— Capitulum I.α —
The core math.
Read every formula below as one question: how far is this output from the crowd's average — and is that gap big enough, and safe enough, to act on? That single idea is the whole apparatus.
Robust-MAD-Z. SigmaScore. Escalation cost. The primitives every Deviate Engine inherits. Patent triggers at 5σ; deviation rises through Cognitive (±20σ), Nature (±20σ), and Physics (±30σ HARD).
— primitive · 01
Robust-MAD-Z
rawMAD = median( |scorei − median(scores)| )
Robust-MAD-Z = 0.6745 · (score − median) / rawMAD
how many standard deviations an output sits from the crowd — measured the outlier-proof way, so one weird sample can't skew the ruler.
M-estimator on the deviation residual. The 0.6745 constant calibrates to a normal distribution. Robust = outlier-resistant. Z = unitless sigma score.
— primitive · 02
SigmaScore
SigmaScore = 0.3 · structural + 0.7 · behavior
Patent trigger ≥ 5σ
a single novelty score that leans mostly on what a system does, not how it looks. Cross 5σ and it's novel enough to review for a patent claim.
Behavior weighted heavier than structure — what the system does matters more than how it's drawn. Any output ≥ 5σ from baseline triggers patent-claim review.
— primitive · 03
Escalation cost
|n| ≤ 10 → 2|n|
10 < |n| ≤ 20 → 1024 · 3(n−10)
20 < |n| ≤ 30 → 1024 · 310 · 5(n−20)
the further you push past the average, the more it costs to go one step further — and the price climbs faster the higher you climb. That budget is the leash.
Cost of deviation grows super-exponentially. Each sigma stage has a higher per-unit ceiling. Governance budget bounds where the engine can operate.
— gate logic
Three gates
- Cognitive — block if any layer's Robust-MAD-Z < 3.0
- Bell — block if |S| ≤ 2 (no entanglement signal)
- Hawking — block if InternalVolume > 10·SurfaceArea (black-box)
- Fine-tuning — block if ViableBand / ParameterSpace > 0.10
- Vacuum — block if BaselineVariance < ε
The three-layer stack.
— Cognitive · engines 01–20 · ±20σ
Mechanism. Rupture. Collision.
Does the idea actually work, does it break an assumption, and does it connect two distant fields? Each score below is a weighted checklist of six signals.
Mechanism = 0.25·CausalLinkDensity + 0.20·FeedbackLoopClarity + 0.20·ConstraintVisibility + 0.15·FailureModeCoverage + 0.10·OperationalSpecificity + 0.10·Testability
Rupture = 0.25·AssumptionRemoval + 0.20·CategoryFrameBreak + 0.20·OrthodoxyViolation + 0.15·ReframeStrength + 0.10·FirstPrinciplesRebuild + 0.10·IrreversibilityOfInsight
Collision = 0.25·DomainDistance + 0.25·StructuralAlignment + 0.20·PracticalTransfer + 0.15·EmergentPropertyCreation + 0.15·MetaphorNonDecorativeness
Gate: any MADZ < 3.0 → BLOCK
— Nature · engines 21–30 · ±20σ
Pheromone. Flight. Spiral.
Nature already solved search. Each engine borrows a proven foraging rule — how ants, hunting birds, slime mold, fireflies and whales find the good stuff.
Ant pheromone: τ(t+1) = (1−ρ)τ + Σ τk
Lévy flight: P(s) ∝ s−1−α
Physarum flow: Q = (D/L)(pi − pj)
Firefly: β(r) = β0·e−γr²
Whale spiral: D′·ebl·cos(2πl)
Biology as architecture: every Nature-layer engine borrows a natural search heuristic — ant colony, Lévy, Physarum, firefly, whale optimization.
— Physics · engines 31–40 · ±30σ HARD
Fine-tuning. Bell. Hawking.
The strictest filter. Borrowed from physics, these gates reject anything too lucky, too disconnected, or too much of a black box — the last check before a patent claim.
FineTuningRatio = ViableBand / ParameterSpace (BLOCK if > 0.10)
FineTuningEntropy = −Σ pi·log(pi)
BellParameterS = E(a,b) − E(a,b′) + E(a′,b) + E(a′,b′) (BLOCK if |S| ≤ 2)
Hawking gate: InternalVolume > 10·SurfaceArea → BLOCK
Vacuum gate: BaselineVariance < εvacuum → BLOCK
Hardest layer. Physics gates are the last filter before patent claim.
Representative Enhanced Brain (REB) · core formulas
The reasoning brain. It weighs evidence (Bayes), remembers and forgets on a schedule (FSRS / Ebbinghaus), and sizes its bets (expected value, Kelly). These are the textbook primitives — the apparatus is how they're governed together.
Bayes update: P(H|E) = P(E|H)·P(H) / P(E)
FSRS retention: R(t,S) = (1 + t/(9S))−1
Ebbinghaus forgetting: R(t) = e−t/S
Expected value: EV = Σ Pi·Vi
Kelly criterion: f* = (bp − q) / b
Compound growth: A = P(1 + r)n
Power law: P(X ≥ x) ∼ x−α
KL divergence: DKL(P‖Q) = Σ P(x)·log(P(x)/Q(x))
Autonomous Digital Twin Brain (ADTB+) · formulas
Same Bayesian spine as the REB, pointed at a different job: measuring information itself — how much a signal tells you (Shannon, mutual information), how far two beliefs diverge (KL), and how capability compounds with experience (Wright's law).
Bayes: P(H|E) = P(E|H)·P(H) / P(E)
Shannon information: I(x) = −log P(x)
Mutual information: I(X;Y) = Σ P(x,y)·log(P(x,y)/(P(x)P(y)))
KL divergence: DKL(P‖Q) = Σ P(x)·log(P(x)/Q(x))
Cross entropy: H(p,q) = −Σ p(x)·log q(x)
Expected value: EV = Σ Pi·Vi
SIR diffusion: dI/dt = βSI − γI
Power law: P(X ≥ x) ∼ x−α
Wright's law: Y = a·Xb
Cumulative learning: output compounds with experience
Prediction Studio · scoring + calibration
How the studio grades its own forecasts and keeps them honest: score each prediction against what actually happened (Brier, Log score), blend a council of engines, throw out the outliers, and reward the engines that were right. The targets at the bottom are the pass bar.
Brier Score: BS = (1/N)·Σi(pi − oi)²
Log Score: LS = −(1/N)·Σi[oi·log(pi) + (1−oi)·log(1−pi)]
Confidence-weighted P: Pfinal = Σ(pi·ci) / Σci
Disagreement: σcouncil = √( (1/N) · Σi(pi − p̄)² )
Trimmed mean: Ptrim = mean( sort(probs)[k : N−k] )
Engine weight: wnew = (1−η)·wold + η·reward
Market-blended: forecast = swarm·agent + market·prior
EVI: E[utility|collect] − E[utility|no] − cost
— targets: Brier < 0.15 · Log Loss < 0.40 · ECE < 0.05/bucket · feedback latency < 1hr
Public surface only. Full claim language, parameter values, and apparatus details are under filing (RIG-DE-001 · Q4 2025). Patent trigger: SigmaScore ≥ 5σ on Robust-MAD-Z.