Tagging the Zeros of the Sign Problem: Data, Model and Principle Zeros of the Average Sign in Determinant Quantum Monte Carlo

When the average sign of a fermionic quantum Monte Carlo simulation reads zero, the zero can mean threedifferent things: the run was too short (a data zero), the chosen Hubbard-Stratonovich decoupling created it (a modelzero), or no decoupling in reach removes it (a principle zero). We carry the tagged-zero vocabulary of an earlier noteover to the sign problem and test it in three seals on the two-dimensional Hubbard model (U = 4, 4x4 periodic lattice,plus 4- and 6-site clusters where the sign is known exactly by enumerating every auxiliary field), each run once onpoints unused during development. Model zero: with identical partition functions (relative difference below 2e-11), thespin decoupling keeps above 0.998 while the charge decoupling drops the average phase to 0.0003. Data zero:with the exact sign 0.035 known, the share of 100 chains that read zero fell from 0.94 to 0.52 at the predicted pricetag n* = 1603 sweeps and to 0.00 beyond it, as the error law says. On the 4x4 lattice a sign extrapolated from beta4-7 to beta 8 ([0.041, 0.145]) was met (0.106); at beta 10 the run read a data zero, as forecast. Principle zero: acrossa family of decouplings (discrete and Gaussian, spin and charge, and a continuous mixing parameter alpha built fromthe identity (n-1)^2 + m^2 = 1, the best member is pure spin, where the sign sits on a cusp: moving alpha either waycosts sign, more steeply on the imaginary side (slopes 7.9 and 5.0 per unit alpha at beta 6), and the decay with betaremains (0.48 to 0.18 from beta 6 to 8). Pure spin is thus empirically optimal among the tested members, and the tagreads "principle, relative to this family", not a complexity-theoretic statement. The half-filling sign, often calledprotected, is protected only by symmetric decouplings: alpha = -0.5 drops it to 0.41. Of 19 sealed forecasts 17 cameout R, one U and one F (cusp slopes not scaling linearly with beta). Checks made after AI comments on a draft showtwo verdicts to be fragile: with conservative binning errors one of four Q1 comparisons falls to 1.9 standard errors, andthe F of O4 becomes U once the shared reference point is treated as correlated; the cusp also appears in theGaussian family. The work is a check on how zeros are read, not a new algorithm; it is meant as a small, reproducibleentry point to the sign problem for readers of the tagged-zero series. (1) Text: CC-BY-4.0 | Code: GNU Affero General Public License v3.0 or later (AGPL-3.0-or-later) (2) Data and code. Seal documents with SHA-256 files, result files, development logs, the interruption log and allanalysis and figure scripts (GNU Affero General Public License v3.0 or later) are inBridge_Paper_Tagging_the_Zeros_of_the_Sign_Problem_scripts_v1.0.0.zip. No external data are used. (3) AI use. Designs, code, analyses and text were produced by Kurado (Claude Opus 5.5, Anthropic) working with M. Watabe, who chose every direction. Draft comments by ChatGPT and Gemini, relayed by M. Watabe, prompted the checksin Section 7; they are AI commentary, not peer review.