Dependency and Scope

0.1 — What this paper does

The observed late-universe energy budget is approximately 68% dark energy, 27% dark matter, 5% visible matter (Planck 2018; the remaining ~0.2% radiation and relativistic neutrinos is not part of the late-universe budget addressed here). The axioms derive the total dark fraction (20/21 ≈ 95.24%, AP10/AP28/AP41) but not the internal split — that is Debt D44. This paper addresses D44. The 68/27 split is not a partition of channels but a clock reading: dark matter is the ongoing defragmentation of records inside black holes; dark energy is the reservoir of content that has completed defragmentation and returned to the 1:1 state. One derived timescale, τ = (6/21) × t_H ≈ 3.9 Gyr, fixes the two dark fractions to within 1.3% of Planck and the visible channel to ≈2%, with no free parameters, and predicts its epoch dependence.

0.2 — Dependencies

AP04 (the loop — black holes as doors to new cycles), AP06 (α, the leakage constant), AP09 (defragmentation — the forgetful functor U: Mon → Set), AP10 (twenty-one channels, three dimensions), AP20 (axioms unconditional — the substrate clock), AP21 (filamentary structure — how matter reaches black holes), AP24 (the six faces of ε), AP28 (the twenty-one-channel geometry; coupling independence), AP41 (fusion as ongoing break; the visible-channel identification). It does not depend on AP30/AP30a (the proton mass, an independent result) or on AP31–AP39 (the ethics chain; no ethical claims are made).

0.3 — Axiom mapping

Axiom S (Symmetry — the involution σ). The fully-defragmented state is the symmetric 1:1 — the unbroken mirror, the condensate at rest. Dark energy is the substrate returned to σ-symmetry; it does not cluster because it has no residual structure.

Axiom B (Break — the unique break ε). Defragmentation reverses the break: structured records (carrying 1×ε structure) dissolve back toward 1:1. The visible channel is the +1×ε; the dark sector is the symmetric remainder.

Axiom R (Record — monotone accumulation). Records persist (the information survives the forgetful functor) but their causal order dissolves. The Hubble time is the total actualisation budget of the substrate (AP20) — the clock against which defragmentation is timed.

Axiom C (Constraint — locality; finite propagation speed c). The σ-boundary (the event horizon) closes the electromagnetic channel for in-transit content, which is why dark matter responds to gravity but not to light.

0.4 — Epistemic status per section

1 (Three components). STRUCTURAL IDENTIFICATION. Visible = 1/21 (derived); dark energy = completed reservoir; dark matter = in-transit process.

2 (Mechanism). STRUCTURAL. Defragmentation (AP09) and the two populations; the cosmological-constant and dark-matter property lists follow from the 1:1 rest state and the residual-structure transit state.

3 (Derivation). DERIVED. The timescale ratio τ/t_H = 6/21, the rate equation, the constant-feeding result (the two dark fractions to within 1.3% of Planck, the visible to ≈2%).

4 (Epoch dependence). PREDICTION. The split as a function of epoch; the dissolution of the coincidence problem; the DESI DR2 confrontation and the effective equation of state w_eff(z) (4.3); the ΛCDM comparison (KS-42.2).

5 (Derivation of τ). DERIVED. Why 6/21 (six faces / twenty-one channels); the same counting logic as AP28; the meaning of 3.9 Gyr.

6 (Dark-matter particles). EMPIRICAL. The sharpest test (KS-42.1).

7 (What this does not do). SCOPE. The formation-history convolution (D48), the equation of state (KS-42.4), substructure, and the full closure of D44.

0.5 — Kill switch summary

KS-42.1 (Dark-matter particle detection). LIVE — EMPIRICAL. A detected particle at the right abundance falsifies the mechanism.

KS-42.2 (Epoch dependence). LIVE — EMPIRICAL. The predicted dark-sector composition at each epoch must match (the key observable is the growth rate σ₈(z)).

KS-42.3 (Defragmentation timescale derivation). LIVE — STRUCTURAL. The τ = (6/21) × t_H derivation rests on one face-erasure as the atomic unit of defragmentation work.

KS-42.4 (Equation of state). LIVE — EMPIRICAL. Dark energy’s intrinsic equation of state is w = −1 (the substrate at rest); the effective w_eff(z) inferred under a non-interacting analysis is predicted to be phantom (w_eff ≤ −1) in the past, relaxing toward −1 today (wₐ < 0). The model is under tension if w₀ settles firmly above −1, and killed outright only if w is non-evolving or evolves the wrong way (4.3).

KS-42.5 (Constant feeding rate). LIVE — STRUCTURAL. The section-3 derivation assumes a constant black-hole feeding rate.

KS-42.6 (Timescale ratio derivation). LIVE — EMPIRICAL. The dark-sector timescale ratio τ/t_H = 6/21 (six faces over twenty-one channels); if the ratio that fits the observed split departs from 6/21 beyond the formation-history correction, the ratio derivation fails.

0.6 — Structural debts owed

D44 (Dark-sector partition) — ADDRESSED. The mechanism is proposed, the timescale derived, the epoch dependence predicted, the two dark fractions fixed to within 1.3% of Planck, the visible to ≈2%. Fully closing D44 requires the formation-history convolution (D48) and observational confirmation of the epoch dependence (KS-42.2).

D47 (Defragmentation timescale) — CLOSED (conditional on KS-42.3). τ = (6/21) × t_H, derived from the ratio of faces (AP24) to channels (AP28). If face-erasure is not the correct atomic unit (KS-42.3), the derivation fails and D47 reopens.

D48 (Formation-history convolution) — OPENED. The constant-feeding model is a first approximation; the exact prediction requires convolution with the observed cosmic black-hole formation history (an astrophysical computation using AP21 plus empirical star-formation data). The 1.2% dark-matter error is attributed to this approximation. The same convolution sets the feeding-rate history that governs the late-time behaviour of the effective equation of state w_eff(z) (section 4.3): the constant-feeding model gives a phantom w_eff ≤ −1 asymptoting to −1, and only a declining feeding rate at late times would let w_eff cross to the w₀ > −1 side DESI’s central value prefers. Deriving that history is what would turn AP42’s phantom-past sign match into a quantitative w₀wₐ contour fit — or confirm the tension.

0.7 — Items held open without claiming debt status

The equation of state w = −1 (KS-42.4) follows from the rest-state reading but is not formally derived from the axioms. Dark-matter substructure (halos, density profiles) requires the detailed defragmentation dynamics, not computed here. These are scope boundaries, named for honesty.

0.8 — Structural relationships

AP41 (The Loop). Identifies the visible channel and reads the dark sector as unbroken symmetry; owes the internal split (D44). AP42 supplies it. The dark-sector partition is read consistently across AP29, AP41, and AP42 — one partition under different ordering and rounding (5/26/69, 1/21 + 20/21, 68/27/5).

AP09 (The Forgetful Functor). Defragmentation as U: Mon → Set — the underlying elements survive while the monoid composition dissolves. The mechanism of the dark sector.

AP24 (The Residual). The six faces of ε — the numerator of the timescale ratio. Defragmentation erases six independent readings.

AP28 (The Constant). The twenty-one independent channels — the denominator. The same counting logic that gives α_G = α_em²¹ (up to the geometric factor 1 + 1/π; AP28) gives τ/t_H = 6/21.

AP03 (The Ratio). The condensate and why the rest state exerts negative pressure and does not dilute. The substrate stiffness itself — λ ≈ 2.15 × 10⁴⁶, the unique constant with dimensions of energy per length — is established in AP15 (see AP21).

AP20 (The Proof). The Hubble time as the total actualisation budget of the substrate — the clock against which defragmentation is timed.

AP21 (The Web). Filamentary structure — how matter reaches black holes; the input to the formation-history convolution (D48).