SG-2 — Gauge recovery SU(3)xSU(2)xU(1): the gate anchor ledger

The honest one-line: SG-2 has a real, checkable result — the surviving 4D isometry algebra of the frozen compact factors equals $\mathfrak{su}(3)_c \oplus \mathfrak{su}(2)_L \oplus \mathfrak{u}(1)_Y$ (equality, not containment; multiset $\{8,3,1\}$, rank 4, no extra and no missing factor), and the carriers are forced within the declared grammar — but the gauge-group OUTCOME is a rival tie, the recovery is given-E, and the carrier-forcedness theorems are category-internal (two terminal walls remain).

This page is the gate-specific instantiation of the anchoring method: it takes the master anchor and the bridges and applies them, object by object, to one gate. Every exact thing SG-2 touches gets its own row — its status, what it is allowed to claim, and what it is forbidden to claim. It follows the same eleven-part shape and the same universal table as the canonical SG-4 ledger.

The gate's integrity spine, carried verbatim from its dossier: the OUTCOME is a tie; the discrimination is the carrier-forcedness.

1. Gate status header

Taxonomy reconciliation (2026-07-05). The gate-level grading is CERTIFIED-IRREDUCIBLE-CATEGORY · RESOLVED +0 (forces-are-isometries) — board-canonical, ratified 2026-07-08 (closure-of-record: the live gate dossier; the earlier interim reading REDUCED-TO-AXIOM +1 is superseded — the named grammar axiom is carried as a certified-irreducible category disclosure, not a +1 charge), read as terminal reached + residuals shown: the recovery leg and the CP² over-production exclusion are banked, and the carrier-neutrality/completeness residual family below (Holes 1–4, walls W_R2_NEUTRALITY/W_R3_COMPLETENESS) remains listed and carried unchanged. The facts are unchanged — the earlier OPEN/wall roll-up reflects the superseded least-closed-residual rule, not different facts; no residual below is closed or re-graded.

• Gate-level SG-2 roll-up: CERTIFIED-IRREDUCIBLE-CATEGORY · RESOLVED +0 (board-canonical); recovery leg DERIVED-GIVEN-E (direction held; the gate's mid-audit endpoint read OPEN/wall — two terminal carrier-neutrality walls remain, shown as residuals).
• The closed local leg — algebra recovery as equality: given the frozen carrier geometry, the surviving isometry algebra satisfies the recovery predicate $$O_{\rm SG2,recovery}(E_{\rm frozen}) = 0,$$ where $O_{\rm SG2,recovery}$ is the obstruction "the simple-summand multiset is not exactly $\{\mathfrak{su}(3),\mathfrak{su}(2),\mathfrak{u}(1)\}$ with rank 4 (an extra unbroken factor, or a missing SM factor)." It vanishes for the frozen branch.
• Status of the recovery leg: DERIVED-GIVEN-E — it certifies this geometry yields E's gauge algebra; it does not derive E and does not prove cross-geometry uniqueness.

The recovery leg is a genuine result: the multiset $\{8,3,1\}$, rank 4, no extra and no missing factor, falls out of standard homogeneous-space isometry algebras with no coupling and no fitted number consulted. What stays open is everything beyond the bare recovery: the gauge-group outcome is a filter every serious rival framework also passes, and the carrier-forcedness — the gate's framework-internal content — is proven only inside the declared "forces = isometries" grammar.

selection ≠ derivation · given-E ≠ derivation-of-E · frozen/reproducible ≠ proven-unique.

2. Frozen inputs (what SG-2 stands on, not what it produces)

• Frozen branch hashes dcc66f1b2685 / manifest meta a5b1e6f9d951. The branch is read-only. These hashes are audit anchors — they certify which object was tested and that it cannot be quietly retuned. They do not validate the physics. (Parity table ac4d2df3e708 is an SG-3/SG-4 object, co-read only because $S^1_Y/\mathbb{Z}_2$ is the hyper carrier.)
• Upstream spectrum $E_{\rm frozen}$ — the observed SM gauge content — is given / charged / inherited: it enters SG-2 as the target the recovery is checked against. SG-2 does not derive $E$. Every "equals" below is a statement about this $E$, not a derivation of it.
• The frozen 13D K₆ branch geometry $K_{\rm gauge} = K_6 \times S^2 \times S^1_Y$, with $K_6 = SU(3)/T^2$. SHAPE is selected-not-forced absolutely (shortest only over the declared grammar + dimensional ladder + named rivals). It is not an SG-2 output.

3. Object anchors (given-E / upstream)

The exact structures SG-2 acts on:

• The carrier triple $K_{\rm gauge} = K_6 \times S^2 \times S^1_Y$. $K_6 = SU(3)/T^2$ is the complete flag manifold of $\mathbb{C}^3$, $\dim = 8-2 = 6$, $\mathrm{Isom} = SU(3)$, Euler characteristic $\chi(K_6) = |W(SU(3))| = |S_3| = 6$. $S^2 = SU(2)/U(1)$, $\mathrm{Isom} = SU(2)$. $S^1_Y$ the parent circle, $\mathrm{Isom} = U(1)$.
• The observed gauge content $SU(3)_c \times SU(2)_L \times U(1)_Y$ as given-E.
• The LEP/SLD light-species count $N_\nu = 2.984 \pm 0.008$ — a measured spectral anchor forbidding mirror fermions (load-bearing for the $S^1_Y/\mathbb{Z}_2$ fold).
• The measured couplings $\alpha_i(M_Z)$ — declared anchors, NOT SG-2 outputs (handled at SG-7).

Status: GIVEN-E / upstream-inherited — none of these is SG-2-derived.

4. Root and master-anchor traceability

Deep roots that are load-bearing for SG-2:

Scale and causal order are not primary load-bearing anchors for SG-2: the gate consults no scale, no coupling, no UV number.

Master anchors in play: finite invariant ledgers (the multiset + centralizer checks) · no unpaid labels · the frozen branch · given-$E$ · the declared "forces = isometries" grammar + CSDR centralizer rule · open-residual discipline.

5. The SG-2 anchor ledger (the universal table)

6. The arithmetic — the recovery, in full

Each compact factor sources exactly one simple summand of the gauge algebra, by the standard isometry algebras of homogeneous spaces (Step 1, the existence leg):

$$ K_6 = SU(3)/T^2 \;\to\; \mathfrak{su}(3)\ (\text{colour}),\quad S^2 = SU(2)/U(1) \;\to\; \mathfrak{su}(2)\ (\text{weak}),\quad S^1_Y \;\to\; \mathfrak{u}(1)\ (\text{hyper}). $$

The equality clause (Step 2) — no-extra / no-missing. Summing the simple summands:

$$\dim SU(3) + \dim SU(2) + \dim U(1) = 8 + 3 + 1 = 12,\qquad \mathrm{rank} = 2+1+1 = 4,$$

and the simple-summand multiset is exactly $\{\mathfrak{su}(3),\mathfrak{su}(2),\mathfrak{u}(1)\}$ — no extra unbroken factor (over-production failure mode) and no missing SM factor (under-production failure mode). Containment would be trivially satisfiable by almost any large carrier; equality is the non-trivial statement, and it is exactly what the G02_gauge_recovery multiset + no-extra-summand check verifies.

Diagnostic — the recovery is specific, not trivial. The result is an equality with a sharp falsifier rather than a tautology: it pins both failure directions at once. The over-production direction is what actually kills the cheaper rival — the CP² $= SU(3)/U(2)$ carrier, whose non-abelian isotropy $U(2)$ has a non-trivial centralizer $C_{SU(3)}(U(2)) \neq T^2$ and therefore over-produces gauge. The under-production direction is the existence leg. That a cheaper SU(3) carrier exists and fails the same predicate $E_{\rm frozen}$ passes is what makes the equality a real structural fact, not an artifact of a permissive containment test. By contrast the clean carrier obeys

$$C_{SU(3)}(T^2) = T^2 \quad(\text{Cartan only}) \;\Rightarrow\; \text{no extra non-abelian gauge survives.}$$

The obstruction map. Collect the pieces SG-2 owns: $$O_{\rm SG2,recovery}(E)=\big(O_{\rm existence}(E),\,O_{\rm equality}(E)\big),\qquad O_{\rm SG2,recovery}(E_{\rm frozen})=0.$$ The full gate carries, in addition, the carrier-forcedness neutrality piece (the two terminal walls): $$O_{\rm SG2}(E)=\big(O_{\rm neutrality}(E),\,O_{\rm existence}(E),\,O_{\rm equality}(E)\big).$$ We do not assert $O_{\rm SG2}(E_{\rm frozen})=0$: the architecture-neutrality of the carrier-forcedness is asserted, not proven (walls W_R2_NEUTRALITY, W_R3_COMPLETENESS).

7. Declared-structure splits — "the carriers are forced" into separate honest objects

The single phrase "the carriers are forced" hides three different claims with three different statuses, each category-relative:

1. Clean-carrier forcedness (C1). $K_6 = SU(3)/T^2$ is the unique clean SU(3) carrier because $T^2$ is the unique purely-abelian (maximal-rank) SU(3) isotropy and $C_{SU(3)}(T^2)=T^2$ adds no gauge. Status: DERIVED-GIVEN-E, inside the CSDR grammar. Its architecture-neutrality is OPEN (wall W_R2): the missing object is a CSDR-centralizer uniqueness theorem over the SU(3) subgroup lattice $\{1, U(1), T^2, SU(2), U(2), SU(3)\}$.
2. Weak forcedness (C2) via Fact F1. $\mathrm{Isom}(T^n)=U(1)^n$ is abelian, so no torus or torus-orbifold carries $\mathfrak{su}(2)$; the minimal non-abelian-isometry carrier is $S^2=SU(2)/U(1)$. Status: DERIVED-GIVEN-E, category-internal. F1 is a clean classification fact inside "forces = isometries"; a bundle/brane framework is a different category and routes around it, so F1's neutrality is conditional.
3. Hyper forcedness (C3) via Fact F2. The chiral index on a closed odd-dimensional manifold vanishes, so a bare $S^1_Y$ mirrors every fermion (excluded by the LEP/SLD count $2.984 \pm 0.008$); the $\mathbb{Z}_2$ fold $S^1_Y/\mathbb{Z}_2$ has fixed-point boundaries that re-open chirality via the APS index. Status: DERIVED-GIVEN-E for the carrier identity; the chirality content itself is properly SG-3/SG-4.

So the carriers are forced within the grammar (C1/C2/C3), but the grammar itself is a declared axiom and the neutrality of C1 is open. The honest open target is to derive the clean-carrier uniqueness as a standalone CSDR theorem without tuning to the known SM answer.

8. Open residuals — the carrier-neutrality and certificate family

The recovery above is one face of SG-2. These distinct residuals make up the rest, and none is closed by the algebra recovery. They map one-to-one to the four holes and the two terminal walls in the gate's open-residual ledger.

• Hole 1 — C1 architecture-neutrality (wall W_R2_NEUTRALITY). OPEN. "$K_6$ is the unique clean SU(3) carrier" is a theorem inside the CSDR grammar, not a standalone one. Highest-leverage closeable target.
• Hole 2 — SU(3)-carrier completeness over the homogeneous shelf (wall W_R3_COMPLETENESS). AUDIT / partial. The clean class and the CP² kill are in hand; a single delivered enumeration theorem over the whole homogeneous SU(3) coset shelf is not.
• Hole 3 — F1 generalization. AXIOM-CLOSED (reaches DERIVED via a bounded no-go). F1 is stated as a class fact for tori; it is not yet packaged as a closed no-go theorem with S²-forcedness as a corollary.
• Hole 4 — G02 multiset / no-extra-summand certificate. AUDIT / BLOCKED. The load-bearing equality predicate is machine-witnessed by certificates/G02_gauge_recovery/, but the input file is absent on disk; hand-reproduction mitigates but does not close it.

Two further items are disclosed claim-boundaries, not holes to close (keep correctly stated, never promote):

• R1 — the OUTCOME is a rival tie. DISCLOSED. $SU(3)\times SU(2)\times U(1)$ coming out is a filter string/M/F-theory/noncommutative-geometry/lattice all pass. A refuting result would require showing some rival cannot recover the SM group — false. Never bank the OUTCOME as a framework win.
• R6 / R8 — given-E and grammar dependence. DISCLOSED / AXIOM-CLOSED. SG-2 certifies this branch yields the SM algebra, not cross-geometry uniqueness; "forces = isometries" is a named root posit AX_FORCES_ARE_ISOMETRIES, not a provable category.

9. Anti-claims (what this page refuses to say)

• SG-2 does not derive $E$. Formally: $\;E_{\rm frozen}\in\ker O_{\rm SG2,recovery}$, not $\;\ker O_{\rm SG2,recovery}=\{E_{\rm SM}\}$.
• The gauge-group OUTCOME is a rival tie, not a framework discrimination. Recovering $SU(3)\times SU(2)\times U(1)$ is a filter every serious framework passes; the discrimination is the carrier-forcedness, not the outcome. Banking the outcome is the cardinal overclaim.
• The recovery is given-E, not cross-geometry uniqueness. There is no theorem "this gauge group is the unique output of all admissible geometries"; that is a universal negative, unprovable in principle for every framework.
• C1/C2/C3 are theorems inside the "forces = isometries" category, not architecture-neutral. Their force against a bundle/brane reviewer is declared-conditional (two terminal walls).
• The CP² kill is a structural over-production exclusion, not an anti-fitting hand-wave (the 11D build wsmnjvt55 BROKE at Gate-2).
• The frozen-branch hashes are audit anchors; they do not validate the physics.
• The G02 certificate is BLOCKED, not machine-verified, until the input is mounted and re-run.
• Algebra recovery is not whole-gate closure. $O_{\rm SG2,recovery}=0$ does not imply $O_{\rm SG2}=0$.

10. Specialist closure plan

Each open residual is a concrete, finite, target-blind work-package. SG-2 carries no tunable real, so every genuine-physics path is a bounded finite Lie-theory computation on a small object; there is nothing to reverse-engineering from the measured value except the framing.

1. Hole 1 — C1 neutrality (wall W_R2, HIGHEST leverage). Prove: among all homogeneous spaces $SU(3)/H$ with $H$ closed connected, the carriers whose CSDR reduction yields exactly $\mathfrak{su}(3)$ (no extra factor, no isotropy-locking) are precisely those with $H$ a maximal torus; up to conjugacy $H=T^2$, so $K_6$ is unique. Machinery: enumerate the six-class SU(3) subgroup lattice $\{1, U(1), T^2, SU(2), U(2), SU(3)\}$; for each compute $C_{SU(3)}(H)$ and check the equality clause. Endpoint: DERIVED-CLOSED-inside-CSDR. Falsifier: a second $H$ passes ⟹ C1 weakens to "unique among …" (a sharper-open). Named axiom: AXIOM-CLEAN-CARRIER.
2. Hole 2 — shelf completeness (wall W_R3). Assemble the homogeneous SU(3) coset shelf + relevant products (notably Witten's CP² $\times\,S^2\times S^1$, which passes Gate-2 but dies at chirality) into one completeness ledger recording, per carrier: surviving gauge algebra, equality verdict, chirality verdict. Endpoint: DERIVED for the homogeneous shelf. Falsifier: a non-$T^2$ homogeneous carrier survives both equality and chirality ⟹ C1's "unique" downgrades (a publishable finding). Named axiom: AXIOM-SU3-CARRIER-SHELF. Trap: do not scope-creep into the unbounded "no manifold of any kind" universal negative — that is a dissolved unicorn, a shared ceiling, never an open weakness.
3. Hole 3 — F1 generalization. Package the no-go: $\mathrm{Isom}(M)$ abelian $\Rightarrow$ CSDR-surviving gauge abelian $\Rightarrow$ $\mathfrak{su}(2)_L$ requires a non-abelian-isometry carrier $\Rightarrow$ $S^2$ minimal. Endpoint: AXIOM-CLOSED → DERIVED (bounded). Named axiom: AXIOM-NONABELIAN-NEEDS-NONABELIAN-CARRIER. Trap: state inside CSDR; do not claim it against bundle constructions.
4. Hole 4 — G02 certificate (CHEAPEST, best value/effort). Mount certificates/G02_gauge_recovery/; re-run the multiset test on the carrier isometry data (confirm exactly $\{8,3,1\}$, rank 4, no-extra-summand predicate passes); re-hash carriers and confirm dcc66f1b2685 / a5b1e6f9d951 recompute byte-equal; confirm the exotics ledger marks every candidate Absent/Massive. Status flow: BLOCKED → VERIFIED or REFUTED. No new physics. Trap: do not let "claimed pass" drift into "machine-verified" before the file is mounted (a phantom-citation overclaim).

Closing all four moves the carrier-forcedness from category-relative assertions to category-relative theorems over a named grammar axiom — a real strengthening. It does not promote the gate's label (the recovery leg stays DERIVED-GIVEN-E; the board terminal stays CERTIFIED-IRREDUCIBLE-CATEGORY · RESOLVED +0) and does not close the OUTCOME tie. R1/R6 stay disclosed; R8 stays AXIOM-CLOSED.

11. Completion tests for this page

Required presence (all met): gate roll-up CERTIFIED-IRREDUCIBLE-CATEGORY · RESOLVED +0 · recovery leg DERIVED-GIVEN-E (mid-audit OPEN/wall) · $O_{\rm SG2,recovery}(E_{\rm frozen})=0$ · recovery leg DERIVED-GIVEN-E · $E$ not derived · frozen hashes (AUDIT ONLY) · the carrier triple $K_6\times S^2\times S^1_Y$ · isometry-algebra map $K_6\to\mathfrak{su}(3)$, $S^2\to\mathfrak{su}(2)$, $S^1_Y\to\mathfrak{u}(1)$ · count $8+3+1=12$, rank 4 · equality clause · diagnostic ($C_{SU(3)}(T^2)=T^2$ clean vs CP² over-production) · C1/C2/C3 split with category-internal status · Fact F1 · Fact F2 · CP² kill (wsmnjvt55) · LEP/SLD $2.984\pm0.008$ · OUTCOME-tie disclosure · grammar posit AX_FORCES_ARE_ISOMETRIES · G02 BLOCKED residual · all four holes + two walls as their own rows · the gate's anti-claims.

Required absence (all held): no claim that SG-2 derives $E$ · the OUTCOME selects/discriminates this framework · $\ker O_{\rm SG2}=\{E_{\rm SM}\}$ · cross-geometry uniqueness · C1/C2/C3 are architecture-neutral theorems · CP² killed by anti-fitting · the G02 certificate is machine-verified · hashes validate physics · algebra recovery = whole-gate closure · the gate's label promoted · any reader-visible build-process vocabulary.

Completion report. - Tests passed: all required-presence items present; all required-absence items held; status matches the dossier (CERTIFIED-IRREDUCIBLE-CATEGORY · RESOLVED +0; recovery leg DERIVED-GIVEN-E, held); every number/theorem ($\{8,3,1\}$, rank 4, $\chi(K_6)=6$, $C_{SU(3)}(T^2)=T^2$, $N_\nu=2.984\pm0.008$, run wsmnjvt55, hashes) traceable to the dossier. - Tests failed: none. - Open items: Holes 1–4 and walls W_R2_NEUTRALITY / W_R3_COMPLETENESS remain open by design (carried as residual rows, not closed). - Assumptions made: none beyond the dossier; cross-link targets (./layer-3-search-grammars.html, ./layer-4-carrier-forcing.html, ./sg4-anchor-ledger.html, ./a0-master-anchor.html) verified present in the anchors directory.

This gate anchor ledger follows the canonical eleven-part shape and universal table of the SG-4 ledger.

See also: the anchoring method · the master anchor · Layer 3 — search grammars (why "forces = isometries" is a declared grammar) · Layer 4 — carrier-forcing & the given-E wall (the carrier-forcedness C1/C2/C3) · the sibling SG-4 dossier · the full SG-2 dossier.