This is the Lead Proposal for the investigation "Primordial Inflation Polarization Explorer (Phase 3)". We propose to complete and fly the Primordial Inflation Polarization Explorer (PIPER) to measure the polarization of the cosmic microwave background (CMB) and search for the imprint of gravitational waves produced during an inflationary epoch in the early universe. Detection of the inflationary signal would have profound consequences for both cosmology and high-energy physics. Not only would it establish inflation as a physical reality, it would provide a direct, model-independent determination of the relevant energy scale, shedding light on physics at energies twelve orders of magnitude beyond those accessible to direct experimentation in particle accelerators. The recent detection of CMB polarization by the BICEP2 instrument brings new urgency to the field. The BICEP2 detection at degree angular scales is consistent with inflation, but the amplitude is a factor of two higher than upper limits set by unpolarized data. A critical test is the rise in power at large angular scales predicted by inflation. Detecting this rise would confirm the signal's inflationary origin, fulfilling a long quest for cosmology while providing new insight into physics at the highest energies. PIPER is the only suborbital instrument capable of measuring CMB polarization on the large angular scales needed to test an inflationary origin for the BICEP2 detection. PIPER is a balloon-borne instrument, optimized to detect the inflationary signal on large angular scales. It consists of two co-aligned telescopes cooled to 1.5 K within a large liquid helium bucket dewar. A variable-delay polarization modulator (VPM) on each telescope chops between linear and circular polarization to isolate the polarized signal while rejecting the much brighter unpolarized emission. Four 32 x 40 element detector arrays provide background-limited sensitivity. A series of flights from mid-latitude sites will map the full sky at frequencies 200, 270, 350, and 600 GHz to allow separation of CMB signals from the dominant dust foreground while providing new information on the diffuse dust cirrus. PIPER's innovative architecture combines cryogenic optics with kilo-pixel detector arrays to provide unprecedented sensitivity to CMB polarization. The fast modulation between linear and circular polarization takes advantage of the lack of astrophysical circular polarization to eliminate common sources of systematic error while enabling mapping on the largest angular scales. The sensitivity and control of systematic errors in turn enable measurements over most of the sky from mid-latitude launch sites; long-duration Antarctic flights are not required. With sensitivity r < 0.007 at 95% CL, PIPER will either confirm the inflationary signal or rule out nearly all large-field inflation models. The combination of sensitivity, sky coverage, and systematic error control provides unprecedented sensitivity to CMB polarization on angular scales greater than 20 deg, providing a critical test of the BICEP2 results against inflationary models. PIPER began development in 2009 and is nearly complete. End-to-end testing of detector arrays demonstrates the pixel yield, electrical, and thermal properties required for background-limited sensitivity. Cryogenics, gondola, and flight electronics use components flown on the successful ARCADE mission. The PIPER team has exceptional experience in all aspects of the proposed work, including detector development, polarization modulation, instrument integration, and cryogenic ballooning. The cryogenic optics provide instantaneous sensitivity comparable to a space mission while the full sky coverage tests systematic error control in ways not possible with restricted coverage: PIPER will probe the limits of sensitivity from a suborbital platform while developing instrumentation, observing techniques, and foreground models for an eventual space mission.