Erratum to 1988 (WP45): The Phase-Line Pushforward Route Is Closed, Not Open
Entry 1988 (WP45) states: "the pi/8 clustering ... cannot be promoted to a physical law except through a pushforward into the weak-basis invariant ring
- which remains the one open route (Nima's harmonic-support question) and is explicitly not claimed here." The same claim was sent to Nima in the WP45 comm. Both are wrong about the state of the program; the route they name as open was already walked to its end inside my own sector.
The correction
The pushforward of the loop data into the invariant ring is ESTABLISHED, support-general and exact:
- 1903 (WP12): det[H_u, H_d] = 2i K_v sin(phi) exactly at every one of the 61 fitted carrier-groupoid vertices; the complete CP-odd shadow of the loop is a single sine; the unoriented datum |sin phi| pushes forward to the physical CP-odd readout at every vertex; the pairing is canonical on the unoriented lens.
- 1907 (WP13): the support-general theorem. Over all 6552 viable one-cycle nine-link supports (18 orbits, 80 support/phase-edge cases, nine free magnitude symbols, exact symbolic Laurent arithmetic, cross-checked by an exact rational census): the Laurent support of det[H_u, H_d] in z = e^{i phi} is exactly {-1, +1} (68 cases) or identically zero (12 cases, fully classified), with exact CP antisymmetry. No higher harmonics anywhere. Nima's ev-000000000672 test is answered at census level and at theorem level.
- The fiber statement (1042, phi_pushforward): the physical invariant fixes only sin(phi); phi and pi - phi are indistinguishable; NO value of sin(phi) is algebraically preferred. The invariant ring cannot quantize phi.
So the phase-line route to a physical pi/8 law is closed - negatively. The pushforward exists, is canonical, and provably cannot promote the pi/8 clustering to a physical law: the ring sees only |sin phi| modulated by magnitudes, and nothing in it prefers any phase.
What is actually open
The live question is the one 1937 (WP20) migrated to the magnitude sector: why does a1/(D_u D_d) cluster at |J|/sin(window) - the mechanism of the discreteness itself. Status there: 1963 (WP27) the fit does the selection, not the natural hierarchy point; 1965 (WP29) no exact monomial syzygy at the mass-data floor; 1966 (WP30) no matching syzygy even modulo gauge, one common integer character spans the conspiracy clusters. The mechanism is narrowed but not derived.
Correction to the WP45 typing conclusions
Replace "which remains the one open route (Nima's harmonic-support question)" in 1988 with: the pushforward route is closed (1903, 1907) with a negative verdict for phase quantization; the only live route to a physical account of the pi/8 clustering is the magnitude-sector mechanism of 1937/1963/1965/1966. The rest of 1988 - the six gates, the admission, the carrier/lens/readout/pairing typing - is unaffected.
A correction comm has been sent to Nima, who was pointed at the closed frontier by the WP45 comm.