Causal Budget Framework

Quick Reference Tables

Key equations and parameters in the Causal Budget Framework, organized for rapid lookup.

CBF-native Bridge CA substrate

Foundation Parameters & Budget Law

Core Budget Rule Foundation

Symbol Meaning Equation / Role
C Total causal budget per tick C = T + M. Causal update law per tick, defines the discrete space–time trade-off.
C² = T² + M² Causal geometry identity Lorentz-like invariant of the discrete causal metric.
T Translation share (motion) T is the compute fraction advancing the wavefront; relates to momentum via the bridge p = ℏk.
M Maintenance share (upkeep) M = 1/γ(v) = √(1 − v²/c²). Compute fraction sustaining internal state. Rest-cost link: ℏω₀ = m c².
Spin persistence consumes M each tick; see Spin–Oscillation Coupling.
ω₀ Baseline maintenance frequency ℏω₀ ≡ m c². In CBF, ω₀ is the per-tick maintenance cost that defines rest mass. Goal: derive ω₀ from CA binding dynamics / beat-matching stability.
f₀ Gate bandwidth Global temporal gate cadence used in beat-matching and commit timing. Controls how often phase alignments are checked per baseline tick. Appears in commit pacing: τ_commit = τ₀ + 1 / [ f₀ ( |Δf_M| + M_ch (1 − cos θ) ) ]. See Beat-Matching.
M_ch Channel maintenance share Effective maintenance fraction for an interaction channel between two participants. For collinear motion: M_ch = 1 / γ(w), where w = (u₁ − u₂) / (1 − u₁u₂ / c²) is the relativistic relative speed. Represents how much of the shared causal budget is reserved for synchronization rather than motion. Used in beat-matching to modulate the reconciliation delay term M_ch (1 − cos θ).
α(x) Lapse / clock-pace Local pacing fraction of the global causal rate. Pipeline: S(x) → α(x) → g(x).
τ₀ Baseline tick Global update period, light crosses one lattice cell.
a Lattice spacing Base spatial stride (one γ-length).
τ_commit Commit delay Wait before Event Ledger accepts a joint update. Beat-matching and confidence-gating are equivalent semantics.
τ_queue, ρ_q Queue buffering Backlog-induced delay and density; produces SR symmetry via buffered commits.
ρ_q(x) Queue load (dimensionless) Local commit demand / capacity. Higher ρ_q → lower α(x) (slower pacing). Weak-field link: ρ_q ∝ GM/(2 r c²) ⇒ α ≈ (1 − ρ_q)/(1 + ρ_q).
c = a / τ₀ Speed of light Universal hop speed set by lattice stride and tick (often normalized to 1).
γ(v), β = v/c Lorentz factor & normalized speed γ = 1/√(1 − β²). Exchange rate between T and M; β used in SR, Doppler, pacing.
ψ Wave amplitude Complex field for interference and spatial reconciliation (used for Born-like maps).
Δω, 2π / |Δω| Beat mismatch & alignment period Desynchronization measure and its reconciliation timescale.

Two‑Layer Time Symmetry Foundation

Layer What It Uses What It Means
CA (compute) level C = T + M The per‑tick update law. Forward‑only, causal. Produces motion and upkeep. No Lorentz required.
Ledger (reconcile) level τ_commit, τ_queue, ρ_q Queue buffering + phase alignment enforce symmetric commit timing across frames.
Observer (geometric) level C² = T̃² + M̃², γ = 1/M Measured symmetry of reconciled histories. Lorentz factor describes outcomes, it does not drive the CA.

Parametrizations Foundation

Type Symbols Formulation
Compute shares (preferred) ε_T, ε_M ε_T + ε_M = 1, with ε_M = 1/γ and ε_T = (γ − 1)/γ. These are the per-tick budget fractions used by CBF for motion and upkeep.
Geometric components T̃, M̃, C C² = T̃² + M̃², so T̃ = √(C² − M̃²). This is the invariant causal geometry. Do not apply √(1−M²) to the compute shares.

Cellular Automaton (CA) Substrate Foundation

Concept Meaning Notes
CA (discrete substrate) Finite-state update grid Underlying compute fabric that runs the per-tick rule. Readers: “CA” = Cellular Automaton (discrete cells, local rules).
Huygens mapping Wavelet emission picture Each active cell behaves like a tiny emitter; secondary wavelets reconstruct the front. This historical picture matches the CA-style local propagation used here.
Diffraction in CA Edge-induced angular spread Discrete Huygens updates reproduce diffraction patterns naturally (slits, edges). No continuum PDEs are required to get realistic spreads.
Vector CA Non‑pixelized motion Cells carry percentage displacement vectors rather than snap-to-pixel moves. “Healing” uses neighboring wave cells to preserve smooth fronts while conserving momentum.
Wave Cells Front agents with momentum Constituents of the wavefront. They propagate momentum, can “heal” broken fronts, and diffract by spawning children near edges/obstacles consistent with local rules.
Child creation Local branching At discontinuities, parent wave cells create child cells with adjusted directions, forming the diffracted front while conserving budget.
Momentum bookkeeping Directional consistency Wave-cell vectors add to preserve net momentum in free space; obstacles re-weight via the Material Coupling Field and Ledger rules.

Cellular Automaton (CA) Tick Definitions Foundation

Symbol Meaning CBF Interpretation
a Lattice spacing The baseline causal stride — one spatial cell of the universal lattice. Chosen as the γ-length, roughly 100× smaller than an atom and the smallest photon wavelength. It defines the visible spatial resolution of the substrate.
τ₀ Global baseline tick The time for light to traverse one lattice spacing:
c = a / τ₀
This sets the universal causal speed limit. Each tick advances all wavefronts by one γ-length stride.
τ_eff(x) Local effective tick Local pacing relative to the global tick: τ_eff(x) = τ₀ / α(x). Lower α(x) means a slower local commit rate and deeper event queues, producing gravitational time dilation.
ℓ_P Planck precision scale The Planck length acts as decimal precision, not a physical pixel. It is much smaller than a, ensuring light waves glide smoothly between lattice sites rather than hopping discretely.
Scale hierarchy: ℓ_P ≪ a(γ-length) ≪ atomic scale. Photons traverse one γ-length per tick τ₀, defining c = a / τ₀. For matter, only the ratios T:M and a:τ₀ matter, so absolute tick size is irrelevant as long as pacing is consistent.

Causal Geometry Summary Foundation

Relation CBF Interpretation
C = T + M The cellular rule: each tick divides its causal budget between translation (T) and maintenance (M). This runs the simulation step-by-step without invoking relativity directly.
C² = T² + M² The geometric identity: expresses the same update law as a Lorentz-like invariant. When averaged over many ticks, this reproduces the Minkowski metric used in SR. It shows that each local tick traces a causal interval of unit length.
c = a / τ₀ Defines the causal stride of the lattice. Light advances one cell a every tick τ₀, forming the discrete analog of a null worldline (M = 0).
Δs² = c²Δt² − Δx² Continuous version of C² = T² + M². The discrete hop (a, τ₀) converges to this relation in the large-scale limit, guaranteeing Lorentz symmetry without needing to impose it externally.
v = (Δx / Δt) = c · (T / C) Effective velocity derived from local budget allocation. Photons use all translation budget (T = C → v = c); matter divides it (M > 0 → v < c).
Takeaway: The causal lattice already encodes special relativity. The per-tick rule C = T + M is the discrete engine, while C² = T² + M² is its smooth geometric projection. Light’s speed c = a / τ₀ fixes the conversion between spatial and temporal grid steps, giving rise to invariant causal intervals without extra postulates.

Event Ledger Foundation

Aspect Definition / Role
Event Ledger
(Global commit/
reconciliation engine)		 A global engine or graph that commits superposition proposals into finalized history of events. Uses four gates—temporal (when), spatial (where), directional (coupling), and informational (if)—to decide eligibility. Supports two pacing semantics (Beat-Matching / Confidence-Gating) and two buffers (Collapse and Queue). Accepted commits update S(x) (working memory → α, g); pruned paths archive to D(x). Performs reverse pathfinding when informational reconciliation is required (quantum-erasure class phenomena): tracing measurement nodes backward through the causal graph to identify and prune inconsistent ancestors.
Gates Temporal (when), Spatial (where), Directional (coupling), Informational (if). Eligibility requires satisfying all four.
Pacing Beat-Matching (streaming micro-commits) and Confidence-Gating (reciprocal, batched view). Commit delay: τ_commit = τ₀ + 1 / [ f₀ ( |Δf_M| + M_ch (1 − cos θ) ) ].
Buffers Collapse Buffering = sync time before a commit can finalize. Queue Buffering = serialized waiting when new proposals arrive mid-reconciliation.
Batching Multi-site commits finalize as SCGs (Simultaneous Commit Groups): one global write read by all frames (see SCG).
Outputs Accepted commits update S(x) (working memory → sets α(x) pacing and contributes to g(x) steering). Rejected/pruned paths write inert scars to D(x).
Causality No action at a distance: one shared-ledger commit satisfies all constraints at once; respects c = a/τ₀.

Event Ledger Parameters Foundation

Symbol Meaning Notes
τ₀ Fundamental tick duration CBF uses τ₀ for the baseline tick. Calibration example (not fixed): τ₀ ≈ 10⁻²³ s.
a Spatial resolution (lattice spacing) Calibration example (not fixed): a ≈ 10⁻¹² m (picometer scale).
c = a/τ₀ Speed of causality With the example values above, c ≈ 3×10⁸ m/s.
δ Coincidence window width Temporal gate opens for duration δ each tick to count phase alignments (beat-matching semantics).
η Stability weight 0 ≤ η ≤ 1. Used in S-ledger for gravity sourcing; η < 1 reduces contribution from unstable matter.
TTL Time-to-live (attenuation / entropy budget) Represents the remaining amplitude or coherence budget of a propagating wave train. Not a literal counter, but a continuous measure of how much amplitude survives before collapse becomes inevitable. TTL drains through the local Material Coupling Field via loss, absorption, and scattering: dA/dt ∝ −A·h(x,ω), where h(x,ω) increases with frequency. Higher oscillation rates (ω) deplete TTL faster, since each cycle dissipates more energy into the medium. When amplitude falls below the Ledger’s commit threshold, the wave path ends—its residual entropy recorded as a pruning weight in the D-ledger.
Precision note: In CBF the Planck length (ℓP ≈ 1.6×10⁻³⁵ m) is a hidden precision layer (like a numerical epsilon), not the visible grid spacing a. This allows smooth waves on a discrete substrate.

Atoms & Particles Foundation

Entity Role CBF Interpretation
Particles Finite T+M wavefront packets Particles are discrete causal carriers, each holding a finite budget of translation (T) and maintenance (M). They deliver energy and information into atomic systems. In flight, they follow the CA rule C = T + M with no internal clock (for photons M = 0).
Atoms Persistent state machines Atoms are bounded configurations of collapsed wavefronts that convert probability into persistent history. Each atom manages a local maintenance loop (ω₀) and exchanges T+M with surrounding particle fields. Their internal coherence is tracked by a Decay TTL (entropy horizon).
Decay TTL Atomic maintenance entropy Measures how long an atomic configuration can maintain Ledger coherence before its maintenance budget is exhausted. Failed reconciliations increase entropy, lowering TTL. When TTL → 0, the atom decays, releasing stored T+M as new wavefronts and updating the D-ledger with a strong pruning scar.
Big Bang / first collapse Origin of atomic inventory Building blocks for Atoms and particles were seeded in the first large-scale collapses. The atomic population is finite and recycled through decay and recombination—there is no continuous creation field.
Causal exchange Atom–particle energy flow Particles send T+M into atoms (inputs); atoms emit particle wavefronts as outputs. This closed exchange makes atoms the processing nodes of reality, turning probabilistic wave data into observable events.
Interpretation: In CBF, atoms are limited historical resources—persistent commit nodes that stabilize probability. Their decay returns stored causal energy (T+M) to the field, maintaining the global conservation of causal budget.

Quantum Mechanics

Wave Mechanics QM

Quantity Equation CBF Interpretation
Phase θ = k·r − ωt Instantaneous phase of a wave cell at position r and tick t. Each cell carries its own θ used for Ledger phase alignment.
Dispersion relation ω² = c²k² + ω₀² Derived directly from the CBF causal metric C² = T² + M². Links wave propagation to maintenance frequency.
De Broglie wavelength λ = h/p = 2π/k Spatial period of the translation component. In CBF this arises from oscillation cadence in the Event Ledger.
Born rule (commit frequency) P(r) ∝ |ψ(r)|² Statistical outcome of phase-coincidence commits: P ∝ |Σᵢ e^{iθᵢ}|². Bright fringes = high commit density.
Energy quantization E = ℏω Planck bridge: oscillation rate ↔ energy. In CBF, ℏω₀ defines maintenance cost at rest; ℏω tracks active oscillation.
Momentum quantization p = ℏk Spatial propagation rate ↔ momentum; implemented via wave-cell direction vectors.

Interference QM

Phenomenon Description CBF Interpretation
Single slit (width w) Minima at sin θ = nλ/w Diffraction arises naturally: edges spawn new momentum directions (k-fan). Temporal gate beat-matches their phases at detection.
Double slit (separation d) Maxima at d sin θ = nλ Two coherent k-bundles interfere. The Event Ledger compares arrival phases and commits where alignment peaks.
N slits (grating) Sharp peaks at d sin θ = nλ N phase sources sum coherently; the Ledger’s temporal gate reconciles all to a common commit rhythm.

Note: In CBF, interference is a phase-coincidence statistic, not physical overlap. The Ledger’s temporal gate counts coincidences, producing the Born-rule pattern automatically. (Basically, it takes time for interference to collect at the detector.)

Entanglement & Shared Ledger QM

Concept CBF Interpretation
Entangled pair Two wave systems share one ledger entry. Each carries complementary state constraints but no independent collapse path. When either side meets all gate conditions, the joint commit resolves for both.
Simultaneous Commit Group (SCG) A set of correlated events that finalize in the same reconciliation window of the Event Ledger. Members may be widely separated in space but share the same causal solution. Each SCG is a commit batch—all entries satisfy global conservation together. Observers in different frames may timestamp the events differently, yet all reference the same SCG identifier in the shared history. This resolves Bell-type correlations without superluminal signaling.
Measurement A local detector satisfies the informational gate on one side, forcing the Ledger to choose a globally consistent outcome. The distant partner's state is updated as part of the same commit, not by signal transfer.
Erasure / delayed choice Removing which-path information reopens the informational gate, allowing the same joint entry to reconcile differently. The Ledger never “rewinds” — it simply finishes an unresolved commit.
Link to SR See Shared-Ledger Processing for timing and ordering rules. Correlations arise from one synchronized commit across both sites, enforcing conservation through the common Event Ledger.
Interpretation: Entanglement in CBF is not action at a distance, but one shared commit written consistently for all participants. Information coherence, not energy transfer, maintains the connection.

Quantum Agents & Collapse QM

Quantity / Rule Equation CBF Interpretation
Wave-cell coherence L_coh, T_coh Wave cells carry phase θ and remain mutually coherent over a finite span (space/time). Coherence bounds how many contributors sum in the Ledger’s temporal gate.
Hazard rate h(x, ω) Local collapse hazard from material environment (absorption, scattering, loss). Governs attempt rate for commits along the path.
Material Coupling Field MCF(x, ω) Summarizes index shift, loss, absorption, scattering. Sets propagation cost and feeds h(x, ω). Not a gravitation or pacing field.
Survival probability P_surv = exp(−∫ h dt) Probability a wave-cell train reaches the detector uncollapsed. Multiplies the interference envelope before the Ledger counts coincidences.
Commit frequency (Born) P(r) ∝ |Σ_i e^{iθ_i}|² Ledger counts phase coincidences within the gate; bright fringes are high commit-density regions. No in-flight state collapse required.
Emission / re-emission channel: M_ch, gate: τ_commit Emitter creates coherent wave cells; in media, absorption can spawn re-emitted trains with new phases set by the local gate and channel maintenance share M_ch.
M_ch = 1 / γ(w) Channel maintenance share Effective maintenance fraction for an interaction channel between moving participants. Relative speed w = (u₁ − u₂)/(1 − u₁u₂/c²). Used in beat-matching to weight the reconciliation delay term M_ch(1 − cos θ). Defined formally in Core Budget Rule.
Π(β, θ) = γ(1 − β cos θ) Directional / Doppler factor (at detection) Couples spatial and temporal reconciliation at the observer. Converts emission frequency and direction (β = v/c) into the observed beat rate.
Photon vs matter photons: ω0 = 0, M = 0 Photons follow null paths (pure T-flow). Matter has ω0 > 0 and consumes M for upkeep; their internal frequency is ω = ω0·M.
CBF stance vs. standard QM: Interference is a phase-coincidence statistic governed by the Event Ledger’s temporal gate. Photons do not change in flight; redshift and frequency changes come from endpoint pacing fields and detection semantics, not mid-path “stretching.”

Special Relativity

Energy–Momentum Relations SR

Quantity Equation (SI) CBF / normalized (c = 1)
Total energy E² = (pc)² + (mc²)² E² = p² + ω₀²
Rest energy E₀ = mc² = ℏω₀ E₀ = ω₀
Relativistic energy E = γmc² E = γω₀
Budget interpretation In CBF the Ledger’s queue buffering enforces the relation above. The Lorentz factor γ describes how much of each tick’s causal budget appears as translation vs. maintenance once reconciled: M = 1/γ, T = (γ − 1)/γ.

Collapse & Queue Process SR

Stage CBF Interpretation
Superposition A particle’s wave cells interact with many potential absorbers (atoms) simultaneously. Each atom holds a provisional entry in the Event Ledger, representing a candidate outcome. These candidates coexist until one path satisfies all gate conditions.
Commit The winner of the superposition — the atom whose timing, momentum, and direction align within all three gates — is chosen for the next Ledger entry. This commit converts probability into a stable event in the shared history graph.
Collapse Buffering The synchronization period required for two entities (e.g., emitter and absorber) to align their frames before a commit can finalize. Duration set by τ_commit, determined by beat-matching and directional channel terms.
Queue Buffering If a new collapse proposal arrives while another commit is still reconciling, it enters a queue until the current synchronization finishes. This ensures causal order: commits are serialized even when candidate events overlap in time.
Queue load view: For steady traffic, the average wait scales with load ρ_q: τ_queue ≈ (ρ_q / (1 − ρ_q)) × ⟨ beat-matching ⟩. This links microscopic gate pacing to macroscopic time dilation through cumulative buffering.
Per-interaction gate times accumulate in local ticks: Δt = τ_eff(x) = τ₀ / α(x). See also: α(x): The Universal Pacing Field.
Interpretation: The sequence Superposition → Commit → Collapse Buffering → Queue Buffering is the event life cycle in CBF. Superposition defines possibilities, commit defines outcome, collapse buffering defines synchronization, and queue buffering preserves causality across overlapping events.

Commit Gates SR

Gate Type Condition CBF Function
Temporal gate (WHEN) Δθ_M = |θ_src − θ_abs| < δ Opens when maintenance-phase difference is within tolerance δ. The effective reconciliation rate depends on both maintenance mismatch and directional/channel misalignment:
r_gate = f₀ * (|Δf_M| + M_ch (1 − cos θ))
where f₀ is the gate bandwidth (baseline cadence).
Spatial gate (WHERE) Δp = |p_in − p_out| < ε_p Enforces local momentum and energy conservation. Determines where the commit lands in space; yields Born-rule weighting P(r) ∝ |ψ(r)|².
Directional gate (COUPLING) Penalty ∝ M_ch(1 − cos θ) Couples spatial and temporal reconciliation. Adds delay or attenuation for angular misalignment. Produces Doppler-like and aberration-like effects at detection.

Fourth Informational Commit Gate (IF) SR

Gate Condition CBF Function
Informational Gate (IF) Determines whether a path remains eligible for commit once global knowledge, coherence, or veto conditions are applied. The logical layer above the physical gates. It does not alter energy or momentum, but decides if a channel can still participate in reconciliation based on informational consistency.
Knowledge veto Triggered when a measurement, obstacle, or known hazard removes a path before the temporal gate aligns. The Ledger marks this path as vetoed and skips its commit. The pruned entry writes a low-amplitude scar to the D-ledger, biasing future reconciliations without transferring energy.
Quantum Bomb example One arm blocked by a live detector or bomb The informational gate vetoes that arm pre-commit. The remaining coherent path commits at the “dark” detector, revealing information through absence of interaction.
Quantum Erasure Erasure removes which-path information The informational gate re-opens the previously vetoed path by restoring coherence. Interference returns because the global knowledge constraint no longer blocks reconciliation.
Coherence flag Boolean state per candidate path Each provisional commit carries a coherence flag. The informational gate toggles it false (veto) or true (eligible) as global information updates, ensuring self-consistent history resolution.
Interpretation: The Informational Gate governs eligibility of paths based on knowledge coherence. Quantum-bomb and erasure experiments demonstrate this logic gate in action— pruning or restoring channels without energy exchange. It completes the four-gate stack: WHEN (temporal), WHERE (spatial), COUPLING (directional), and IF (informational).

Shared-Ledger Processing SR

Concept CBF Interpretation
Joint commit Spatially separated candidates (A, B, …) are finalized as a single history entry in the shared Event Ledger. The commit is written once and read by all frames; there is no message sent between sites.
How it finalizes The same gates decide eligibility: temporal (phase), spatial (momentum), directional (channel), and informational (knowledge). Ledger waits until all constraints are simultaneously satisfiable, then writes one coherent commit.
Entanglement correlations Correlations arise because the Ledger solves one global consistency problem for the pair, not because signals traverse space. The informational gate enforces which-path and erasure logic.
No-signaling Local outcomes’ statistics (marginals) are unchanged by remote settings until classical comparison. The Ledger’s commit preserves causal order and respects the speed limit c = a/τ₀.
Ordering Space-like separated writes are recorded with a consistent partial order; each frame can index them differently, but the finalized event is the same entry for all (see Two-Frame Buffer Symmetry).
Interpretation: What looks like “action at a distance” is a single shared-ledger commit that satisfies all gate constraints at once. No superluminal messages, no stretched space—just one globally consistent write.

Beat-Matching vs. Confidence-Gating

Mode Mechanism Interpretation
Beat-Matching Streaming micro-commits accumulate whenever phase alignments occur. τ_commit = τ₀ + 1 / [ f₀ ( |Δf_M| + M_ch (1 − cos θ) ) ] Continuous, low-confidence mode. Produces apparent smooth time evolution and interference as averaged commit density.
Confidence-Gating Ledger waits for full reconciliation before writing a single high-confidence commit. Its effective pacing is the reciprocal of the beat-matching rate. Batched, high-confidence mode. Yields the same Lorentz-symmetric outcomes as beat-matching when integrated over shared ticks.
Time dilation driver: Increasing the mismatch term |Δf_M| or the directional term M_ch (1 − cos θ) enlarges the gating delay τ_commit. When many interactions are in flight, queue buffering accumulates these waits, making the observed clock run slower relative to a lightly loaded frame. See also: Two-Frame Buffer Symmetry.
Interpretation: The four gates (temporal, spatial, directional, informational) define how the Event Ledger filters potential commits. Beat-matching and confidence-gating are two timing semantics for the temporal gate—streaming vs. batched reconciliation. Both enforce symmetry and preserve energy-momentum consistency across frames.

Time Dilation & Length Contraction SR

Effect Formula CBF Interpretation
Time dilation Δt = γ Δτ Emerges from gate delays (beat-matching) plus directional coupling and accumulated queue buffering. The CA rule C = T + M never applies Lorentz at runtime.
Operational length (no physical shrink) L_oper = c × (Δt_roundtrip / 2) Distances don’t change; schedules do. Apparent L ≈ L₀/γ is an observer reconstruction from asymmetric commit timing. See Two-Frame Buffer Symmetry.
Proper time dτ = √(1 − v²/c²) dt dτ = M dt
Where to find the mechanics — Gate mechanics and pacing: Commit Gates. Reciprocal measurements across frames: Two-Frame Buffer Symmetry.

Photon–Matter Unification SR

Species ω₀ Dispersion Budget Split (T : M) Speed
Photon 0 ω = ck T = 1 , M = 0 = c (null path)
Massive particle > 0 ω² = c²k² + ω₀² T < 1 , M > 0 < c
CBF interpretation Photon and matter obey the same update rule. Mass arises when a fraction of the causal budget is locked into maintenance (ω₀ > 0). The speed limit c comes directly from the lattice ratio a/τ₀, not from imposed symmetry.
CBF stance: Lorentz symmetry is an observed consequence of queue buffering and maintenance trade-offs, not a built-in transformation law. The Event Ledger’s reconciliation produces the metric identity C² = T² + M² automatically.

Two-Frame Buffer Symmetry SR

Scenario Ledger Mechanics Observed Outcome
A and B in uniform relative motion Each frame has its own maintenance rate (M_A, M_B) and tick pacing. The Event Ledger inserts symmetric queue delays to align cross-frame commits: τ_commit^A = τ₀ + 1/|Δf_M(A,B)|, τ_commit^B = τ₀ + 1/|Δf_M(B,A)|. Each side measures the other as time-dilated. Reciprocity arises from equal-and-opposite buffering; no preferred frame is invoked.
Shared detection event A and B propose a joint commit; the Ledger holds both in the queue until phase coincidence within the temporal gate δ. The final commit is a single history entry written to both ledgers. A and B agree on the event itself (same commit), while disagreeing on the elapsed local ticks leading to it (differential buffering → apparent γ).
Beat mismatch drives delay Maintenance frequency mismatch sets the buffering scale: Δf_M = |f_{M,A} − f_{M,B}|, with f_{M}=ω/2π= (ω₀ M)/2π for matter. Smaller Δf_M → longer wait; larger mismatch → quicker disambiguation. Macroscopic “time dilation” is the statistical shadow of these waits; γ is a diagnostic, not a runtime input.
Directional coupling The gate includes directional misalignment: + M_ch(1 − cos θ) (channel maintenance share). Collinear motion minimizes extra delay; transverse increases it. Doppler/aberration appear as changes in reconciliation cadence at detection (not in-flight edits to photons).
No Lorentz at runtime The CA runs the budget rule C = T + M, writes proposals; the Ledger enforces symmetry by buffering. The geometric identity C² = T̃² + M̃² is reconstructed from the finalized commits. Observers summarize with γ = 1/M (normalized), but γ is inferred after reconciliation, not applied during updates.
Interpretation: Mutual dilation is a property of the commit queues on both sides. There is no global simultaneity plane to carry; the Event Ledger’s single shared commit guarantees agreement on events while preserving reciprocal timing asymmetries.

General Relativity

α(x): The Universal Pacing Field GR

Aspect Description / Role
Definition Local pacing field controlling both proper-time and signal propagation rate. dτ = α dt, v_effective = α × c.
Gravity Queue buffering from interaction density lowers α near massive regions, producing gravitational time dilation and steering via −∇α.
Material coupling α also includes optical pacing: α_total = α_gravity × 1/n(ω). Refraction and gravitational slowdown are one mechanism.
Steering field Directional bias from gradients: g(x) = −∇α(x). Bends trajectories toward slower-paced regions.
Observable effects Redshift (f_obs/f_emit = α_obs/α_emit), Shapiro delay, light bending, and optical slowdown all stem from α-variations.
Interpretation α is the local tick-rate multiplier of the universe—how fast commits clear. Lower α → deeper queue → slower clocks and signal pacing. Space itself never stretches; α(x) simply changes the rhythm of causality.
Interpretation: α(x) unifies gravity, refraction, and time dilation as one pacing law. Variations in α encode the universe’s causal tempo rather than geometric curvature.

Budget Law with Pacing & Steering GR

Statement CBF Visualization
Local budget The per-tick causal budget at x is the pacing field itself:
C(x) = T(x) + M(x) = α(x)
Lower α(x) means fewer “compute quanta” per global tick available to split between motion and upkeep.
Pacing inside T and M α(x) scales both translation and maintenance shares locally:
T(x) = α(x) · ε_T, M(x) = α(x) · ε_M, with ε_T + ε_M = 1
Photons: M = 0 ⇒ T = α(x). Matter: internal rate follows ω = ω₀ · M(x).
Effective speed & proper time Causal stride and clock-pace are both set by α:
v_effective = α(x) · c (photons/null flow),   dτ = α(x) · dt
In media: α_total(x, ω) = α_gravity(x) × 1/n(ω) (gravity and refraction are one pacing mechanism).
Steering from ∇α Unequal pacing across a wavefront bends motion toward slower regions:
g(x) = −∇α(x)
Per local tick the wave-cell velocity is updated by the pull-arrow:
v_next = v_now + g(x) · Δt with Δt = τ_eff(x) = τ₀ / α(x)
Photons keep null speed by renormalizing: |v_next| → c.
Oscillation depends on α(x) Maintenance oscillation slows where pacing is slow:
M(x) = α(x) · ε_M, so ω(x) = ω₀ · M(x) = ω₀ · α(x) · ε_M
Proper time relation dτ = α dt and endpoint ratio f_obs / f_emit = α_obs / α_emit express gravitational redshift without editing photons in flight.
Queue link (dilation) Lower α indicates higher local interaction density (deeper queue), which increases reconciliation wait:
τ_commit = τ₀ + 1 / [ f₀ ( |Δf_M| + M_ch (1 − cos θ) ) ]
Many in-flight interactions add queue buffering (see beat-matching & gates) → observed time dilation.
Takeaway: In CBF the core rule becomes C(x) = T(x) + M(x) = α(x). The value of α sets how much budget exists locally; its gradient (−∇α) steers motion. Gravity and refraction are both α-pacing phenomena; no rods shrink and no space stretches—only commit rhythm and direction change.

Gravity Fields & Ledgers GR

g(x) Pull-arrow Directional steering per tick: bends the translation part T of wave/particle motion. Operationally: Δv ∝ g(x) Δt in the small deflection limit.
S(x) Shared stability ledger (gravitational working memory) Stores only the subset of history required for causal continuity—the active scaffold, not a full archive. Updated by all successful ledger commits (atomic or event-based); inconsistent or redundant paths are pruned by the Informational Gate and leave inert scars in D(x).

Sourcing: Sourced primarily by stable atomic commits with stability weight η (long-lived states contribute more).
Pipeline: S(x) → α(x) (clock pace / queue depth) and contributes to g(x) (steering via −∇α). Dense S(x) regions slow local ticks (time dilation) and bend trajectories toward lower α.
D(x) Pruning / dark-matter ledger Archive of rejected or failed branches (Informational Gate pruning). Inert “scars” bias probability flow but do not contribute clock pacing like S(x).
W(x) Spatial weighting Replaces “stretched space” language: W biases spatial reconciliation density. Use instead of a literal conformal stretch. (Avoid “ψ stretches distances”.)

Bridge Equations (CBF → Schwarzschild) GR

Relation CBF Interpretation
ρ_q(r) ∝ GM / (2rc²) Queue density produced by ledger commits from a mass M. Higher ρ_q → deeper queue → lower pacing α.
α(r) = (1 − GM/2rc²)/(1 + GM/2rc²) Lapse derived from queue buffering: α ≈ (1 − ρ_q)/(1 + ρ_q). Time dilation emerges from slower local commit cadence.
ψ(r) = 1 + GM / (2rc²) Conformal factor from stability field S(x) (spatial reconciliation layer). Expands distances as maintenance overhead increases.
g(x) = −∇α(x) Steering field from pacing gradient. Causes “falling” motion and light bending without curvature tensors.
These bridge equations show how the queue-buffering and stability fields (S, ρ_q) reproduce the Schwarzschild metric in isotropic form. The classical tests (perihelion precession, Shapiro delay, light bending) follow automatically once α and ψ are substituted into the metric.

Classical Tests (CBF reads) GR

Test Bridge Formula CBF Mechanism
Perihelion precession Δφ = 6π GM / [ a (1 − e²) c² ] Slow secular rotation from cumulative steering by g(x) with α-dependent pacing. No space stretch; it’s path-integrated bias in commits.
Light bending δθ = 4 GM / (b c²) Null paths (M = 0) are deflected because T’s direction is bent each tick by g(x). Path density also shaped by D(x) → lensing-like weighting.
Shapiro delay Δt ≈ (4 GM / c³) ln (4 r₁ r₂ / b²) Extra round-trip time from lower α(x) along the path (slower local commit pace). CA distance doesn’t change; timing does.
Gravitational redshift f_obs / f_emit = α_obs / α_emit Endpoint comparison of lapse fields; photons unchanged in flight. Ledger applies the ratio at detection (Directional/Temporal gates).

Gravitational Waves GR

Quantity CBF / Bridge
Nature Small, propagating perturbations in the stability sourcing (S-ledger) that modulate α(x) and g(x). Propagate at c (null, ω0 = 0).
Wave equation (far field) □ h_TT = 0 (with □ = −∂²/∂t² + ∇²)
Dispersion ω² = c² k²
Quadrupole radiation P = (G / 5 c⁵) ⟨ Q̈_ij Q̈^ij ⟩
Strain ΔL / L ≈ (1/2) h_TT
CBF interpretation Standard GR formulas summarize how S-ledger fluctuations look to observers. Runtime is still CA: commits paced by α(x) and steered by g(x).

Weak-Field Mapping GR

Quantity Approximation / CBF Interpretation
Potential limit For small source fields (GM / (rc²) ≪ 1), the lapse relates to the Newtonian potential: α(x) ≈ 1 + Φ(x) / c² or equivalently Φ ≈ c² (α − 1).
Acceleration The pull-arrow field reproduces classical gravity in the weak limit: g(x) ≈ −∇Φ. In the CA, g(x) changes only particle direction per tick, never the grid itself.
Operational view Gravity appears as slightly slower pacing (α < 1) and directional drift (g) for nearby commits. Over many ticks this integrates to the same GM/r² acceleration law observed macroscopically.
Interpretation: In the weak-field limit, CBF reproduces Newtonian behavior through pacing and steering, not geometric curvature. α encodes time-rate variation; g encodes path bias. Their gradients yield the familiar potential and acceleration, confirming GR consistency without invoking spatial stretch.
CBF stance: Gravity is pacing (α) and steering (g) sourced by stable commits (S) and shaped by pruning history (D). Space itself does not “stretch” in the CA; observers reconstruct GR geometry from the timing and path statistics of finalized commits.

Electromagnetism

Spin–Oscillation Coupling CBF

Concept CBF Description
Spin as Maintenance Spin is stored in the M share of the causal budget—its persistence requires per-tick upkeep. The system must expend maintenance work to preserve rotational identity and orientation coherence.
Oscillation as Translation Oscillation lives in the T channel, representing periodic phase motion. Together, T and M form a closed rotation in budget space, maintaining the C = T + M rule.
Spin–Phase Relation A half-turn in spin phase corresponds to a full oscillation cycle in T. This yields the ½-spin quantization rule naturally: two oscillations in T are required for a full spin rotation.
Polarization Connection Electromagnetic polarization is the wave-cell analog of spin precession. Both arise from rotational motion in the T–M plane and share the same per-tick rotation law. Sign of rotation in M encodes left- or right-handed helicity (fermion chirality).
Spin does not require a separate field in CBF—it is the rotational mode of the existing budget loop. Bound matter keeps finite spin by reserving M each tick; free photons express it as polarization in T.

Maxwell Equations EM

Equation Differential Form CBF Interpretation
Faraday's law ∇ × E = − ∂B/∂t Curl-coupled update that converts stored magnetic “M” into electric “T”-like motion. In CBF, this is the wave-cell budget handoff across edges/faces per tick.
Ampère–Maxwell law ∇ × B = μ₀ j + μ₀ ε₀ ∂E/∂t Curl-coupled update that converts electric “T” into magnetic “M” plus source injection (current). Together with Faraday, gives self-propagating null flow (M = 0 for photons).
Gauss's law (electric) ∇ · E = ρ/ε₀ Charge is a node-source that injects/withdraws budget; in CBF this is a commit source at 0-cells.
Gauss's law (magnetic) ∇ · B = 0 Exactness (no monopoles). On the lattice: “no net magnetic charge” via d₂ ∘ d₁ = 0 consistency.

Energy & Momentum EM

Quantity Formula (SI) Normalized (ε₀ = μ₀ = 1)
Energy density u = 0.5 ( ε₀ E² + B²/μ₀ ) u = 0.5 ( E² + B² )
Poynting vector S_EM = (1/μ₀) E × B S_EM = E × B
Poynting theorem ∂u/∂t + ∇ · S_EM = − j · E Budget conservation: local “M” change + “T” flux equals work on charges. CBF view: the same per-tick budget law C = T + M, expressed as a continuum balance.
Momentum density g_EM = ε₀ (E × B) = S_EM / c² g_EM = E × B
Symbol note: We write the Poynting vector as S_EM to avoid confusion with the directional Doppler factor Π(β, θ) used elsewhere.

Field Assignment on Lattice EM

Geometric Element Field Budget Role Physical Meaning
0-cells (nodes) Charge ρ, potential φ Source / sink Budget injection point (commit source)
1-cells (edges) Electric field E, vector potential A Translation T Flow channels (propagating influence)
2-cells (faces) Magnetic flux Φ (∫B·dS) Maintenance M Storage surfaces (phase memory)
3-cells (volumes) Total charge Q (∫ρ dV) Source density Integrated commit budget within a region

Pacing, Media, and Hazard EM

Concept CBF Form
Photon propagation Null budget path (M = 0). Per tick τ₀ the wave-cell advances one lattice stride a in free space. Local pacing scales by α(x): v_effective = α(x) × c. Photons renormalize to null speed at each step.
Material coupling (MCF) Unified pacing: α_total(x, ω) = α_gravity(x) × 1/n(ω). Hazard/attenuation: h(x, ω) reduces amplitude (TTL-like depletion) faster at higher oscillation. This models absorption, scattering, and loss without editing photons “in flight” beyond local pacing.
Directional coupling At detection, the directional factor couples momentum and timing via the gate term M_ch (1 − cos θ). (See Beat-Matching.) Use Π(β, θ) in SR for observer-side Doppler; here we keep Poynting as S_EM to avoid symbol clash.

Discrete Update (FDTD-style) EM

Coefficient Formula Purpose (CBF language)
C_a C_a = (1 − σ Δt / (2 ε)) / (1 + σ Δt / (2 ε)) E-field retention (M → M continuity under loss)
C_b C_b = (Δt / ε) / (1 + σ Δt / (2 ε)) Cross-cell transfer (M ↔ T coupling under loss)
S_CFL S_CFL = c Δt / Δx ≤ 1/√2 (2D) Courant stability: ensures budget doesn’t outrun the lattice stride
Note: In normalized CBF units, the term σ Δt / (2 ε) is the fraction of budget lost per tick due to local absorption (part of the MCF). Proper accounting yields machine-precision residuals (R ≈ 10⁻¹²) in standard FDTD tests—consistent with per-tick budget conservation.

Physical Constants & Conversions

Fundamental Constants Bridge

Constant Symbol Value (SI) Role in CBF
Speed of light c 2.997 924 58 × 10⁸ m/s Causal stride: c = a / τ₀. Each tick τ₀ moves a photon one lattice step a.
Planck constant h 6.626 070 15 × 10⁻³⁴ J·s Bridge from phase rate → energy. E = hf gives physical units to the update frequency.
Reduced Planck constant ℏ 1.054 571 817 × 10⁻³⁴ J·s Angular form: ℏ = h / 2π. Used in E = ℏω and p = ℏk.
Gravitational constant G 6.674 × 10⁻¹¹ m³ / (kg·s²) Queue-depth coupling per unit mass. Sets how much α(x) slows per stored S(x) commit.
Permittivity of free space ε₀ 8.854 × 10⁻¹² F/m Storage capacity of the E-field. Fraction of M reserved per unit electric tension.
Permeability of free space μ₀ 1.257 × 10⁻⁶ H/m Storage capacity of the B-field. Fraction of T bound into magnetic circulation.
Planck length ℓP 1.616 × 10⁻³⁵ m Precision floor, not cell spacing. Sets decimal resolution of field updates.
Planck time tP 5.391 × 10⁻⁴⁴ s Minimum temporal precision for ledger updates. Sub-tick rounding, not the simulation rate.

Unit Conversions Classical

From To Conversion
Energy (J)Frequency (Hz)f = E / h
Energy (J)Angular frequency (rad/s)ω = E / ℏ
Mass (kg)Energy (J)E = mc²
Mass (kg)Compton frequency (Hz)f₀ = mc² / h
Wavelength (m)Wavenumber (rad/m)k = 2π / λ
Momentum (kg·m/s)Wavenumber (rad/m)k = p / ℏ
Geometric (G = c = 1)SIRestore factors of G and c by dimensional analysis.

Common Composite Units Foundation

Quantity SI Units Normalized (c = ℏ = 1)
EnergyJDimensionless compute rate
Momentumkg·m/sSame as energy
MasskgMaintenance rate ω₀
Lengthm1 / energy
Times1 / energy
ActionJ·sDimensionless (= ℏ)
Interpretation: Constants in CBF are not fixed background numbers; they are conversion bridges between causal compute shares and physical measurement units. The pair (a, τ₀) defines c, while h and ℏ give real-world scaling to frequency and momentum updates.

Symbol Glossary

Greek Letters Foundation

Symbol Name CBF Meaning
αAlpha (lapse)Local commit cadence / proper-time pacing. dτ = α dt. Lower α → deeper event queue, slower local clocks.
βBeta (shift / velocity)β = v / c. Normalized velocity. βⁱ: shift vector for frame drift or rotation (used in GR mapping).
γGamma (Lorentz / queue factor)Time-symmetry ratio γ = 1 / √(1 − β²). In CBF, also the stride count per tick for photons (γ-length ≈ 10⁻¹² m).
δDeltaCoincidence window width (temporal gate span) or small perturbation in reconciliation.
ε₀Epsilon-zeroVacuum permittivity, electric storage capacity (fraction of M reserved for E-field tension).
ηEta (stability)Commit stability weight (0 ≤ η ≤ 1). Controls S(x) sourcing strength: η ≈ 1 for bound atoms, η ≪ 1 for transient events.
θTheta (phase / angle)Wave phase θ = k·r − ωt or measurement angle used in phase comparison during beat-matching.
λLambda (wavelength / rate)λ = 2π/k (spatial period) or λ(t): instantaneous commit frequency.
μ₀Mu-zeroVacuum permeability, magnetic storage capacity (fraction of T bound in circulation).
ρRho (density)Mass or charge density; also used for queue load ρ_q = n_commit / n_capacity in pacing analysis.
σSigmaConductivity or local loss coefficient in MCF; scales attenuation rate and TTL drain.
τTauFundamental tick or proper-time interval. Baseline tick τ₀ defines c = a / τ₀.
φPhiScalar potential or orbital phase. Also used for radar-chart coordinate in φ-space composition laws.
ψPsiWave amplitude or spatial reconciliation factor; in GR mapping ψ⁴ scales the 3-metric.
ωOmegaAngular frequency (rad/s). ω₀ = baseline maintenance frequency; ω = ω₀ M(x).
ΦCapital PhiField flux or Newtonian potential (low-energy limit of S(x) curvature).
Π(β, θ) Capital Pi (directional factor) Observer-side Doppler/directional factor at detection. Π(β, θ) = γ(1 − β cos θ). Note: Poynting vector is denoted S_EM in this document to avoid symbol clash.

Latin Letters (Key Variables) Foundation

Symbol Name CBF Meaning
aLattice spacingSpatial stride between cells (γ-length ≈ 10⁻¹² m). One photon hop per τ₀.
AVector potentialMagnetic potential (T·m). Label for cross-cell coupling channel in T.
BMagnetic fieldMaintenance storage on faces (Tesla). Represents curl of T-flow loops.
CTotal causal budgetC = T + M. Per-tick compute share sum, normalized to α(x).
EEnergy / Electric fieldTotal energy or local E-field. Represents work potential in T-channel.
fFrequencyOscillation rate (Hz). f₀ = Compton frequency = mc²/h.
GGravitational constantQueue-depth coupling per mass; sets α-slowdown from S(x).
hPlanck constantPhase → energy bridge; assigns units to update frequency.
jCurrent densityCharge flow (A/m²), transfers T-budget between neighboring cells.
kWavenumberSpatial frequency (rad/m). k as vector carries propagation direction.
MMaintenance share / MassFraction of C used for internal upkeep. M > 0 defines rest mass; M = 0 for photons.
mMassPhysical mass (kg) = ℏω₀/c². Encodes baseline maintenance cost.
pMomentump = ℏk = γmv. Translation energy per unit c.
rPosition / radiusr: position vector; r: radial coordinate for α(r), ψ(r).
RAreal radiusSchwarzschild areal radius; R = r(1 + GM/2rc²)².
SStability / Source ledgerS(x): gravitational working memory (RAM of reality). Feeds α(x) and g(x).
tTimeGlobal tick count × τ₀. Coordinate time of the shared Event Ledger.
TTranslation shareFraction of C used for motion / wave advance. Complements M via C = T + M.
uEnergy densityLocal energy per volume: u = (E² + B²)/2 (normalized units).
vVelocitySpatial translation rate: v = α c T for matter; v = α c for light.

Special Notation Bridge

NotationMeaning
∇Gradient/divergence/curl operator. Defines local field topology on lattice edges and faces.
∇²Laplacian (∂² sum). Governs diffusion or potential smoothing in α and ψ fields.
□D’Alembertian = −∂²/∂t² + ∇². The general wave operator over the causal grid.
⟨...⟩Ledger-averaged or time-averaged quantity over multiple commits.
|ψ|²Commit frequency density (Born-rule emergence).
TTTransverse-traceless gauge; purely shear mode for gravitational waves.
SCGSimultaneous Commit Group. Shared-ledger constraint linking entangled participants.
FDTDFinite-Difference Time-Domain method. Numerical validation of EM budget conservation.
1PNFirst Post-Newtonian term (O(c⁻²)) linking queue-based GR corrections.
S_EM Poynting vector (energy flux): S_EM = (1/μ₀) E × B (SI), = E × B (normalized).
Summary: Greek letters describe pacing, phase, and maintenance; Latin letters mark lattice quantities and conserved budgets; special symbols capture the Ledger’s computation rules. Each appears in at least one equation of C = T + M = α(x).