Abstract
Let p₁ Kᵢ := (pᵢ·√pᵢ₊₁ − pᵢ₊₁·√pᵢ) / (pᵢ + pᵢ₊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 Kᵢ = [√(pᵢ·pᵢ₊₁) / (pᵢ + pᵢ₊₁)] · (√pᵢ₊₁ − √pᵢ) = gᵢ·√(pᵢ·pᵢ₊₁) / [(pᵢ + pᵢ₊₁)(√pᵢ + √pᵢ₊₁)],
√Rᵢ₊₁ − √Rᵢ = γ·gᵢ / 2, gᵢ = γ(tan θᵢ₊₁ − tan θᵢ),
and the exact hyperbolic representation
Kᵢ = gᵢ·(pᵢ·pᵢ₊₁)^(1/4) / [2(pᵢ + pᵢ₊₁)·cosh( (1/4)·log(pᵢ₊₁/pᵢ) )].
The infinite fan of rays through consecutive primes is also formalized: if ωᵢ = θᵢ₊₁ − θᵢ, then
Σ (i = m to ∞) ωᵢ = arctan( γ / (pₘ − r) ), sin ωᵢ = γ·gᵢ / (aᵢ·aᵢ₊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 Kᵢ ≤ K* is equivalent to an explicit prime-gap condition gᵢ ≤ G*(pᵢ), where G*(7) = 4 and G*(p) = 4K*·√p + 4K*² + O(p^(−1/2)). Thus (7, 11) lies exactly on the proposed extremal frontier. The remaining logical gap is unambiguous: prove gᵢ ≤ G*(pᵢ) 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 Kᵢ ≤ 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 pᵢ₊₁ + pᵢ and generates pᵢ₊₁^m + (−1)^(m+1)·pᵢ^m for m = 1, 2, 3, ... . The name OrHi constant is reserved exclusively for the oriented half-power observable Oᵢ = √pᵢ₊₁ − √pᵢ, with reference value C_OrHi = √11 − √7. An exhaustive sieve through 10⁸ 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 Nᵢ = (√pᵢ₊₁ − √pᵢ) / (pᵢ + pᵢ₊₁), prove its exact rationalized identity, its monotonicity for fixed gap, and the unconditional limit Nᵢ → 0 along the full sequence of consecutive primes.
**Keywords:** consecutive primes; prime gaps; normalized functional Kᵢ; 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.