Continuity note: rapidity and bounds
Treat alignment as a bounded channel carried with magnitude. We linearize alignment in rapidity space, combine there, then decode back so the alignment stays bounded while classical magnitudes behave exactly like ordinary math.
# clamp keeps alignment strictly inside (-1, +1)
sign(z) = 1 if z >= 0 else -1
clamp_a(a, eps_a):
if abs(a) >= 1 - eps_a: return sign(a) * (1 - eps_a)
else: return a
# encode and decode
u(a) = atanh( clamp_a(a, eps_a) )
a_of(u)= tanh(u)
# numeral and collapse (recap)
x = (m, a)
phi(m, a) = m
Micro example: sums (streaming-safe)
Weighted rapidity averaging keeps addition associative in batch and stream.
# inputs
x1 = (m1, a1)
x2 = (m2, a2)
gamma >= 0, eps_w > 0
# magnitude path (classical)
m_sum = m1 + m2
# alignment path (rapidity mean)
w1 = abs(m1)^gamma
w2 = abs(m2)^gamma
u1 = u(a1)
u2 = u(a2)
u_sum = (w1*u1 + w2*u2) / max(w1 + w2, eps_w)
a_sum = a_of(u_sum)
# output
s_add(x1, x2) = (m_sum, a_sum)
Micro example: products (bounded combine)
Two portable combine laws for the alignment channel:
M1 (simple): a' = a1 * a2
M2 (rapidity): u' = u(a1) + u(a2) ; a' = a_of(u')
m_mul = m1 * m2
a_mul = M2(a1, a2) # default (associative in u-space)
s_mul(x1, x2) = (m_mul, a_mul)
Micro example: integer powers
m_pow = m^k
a_pow = a_of( k * u(a) ) # = tanh( k * atanh(a) )
s_pow( (m, a), k ) = (m_pow, a_pow)
Micro example: banded comparisons (equality-first)
Compute numeric scores for greater-than and equality; then map to 4 bands.
# deltas
dv = (m_x - m_y) / max( abs(m_x) + abs(m_y), V_eps )
du = g_t * ( u(a_x) - u(a_y) ) # optional gate g_t in [0,1]; alignment-only
# scores
s_gt(x,y) = tanh( beta_v*dv + lambda_u*du )
raw_eq = beta_v*abs(dv) + beta_u*abs(du)
s_eq(x,y) = 1 - 2*tanh(raw_eq)
# banded report (precedence: equality first)
if s_eq >= tau_eq -> "equal"
else if s_gt >= +tau_hi -> "x > y"
else if s_gt <= -tau_hi -> "x < y"
else -> "undecided"
Calibration notes (brief): choose V_eps, beta_v, lambda_u, beta_u, tau_hi, tau_eq once per study and publish them in the manifest.
Lifting recipe (classical to symbolic)
To lift any classical expression f to stability-aware form:
+ -> s_add - -> s_sub * -> s_mul / -> s_div ^ -> s_pow
min -> min max -> max abs -> abs round -> round
phi( f_lifted((m,a), ...) ) = f_classical(m, ...) holds by construction.
Optional environment gate (alignment-only finalizer)
a_env = clamp_a( g_t * a_op , eps_a ) # g_t in [0,1]
# magnitudes unchanged; collapse parity preserved
Soft reminder (observation-only)
These operators expose stability and confidence for analysis and decision support. They do not replace domain models, operational controls, or safety processes.
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