve different effective degrees of freedom from the same underlying registration structure.

ACRF-4 enlarged state: the pair (x, K), where x ∈ R⁴ is the instantaneous four-coordinate registration state and K is a retained relational matrix carrying history-dependent structure not reconstructible from x alone.

Sources: Canonical Foundations of Registration Theory (v1.0), System-Relative Dimensionality (v1)

Exact finite-model structure

Exact Theorem

The canonical ACRF-4 decomposition.

For canonical ACRF-4, the permutation representation on the four registration coordinates decomposes exactly:

R⁴ = U ⊕ H4, dim(U) = 1, dim(H4) = 3

U = span{1}, H4 = { r ∈ R⁴ : 1ᵀr = 0 }

R⁴ / U ≅ H4 ≅ R³

1.The four registration coordinates are interchangeable: U is the one common mode.
2.H4 is the three-dimensional sector of relative registration degrees of freedom.

This does not directly identify the four coordinates with physical time, space, or energy; it does not by itself derive physical spacetime; and it does not automatically provide a preferred physical basis — host calibration is downstream.

Sources: Structural Trinity Conjecture (v2.0) (v2.0), Trinity Symmetry / ACRF-4 Registration Quotient Correspondence

Current/retained architecture

Exact Model Result

An exact reciprocal channel between the current sector and a retained-edge sector.

A second exact result concerns the rank-three retained edge sector built from the current sector H4. Writing the edge map (Lr)ᵢⱼ = rᵢ + rⱼ, the image E = Im L is again three-dimensional, and the identity L*L = 2I gives an exact isometry between the two sectors:

‖Lr‖² = 2‖r‖² ⇒ W = L / √2 is an isometry: H4 ≅ E

𝒥 = [ 0 W* ; W 0 ], 𝒥² = I

This does not claim H4 = physical space, E = physical time, or 𝒥 = a Lorentz transformation. Those identifications are not made by the source and would require further proof.

Sources: Reciprocal Current–Retention Conjugacy in ACRF-4 (v1.0)

Consequence and locality

Conditional Result

A finite-support theorem for the tangent propagator, conditional on one-step locality.

If one-step consequence obeys locality on the registration graph —

i not adjacent to j ⇒ Jⱼᵢ = 0

d_reg(i,j) > n ⇒ (Jⁿ)ⱼᵢ = 0

ℓ_reg(n; i) ≤ n

This is standard locality mathematics once one-step locality is assumed; it does not by itself derive the physical speed of light. The unresolved task is deriving the forward-generated shared-event transfer carrier and boundary-growth dynamics, rather than assuming them.

Sources: Reciprocal Current–Retention Conjugacy in ACRF-4 (v1.0)

Operational reconstruction

Registration order is not the same thing as time, and relative distance is not automatically physical space.

Registration order, operational time, coordinate time, and proper time are kept distinct — the program does not write N = t. Within a recurrent ideal clock lineage, an operational clock time can be reconstructed proportionally to counted elementary succession at tick events, and under repeatable congruent relational intervals with additive concatenation, an operational distance can be reconstructed in the corresponding homogeneous sector:

t_C = α_C N

x = β d_reg

c = β / α

These reconstructions hold within the stated model assumptions for a single host. A universal physical c across arbitrary hosts remains a deeper calibration and reconstruction issue, not solved by this step; this section is not yet linked to a specific verified publication in this registry and should be read as the program's stated reconstruction target.

Sources: System-Relative Dimensionality (v1)

Open proof obligations

Open Problem

The program's current working list of unresolved obligations.

1.Resolution existence — that a system-relative resolution map exists and is well-defined for a given host.
2.Effective rank and dimension agreement — that independently registering systems resolve compatible effective dimension.
3.Cross-registration consistency — that different hosts' resolutions of a shared event remain mutually consistent.
4.Macroscopic stabilization — that persistent, localized, classical-body-like records can be derived rather than conceptually proposed.
5.Temporal asymmetry / Lorentzian signature — a non-circular derivation of a single time-facing sign, not assumed as input.
6.Metric and continuum reconstruction — a completed, general theorem reconstructing a physical metric from registration structure.
7.Observer diversity / measurable residuals — testable, host-dependent residuals that distinguish this program from a fixed-background theory.

This list is the program's own working enumeration of unresolved obligations, not a verbatim quotation from a single controlling document. Where the status ledger gives more precise status language for a specific item, the ledger controls — see Status.

The open proof obligations above are also presented as navigable objects, each linked to related research threads and sources, on the Research page.