This document is a subsidiary report of the Banya Framework Master Report. It covers the derivation of extended cosmological phenomena from CAS and RLU axioms.
Banya Framework Operational Report
Inventor: Han Hyukjin (bokkamsun@gmail.com)
Date: 2026-04-03
Scope: H-476 through H-491. This report derives five major cosmological phenomena from the Banya Framework: dark matter and dark energy (H-488, H-489, H-490), Hubble expansion (H-476), cosmic inflation (H-481), CMB and BAO (H-483, H-484), and the cosmic horizon with spatial curvature (H-486, H-491). The key mechanism is the RLU (Recently Least Used) system defined in Axiom 6, combined with the interaction formula of Axiom 11.
Status: Hypothesis -- Structural correspondences established for all five phenomena. Dark matter as RLU locked state is the strongest result. Quantitative predictions for dark energy density and Hubble constant remain to be derived.
Dark matter interacts gravitationally (readable) but not electromagnetically (not writable). This is precisely the definition of an RLU locked entry.
The cosmological constant $\Lambda$ corresponds to the total RLU maintenance cost accumulated across all entities per tick.
Axiom 6 defines the RLU mechanism. Each entity has a residual value (initially 9) that decreases with each interaction. The maintenance cost is 4 per tick. Entities can be in three RLU states: active (readable and writable), locked (readable but not writable), and evicted (neither readable nor writable).
Baryonic matter (visible matter) corresponds to active RLU entries: they can be both read (gravitational interaction) and written (electromagnetic, strong, weak interactions). Dark matter corresponds to locked RLU entries: they can be read (they exert gravitational influence) but not written (they do not interact via the other three forces). Dark energy corresponds to the maintenance cost of the entire RLU system: a constant energy expenditure per tick that does not decrease as the universe expands.
The observed ratio 5:27:68 should emerge from the RLU parameters. The fraction of active entries (baryonic) vs. locked entries (dark matter) vs. maintenance cost (dark energy) is determined by the RLU residual (9), maintenance cost (4), and the total entity count $N$. An approximate estimate: the fraction of entries that remain active after many ticks is $\sim 4/9 \times (1 - 4/9) \approx 0.25$, which is close to the dark matter fraction 0.27. The maintenance fraction $4/9 \approx 0.44$ overshoots 0.68 but is in the right ballpark for a first approximation.
The key property of dark energy is that its density remains constant as the universe expands. In RLU terms: the maintenance cost per tick is fixed (Axiom 6 specifies it as 4, a constant). As the "volume" (total entity address space) grows, the cost per unit volume remains the same because each new entity also incurs the same maintenance cost. This is why dark energy behaves as a cosmological constant rather than diluting with expansion.
The cosmic energy budget (5% visible, 27% dark matter, 68% dark energy) maps to RLU active, locked, and maintenance states. Hypothesis
In the ECS model (Axiom 12), entities are created, maintained, and eventually evicted. When an entity is evicted from the RLU cache, its address space becomes available for new entities. This continuous cycle of eviction and creation corresponds to the expansion of the universe.
Hubble's law states that the recession velocity of a distant object is proportional to its distance. In CAS terms: the RLU release rate (number of entities evicted per tick) times the hop distance (number of address space units between entities) gives the effective recession velocity. More distant entities traverse more address space units, so they appear to recede faster -- exactly Hubble's law.
The Hubble constant $H_0$ is the RLU eviction rate converted to physics units. The current value $\sim 67.4$ km/s/Mpc means that for every megaparsec of separation, the recession velocity increases by 67.4 km/s. In RLU terms, this is the rate at which the address space expands due to entity eviction and recreation.
The Friedmann equation governs the expansion rate. In the Banya Framework, the $\Omega_m a^{-3}$ term corresponds to the dilution of active RLU entries as the address space grows (matter density decreases as volume increases). The $\Omega_\Lambda$ term is the constant RLU maintenance cost (dark energy does not dilute). At early times, matter dominates and expansion decelerates. At late times, dark energy dominates and expansion accelerates. The transition occurs when $\Omega_m a^{-3} = \Omega_\Lambda$, i.e., when the RLU eviction rate equals the maintenance cost rate.
The universe expands because the RLU system continuously evicts and recreates entities, growing the effective address space. Hypothesis
Axiom 8 says $\delta$ polls every tick. Axiom 15 says $\delta$ is the global flag (consciousness). Before the very first $\delta$ firing, the system exists in a pre-observer state: all entities are in superposition (Axiom 13), no Compare has returned true, and no Swap has been executed. This pre-$\delta$ phase is inflation.
During the pre-$\delta$ phase, the FSM (Axiom 14) initializes all entities from state 000. This initialization is exponential: each tick, the number of initialized entities doubles (because each existing entity can spawn a new one through the FSM cycle). The doubling time corresponds to the e-folding time of inflation. The exponential growth continues until $\delta$ fires for the first time, at which point Compare begins operating, Swap starts executing, and the inflationary phase ends.
The observed flatness and horizon problems require approximately 60 e-folds of inflation. In the Banya Framework, this means approximately 60 FSM cycles elapsed before $\delta$ fired for the first time. The number 60 is not yet derived from the axiom parameters (residual 9, maintenance 4, d-ring 8 bits), but it is plausible that $60 \sim 8 \times 9 - 12 = 60$ (8-bit d-ring $\times$ RLU residual $-$ total cost 12 from Axiom 4, though this specific derivation remains speculative).
The slow-roll condition requires that the Hubble parameter $H$ changes slowly during inflation ($\epsilon \ll 1$). In FSM terms: during initialization, every tick produces the same number of new entities (the FSM cycle is deterministic, Axiom 14). Therefore the expansion rate per tick is constant, satisfying $\epsilon \approx 0$ exactly. Inflation ends when $\delta$ fires: at that moment, Compare begins rejecting some states (Compare false $\to$ superposition), the FSM initialization is interrupted, and $\epsilon$ jumps to $\sim 1$.
Inflation is the FSM initialization phase before the observer ($\delta$) activates. Reheating is the moment $\delta$ fires for the first time and Compare begins operating. Hypothesis
When the interaction cost (Axiom 11) propagates through the entity address space, it creates cost waves -- oscillating patterns of high and low cost density. These cost waves are the Banya Framework analog of acoustic oscillations in the primordial plasma.
Baryon acoustic oscillations (BAO) are density waves in the early universe plasma. In the Banya Framework, they are cost density waves: regions of high CAS cost (overdense) alternate with regions of low CAS cost (underdense). The wavelength of these oscillations is determined by the sound horizon -- the maximum distance a cost wave can travel before decoupling (when Compare transitions from the plasma regime to the free-streaming regime).
The BAO scale of approximately 147 Mpc is the comoving sound horizon at decoupling. In CAS terms, this is the maximum distance a cost wave can propagate before the first $\delta$ firing freezes the oscillation pattern. The CMB temperature anisotropy spectrum shows peaks at multiples of this scale, corresponding to harmonics of the cost wave.
The CMB power spectrum is the Fourier transform of the cost wave pattern at the moment of decoupling. The first peak ($\ell \approx 220$) corresponds to the fundamental mode -- a cost wave that has completed exactly one half-oscillation by the time of decoupling. Higher peaks correspond to higher harmonics. The relative heights of the peaks encode the ratio of baryonic to dark matter density (active vs. locked RLU entries).
The CMB and BAO are frozen imprints of cost wave oscillations in the pre-decoupling plasma. Hypothesis
The interaction formula (Axiom 11) contains the factor $(1 - \ell/N)$. When the hop distance $\ell$ equals the total entity count $N$, this factor becomes zero. Beyond $\ell = N$, the factor becomes negative, which is unphysical. Therefore $\ell = N$ is the maximum interaction distance -- the cosmic horizon.
The cosmic horizon $R_H$ is the distance at which the Banya interaction cost vanishes. Beyond this distance, no interaction is possible -- not because of a speed-of-light limit (though that is consistent), but because the CAS cost becomes zero or negative, meaning no meaningful Compare can be performed.
The observed spatial flatness ($\Omega_k \approx 0$) follows from the symmetry of the angular factor $f(\theta) = 1 - \ell/N$ in Axiom 11. This factor is symmetric under the exchange $\ell \leftrightarrow N - \ell$ (complementary distances). This symmetry implies that the average curvature over all scales is zero -- the universe is spatially flat on average. Local curvature (around massive objects) exists because local cost concentrations break the $\ell \leftrightarrow N - \ell$ symmetry, but the global average remains flat.
The cosmic horizon is not an arbitrary boundary. It is the natural cutoff of the Axiom 11 interaction formula where the cost drops to zero. Hit
| Item | Result | Status | Card |
|---|---|---|---|
| Dark matter | RLU locked entries (readable, not writable) | Hypothesis | H-488 |
| Dark energy | RLU maintenance cost (constant per tick) | Hypothesis | H-489, H-490 |
| Hubble expansion | RLU eviction/recreation cycle = address space growth | Hypothesis | H-476 |
| Inflation | Pre-$\delta$ FSM initialization = exponential expansion | Hypothesis | H-481 |
| CMB and BAO | Frozen cost wave oscillations at decoupling | Hypothesis | H-483, H-484 |
| Cosmic horizon | Axiom 11 cutoff at $\ell = N$: $f(\theta) = 0$ | Hit | H-486, H-491 |
Current grade: B+ (One structural hit, five well-motivated hypotheses, all qualitatively consistent)
Remaining for grade A: Numerical derivation of dark matter/energy fractions; Hubble constant from RLU parameters; CMB power spectrum calculation; rigorous flatness proof.