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

Let p_1 < p_2 < ··· be the sequence of primes, let g_i = p_(i+1) - p_i, and define K_i := [p_i√p_(i+1) - p_(i+1)√p_i] / [p_i + p_(i+1)]. 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. To prevent notational collisions, the Heron radicand is denoted by R_i and the hyperbolic detector by D_i(σ). We prove K_i = [√(p_i p_(i+1)) / (p_i + p_(i+1))] × [√p_(i+1) - √p_i] = [g_i√(p_i p_(i+1))] / [(p_i + p_(i+1))(√p_i + √p_(i+1))], √R_(i+1) - √R_i = γg_i / 2, g_i = γ[tan θ_(i+1) - tan θ_i], and the exact hyperbolic representation K_i = [g_i(p_i p_(i+1))^(1/4)] / [2(p_i + p_(i+1)) cosh((1/4) log(p_(i+1)/p_i))]. The infinite fan of rays through consecutive primes is also formalized. If ω_i = θ_(i+1) - θ_i, then Σ_(i=m)^∞ ω_i = arctan[γ / (p_m - r)], and sin ω_i = γg_i / (a_i a_(i+1)), yielding a global weighted identity for the gaps and a rigorous chord-sector-segment construction. We then obtain an exact reduction of the extremal problem. If K* = K(7,11), the inequality K_i ≤ K* is equivalent to an explicit prime-gap condition g_i ≤ G*(p_i), where G*(7) = 4 and G*(p) = 4K*√p + 4(K*)² + O(p^(-1/2)). Thus (7,11) lies exactly on the proposed extremal frontier. The remaining logical gap is unambiguous: prove g_i ≤ G*(p_i) for every consecutive-prime pair. Geometry, trigonometry, hyperbolic identities, and finite computation are not substituted for this universal arithmetic step. We also audit the new handwritten procedure based on F(x,g) = √(x+g) - √x and on power comparisons. It is proved rigorously that, for fixed gap g, both F(x,g) and K(x,x+g) decrease as x increases. In particular, (7,11) is the maximum inside the stratum g = 4. However, F increases with g when x is fixed, so the passage from g = 4 to all prime gaps requires an additional arithmetic inequality. We further prove that a global bound K_i ≤ K(7,11) would imply Legendre’s conjecture. Thus the new procedure strengthens and localizes the remaining gap, but power identities alone do not eliminate it. We also formalize the handwritten pattern that starts from the sum p_(i+1) + p_i and generates p_(i+1)^m + (-1)^(m+1)p_i^m, for m = 1,2,3,... The name OrHi constant is reserved exclusively for the oriented half-power observable O_i = √p_(i+1) - √p_i, with reference value C_OrHi = √11 - √7. 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. We also introduce the normalized quotient N_i = [√p_(i+1) - √p_i] / [p_i + p_(i+1)], prove its exact rationalized identity, its monotonicity for fixed gap, and the unconditional limit N_i → 0 along the full sequence of consecutive primes. A universal proof using Nagura is added: for consecutive primes with leading binary exponent n > 1, equality T = R occurs only at (7,11). The sign classification is also proved and the five recent photographs are integrated. These theorems are not identified with a proof of the global maximum of K or of Andrica’s conjecture. 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.