An Integrated Program for the Supremum of a Normalized Prime-Gap Functional Andrica, Heron, Angular Fan, Alternating Power Ladder, the OrHi Constant, and Exact Arithmetic Reduction

<p dir="ltr">Let p_1 < p_2 < ··· be the sequence of primes, let</p><p><br></p><p dir="ltr">g_i = p_(i+1) - p_i,</p><p><br></p><p dir="ltr">and define</p><p><br></p><p dir="ltr">K_i := [p_i√p_(i+1) - p_(i+1)√p_i] / [p_i + p_(i+1)].</p><p><br></p><p dir="ltr">This manuscript integrates in one chain the algebraic identities of the functional, its exact relation with the Andrica difference, a Euclidean construction with 0 < r < 1, Heron’s formula, semiperimeter variation, an angular parametrization, and a hyperbolic detector abstracted and renamed from a parallel zeta-function program.</p><p><br></p><p dir="ltr">To prevent notational collisions, the Heron radicand is denoted by R_i and the hyperbolic detector by D_i(σ).</p><p><br></p><p dir="ltr">We prove</p><p><br></p><p dir="ltr">K_i = [√(p_i p_(i+1)) / (p_i + p_(i+1))] × [√p_(i+1) - √p_i]</p><p><br></p><p dir="ltr">= [g_i√(p_i p_(i+1))] / [(p_i + p_(i+1))(√p_i + √p_(i+1))],</p><p><br></p><p dir="ltr">√R_(i+1) - √R_i = γg_i / 2,</p><p><br></p><p dir="ltr">g_i = γ[tan θ_(i+1) - tan θ_i],</p><p><br></p><p dir="ltr">and the exact hyperbolic representation</p><p><br></p><p dir="ltr">K_i = [g_i(p_i p_(i+1))^(1/4)] /</p><p dir="ltr"> [2(p_i + p_(i+1)) cosh((1/4) log(p_(i+1)/p_i))].</p><p><br></p><p dir="ltr">The infinite fan of rays through consecutive primes is also formalized.

If</p><p><br></p><p dir="ltr">ω_i = θ_(i+1) - θ_i,</p><p><br></p><p dir="ltr">then</p><p><br></p><p dir="ltr">Σ_(i=m)^∞ ω_i = arctan[γ / (p_m - r)],</p><p><br></p><p dir="ltr">and</p><p><br></p><p dir="ltr">sin ω_i = γg_i / (a_i a_(i+1)),</p><p><br></p><p dir="ltr">yielding a global weighted identity for the gaps and a rigorous chord-sector-segment construction.</p><p><br></p><p dir="ltr">We then obtain an exact reduction of the extremal problem.</p><p><br></p><p dir="ltr">If</p><p><br></p><p dir="ltr">K* = K(7,11),</p><p><br></p><p dir="ltr">the inequality</p><p><br></p><p dir="ltr">K_i ≤ K*</p><p><br></p><p dir="ltr">is equivalent to an explicit prime-gap condition</p><p><br></p><p dir="ltr">g_i ≤ G*(p_i),</p><p><br></p><p dir="ltr">where</p><p><br></p><p dir="ltr">G*(7) = 4</p><p><br></p><p dir="ltr">and</p><p><br></p><p dir="ltr">G*(p) = 4K*√p + 4(K*)² + O(p^(-1/2)).</p><p><br></p><p dir="ltr">Thus (7,11) lies exactly on the proposed extremal frontier.</p><p><br></p><p dir="ltr">The remaining logical gap is unambiguous: prove</p><p><br></p><p dir="ltr">g_i ≤ G*(p_i)</p><p><br></p><p dir="ltr">for every consecutive-prime pair.</p><p><br></p><p dir="ltr">Geometry, trigonometry, hyperbolic identities, and finite computation are not substituted for this universal arithmetic step.</p><p><br></p><p dir="ltr">We also audit the new handwritten procedure based on</p><p><br></p><p dir="ltr">F(x,g) = √(x+g) - √x</p><p><br></p><p dir="ltr">and on power comparisons.</p><p><br></p><p dir="ltr">It is proved rigorously that, for fixed gap g, both</p><p><br></p><p dir="ltr">F(x,g)</p><p><br></p><p dir="ltr">and</p><p><br></p><p dir="ltr">K(x,x+g)</p><p><br></p><p dir="ltr">decrease as x increases.</p><p><br></p><p dir="ltr">In particular, (7,11) is the maximum inside the stratum</p><p><br></p><p dir="ltr">g = 4.</p><p><br></p><p dir="ltr">However, F increases with g when x is fixed, so the passage from g = 4 to all prime gaps requires an additional arithmetic inequality.</p><p><br></p><p dir="ltr">We further prove that a global bound</p><p><br></p><p dir="ltr">K_i ≤ K(7,11)</p><p><br></p><p dir="ltr">would imply Legendre’s conjecture.</p><p><br></p><p dir="ltr">Thus the new procedure strengthens and localizes the remaining gap, but power identities alone do not eliminate it.</p><p><br></p><p dir="ltr">We also formalize the handwritten pattern that starts from the sum</p><p><br></p><p dir="ltr">p_(i+1) + p_i</p><p><br></p><p dir="ltr">and generates</p><p><br></p><p dir="ltr">p_(i+1)^m + (-1)^(m+1)p_i^m,</p><p><br></p><p dir="ltr">for</p><p><br></p><p dir="ltr">m = 1,2,3,...</p><p><br></p><p dir="ltr">The name OrHi constant is reserved exclusively for the oriented half-power observable</p><p><br></p><p dir="ltr">O_i = √p_(i+1) - √p_i,</p><p><br></p><p dir="ltr">with reference value</p><p><br></p><p dir="ltr">C_OrHi = √11 - √7.</p><p><br></p><p dir="ltr">An exhaustive sieve through 10^8 verifies this value as the maximum among 5,761,454 consecutive pairs in the range; this is finite evidence and not a universal proof.</p><p><br></p><p dir="ltr">We also introduce the normalized quotient</p><p><br></p><p dir="ltr">N_i = [√p_(i+1) - √p_i] / [p_i + p_(i+1)],</p><p><br></p><p dir="ltr">prove its exact rationalized identity, its monotonicity for fixed gap, and the unconditional limit</p><p><br></p><p dir="ltr">N_i → 0</p><p><br></p><p dir="ltr">along the full sequence of consecutive primes.</p><p><br></p><p dir="ltr">A universal proof using Nagura is added: for consecutive primes with leading binary exponent</p><p><br></p><p dir="ltr">n > 1,</p><p><br></p><p dir="ltr">equality</p><p><br></p><p dir="ltr">T = R</p><p><br></p><p dir="ltr">occurs only at</p><p><br></p><p>(7,11).</p><p><br></p><p dir="ltr">The sign classification is also proved and the five recent photographs are integrated.</p><p><br></p><p dir="ltr">These theorems are not identified with a proof of the global maximum of K or of Andrica’s conjecture.</p><p><br></p><p dir="ltr">Keywords: consecutive primes; prime gaps; normalized functional K_i; Andrica function; Heron’s formula; semiperimeters; complex plane; critical strip; trigonometric parametrization; angular fan; chords; circular sectors; hyperbolic detector; sinh and cosh; extremal reduction; uniform arithmetic inequality; supremum; computational verification; OrHi constant; half-power; alternating signs.</p>