# Actual flagship singlets and local invariant tensors

September 20, 2026. Producer: `flagship_singlet_invariant_audit.py`; exact replay certificate: its adjacent JSON.

This extends `LOCAL_NINE_SPURION_ALGEBRA.md` and the retraction `retraction_the_local_gut_multiplet_is_complete_so_the_coefficients_are_locked.py`. The previous nine-weight orbit was a mathematical closure of five displayed weights. We now identify its four extra weights in the actual projected model: **n_14, n_15, n_16, n_17**, all at k=4, n=(-1,1,0,0,0,0), with no oscillators. Their weights match exactly, not just by dimension. The local fundamental branches as 9 -> 5 + four singlets under the selected A4.

The complete dump contains **628 weight rows, 274 fields, 65 singlet fields and 132 singlet weights**. All 132 singlet weights pair to zero with all four simple roots of that A4. This is a Cartan-charge statement; a zero-weight state in a different representation is not automatically an invariant vector. Do not infer a universal vacuum no-go from this charge test alone.

The local fundamental bilinear invariant tensor is computed by solving E_i X = X E_i for the 16 adjacent raising/lowering matrix units of A8. The exact rational constraint rank is 80 on 81 variables, so the invariant space is one-dimensional, generated by the identity. Its visible triplet and doublet blocks agree. This substantiates the local fundamental bilinear relation. It does not determine all higher-point CFT tensors, Wilson-line-dressed amplitudes or interactions between differently localized fields.

The stored selection-rule coupling lists contain **64 mu rows and 144 triplet rows** through total order five. For every row, the certificate enumerates explicit weight indices whose sixteen momenta sum to zero. These are necessary gauge-invariant channels. A permitted channel is not proof of a nonzero amplitude or its numerical coefficient. The distinction follows the primary [orbifolder description](https://arxiv.org/abs/1110.5229) and [heterotic coupling selection-rule analysis](https://arxiv.org/abs/1107.2137).

## Data and reproduction

`flagship_all_weights.txt` is a fresh all-field dump from the installed orbifolder 1.2.1 engine. `flagship_all_weight_dump.cpp` extends the existing local wrapper to remove its triplet/doublet-only filter and print all momenta at 17 digits. The exact model text and cached coupling lists are in `flagship_singlet_coupling_inputs.json`. The parser reconstructs sixth-denominator rationals only when every component is within 1e-12; the largest actual error is 4.4e-16. It rejects unsupported denominators rather than applying an arbitrary denominator cap.

Run `python3 analysis/flagship_singlet_invariant_audit.py` to replay the frozen inputs without installing the external engine; append `--write` to refresh the certificate. To regenerate the dump, build the wrapper against orbifolder 1.2.1 headers and objects (orbifolder, groupTheory, linalg, io), Boost and GSL. Run it with the saved model as its argument in a directory containing the model's Geometry and GaugeInvariance resources. Our local run linked the existing objects in `/home/wiljd/orb/build`, used the GSL libraries in `/home/wiljd/orb/sysroot/usr/lib/x86_64-linux-gnu`, and ran the flagship `m1.txt` in `/home/wiljd/orb/scan/cp/w_Z6I_27`. These are local provenance paths, not portable dependencies of the Python replay.

The physical next target is now sharper: determine whether any actual projected interaction outside the co-local fundamental spurion library generates a triplet/doublet-distinguishing endomorphism. The matrix target from the prior packet remains useful, but no physical occurrence is claimed here.

## Actual closure of the parallel rank-four target

The incoming `PASS20260920_HYPERCHARGE_DUAD_CENTER_COKERNEL.md` and W33 `w33_hypercharge_duad_center_cokernel.py` own the geometric target: singlet charges must span the rank-four hypercharge-orthogonal Abelian sector. The preexisting `data/w33_flagship_a5_hypercharge_organizer.json` supplies its exact E8 basis. Applying that basis to this fresh dump gives an explicit **65x4 charge matrix of rank four**. Fields **n_3, n_6, n_8, n_9** already span it, and every field has zero hypercharge.

The selected nonabelian A4 and the extra-U1 organizing A4 are **orthogonal four-dimensional spaces** in the second E8. Thus the earlier zero selected-A4 Cartan charges do not mean zero extra-U1 charges. The producer verifies that orthogonality explicitly.

There is also an exact positive solution of the four projected zero-FI D equations. Write t_i=|VEV_i|² in arbitrary common units: set every t_i=1/100 except t_32=19/100, t_42=3/20 and t_60=1/25. Then sum(t)=1 and Q^T t=0 exactly. The positive mass-Gram matrix Q^T diag(t) Q has leading principal minors

179/450, 6061/67500, 365569/40500000, **204557/81000000**.

Consequently this projected Higgs sector has no remaining massless extra-Abelian direction for positive gauge couplings and a nonsingular kinetic metric. These are dimensionless Gram data, not physical boson masses. The actual singlet matrix and one positive projected-D solution close the rank target. Hidden/nonabelian D equations, remaining U1 equations, FI terms and F equations are not solved; this is not yet a physical vacuum.

## Minimal projected candidates

`flagship_minimal_projected_higgs_circuits.py` enumerates all 252 five-subsets of the ten nonzero charge types and finds **14 positive full-rank circuits**. Five active types are minimal: a rank-four matrix with at most four rows has no nonzero left-kernel vector. One representative is (n_3,n_6,n_9,n_10,n_18) with primitive squared-VEV ratios **(1,1,3,3,3)**. These small supports give concrete inputs for the remaining vacuum equations. The certificate lists all field alternatives within each charge type; equal projected charges do not imply equal hidden charges, localization or F terms.

A further full-Cartan check rejects **all 14 canonical representative choices at zero FI**, even when each field may distribute its fixed total squared VEV over all its weight components. The circuit certificate stores exact Farkas vectors y with A^T y nonnegative and b^T y negative for each system A x=b, x>=0. Thus the projected solutions are not being promoted to full D-flat vacua. Other physical fields with the same projected charge, additional active fields, or nonzero FI terms are not excluded.