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
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
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
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 in preparation (RIG-DE-001 · Q4 2025). Patent trigger: SigmaScore ≥ 5σ on Robust-MAD-Z.