Two model results: missing temperatures and change sensing

Conditional mathematical results · Physical hypotheses remain unproved.

Study 2 includes two completed mathematical analyses: reconstructing missing temperatures in a thermal network, and detecting changes through two swelling-response paths. Their value is a precise statement of when each idea works inside its model—and what makes it fail. Neither analysis includes a physical experiment or demonstrates a new device. The equations use established mathematical and transport principles.

N0755: missing temperatures require enough known information

Energy conservation alone cannot recover arbitrary missing measurements. The supplied result concerns a finite, steady thermal network with known positive conductances and known net powers at unobserved nodes. Every connected component containing an unknown temperature must also contain a measured-temperature anchor.

Under those assumptions, the grounded network equations have one unique solution. If only the anchor temperatures are uncertain, reconstructed temperatures do not exceed the largest anchor error in absolute value. This error bound assumes powers and conductances are exact; their uncertainty is a separate issue.

A constructed chain with unit conductances, a zero-temperature anchor and known end power 4 gives successive temperatures 4, 8 and 12 in consistent arbitrary model units. The limiting cases explain the scope: a component without an anchor can shift by an arbitrary constant; unknown heat losses or powers remove information needed for reconstruction; a weak thermal link can amplify power error. A small balance residual is not proof that a physical model is correct. Read the full derivation, examples and failure cases.

N0751: a change sensor has a finite response band

The second model subtracts two stable, first-order swelling responses driven by the same input. Their equilibrium gains are matched, while their positive response times differ. At equilibrium a constant input cancels; a step produces a temporary differential response.

The equations also show that this is a finite-band change detector. Both zero-frequency and arbitrarily high-frequency responses vanish. With gains and step amplitude set to 1, and time constants 1 and 4, the modeled transient peaks near time 1.8484 with amplitude 0.47247. The maximum frequency gain is 0.6 at angular frequency 0.5. These are model values, not sensor calibration measurements.

Equal time constants eliminate the ideal change signal. Unequal equilibrium gains leave a static false signal. Temperature drift, hysteresis, input-delivery delay or changing geometry can invalidate the two-path approximation. The smooth-input derivative approximation has an explicit error bound, but does not apply to arbitrary sharp changes. No chemical specificity or noise floor has been demonstrated. See the response equations and counterexamples.

What was checked

An internal review read both derivations and reran an isolated copy of the supplied standard-library code once. Four exact network examples and the supplied summary matched exactly. Swelling checks covered two numerical step sizes and 241 frequencies; regenerated floating-point tables agreed within the documented tolerance, with tiny last-digit differences recorded. These finite checks supplement the written derivations. They are not formal verification, external peer review or certification of novelty. The archive preserves the originals, code, result tables and provenance.

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