Discover the operators hiding inside symbolic math.

SSDS finds the mathematical pattern behind a phenomenon — even when it's scattered across equations that don't look alike.

Read the documentation →

Example / recurring structure

f₁ = v₀(φ₁₄ + 1)^φ₁₅
f₂ = v₀(φ₁₄ + 1)^φ₁₅

→ reusable structure

SSDS treats repeated symbolic structure as a candidate mathematical object rather than only another expression to fit.

01 / Philosophy

The Philosophy behind

Every phenomenon that behaves predictably has a mathematical fingerprint — a pattern in how its quantities relate, scale, or change together. Math is how we notice that fingerprint and write it down. SSDS looks across a domain's equations for that same pattern showing up more than once, in different forms, because a pattern that keeps reappearing is evidence it's tracking something real — not a coincidence of notation.

02 / Problem

The same pattern gets rediscovered and rewritten every time, and nothing connects the dots.

Different equations in the same domain often describe the same underlying behavior — but because each was derived separately, the pattern connecting them stays invisible unless someone notices the resemblance by hand.

SSDS looks across a domain's whole equation collection at once, asking not "does this expression fit," but "is this same pattern quietly showing up more than once, because it reflects something real about how this domain behaves."

A repeated symbolic structure may be more than a repeated expression.SSDS research premise

03 / Product

What SSDS actually produces.

The output is not only a fitted equation. SSDS is designed to produce a characterized, reusable candidate abstraction.

A phenomenon's recurring pattern

A pattern in how the domain's quantities relate, found because it kept showing up across separately-derived equations.

v₀ · (φ₁₄ + 1)^φ₁₅

Behavioral characterization

Mathematical behavior and invariants are extracted to assess whether the structure has meaningful reusable properties.

Candidate operator

Structures that survive novelty and reusability criteria are formalized as candidate operators.

O(x; θ) = recurring structure
status: candidate abstraction

Semantic interpretation

A language model receives the formal definition and discovered properties and proposes possible interpretations and roles within the domain's emerging mathematical framework.

04 / Discovery

Search for the structure between equations, not only the equation itself.

This changes the object of interest. A symbolic regression system can search for an expression describing data; SSDS asks whether expressions contain recurring structures that can be abstracted and reused.

Input equations

f₁(x) = a · S(x, θ)

f₂(x) = b · S(x, θ)

f₃(x) = c · S(x, θ)

f₄(x) = d · S(x, θ)

→

Candidate abstraction

S(x, θ)

repeated symbolic structure
+ behavioral properties
+ invariants

formalized for reuse across equations

05 / Reuse

One discovered structure can describe multiple equations.

The abstraction becomes useful precisely because it can be referenced across related expressions rather than rediscovered independently each time.

Equation A

y₁ = α · S(x, θ)

Equation B

y₂ = β + S(x, θ)

Equation C

y₃ = γ · S(x, θ)²

Operator

S(x, θ) → shared candidate structure

EVIDENCE

What SSDS has actually produced.

Three runs illustrate different capabilities: characterizing recurring structure, exposing hierarchical nonlinear architecture, and recovering a named physical quantity — the Lorentz factor — as one member of a discovered operator family.

Transformational self-similarity

13 relationships 4 transformation families

RECURRING STRUCTURE

F = v₀ exp[-(v₂ − v₃)² / (2v₁²)]

SSDS identified this recurring structure as a parameterized family and generated 13 mathematical relationships across first derivatives, second derivatives, integrals, and products.

The Gaussian itself was already present in the turbulence equation bank. The result is therefore not the discovery of the Gaussian, but the connected system of transformations reconstructed around it.

RECOVERED REPRESENTATION

F = Ae^−z²/2z = (x − μ) / σ

The family reduces to a dimensionless displacement z, while A, μ, and σ control magnitude, position, and scale.

FIRST DERIVATIVE∂F/∂v₀, ∂F/∂v₁, ∂F/∂v₂, ∂F/∂v₃

Differentiating each raw parameter separately tests amplitude, width, and the two location terms rather than one spatial variable.

SECOND DERIVATIVE∂²F/∂vᵢ², for each parameter

The second-derivative results are parameter-specific expressions from the four-parameter operator, not one universal z² − 1 form.

INTEGRATIONError-function forms

Produces a cumulative counterpart of the recovered structure.

MULTIPLICATIONF² = A²e^{−(x−μ)²/σ²}

Suggests the Gaussian form persists under multiplication, though this relationship remained open and unverified in the run rather than fully closed.

What the run shows

SSDS reconstructed a transformationally self-similar symbolic system: differentiation changes algebraic weighting while preserving the exponential envelope, integration produces a cumulative counterpart, and multiplication produces a rescaled member of the same family.

Scope: This does not establish a new law of turbulence. It demonstrates structure recognition followed by mathematical characterization and relationship reconstruction. The A/z/μ/σ form is an illustrative simplification; the run differentiated the raw four-parameter operator.

Hierarchical nonlinear architecture

Novelty: 88.82 related members

HIGHEST-RANKED SURVIVING STRUCTURE

F = φ₅ + (φ₂e^−v₃ + φ₃ + φ₄ − e^{v₀(−v₁v₂ + φ₁)})²

SSDS recovered this structure from multiple expressions whose variables and numerical parameters differed. Its architecture is layered:

01Interactionφ₁ − v₁v₂

→

02Exponentiale^{v₀(φ₁−v₁v₂)}

→

03Combinationlinear terms + exponential

→

04Magnitude(...)²

→

05Offset+ φ₅

INTERACTION VARIABLEv₁v₂

The response depends on a combined interaction rather than treating the variables independently.

NESTED TRANSFORMATIONexp → combine → square

Multiple nonlinear transformations are composed into a single mathematical architecture.

BROADER RUN5 surviving structures

The run also produced logarithmic-square and square-root/fractional-power families.

What the run suggests

SSDS can move beyond isolated expressions toward a hierarchical description of how nonlinear transformations are composed—a candidate mathematical architecture for interpreting amplification, suppression, interaction, and scale.

STRUCTURAL RECURRENCE✓ Identified

NOVELTY FILTERING✓ Survived

RELATED MEMBERS✓ 2 found

CLOSURE TESTS⚠ Incomplete

Scope: Integration, product closure, power closure, and several parameter-shift tests timed out or failed for the highest-novelty structure. The result is therefore best described as a recurring hierarchical symbolic architecture and candidate mathematical representation, not a verified new mathematical law.

Recovering the Lorentz factor

Novelty: 55.410 related members7 invariants · 3 families

RECURRING STRUCTURE

v₀(1 − v₁²φ₄)^φ₁

Run on a special-relativity equation bank, SSDS generalized this structure from 10 member expressions that appeared across the bank in forms such as v₀/√(1 − v₁²), v₀√(1 − v₁²), v₀√(−v₁²/v₂² + 1), and v₀/√(−v₁²/v₂² + 1) — square roots, reciprocal square roots, and their rescaled variants, all folded into one exponent parameter φ₁.

The structure was not handed a physics label. It emerged purely from recurring symbolic form, and its exponent turns out to select between two named relativistic quantities depending on its value.

BEHAVIORAL PROFILE

bounded, boundedness = 1.00periodic · variance 21.8 · sensitivity 631.6

SSDS's numerical sampling classified the family as bounded and periodic over the tested input range — consistent with a quantity that stays finite as long as the velocity ratio stays under the light-speed limit.

RECOVERED REPRESENTATIONγ = 1/√(1 − β²)

φ₁ = −1/2Lorentz factor γ

φ₁ = +1/2Reciprocal 1/γ

v₁ = βVelocity ratio v/c

AMPLITUDE DERIVATIVE∂Op/∂v₀ = Op/v₀

Differentiating with respect to the leading coefficient returns a scaled copy of the operator itself — closed.

VELOCITY DERIVATIVE∂Op/∂v₁ = 2v₁φ₁φ₄·Op/(v₁²φ₄ − 1)

Differentiating with respect to the velocity-ratio term closes to an operator-scaled expression — closed.

SCALE-PARAMETER DERIVATIVE∂Op/∂φ₄ = v₁²φ₁·Op/(v₁²φ₄ − 1)

The same closure pattern holds for the auxiliary scale parameter φ₄ — closed.

EXPONENT DERIVATIVE∂Op/∂φ₁ = Op·log(1 − v₁²φ₄)

Differentiating the exponent itself introduces a logarithmic factor of the base — closed.

PARAMETER RECURSION (−1)Op(φ₁−1) = Op(φ₁)/(1−v₁²φ₄)

Decreasing the exponent by one divides the expression by the base term — closed.

PARAMETER RECURSION (+1)Op(φ₁+1) = Op(φ₁)·(1−v₁²φ₄)

Increasing the exponent by one multiplies by the same base term — closed.

POWER RELATIONOp² = (v₀(1−v₁²φ₄)^φ₁)²

The squaring relationship was tested but marked open, not closed, in this run.

What the run shows

Six of the seven invariants — all four derivatives and both parameter-recursion shifts — closed as operator laws, spanning 2 of the 3 discovered families. SSDS treated the Lorentz factor not as a single formula to fit, but as one instance of a parameterized operator family whose exponent continuously (via differentiation) and discretely (via integer shifts) connects it to the reciprocal relativistic quantity 1/γ.

Scope: The power/squaring invariant for this operator remained open — it was tested but did not pass closure verification in this run, unlike the six derivative and recursion invariants. The physical labels (γ, β) are an interpretive overlay applied here for readability; SSDS's raw output is the parameterized symbolic operator v₀(1−v₁²φ₄)^φ₁ and its invariants, not the relativity terminology itself.

01Characterize

Reconstruct mathematical relationships around recurring structures.

02Abstract

Identify higher-level architectures shared across expressions.

03Reuse

Represent structurally related equations through common candidate abstractions.

06 / The core move

From expression, to structure, to abstraction.

Every candidate operator SSDS proposes follows the same derivation. Reading it top to bottom is reading how SSDS thinks.

01Expression

a · (x + 1)^b

One equation, taken at face value.

↓ generalize the constants that vary

02Structure

(φ + 1)^ψ

The shape that remains once the specific coefficients are treated as free parameters.

↓ check whether the shape reappears

03Recurrence

appears across 17 of the equations examined

A structure earns attention by recurring — a single instance is just an expression.

↓ formalize with named parameters

04Abstraction

S(x; φ, ψ)

A named, parameterized object that can be referenced instead of rederived.

↓ substitute back in

05Reuse

S(x; φ, ψ)

appears as a shared operator inside equations that once looked unrelated.

07 / System

The pipeline from equations to candidate operators.

Six stages, each narrowing the set of candidates. Open a stage to see what enters, what happens to it, and what leaves.

Equation collection

Assemble the domain's equations.›

inputa domain's known equations

processparse into a common symbolic representation

outputan equation set ready for comparison

Structural discovery

Find shapes that recur across equations.›

input120 equations

processidentify recurring symbolic structures

outputcandidate structural families

Characterization

Extract behavior and invariants.›

inputcandidate structural families

processextract behavioral properties and algebraic invariants

outputcharacterized candidates

Filtering

Discard what isn't novel or reusable.›

inputcharacterized candidates

processdiscard structures that fail novelty or reusability criteria

outputsurviving candidates

Formalization

Represent survivors as operators.›

inputsurviving candidates

processexpress each as a named, parameterized operator

outputO(v, φ, ψ)

Interpretation

Propose what the operator might mean.›

inputformal operator definition + properties

processprompt a language model for semantic hypotheses and domain roles

outputcandidate interpretations, not conclusions

08 / Research

Different search object. Different research question.

SSDS is concerned with recurring symbolic structure and candidate operator construction, rather than only expression-level discovery.

Capability

SSDS

Expression-level SR

Research purpose

Search space

Operator / structure space

Expression space

Find reusable structure

Cross-equation structure

Explicitly investigated

Not the primary object

Identify recurrence

Algebraic characterization

Behavior + invariants

Not the primary output

Assess candidate abstraction

Reusable operators

Candidate operators

Individual expressions

Build domain abstractions

Status

Research prototype · candidate abstractions

SSDS is an exploratory research system. A recurring structure is not automatically a new mathematical law, and generated semantic interpretations are hypotheses rather than validated domain theories. The system is intended to support mathematical exploration and hypothesis generation, not replace mathematical validation.

Continue the research

Read how the discovery pipeline is implemented.

Documentation covers the discovery process, operator characterization, and the format used to represent candidate abstractions.

Open documentation →