dT_dx ↔ x
Tier II — Similarity CONJECTURAL
- Object
- heat_conducting_medium
- Property
- TemperatureGradient
- Context
- near_equilibrium
- Object
- rigid_body
- Property
- Position
- Context
- inertial_frame
- Constraint
- displacement_from_equilibrium
Sieve rationale
Structural equation-level identity (sign=+1): the canonical forms of EQ-FOURIER-HEAT and EQ-HOOKE are syntactically equal under rename, and dT_dx maps to x under the discovered bijection
Physical constraint filter
At least one variable is not a bond-graph port variable (role_a=gradient, role_b=state); Tellegen pairing is only defined for effort/flow.
Onsager reciprocity applies only to pairs of linear transport laws; at least one equation is not a linear PDE.
Energy conservation check is narrowly targeted at the Newton+Hooke → SHO composite in Medium m2; other composites require the Phase 7 Lyapunov/Noether machinery (not yet built).
Emergent properties
Emergent Buckingham Π groups
Dimensionless combinations that mix parameters from both equations — only appear in the composite system.
AI review board
3 REJECT vote(s); the coupling is demoted to conjectural per §12.2.
This coupling maps a temperature gradient (dimension [temperature]/[length]) to a displacement/position (dimension [length]) via an identity transfer function, which is physically nonsensical — an identity map requires dimensional compatibility, and these quantities differ by two fundamental dimensions. The coupling is based purely on a syntactic bijection between the canonical forms of Fourier's law and Hooke's law, which, while structurally interesting, does not establish a physically meaningful relationship between these specific variables. The physical constraint filter passed no checks (all returned not_applicable), meaning there is zero physical evidence supporting this coupling. The claimed emergent pi groups are simply restating the original equations' dimensionless forms and do not arise from any genuine cross-domain interaction. A legitimate analogy between these two linear constitutive laws does exist (thermal resistance ↔ spring compliance), but it would require a properly defined analogical transfer function with explicit dimensional scaling, not an identity map between a gradient quantity and a positional state variable.
Although Fourier’s law (q = -k dT/dx) and Hooke’s spring law (F = -k x) share a linear form, mapping a temperature gradient to a mechanical displacement via identity is not physically or dimensionally meaningful (K/m vs m) and the semantic descriptors do not denote the same quantity; a valid analogy would pair gradient with strain (∂u/∂x) or temperature/force-like potentials under a named transform, not identity. The physical-constraint filter’s non-applicability outcomes and failure to pass are consistent with this mismatch, and the cited Buckingham-π groups are trivial restatements of the constitutive relations rather than genuine emergent properties.
The proposed coupling is physically nonsensical as it equates a temperature gradient (dT_dx) with a displacement (x). These quantities possess fundamentally different physical dimensions ([temperature]/[length] versus [length]), rendering a direct identity mapping a violation of dimensional analysis. The noted structural similarity between Fourier's Law and Hooke's Law is a purely mathematical analogy, not representative of a valid physical cross-domain interaction. Consequently, the coupling lacks physical plausibility, and any claimed emergent properties are merely artifacts of this invalid algebraic substitution.