Statement
Path A residual closure gates: H4 exhaustiveness + G1 anisotropic-Bragg Hartree + G2 phase/amplitude-pattern scan + G3 full-PDE ensemble comparison
Pillar: P4
Tier: T5
Details
[OPENED 2026-06-04 — Math424-AddA Steps 1–4 one-loop closure] Context: Math424-AddA established Reading-H minimality among the enumerated admissible readings (T5 CLOSED@1-LOOP PROVED CONDITIONAL on H2 single-shell equal-amplitude zero-phase ansatz, H3 isotropic trial gap, H4 enumeration). The H_RH-BCC-channel axiom is NOT yet promoted: its residual content is exactly (H4 + H2/H3 adequacy). Four gates remain: (Step 5 / H4) admissible-class exhaustiveness — no reading outside {LAM, HEX, FCC, BCC, R_H} satisfies cosmological isotropy + BCC-channel stability + gauge-sector emergence simultaneously (Math424-AddB no-go companion); (G1) anisotropic-Bragg Hartree refinement of the ordered-phase trial propagator (Math424-AddA-AddA candidate); (G2) phase/amplitude-pattern scan within the single shell (zero-phase maximises the coherent quartic sum, but a full scan is required for scope removal); (G3) full-PDE multi-shell ensemble free-energy comparison at the canonical point (connects the single-shell theorem to the Math400-AddF N=64 channel result).
[UPDATED 2026-06-04 — Math425 Cauchy–Schwarz no-condensation theorem]: operator hostile audit ACCEPTED (3 attacks + sentence defect; patches P1–P3 applied to Math424-AddA). Gate re-scoping: G2 CLOSED for mu^2 > 0.0571 — production convention; the −0.157 boundary was the offset-variable image, superseded by Math426 (CS theorem: J = <phi^4>^2/(<phi^2><phi^6>) ≤ 1 < 4rv/u^2 = 3.838 at canonical, ANY phases/amplitudes/spectrum; adversarial sup found 0.848 BCC / 0.808 HEX / 0.785 multi-shell, 18/18 asserts). H2 single-shell restriction DISSOLVED at mean-field level on mu^2 > mu^2*. Residual gates: G1' anisotropic-Bragg Hartree (ONLY mu^2 < −0.157 region); G3' Hartree-level ensemble comparison (MF component closed); G4 (NEW): Math400-AddF N=64 BCC "true local min" vs CS theorem reconciliation (discretisation/convention/constraint audit — a stationary point of the same continuum MF functional at canonical is forb
Source of truth
Docs/status/OPEN-QUESTIONS.md Active section, Q-2026-06-04-Math424-AddA-PathA-Residual-Gates row (raw).
Issue body is auto-synced from canonical source. Manual edits to the body will be overwritten on next sync. Discussion in comments is welcome and preserved.
Statement
Path A residual closure gates: H4 exhaustiveness + G1 anisotropic-Bragg Hartree + G2 phase/amplitude-pattern scan + G3 full-PDE ensemble comparison
Pillar: P4
Tier: T5
Details
[OPENED 2026-06-04 — Math424-AddA Steps 1–4 one-loop closure] Context: Math424-AddA established Reading-H minimality among the enumerated admissible readings (T5 CLOSED@1-LOOP PROVED CONDITIONAL on H2 single-shell equal-amplitude zero-phase ansatz, H3 isotropic trial gap, H4 enumeration). The H_RH-BCC-channel axiom is NOT yet promoted: its residual content is exactly (H4 + H2/H3 adequacy). Four gates remain: (Step 5 / H4) admissible-class exhaustiveness — no reading outside {LAM, HEX, FCC, BCC, R_H} satisfies cosmological isotropy + BCC-channel stability + gauge-sector emergence simultaneously (Math424-AddB no-go companion); (G1) anisotropic-Bragg Hartree refinement of the ordered-phase trial propagator (Math424-AddA-AddA candidate); (G2) phase/amplitude-pattern scan within the single shell (zero-phase maximises the coherent quartic sum, but a full scan is required for scope removal); (G3) full-PDE multi-shell ensemble free-energy comparison at the canonical point (connects the single-shell theorem to the Math400-AddF N=64 channel result).
[UPDATED 2026-06-04 — Math425 Cauchy–Schwarz no-condensation theorem]: operator hostile audit ACCEPTED (3 attacks + sentence defect; patches P1–P3 applied to Math424-AddA). Gate re-scoping: G2 CLOSED for mu^2 > 0.0571 — production convention; the −0.157 boundary was the offset-variable image, superseded by Math426 (CS theorem: J = <phi^4>^2/(<phi^2><phi^6>) ≤ 1 < 4rv/u^2 = 3.838 at canonical, ANY phases/amplitudes/spectrum; adversarial sup found 0.848 BCC / 0.808 HEX / 0.785 multi-shell, 18/18 asserts). H2 single-shell restriction DISSOLVED at mean-field level on mu^2 > mu^2*. Residual gates: G1' anisotropic-Bragg Hartree (ONLY mu^2 < −0.157 region); G3' Hartree-level ensemble comparison (MF component closed); G4 (NEW): Math400-AddF N=64 BCC "true local min" vs CS theorem reconciliation (discretisation/convention/constraint audit — a stationary point of the same continuum MF functional at canonical is forb
Source of truth
Docs/status/OPEN-QUESTIONS.mdActive section, Q-2026-06-04-Math424-AddA-PathA-Residual-Gates row (raw).Issue body is auto-synced from canonical source. Manual edits to the body will be overwritten on next sync. Discussion in comments is welcome and preserved.