A Refuted Golden-Ratio Floating-Point Conjecture, Exactly Reversible Integer Maps, and a Fixed-Width Lifting Wavelet

This is an expository paper with a negative result at its centre. It began from a geometric picture of reversible computation, a state that expands by the golden ratio φ = (1+√5)/2 and folds back by 1/φ, and from the conjecture that scaling by φ gives better round-trip accuracy in binary floating point than scaling by other factors. That conjecture is refuted. For six scale factors and inputs in [1,2), scaling by 2 restores every value, and φ is not consistently better than √2, e, π or 3: it fails to restore 15% of double-precision values in one variant and 37% in another. What survives is algebra that is not new. φ is a unit of the ring Z[φ], so multiplication by φ is the integer map (a,b)↦(b,a+b) with integer inverse (a,b)↦(b-a,a), exactly reversible on unbounded integers and on fixed-width words. In the two real conjugate coordinates of the ring the same map expands one coordinate by φ and contracts the other by 1/φ, which is the precise form of the starting picture. The map is the linear case of the lifting, or Feistel, step (a,b)↦(b,a+f(b)), invertible for every f, which underlies integer wavelets, block ciphers and reversible networks. Every integer matrix of determinant ±1 shares the property; φ is the example of smallest spectral radius. One case study is measured. The reversible 5/3 lifting wavelet restores data exactly in wrapping fixed-width arithmetic, and the question is what a fixed word costs. On the 24 Kodak images and ten MIT-BIH electrocardiogram records, with the update functions evaluated without intermediate overflow, storing the coefficients in the input's own word width costs a median of 0.025 bit per sample on the images and nothing measurable on the records, in adaptive order-0 code length per subband, relative to unbounded width. Naive wrapping costs 1 to 2 bits per sample. Modular-arithmetic wavelets are prior art, which is cited; the measurement is of how the update functions must be evaluated for the idea to cost nothing. Mathematical reversibility is not thermodynamic reversibility, and nothing here implies that a machine built this way dissipates less energy.