Research
Current research frontier
The centerpiece result: an exact ACRF-4 current/retained reciprocal structure, a conditional finite-support theorem built on it, and — the major unresolved physical step — no forward-generated shared-event transfer carrier yet. Organized as established, conditional, missing bridge, and next derivation target.
Established
ACRF-4 admits an exact 1 ⊕ 3 decomposition
R⁴ decomposes exactly as U ⊕ H4, with U = span{1} of dimension 1 and H4 = { r : 1ᵀr = 0 } of dimension 3, and R⁴ / U ≅ H4 ≅ R³.
Scope: A structural linear-algebra fact about the canonical finite ACRF-4 registration model. It does not by itself identify the four ACRF-4 coordinates with physical quantities.
- Does not constitute a derivation of physical spacetime from ACRF-4.
- Host calibration is downstream and does not supply a preferred basis automatically.
Last reviewed 2026-08-27
ACRF-4 admits an exact reciprocal current–retention channel
Within canonical ACRF-4, the three-dimensional current sector H4 and the three-dimensional retained-edge image E = Im L, with (Lr)ᵢⱼ = rᵢ + rⱼ, couple reciprocally under the model's linearized tangent dynamics. Since ‖Lr‖² = 2‖r‖², W = L/√2 is an isometry between the two sectors, and the normalized cross-sector operator J = [0 W*; W 0] satisfies J² = I.
Scope: A structural result about linearized tangent dynamics of the canonical ACRF-4 model, not an unconditional algebraic fact like the 1 ⊕ 3 decomposition above.
- Does not identify H4 with physical space or the retained sector with physical time.
- Does not derive an expanding ledger, Lorentz symmetry, the physical value of c, or an inter-host transfer map.
Source
- Reciprocal Current–Retention Conjugacy in ACRF-4 — no DOI on record
Last reviewed 2026-08-27
Conditional
Finite-step consequence support under one-step locality
If the native ACRF-4 update couples only primitive neighbors (one-step locality), the n-step full-state tangent propagator has no support beyond registration depth n: d_reg(i,j) > n ⇒ (Jⁿ)ⱼᵢ = 0, so the consequence-front length obeys ℓ_reg(n;i) ≤ n.
Scope: Standard locality mathematics applied to the ACRF-4 update graph, conditional on the one-step locality assumption stated in the source; the result itself is not claimed as novel.
- Does not by itself derive a physical speed of light.
- Canonical ACRF-4 itself is not thereby shown to satisfy sparse primitive-neighbor locality: its mean term is globally consequential and its adaptive retained-coupling target gives every pair positive support after one update.
- The forward-generated shared-event transfer carrier required for physical propagation remains unresolved.
Source
- Reciprocal Current–Retention Conjugacy in ACRF-4 — no DOI on record
Last reviewed 2026-08-27
Missing bridge
The finite-support theorem bounds how far a single registration event’s consequence can reach within ACRF-4. It does not supply a forward-generated shared-event transfer carrier between hosts, schematically Hi → Bij → Hj. This is the major unresolved physical step standing between the exact ACRF-4 structure above and a derivation of physical spacetime.
Reconstruction of physical spacetime from registration dynamics
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. Deriving physical spacetime still requires a forward-generated boundary-growth and inter-host transfer structure whose complete dynamics remain unresolved.
Last reviewed 2026-08-28
Next derivation target
Once a transfer carrier and boundary-growth rule are established, the next targets are an operational causal reconstruction — a universal physical value of c, and full Lorentz symmetry without further assumptions — neither of which is claimed as complete here.
A universal physical value of c derived from the framework
Not yet established as a completed universal derivation.
Last reviewed 2026-08-27
Full Lorentz symmetry without remaining assumptions
A derivation of full Lorentz symmetry that does not depend on further unstated assumptions has not yet been completed.
Last reviewed 2026-08-27
See the research map for how this fits into the full program, or Status for the complete ledger.