# Verification report

All strictly positive C1 functions equimeasurable with their first derivative on [0,L] are parametrized by a positive initial value and a continuous measure-preserving map T with spectral radius r(K_T)<1. The twice-differentiable subclass consists only of exponentials and a phased sine family. For the tent map r(K_T)=L/2, giving the exact threshold L<2, sharp approximation errors and dense dyadic losses of higher differentiability. A rescaled classical Fabius distribution gives a smooth nonnegative example at L=2. A positive-operator construction supplies functions equimeasurable with every prescribed higher derivative on sufficiently short intervals.

Rubel's Problem7.77, Hayman-Lingham2018 p185: these complete theorems answer the explicit request for further examples and classify strictly positive C1 functions. The general classification of vanishing or sign-changing functions is not solved. The whole frozen record remains open.

# Detailed originating-researcher self-audit

1. Equimeasurability is equality of pushforward Lebesgue measures, not pointwise equality. Continuous functions on a compact interval have range equal to the support of their pushforward measure: every neighborhood of an attained value has positive-measure preimage, also at an endpoint. Thus positive f and equimeasurable continuous f' have the same strictly positive minimum. It follows that f is strictly increasing with positive derivative and an inverse.
2. T=f^-1 composed with f' is continuous into the same interval. The strict-sublevel calculation m{T<y}=y for every y in[0,L] determines the uniform pushforward measure, including no atoms at the endpoints. No independent assertion of measure preservation is assumed in this direction.
3. K_T is a positive bounded operator. From f=a+K_Tf, the equivalent weighted norm |h|/f bounds its norm by1-a/max(f)<1. The spectral radius is unchanged by equivalent norms. Conversely r(K_T)<1 implies operator-norm convergence of the Neumann series by the spectral-radius formula; positivity and the fundamental theorem of calculus give f>=a and f'=fT. Uniqueness follows from invertibility of1-K_T. This is an exact parametrization but not a closed formula for arbitrary spectral radii.
4. For continuous measure-preserving T, a closed interval J lies inside the inverse image of its interval image. Hence |T(J)|>=|J|. Differentiability with |T'|<1 contradicts this using a sufficiently small centered interval. Derivatives have the Darboux property even without continuous derivative. Therefore T' has one sign, T is strictly monotone by the mean value theorem, and preservation of initial-interval lengths forces identity or reflection.
5. Twice differentiability of positive f ensures differentiability of T=f^-1 f'. Identity gives an exponential. Reflection gives f''=-f and midpoint data f'=f>0, hence c sin(x+pi/4-L/2). Positivity of both f and f' throughout the closed interval forces L<pi/2. This excludes zero endpoint values; the classical sine on length pi/2 is not a counterexample to the restricted theorem.
6. For the tent map, each inverse branch contributes half the length. K^n1 is nonnegative and increasing. For n>=1 its derivative is symmetric, giving h_n(x)+h_n(L-x)=h_n(L). The endpoint recurrence is valid for n>=2 and begins at h_1(L)=L. Positivity then gives ||K^n||=L^n/2^(n-1), so r(K)=L/2 for every L>0. This proves necessity as well as sufficiency of L<2 for a strictly positive solution with prescribed positive initial value.
7. Each positive series tail is increasing and maximized at L, so geometric summation gives exact, not merely upper, uniform errors. T is onto and S_N'=S_(N-1)T for N>=1, giving the sharp derivative error. Standard-library rational code independently binds S8's endpoint54613/32768 to exact endpoint5/3 and errors1/98304,1/24576. Its separate coarser bounds remain valid.
8. At a point first reaching the tent midpoint after m iterates, the preceding m branch choices are constant locally. Repeated chain differentiation supplies C_m f(T^(m+1)x), with |C_m|=2^(m(m+1)/2). The last iterate has a corner at value L with slopes of opposite sign and magnitude2^(m+1); f'(L)=a. Thus the derivative of orderm+2 has nonzero absolute jump2a*2^((m+1)(m+2)/2). Lower derivatives are continuous locally. First-hit points are exactly interior dyadic multiples of L and dense. This establishes lack of analyticity on every open subinterval without claiming failure of every derivative at every point.
9. The Fabius random series is supported in[0,1], is nonatomic because it includes an independent uniform summand, and is symmetric. Its distributional recursion yields F'=2(F(2x)-F(2x-1)) for the extension constant outside[0,1]. Continuity bootstraps to global C-infinity. f(x)=F(x/2) on[0,2] satisfies f'=fT2 by the two branches and symmetry. It has f(0)=f'(0)=0 and f(2)=1, so is neither a nontrivial exponential nor a sine solution of f''=-f. This classical construction is credited and is not claimed as a newly discovered smooth function.
10. For derivative order r the positive r-fold integral operator has norm L^r/r!, since its application to1 is x^r/r!. The sufficient condition q<1 yields the unique C^r solution with the stated initial data and positivity. At the tent midpoint its r-th derivative has unequal one-sided derivatives since f'(L)>0. This is a sufficient short-interval construction, not an optimal all-length classification for r>1.
11. Finite rational checks verify recurrences, continuity, initial data, moments within certified errors,36 norm powers,21 monomials and affine slope weights. They do not prove the infinite-series theorem by extrapolation. The complete analytic arguments are in the paper. No independent peer review or formal verification is claimed.

No gap was found in these scoped written arguments. The full unrestricted source problem is not closed.

# Bounded literature review,25-26October2026

Primary sources used: Hayman-Lingham arXiv1809.07200v2, Problem7.77, p185; Fabius1966, DOI10.1007/BF00536652, institutional repository https://ir.cwi.nl/pub/8160. Queries included Rubel differential rearrangements; derivative equimeasurable function Rubel; function equimeasurable with its derivative; function and its derivative same distribution; Rubel rearrangement function derivative sine; derivative is a rearrangement function; equimeasurable derivative tent; differential rearrangement mathematical functions; Rubel measure-preserving functions; Fabius function equimeasurable derivative Rubel; f'(x) tent map function. These searches found no matching treatment of the exact positive-class parametrization and tent threshold. This is a bounded search, not an exhaustive novelty certificate. Related derivative-rearrangement inequalities do not by themselves answer equality of the distributions of f and f'.

The Fabius function, tent map, contraction/Neumann argument and spectral-radius formula are not new. Their use is credited. The paper's claimed contributions are precisely the written parametrization, differentiable-map rigidity, exact norm powers and regularity of the positive tent family, and the differential-rearrangement application. The general classification is explicitly outside scope; the source record must not be marked fully solved.

All five final PDF pages were directly visually inspected. No clipping, missing symbols, unresolved citations or layout defects were found. The built-in compiler and Tectonic succeeded. Author name is prominent; affiliation/contact is a footnote; AI assistance appears in prose.