SSM-Infinity — Alignment Lanes & Hyperbolic Merging

How infinite values preserve posture through symbolic atanh/tanh fusion.

Every value in SSM-Infinity — including (+∞) and (-∞) — carries a bounded alignment lane:

a ∈ (-1, +1)

The alignment lane is not “probability,” not “weight,” and not “uncertainty.”
It is a symbolic posture representing how the value enters the arithmetic.
Through this lane, even infinite values can take part in lawful, reproducible merging.

SSM-Infinity uses the same alignment kernel as the rest of Shunyaya Symbolic Mathematics:

a_c := clamp(a, -1+eps, +1-eps)
u := atanh(a_c)
U := U1 + U2           # merging posture in hyperbolic space
a_out := tanh(U)

This hyperbolic method ensures three key properties:

  1. Determinism — no randomness, no branching ambiguity.
  2. Reversibility — alignment behaviour is mathematically reversible.
  3. Boundedness — output always remains inside (-1, +1).

1. Why atanh/tanh? Symbolic Geometry of Stability

Classical merging (average of weights) does not work for infinities.
It collapses easily or becomes unstable.

Hyperbolic merging is chosen because:

  • atanh() linearizes the bounded interval
  • tanh() re-bounds the final posture
  • It is order-invariant
  • It is associative (within safe precision)
  • It is stable for extreme inputs, including infinite-class values

This gives SSM-Infinity its signature behaviour:

(+∞, a1) + (+∞, a2) → (+∞, a_out)
(+∞, a1) + (-∞, a2) → zero-class     # collapse into stable symmetric class

2. Alignment Lane in Infinite Operations

Addition (same direction)

a_out = tanh(atanh(a1) + atanh(a2))

Addition (opposite direction)

The sign conflict forces a zero-class collapse, discarding alignment:

(+∞) + (-∞) → zero-class

Multiplication & Division

The alignment lane simply travels with the sign:

(+∞, a) * (-5) → (-∞, a)
(12) / (-∞, a) → zero-class

Exponentiation

The lane remains unchanged except for zero-class outcomes:

(+∞, a) ** (-k) → zero-class

In SSM-Infinity, alignment always respects the symbolic class rules.

3. Relevant Code (Hyperbolic Merge)

Below is the exact merging logic from the engine:

# hyperbolic merge (excerpt from ssm_infinity_core)

def merge_align(a1, a2):
    # clamp inputs
    a1_c = max(min(a1, 1-1e-12), -1+1e-12)
    a2_c = max(min(a2, 1-1e-12), -1+1e-12)

    # map into hyperbolic posture space
    u1 = math.atanh(a1_c)
    u2 = math.atanh(a2_c)

    # combine and re-bound
    return math.tanh(u1 + u2)

This is the same kernel used across the entire Shunyaya ecosystem —
SSMDE, SSMNET, SSMEQ, SSMClock, and now SSM-Infinity.

4. Why It Matters

Thanks to hyperbolic alignment merging:

✔ Infinite arithmetic becomes predictable
✔ Symbolic class boundaries stay clean
✔ Extreme values never destabilize the computation
✔ The entire system becomes reproducible and testable (22/22 tests passed)

In other words:

SSM-Infinity does for ∞ what SSM-Zero did for 0 — it makes it lawful.

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Disclaimer

Shunyaya Symbolic Mathematical Infinity (SSM-Infinity) is a symbolic research framework — not a predictive model or numerical estimator.