Proof of Concept — Bounded Classical Laws — Shunyaya Symbolic Mathematics (SSM)

From bare magnitudes to calm-vs-stressed physical laws

Classical laws can tell you not only what number they produce, but also how calmly they are being satisfied in a given situation.

By extending classical values from m to (m, a) with a in (-1,+1), Shunyaya Symbolic Mathematics (SSM) keeps the usual scalar results intact while adding a human-readable sense of posture (stable, drifting, unreliable). In simple terms, this can support:

  • Fewer surprises in devices and field systems
  • Calmer, faster experimentation in labs
  • Clearer safety margins in mechanical and electrical design
  • Steadier operational decisions, separating law correctness from situational stress

These are simple, reproducible examples; all results are observation-only and must be independently validated before any production or critical use.

SSM one-line primer

Extend numbers from m to (m, a) with a in (-1,+1) and collapse parity:

phi((m,a)) = m

The classical value is always preserved.
SSM adds a bounded alignment lane a beside it.

SSMS one-line primer

A shared, plain-text symbolic layer where classical operators (+, −, *, /) lift to structured, alignment-preserving verbs.

It keeps formulas portable across documentation, code, and hardware, without ambiguity.

What this proof of concept contains

Each bounded law POC shows:

  • The classical law
  • The SSM / SSMS-lifted form
  • A tiny script printing Classical vs SSM side-by-side
  • A short explanation of how the alignment lane a helps reasoning

Each law is a bounded conservative extension:

  • The underlying physics remains unchanged
  • phi((m,a)) = m ensures classical magnitudes are preserved
  • The alignment lane a reports how calm vs stressed this particular instance appears

Laws progressively covered include:

On this site, each law appears as its own page:
“POC Bounded Classical — Law LXX: [Law Name] — SSM”.

How to navigate these pages

  • This overview page introduces the concept.
  • Each Law LXX page:
    • States the classical law in ASCII
    • Shows the SSM / SSMS-lifted form
    • Explains the meaning of the alignment lane a
    • Includes a tiny script that prints Classical vs SSM (m, a) and a simple band (A+, A0, A−)

Typical reading flow:

  • When both methods agree:
    “Classically OK and posture calm.”
  • When posture disagrees (same magnitude, different a or band):
    SSM surfaces useful information that classical arithmetic cannot, without changing the physics.

Formulas and pooling rules (ASCII only)

Sum pooling (alignment lane)

Used when combining multiple contributions:

  • eps_a := 1e-6
  • eps_w := 1e-12
  • gamma := 1

clamp_a(z, eps_a) := max(-1+eps_a, min(+1-eps_a, z))

Weights and rapidities:

  • w_i := |m_i|^gamma
  • u_i := atanh(clamp_a(a_i, eps_a))

Aggregate:

  • U := sum_i (w_i * u_i)
  • W := sum_i (w_i)

Collapse:

  • m_out := sum_i m_i
  • a_out := tanh(U / max(W, eps_w))

Product chaining (alignment lane)

Used in laws like V = I * R or P = V * I:

  • clamp_a(z, eps_a) := max(-1+eps_a, min(+1-eps_a, z))

Lifted product and quotient:

  • (m1,a1) * (m2,a2) := (m1*m2, tanh(atanh(a1) + atanh(a2)))
  • (m1,a1) / (m2,a2) := (m1/m2, tanh(atanh(a1) - atanh(a2)))

These ensure:

  • Collapse parity
  • Order invariance
  • Boundedness
  • Conservative extension of classical arithmetic

How subtraction and division affect alignment

To keep the extensions intuitive:

  • Subtraction amplifies drift when terms oppose.
    If (m1,a1) is calm but (m2,a2) is shaky, the difference inherits the shakier posture.
  • Division behaves like a difference of rapidities.
    Opposing lane signs in numerator/denominator increase |a|, revealing unstable ratios.

These behaviors match the SSMS treatment of s_sum, s_diff, and s_div.

Relation to SSMS operators

The helper functions used in these POCs correspond directly to SSMS primitives:

  • ssm_align_product → analogous to s_mul
  • ssm_align_sum → analogous to s_sum
  • ssm_align_div → analogous to s_div

Posture semantics (for example, a_semantics = "drift-positive" or "stability-positive") are display-layer only.
The underlying two-lane math remains invariant.

How to derive a from real data (optional hook)

These POCs use hand-assigned a values to keep examples simple.
In real deployments, a can be computed from:

  • Time-series variance or jitter (e.g., window variance of I(t) or V(t))
  • Across-trial variability (e.g., consistency of repeated F = m * a measurements)
  • Residual drift (difference between theoretical vs observed law balance)

Once computed, real-world a values can flow directly into other Shunyaya components such as:

  • SSM-Audit
  • SSMDE (data envelopes)
  • Dashboards and monitoring systems

How to run everything (scripts)

The reference scripts (in the associated project code) are designed to be:

  • Plain ASCII
  • Small and easy to inspect
  • Runnable with standard Python 3.10+

A typical “run all laws” entry point prints one summary line per law, for example:

  • L01 | law=ohms | m_classical=2300.0 | m_ssm=2300.0 | a=+0.78 | band=A-
  • L02 | law=f_ma | m_classical=1960.0 | m_ssm=1960.0 | a=+0.32 | band=A0

Where:

  • m_classical = classical magnitude
  • m_ssm = SSM’s magnitude (always matches under collapse)
  • a = alignment lane (posture)
  • band = human label (A+, A0, A−)

Law POC template (consistent across pages)

Each Law POC page follows the same structure:

  1. Classical law + units
  2. SSM / SSMS-lifted form (ASCII)
  3. Micro-scenario with checkable numbers
  4. Python script printing Classical and SSM (m,a)
  5. Interpretation: what the band means in that scenario

Shared knobs:

  • eps_a := 1e-6
  • eps_w := 1e-12
  • gamma := 1
  • a_semantics := "stability-positive" or "drift-positive"

Core rules:

  • a_out := tanh(U / max(W, eps_w))
  • a_out := tanh(atanh(a1) + atanh(a2))
  • phi((m,a)) = m

This keeps each law intact while surfacing posture.

Safety and scope

  • Observation-only. This is not a safety case or approval mechanism.
  • Reproducible. Tiny scripts and ASCII formulas for easy checking.
  • Portable. Same meaning across documents, code, and (future) hardware.
  • Respectful of classical laws. SSM never alters the physics; it only adds a bounded lens on top.

Navigation

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Next: POC Bounded Classical — Law L01: Ohm’s Law — SSM

Explore Further:
Symbolic-Mathematics-Bounded-Classical-Laws-POC

Disclaimer
All examples on this site are observation-only and must be independently validated before any production, safety-critical, or regulatory use.