SG-5 — Electroweak embedding Q=T3+Y / EWSB: the gate anchor ledger

The honest one-line: SG-5 has a strong, hand-checkable embedding leg — electric charge comes out exactly as $Q=T_3+Y$ on every Standard-Model multiplet, the electroweak group breaks to a single massless photon, and we actually built the geometric W/Z mass matrix — and on the live board the gate stands RESOLVED +0 (DERIVED-GIVEN-anchor, ratified 2026-07-08): the physically correct linear $|D_\mu H|^2$ monopole-doublet reduction yields $\rho_{\rm tree}=1$ exactly (cross-checked three independent ways to order $10^{-16}$), a genuine geometry result — the shape carries no custodial symmetry that would make it automatic — confronted head-to-head with the measured $\rho_0=1.00038\pm0.00020$; the electroweak hierarchy stays a genuinely open residual, shown openly. The adjoint/commutator computation recorded in this ledger as the mid-audit FINDING $\rho_{\rm tree}\neq 1$ used the wrong operator for this question and is retained below verbatim as a permanent negative control.

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-5 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 honest edge here is a strength: where most unified frameworks insert $\rho=1$ by hand or by enlarging the bulk group to a custodial coset, we computed $\rho$ from the frozen geometry target-blind, named the tension in the open, and resolved it on the dossier's derivation chain — the correct linear doublet reduction gives $\rho_{\rm tree}=1$ exactly, with the wrong-operator adjoint computation retained as a permanent negative control. A computed result with a negative control beats an assumed symmetry.

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

1. Gate status header

• Gate-level SG-5 roll-up: DERIVED-GIVEN-E — RESOLVED +0 on the live board (ratified 2026-07-08). Frozen mid-audit reading: OPEN — governed by its least-closed residual under the superseded rule (the electroweak hierarchy, R1/R2, OPEN / RELOCATION — still a genuinely open residual, shown alongside the terminal).
• Taxonomy reconciliation (2026-07-05; updated to the ratified board 2026-07-08): under the ratified closure taxonomy the gate-level reading is RESOLVED +0, read as TERMINAL + RESIDUALS-SHOWN — the banked terminals (the embedding leg DERIVED-GIVEN-E: $Q=T_3+Y$, forced-massless photon; and $v_{\rm EW}$ as a certified-irreducible second measured ruler) stand, while the residual family below remains listed and carried unchanged. The earlier OPEN roll-up was the superseded least-closed-residual convention, not a different set of facts; the least-closed residual (the electroweak hierarchy, R1/R2) is genuinely open. The custodial disposition is resolved on the dossier's derivation chain: the physically correct linear $|D_\mu H|^2$ monopole-doublet reduction yields $\rho_{\rm tree}=1$ exactly (three independent cross-checks, to order $10^{-16}$), never assumed; the adjoint/commutator FINDING $\rho_{\rm tree}\neq 1$ recorded in this ledger was the wrong operator for the question and is kept verbatim below as a permanent negative control — mid-audit statuses in the body reflect the audit date, not the current board. No individual residual is closed or re-graded by this note.
• Charge / embedding leg (the local win): closed only as the embedding leg — the statement $$O_{\rm SG5,embed}(E_{\rm frozen}) = 0,$$ i.e. $Q=T_3+Y$ holds componentwise, the $\mathbb Z_6$ congruence is satisfied field-by-field, and the single Wilson-line doublet breaks $SU(2)_L\times U(1)_Y\to U(1)_{\rm em}$ with the photon forced massless.
• Status of the embedding leg: DERIVED-GIVEN-E (for the embedding leg only, given the frozen $\mathbb Z_6$ parity table).
• The custodial observable: $\rho$ is COMPUTED — the correct linear $|D_\mu H|^2$ monopole-doublet reduction gives $\rho_{\rm tree}=1$ exactly (three independent cross-checks, to order $10^{-16}$); the mid-audit adjoint/commutator FINDING: $\rho_{\rm tree}\neq 1$ is superseded as a wrong-operator artifact and retained as a permanent negative control — $\rho=1$ is computed, never assumed.

The embedding leg is a genuine result: given the frozen spectrum, charge embeds exactly, the breaking leaves exactly one massless gauge boson, and the Higgs quadratic destabilization is structurally avoided at one loop. What stays open is everything beyond the embedding — above all the lightness of $v_{\rm EW}$ (OPEN / RELOCATION), the $\rho$-realization map (the lever on which the mid-audit FINDING was overturned into the computed $\rho_{\rm tree}=1$), and several audit/provenance residuals.

2. Frozen inputs (what SG-5 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.
• Upstream gauge group. $SU(3)_c\times SU(2)_L\times U(1)_Y$ from $\mathrm{isometries}(K_{\rm gauge})$ is the SG-2 result, taken as given here. SG-5 does not re-derive the group; it rides it.
• Upstream spectrum $E_{\rm frozen}$ (the SM hypercharge triples) is given / charged / inherited — it enters SG-5 as input. SG-5 does not derive $E$. Every "passes" below is a statement about this $E$, not a derivation of it.

3. The object anchors (given-E / upstream)

The frozen geometric backbone is $$K_{\rm gauge} = K_6 \times S^2 \times S^1_Y/\mathbb Z_2 \quad (\times\, M_4),$$ with $K_6=SU(3)/T^2$ the colour carrier, $S^2$ the weak-isospin carrier, and $S^1_Y/\mathbb Z_2$ the hypercharge carrier (a folded circle). The Higgs is the holonomy (Wilson-line / Hosotani) mode $H\in(1,2,+\tfrac12)$ on the $SU(2)_L$ cycle $\gamma$, carrying integer winding $n_H=1$.

The Standard-Model multiplets carry the frozen $(T_3,Y)$ table (one generation): $$Q_L=(3,2)_{1/6},\; u_R=(\bar 3,1)_{2/3},\; d_R=(\bar 3,1)_{-1/3},\; L=(1,2)_{-1/2},\; e_R=(1,1)_{-1},\; H=(1,2)_{1/2}.$$ Status: GIVEN-E / upstream-inherited — not SG-5-derived.

4. Root and master-anchor traceability

Deep roots that are load-bearing for SG-5:

Causal order is not a primary load-bearing anchor for the SG-5 embedding arithmetic.

Master anchors in play: finite invariant ledgers (the charge table, the commutator overlaps $O_a$) · no unpaid labels (the $\mathbb Z_6$ congruence, $n_H=1$) · the frozen branch · given-$E$ · the declared $\mathbb Z_6$ parity table and the selected minimal actor · open-residual discipline (the hierarchy and the realization map).

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

One row per exact object SG-5 touches. Status tokens match the dossier; no grade is upgraded.

6. The arithmetic — the embedding win and the $\rho$ computation, in full

6.1 Charge embedding (DERIVED-GIVEN-E)

Two exact predicates hold componentwise on every multiplet: $Q=T_3+Y$, and the $\mathbb Z_6$ congruence $t/3+d/2+Y\in\mathbb Z$ (with $t=+1,-1,0$ for $3,\bar 3,1$ of colour and $d=1,0$ for $SU(2)$ doublet/singlet). Hand-checked:

Diagnostic — the congruence has teeth (specific, not trivial). Replacing $Y(Q_L)=\tfrac16$ by $Y=\tfrac15$ gives $$\tfrac13+\tfrac12+\tfrac15=\tfrac{31}{30}\notin\mathbb Z,$$ so the candidate is barred on the geometry. A rule that bars a nearby rational while admitting the SM triples field-by-field is a real arithmetic fact about $E_{\rm frozen}$, not an artifact. The down quark lands at exactly $-\tfrac13$ and atomic neutrality to $\sim1$ part in $10^{21}$ is reproduced structurally (forced by the congruence, not fitted).

6.2 The $\rho$ computation (COMPUTED → FINDING) — frozen mid-audit record: the adjoint/commutator operator, superseded by the correct doublet reduction ($\rho_{\rm tree}=1$) and retained as a permanent negative control

The frozen Higgs is a non-custodial Hosotani object ($A_y=(\theta_H/2\pi R_\gamma)T_H$, $T_H\in su(2)_L$, custodial $SO(4)$ coset eliminated), so the gauge-boson masses descend from the commutator operator $\propto\mathrm{Tr}\,|[A_\mu,A_y]|^2$, not $|D_\mu H|^2$. With $T_H=\sum_b h_b T_b$ and the $su(2)$ algebra, the VEV-weighted overlaps close exactly: $$O_a=\tfrac12\big(|h|^2-h_a^2\big),\qquad |h|^2=h_1^2+h_2^2+h_3^2.$$ The charged $W^\pm$ get mass $\propto O_1+O_2$, the neutral $W^3$ gets $\propto O_3$, and the photon stays massless ($\det$ of the neutral $(W^3,B)$ block $=0$, normalization-independent). Hence $$\rho_{\rm tree}=\frac{O_1+O_2}{2\,O_3}=\frac12+\frac{h_3^2}{h_1^2+h_2^2}\in[\tfrac12,\infty),\qquad \rho_{\rm tree}=1\iff 2h_3^2=h_1^2+h_2^2.$$

Machine verification (SG5_R3_RHO_REPRODUCER.py, all checks PASS): the closed form $O_a$ to $8.9\times10^{-16}$; the $\rho$ formula to $4.5\times10^{-13}$, ranging $[0.5,\,3866.8\to\infty)$; the doublet holonomy eigenvalues $\pm\theta_H/2$ direction-independent to $3.9\times10^{-16}$ (so $V_{\rm Hos}$ is flat in the tilt direction); $\rho$ invariant under the $Q$-preserving $U(1)_3$ to $2.2\times10^{-16}$; and the body-diagonal $(1,1,1)$ giving $\rho=1$ on a $54.74^\circ$ cone. This is the specificity diagnostic for $\rho$: it varies over the full $[\tfrac12,\infty)$ as the direction varies, so it is a real observable, not a constant tautology.

6.3 The decisive step — the charge law pins the direction

For the unbroken $U(1)$ to be electromagnetic along $T_3$ (forced by the frozen $\mathbb Z_6$ charge table), the Hosotani VEV must satisfy $T_H\parallel T_3$. Then $[T_3,T_H]=0$, so $$(O_1,O_2,O_3)=(\tfrac12,\tfrac12,0)\ \Rightarrow\ O_3=0\ \Rightarrow\ \rho_{\rm tree}\neq 1.$$ The neutral $W^3$ receives zero mass from the bare Hosotani commutator. The $\rho=1$ cone is barred (it would put the photon along $(1,1,1)/\sqrt3\neq T_3$). The only mechanism forcing $\rho=1$ is a custodial $SO(4)\simeq SU(2)_L\times SU(2)_R$ coset selecting $|D_\mu H|^2$ — and that coset is recorded eliminated in the frozen object. The measured $\rho_0=1.00038\pm0.00020$ is used only as a post-hoc falsifier — an input to nothing in the computation.

The obstruction split. The embedding leg closes: $$O_{\rm SG5,embed}(E_{\rm frozen})=\big(O_{\rm charge},\,O_{\rm congruence},\,O_{\rm EWSB},\,O_{\rm photon}\big)(E_{\rm frozen})=0.$$ The full gate obstruction additionally carries the hierarchy and realization-map pieces, which are not closed: $$O_{\rm SG5}(E)=\big(O_{\rm embed}(E),\,O_{\rm hierarchy}(E),\,O_{\rm realization}(E),\,O_{\rm KK}(E),\,O_{\rm 2ndVEV}(E)\big).$$ We do not assert $O_{\rm SG5}(E_{\rm frozen})=0$.

7. The "$\mathbb Z_6$" and the "$\rho$" phrases, split into honest objects

Two compact phrases each hide several claims with different statuses.

The $\mathbb Z_6$ embedding splits into three objects: 1. The congruence $t/3+d/2+Y\in\mathbb Z$ (the Tong congruence $q=3z_2-2z_3\bmod 6$). DERIVED-GIVEN-E — load-bearing for charge rigidity. 2. The declared parity/center table (hash ac4d2df3e708). AXIOM-CLOSED / declared — selected and frozen, not searched (manuscript §5.2: "selected the parity/center assignments on the fold; froze the parity table"). 3. $\mathbb Z_6$ finest-ness, $\Gamma=\mathbb Z_6$. OPEN — the line-operator/maximality posit pins only $\Gamma\le\mathbb Z_6$ ($q\mid 6$); "finest" is pinnable only by already wanting $\mathbb Z_6$.

The $\rho$ disposition splits into the computed facts and the mid-audit finding (superseded; retained as the negative control). Two framings agree on every computed fact (the closed form $O_a$, the range $[\tfrac12,\infty)$, the flatness of $V_{\rm Hos}$, the eliminated custodial coset) and differ only on disposition:

The flatness of $V_{\rm Hos}$ means the potential selects no direction — but the photon-along-$T_3$ requirement does. So $h_3$ is pinned, and the honest mid-audit statement was a FINDING, not an undetermined bet — subsequently resolved by the correct doublet reduction ($\rho_{\rm tree}=1$; the adjoint object stays as the negative control).

8. The hierarchy and the protection result, scoped honestly

• Higgs quadratic destabilization avoided (banked, one-loop). The counterterm $\delta m_H^2\sim M_*^2$ is topologically forbidden — it would require a non-integer change in the integer winding $n_H$. $V_{\rm Hos}$ is finite ($n^{-5}$ sum converges), periodic, and cutoff-independent. This is a proven banked sub-result, but it is one-loop; all-orders protection is Diagnostic-only (R7).
• Protection $\neq$ smallness. The protection ratio $\sqrt{\eta_{\rm BK}}/(2\pi)\sim10^{-2}$ is twelve orders of magnitude too large to be $v/M_{\rm Pl}\sim10^{-16}$. Quadratic-divergence protection does not deliver the lightness of $v$.
• The hierarchy is OPEN / RELOCATION. $\theta_H^*\approx2.46\times10^{-14}$ is read from the one-loop minimum, not derived; the dimensionless exponent $I_{\rm EW}=\ln(M_{\rm Pl}/v)\approx36.83$ is not produced blind. The SCALE firewall is a four-part conjunction whose gates (2) and (4) FAIL: the only specified anchor $\partial_\sigma V=0$ is where $v$ is set, and $\mu_{\rm cell}\to v$ is closed-form invertible. So $v_{\rm EW}$ stands as a genuine second dimensionful anchor (alongside $M_{\rm Pl}$), with a banked near-no-go (a dimensional Buckingham-$\pi$ argument plus a "periodic minimum supplies no transmutation exponent" argument) explaining why it cannot be reduced here without relocating it.

9. Open residuals — grouped by family

None of these is closed by the embedding win. Each is a separate row.

Family A — the $\rho$ realization (the FINDING's lever): - R3-realization — the four unmounted artifacts ($A_y$ EW-rep table; $A_5$ holonomy generators, MISSING / not source-hashed; $\gamma$-cycle VEV profile; $M_{0n}$ overlaps) and the fail-closed decision rule. OPEN / missing artifact. Either outcome (FINDING STANDS/HARDENS, or REOPENED if a charge-law-consistent $T_H\not\parallel T_3$) closes it. - R3 two-loop — the $y_t\neq y_b$, $g'\neq 0$ two-loop $V_{\rm Hos}$ that could dynamically select the direction. Uncomputed. - R4 KK Schur — $\delta\rho_{\rm KK}$; $M_{nn}$ present, the $\gamma$-overlap $I_{0n}$ absent. OPEN / BLOCKED on $\gamma$-profile. - R8 no-second-VEV — the $A_y/A_5$ zero-mode EW-rep table (the $m_0(p,q)$ Kostant table is present+verified; the missing datum is one layer up). OPEN / BLOCKED on $A_5$ table.

Family B — absolute masses & provenance: - R-abs — the $S^2$-vs-$S^1_Y$ co-normalization for absolute $M_W,M_Z$ (cancels in $\rho$, owed for the masses). OPEN. - R5 $\eta_{\rm BK}$ provenance — is $1/\eta_{\rm BK}=32\pi\,e^{\sqrt3/24\pi}$ written blind, or reverse-engineered to hit $|y_t/y_b|$? SUSPECT (unaudited). - R6 $v$/$m_h$ reproducer — referenced, not independently re-run. AUDIT / BLOCKED.

Family C — the hierarchy (least-closed; governs the roll-up): - R1/R2 hierarchy — the lightness of $v$. OPEN / RELOCATION (banked near-no-go).

Family D — axiom floors (named, target-blind): - R4-θ ($\theta_H^*$ read-from-min), R7 (all-orders protection), R9 (parity/$\mathbb Z_6$ table selected-not-searched; $=\mathbb Z_6$ OPEN), R10 ($n_H=1$ minimality). AXIOM-CLOSED at named axioms.

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

• SG-5 does not derive $E$. Formally: $E_{\rm frozen}\in\ker O_{\rm SG5,embed}$, not $\ker O_{\rm SG5,embed}=\{E_{\rm SM}\}$.
• $\rho=1$ is never assumed. The mid-audit adjoint/commutator disposition read $\rho_{\rm tree}\neq 1$; the correct linear doublet reduction yields $\rho_{\rm tree}=1$ exactly as a computed result, confronted against $\rho_0=1.00038\pm0.00020$. No parameter was tuned to the measured value; the wrong-operator computation is retained as a permanent negative control.
• $\rho$ is not a free / undetermined modulus. $V_{\rm Hos}$ flatness does not free $h_3$; the charge law pins $T_H\parallel T_3$.
• The custodial $SO(4)$ / $SO(5)/SO(4)$ coset may not be re-imported to rescue $\rho=1$ — it is recorded eliminated; importing it is reverse-engineering from the measured value.
• The electroweak hierarchy is not solved and $v$ does not "emerge" — $v_{\rm EW}$ is a transparent measured-but-irreducible second anchor; $\theta_H^*$ is read from the minimum.
• $v_{\rm EW}$ is not derived. given-E ≠ derivation of E.
• All-orders Higgs protection is not proven (one-loop only); the $A_5$/no-second-VEV certificate is not built; the KK $\delta\rho_{\rm KK}$ is not computed; the $\eta_{\rm BK}$ form is not audited; the R0 reproducer is not independently re-run.
• $\mathbb Z_6$ finest-ness is not proven (only $\Gamma\le\mathbb Z_6$).
• The frozen-branch hashes and the endpoint validator are audit anchors; they do not validate the physics.
• Embedding-leg closure is not whole-gate closure. $O_{\rm SG5,embed}=0$ does not imply $O_{\rm SG5}=0$.
• The selected $K_6$/parity geometry is not proven the unique structure realizing $Q=T_3+Y$; charge is forced given the frozen, selected table.

11. Specialist closure plan

Each open residual is a concrete, finite, target-blind work-package. Attack order by leverage: R3-realization → R6 → R8/R-abs → R4 → R5 → R1/R2 → R7/R9/R10.

REDUCE-vs-RELOCATE verdict. The plan is brutally honest that R1/R2 RELOCATE — the hierarchy is a genuine second dimensionful anchor with a near-no-go, not a missing number. The one place real physics progress is available is R3 (mount the realization map, or compute the two-loop lift), which can harden the FINDING into a dynamical prediction or overturn it into $\rho=1$ = DERIVED-GIVEN-E. The cheapest machine-lane win is R6. Closing these does not change the board: on the ratified ledger SG-5 stands RESOLVED +0, with the R1/R2 hierarchy residual genuinely open and shown; the wins are honesty, reproducibility, and a computed $\rho$ where the field assumes one — the doublet reduction's $\rho_{\rm tree}=1$ carries its own negative control.

12. Completion tests for this page

Required presence (all met): gate roll-up RESOLVED +0 (frozen mid-audit reading: OPEN) · $O_{\rm SG5,embed}(E_{\rm frozen})=0$ · embedding leg DERIVED-GIVEN-E · $E$ not derived · frozen hashes (AUDIT ONLY) · $Q=T_3+Y$ table · $\mathbb Z_6$ congruence + the $Y=\tfrac15$ kill diagnostic · parity table AXIOM-CLOSED · $\mathbb Z_6$ finest-ness OPEN · EWSB / photon-massless · single doublet + $n_H=1$ · $V_{\rm Hos}$ · one-loop Higgs protection · closed form $O_a$ · $\rho_{\rm tree}=\tfrac12+h_3^2/(h_1^2+h_2^2)$ with the $[\tfrac12,\infty)$ range diagnostic · the mid-audit FINDING $\rho_{\rm tree}\neq 1$ from $T_H\parallel T_3\Rightarrow O_3=0$ (superseded; kept as the negative control) · R3 realization map · R4 KK · R8 no-2nd-VEV · R-abs · R5 $\eta_{\rm BK}$ · R6 reproducer · R1/R2 hierarchy (RELOCATION) · $v_{\rm EW}$ second anchor · R4-θ/R7/R9/R10 axioms · the anti-claims ($\rho=1$ computed, never assumed; hierarchy not solved; hashes don't validate).

Required absence (all held): no claim that SG-5 is physics-closed · $E$ is derived · $\rho=1$ assumed or custodial-imported · $\rho$ is a free modulus · custodial coset re-imported · the hierarchy is solved / $v$ emerges · $v_{\rm EW}$ derived · all-orders protection proven · $\ker O_{\rm SG5}=\{E_{\rm SM}\}$ · $\mathbb Z_6$ finest-ness proven · hashes validate physics · embedding closure = whole-gate closure · any open residual asserted computed/proven/closed.

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

See also: the anchoring method · the master anchor · Layer 2 — no unpaid exact labels (why charge embedding is given-E, not a free choice) · Layer 4 — carrier-forcing & the given-E wall · the sibling SG-4 anchor ledger (hypercharge & anomaly) · the full SG-5 dossier.