Established in scope
Existence and uniqueness of the positive root are proved
In the question: Alpha: derived terms, unique root and physical EM identification
Mathematical theorem for the fixed function; not a cubic polynomial because the seam response depends on x. No parameter or physical Thomson-transfer derivation; ledger Numerical unchanged.
Read the exact result, assumptions and evidence
Result
Log-convexity of the stated seam response and the sign before x=2c³ prove exactly one positive simple root globally. For M=41 its reciprocal is approximately 137.0359992168407. The analytic proof is distinct from the 10 rerun symbolic/interval checks. It holds for every M>0 and therefore does not select M=41.
Required assumptions
- Specified EM function with positive seam response
F(x)=x³−2c³x²+(4/5)c⁶M log(p+q(1−q)^(−5/4)), c=1/(8π), p=1/(6π), q=48c⁴exp(−2x), M>0; original budget M=41. The function and inputs are premises here.
Original evidence
- PROOF.txt lines 1–76
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/em-positive-root-convexity-20261013/PROOF.txt
c1ad319037418e7745c287132c4065cf6d963af797c5a4c81f12273e486d7b50
finding:alpha-global-positive-root-unique