Law 0 – Computing the Alignment Lane

How to turn raw drift, jitter, and disagreement into a single bounded alignment lane a in (-1,+1) that any system can reuse.

4. Computing the alignment lane: recipes and invariants

Shunyaya Symbolic Mathematical Law (Law 0) gives every important value a second lane a in (-1,+1). This section explains how that lane is computed and combined in a way that is:

  • bounded (never leaves (-1,+1)),
  • order-independent (fusion does not depend on input order),
  • stream-safe (you can update as new data arrives),
  • compatible across domains (sensors, KPIs, AI, physics laws).

Under Law 0, the classical magnitude m remains exactly what it was. The alignment lane a becomes a compact, quantitative summary of how reality was behaving around that m.

4.1 One value, two lanes (quick recap)

For any quantity you care about:

  • Classical view:
    • m = 230.0 (voltage)
    • m = 3.167 (velocity)
    • m = 1.540 (refractive index)
  • Law 0 view:
    • (m, a) = (230.0, +0.03) → “230.0 and calm”
    • (m, a) = (230.0, +0.71) → “230.0 but stressed / noisy”
    • (m, a) = (3.167, +0.28) → “3.167 with borderline posture”

Collapse parity remains:

  • phi((m, a)) = m

Any system that only understands m can ignore a and behave exactly as before.

The alignment lane is an extra, structured signal about how reality behaved around m. Law 0 makes it possible to carry that signal next to every important number, without rewriting any classical formulas.

4.2 Weighted pooling of multiple contributions

In real systems, posture is rarely driven by a single factor. Many indicators contribute:

  • multiple sensors or instruments,
  • multiple samples over time,
  • multiple terms inside a formula,
  • multiple kinds of drift or noise.

To combine these into a single lane a, Shunyaya Symbolic Mathematics uses a hyperbolic pooling rule. The idea:

  1. Clamp each raw alignment into the open interval.
  2. Move to a rapidity space via atanh.
  3. Weight and sum contributions.
  4. Return to the bounded lane via tanh.

A canonical recipe in ASCII pseudo-code:

# Inputs:
#   (m_i, a_i_raw)    i = 1..N   # magnitudes and raw alignment lanes
#   gamma             # exponent for weighting by |m_i|
#   eps_a, eps_w      # small positive epsilons

clamp_a(a_raw, eps_a):
    return max(-1 + eps_a, min(+1 - eps_a, a_raw))

pool_alignment(m_1..m_N, a_1_raw..a_N_raw, gamma, eps_a, eps_w):
    U = 0.0
    W = 0.0
    for i in 1..N:
        a_i_c = clamp_a(a_i_raw, eps_a)
        u_i   = atanh(a_i_c)
        w_i   = (abs(m_i))**gamma
        U    += w_i * u_i
        W    += w_i
    denom = max(W, eps_w)
    a_out = tanh(U / denom)
    return a_out

Key properties:

  • Bounded.
    Because of the clamp and tanh, a_out always stays in (-1,+1).
  • Order-independent.
    The sums of w_i * u_i and w_i are commutative. The fusion does not depend on the order in which contributions arrive.
  • Magnitude-aware (if desired).
    Weights w_i = |m_i|**gamma allow larger magnitudes to influence posture more, when that matches the domain. Setting gamma = 0 recovers equal weighting.
  • Stream-safe.
    You can keep U and W as running totals and update a_out as new contributions arrive:
    • append a new sensor reading,
    • extend a time window,
    • add a new term to a law’s computation.

The same pooling pattern can be reused:

  • across time (sliding or cumulative windows),
  • across sensors or models,
  • across terms in a formula when computing a law’s output lane.

Law 0 does not force a single choice of weights or posture indicators. It provides a disciplined way to fuse them into one bounded alignment lane.

4.3 Product and division style chaining (for laws)

Classical laws often combine terms via products and ratios (*, /).

Law 0 extends them by propagating alignment through a hyperbolic-addition style rule. For two terms with clamped alignment lanes a1_c and a2_c:

  • Product-like combinations
    (for example, when forming m_out = m1 * m2): a_mul = tanh(atanh(a1_c) + atanh(a2_c))
  • Division-like combinations
    (for example, when forming m_out = m_num / m_den): a_div = tanh(atanh(a_num_c) - atanh(a_den_c))

These rules:

  • keep a_mul and a_div inside (-1,+1),
  • treat alignment as something that adds or subtracts in rapidity space,
  • provide a consistent way to propagate posture when laws multiply or divide classical quantities.

In practice, bounded classical laws (Ohm, Bernoulli, continuity, Faraday, and others) use combinations of:

  • pooling to summarise multiple contributions into a single lane, and
  • product/division chaining to reflect how posture flows through formulas.

The goal is simple: whenever classical magnitudes are combined, the posture lane should combine in a traceable, reproducible way.

4.4 Banding for everyday reading

Most users do not need to think in atanh and tanh. They only need to read the lane.

A simple, policy-driven banding scheme might be:

|a| < 0.20           ->  A+   (calm)
0.20 <= |a| < 0.50   ->  A0   (borderline)
|a| >= 0.50          ->  A-   (stressed)

This turns any (m, a) into a human-friendly posture tag:

  • (m=3.17, a=+0.08) → A+ → “3.17 and calm”
  • (m=3.17, a=+0.28) → A0 → “3.17 but borderline”
  • (m=3.17, a=+0.62) → A- → “3.17 but stressed / noisy”

Important points:

  • These bands are policy choices, not fixed by the math.
  • Different domains may tweak thresholds, but the structure stays the same:
    • a in (-1,+1),
    • phi((m, a)) = m,
    • a small number of bands to separate calm, borderline, and stressed regimes.

Once banding is in place, every time series, dashboard, or experiment can show:

  • not just “what number did we see?”,
  • but also “how healthy or risky was the reality that produced it?”.

Together, the dual-lane representation, pooling rule, product/division chaining, and banding form the computational backbone of Shunyaya Symbolic Mathematical Law (Law 0): a reproducible way to let each classical value speak about how reality was behaving when that value appeared.

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Disclaimer (summary).
Shunyaya Symbolic Mathematical Law (Law 0) is an observation-only framework and must not be used directly for design, certification, or safety-critical decisions.