1.1 — Numeral and invariants (non-negotiable)
- Two-lane numeral. Each datum is
x := (m,a)wheremis the classical value anda in (-1,+1)is the stability lane. - Collapse parity. Classical numbers never change:
phi((m,a)) = m. - Clamp rule. Always bound inputs before any
atanh:a := clamp(a, -1+eps_a, +1-eps_a) # default eps_a = 1e-6 - Order invariance (target). For fixed knobs, composition yields the same result (within epsilon) whether processed in batch, as a stream, or in any permutation.
- Determinism. Same inputs + same knobs => same outputs (byte-for-byte where applicable).
- Rapidity (internal). Map into rapidity space to combine safely, then map back:
u := atanh(a) # combine lanes in u-space a := tanh(u) # decode, remains bounded - Zero-class display (policy). If
m_out == 0, display can be(0, +1)(strict center) or(0, a)(informative); declare once per study.
1.2 — Lane mappers (finance-ready library)
Each mapper is declarative (manifest) and bounded (outputs in (-1,+1) before clamp). Pick the smallest mapper that answers the business question; add blends only if needed. Apply the final safety clamp unless stated otherwise.
# Coverage (completeness, tails) # q in [0,1] a_cov := 2*q - 1 # Agreement (reconciliation quality) # d := |m1 - m2|, b > 0 is the declared bound a_agree := tanh( 1 - d / b ) # Residual stability (forecast vs actual) # r := actual - forecast, s > 0 (scale), k > 0 (sensitivity) a_resid := tanh( k*( 1 - |r|/s ) ) # Ratios (lane lift for m := f/g) a_div := tanh( atanh(a_f) - atanh(a_g) ) # Products (lane lift for m := f*g) a_mul := tanh( atanh(a1) + atanh(a2) ) # Blends (combine two lanes into one), alpha in [0,1] a := tanh( alpha*atanh(a_A) + (1-alpha)*atanh(a_B) ) # Final safety clamp (always) a := clamp(a, -1+eps_a, +1-eps_a)
ZEOZO-Core (entropy drift, preferred mapper; scale-free).
Edge-normalize once; maintain bounded energy; log-compress; then map to a lane.
# Edge normalization (robust)
med := median(x)
rad := median(|x - med|); rad := max(rad, eps)
y_t := (x_t - med) / rad
# Energy and log-compress
E_t := (1 - lam)*E_{t-1} + lam*(y_t^2)
Z_t := log(1 + E_t)
# Option A (drift-as-risk lane), with Z_ref > 0, c > 0
a_zeozo := tanh( c*( 1 - Z_t / Z_ref ) )
# Option B (alignment-track lane)
A_t := (1 - mu)*A_{t-1} + mu*Z_t
Delta_t := | Z_t - A_t |
a_gap := tanh( k_gap*( A_t - Z_t ) ) # positive when calm dominates
# Defaults (declare once)
lam = 0.10
mu = 0.04
eps = 1e-6
# Initialize deterministically
E_0 := 0
A_0 := Z_0
SYASYS-Core (calm-gated alignment; optional executive “safe-unlock”).
Accumulate calm dose; gate advisories only when drift is low and calm is sufficient.
# Reuse Z_t, A_t, Delta_t from ZEOZO
clip(u, lo, hi) := min(max(u, lo), hi)
# Calm dose
Q_t := rho*Q_{t-1} + (1 - rho)*clip( A_t - Z_t , 0 , 1 )
# Gate (bounded in [0,1))
SyZ_t := ( 1 / ( 1 + Z_t + kappa*Delta_t ) ) * ( 1 - exp( - muR*Q_t ) )
# Lane from gate (into (-1,+1))
a_syasys := 2*SyZ_t - 1
# Defaults
rho = 0.90
kappa = 0.50
R = 8.0
muR := ln(2)/R
Monotonicity note. If a source score increases (holding scale/bounds fixed), the induced lane should not decrease; the defaults above are isotonic after clamps.
1.3 — Fusion (streaming/batch equivalence)
Goal: one canonical rule that makes batch == stream == shuffled (within rounding tolerance).
# Streaming accumulator (canonical)
# For observations with lanes a_i and weights w_i >= 0:
U := 0
W := 0
for each (a_i, w_i):
a_i := clamp(a_i, -1+eps_a, +1-eps_a)
U := U + w_i*atanh(a_i)
W := W + w_i
# Decode once at the end
a_out := tanh( U / max(W, eps_w) ) # default eps_w = 1e-12
# Default weights: w_i := |m_i|^gamma with gamma := 1.0
# Uniform option: w_i := 1
Why order-invariant (sketch). (U,W) are simple sums; addition is associative/commutative, so any permutation yields identical U,W. The same decode function is applied once, hence a_out is permutation-invariant (within rounding).
Groupby / rollups. Apply the same (U,W) rule within each group; nested aggregations preserve results because addition composes.
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