Kernel, streaming fuse, bands, and selection
Datum and collapse
Carry two lanes for every quantity you already compute.x := (m, a) with a in (-1,+1) and collapse parity phi((m,a)) = m.
Clamp
Bound alignment before any map or composition.
a_c := clamp(a, -1+eps_a, +1-eps_a) # default eps_a = 1e-6
Rapidity map
Compose safely in u-space, then map back to a bounded lane.
u := atanh(a_c)
a' := tanh(u') # after composition in u
Lane for mul/div (M2 policy, default)
Multiply or divide m as usual. Compose the lane with signed rapidities.
a' := tanh( atanh(a1) +/- atanh(a2) )
# "+" for product, "−" for ratio, declared in manifest
Optional M1 policy (multiply-only, rare, opt-in)
Only when declared and both operands share one lens with small |a|.
a' := clamp(a1 * a2, -1+eps_a, +1-eps_a)
Streaming fuse, order-independent
Batch equals stream equals shuffled within fixed epsilon.
U += w*atanh(a)
W += w
a_out := tanh( U / max(W, eps_w) ) # default eps_w = 1e-12
Weights
Pick once and pin in the manifest.
w := |m|^gamma # default gamma = 1
# or
w := 1 # uniform
Lens → align → RSI (single bounded chooser)
Turn a declared contrast into a bounded chooser used for selection.
# given contrasts e_in, e_out and scale c>0
a_in := tanh(-c*e_in)
a_out := tanh(+c*e_out)
U_in := SUM w*atanh(a_in)
V_out := SUM w*atanh(a_out)
W_in := SUM w
RSI := tanh( (V_out - U_in) / max(W_in, eps_w) ) # RSI in (-1,+1)
Calm gate (alignment-only, read-only actuation)
Scale selection pressure without touching values.
RSI_env := g_t * RSI # g_t in [0,1] from telemetry
Bands (global defaults, normative)
Publish once and use everywhere for legibility and policy.
A++: a >= +0.90
A+ : +0.60 <= a < +0.90
A0 : -0.60 < a < +0.60
A- : -0.90 < a <= -0.60
A--: a <= -0.90
Division policy, normative
Keep division safe and explicit.
division_policy := "strict" # default
# alternatives "meadow" or "soft" require explicit declaration + numeric guards
# combine policy for alignment remains M2 unless declared otherwise
Non-negotiable invariants, normative
(i) Collapse parity. phi((m,a)) = m
(ii) Order invariance. Batch == stream == shuffled within epsilon
(iii) Determinism. Same manifest implies same outputs
(iv) Lane purity. Policies, bands, and gates act on a only. m is never mutated
(v) Division near zero. Default strict, alternatives declared
(vi) Cross-vendor parity. Identical manifests yield comparable a and RSI
Dtype and epsilon note
Prefer float64 for u-space accumulation. If constrained to float32, set eps_w >= 1e-8 and keep |a| < 1 - eps_a with eps_a >= 1e-6 to avoid atanh(±1) and division by zero in a_out := tanh(U / max(W, eps_w)).
Manifest crib, minimum fields
Pin the knobs that govern reproducibility and parity.
eps_a, eps_w, gamma, weights_policy,
combine_policy=M2, division_policy=strict,
bands{...}, lens_id, lens_params, g_t_source,
dtype, knobs_hash
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SSM-AI — Table of Contents
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