The Riemann Hypothesis as the Avoidance of an Information Entropy Collapse

The Riemann hypothesis is nature's way of avoiding an information entropy collapse: the entropy over the modes of the prime fluctuation is conserved in log x only if every zero carries the same weight. Shannon's bound makes the variance the currency of the problem: the smooth count of the primes is a first moment, a first moment holds no distribution, and so any statement about the size of the prime fluctuation is a statement about a discarded second moment, which for the primes is the set of zeros of the zeta function. The step is the one Perelman took for Ricci flow; he established the invariant, and here its cause is given: unitarity. The quantum connection is the reason a lock exists in the first place. The prime fluctuation is a state acted on by the matrix M(t) = diag(e^(δ_n t) e^(iγ_n t)); the variance is its norm, and it is conserved, with a fixed share for every mode, only when M is unitary, that is, when every zero carries the weight ½, the single fixed point of the two fold symmetry of the functional equation. The Riemann hypothesis is then the statement that the prime fluctuation is a closed quantum system. We measure the second moment directly on 100,000 zeros and on 10,000 zeros each near zero number 10^12, 10^21 and 10^22. Six results follow. 1. With every zero carrying the same weight, the variance of the scaled prime fluctuation is the same in every window of log x: 0.0460 with a spread of 0.0010 over eight windows up to log x = 30, against the closed form 0.0462. 2. A single mode moved off the line by δ grows in variance as x^(2δ). The measured growth factors agree with e^(6δ) per window to within half a percent. 3. The zeros interfere to produce the primes. At the 25 prime powers up to 60 the measured spike is 0.9995 of the predicted height with a scatter of 0.0006, and the smallest spike is 208 times the largest value at any other integer. 4. The zeros are stiff. Neighbours closer than a tenth of the mean spacing occur 120 times less often than without stiffness, the power of the complex amplitude in reciprocal space follows the unitary ramp, and the count variance saturates at a level fixed by the primes: 0.36 at height 4 × 10^4 and 0.52 near 10^21, as predicted. 5. The primes hold each zero in a harmonic well. The measured stiffness agrees with K = (2/|ρ|^2)⟨(log x)^2⟩ to within one percent for the low zeros. 6. One ancilla qubit, read in X and Y, returns both the primes and the ramp from a register that holds the zeros with equal weight. Forcing one mode off the line is not unitary, and the entropy over modes collapses from log N toward zero. The restoring force acts on the position of each zero. The stiffness between neighbours is a two point property, carried by the square of the prime sum and not by its linear response: that sum, with no zero in it, rebuilds the ramp of the first 100,000 zeros to within 0.006, including their position under the unitary ramp. By sieve to x = 10^9 the true primes hold one variance at every scale, 0.0461, where random primes are 338 times louder and growing, and the Möbius sum holds 0.0289 where random signs give 0.61. The Davenport and Heilbronn function, which has the same symmetry and no product over primes (no Euler product), shows the collapse with nothing moved: below height 2,000 one zero in twelve is off the line, its variance grows by a factor of 526, and its entropy over modes is not conserved. Its two components, the Dirichlet L functions modulo 5, each have a product over primes and each hold the lock: all 4,057 zeros found below that height are on the line, the variance stays at its closed form 0.2032, and the entropy is constant. The Davenport and Heilbronn function is the open system: a mode off the line leaks by its δ, and its entropy collapse is the removal, from the lowest δ up, of layers of modes that each keep their own lock. The unitary is read from the explicit formula itself, and the measured results fix what any operator of the spectral program must satisfy.