SG-9 — Proton safety: the gate anchor ledger

The honest one-line: SG-9 has two genuinely strong, checkable legs — through operator dimension 7 the dangerous-operator census has an empty escapee bin, and no Kaluza–Klein gauge mode can be a leptoquark at any level — and on the live board the gate stands RESOLVED +0 (DERIVED-GIVEN-anchor, ratified 2026-07-08; the neutrino / $M_R$ sector is scoped separately and stays OPEN); in the frozen mid-audit record the operator-safety win is carried Reduced-to-Axiom on the $d\le7$ slice only, and none of it derives the spectrum, predicts a lifetime, or makes the proton absolutely stable.

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-9 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.

1. Gate status header

• Gate-level SG-9 roll-up: DERIVED-GIVEN-E — RESOLVED +0 on the live board (ratified 2026-07-08; neutrino / $M_R$ sector scoped separately, OPEN). Frozen mid-audit reading: OPEN (hardened).
• Taxonomy reconciliation (2026-07-05): under the ratified closure taxonomy (board 2026-07-08) the gate-level grading is RESOLVED +0, with the mid-audit roll-up OPEN (hardened) read as terminal reached on the strong checkable legs, with the residual family below shown and carried unchanged. The residual family listed in this ledger remains listed and carried exactly as before (facts unchanged — the earlier OPEN roll-up under the superseded least-closed-residual rule reflects the same facts, not different ones). No individual residual is deleted, closed, or re-graded.
• Operator-safety leg (the central content): closed only on the bounded slice as an empty escapee bin through mass-dimension 7 — the statement $$\mathrm{Escapee}^{\ \rm perturbative,\ local,\ SM\text{-}zero\text{-}mode}_{d\le 7} = \varnothing ,\qquad O_{\rm SG9,\,d\le7}(E_{\rm frozen}) = 0 .$$
• Status of that leg: Reduced-to-Axiom on the $d\le7$ slice (the declared-class Wilson zeros themselves are DERIVED-GIVEN-E; the physical-completeness step on the bounded slice is AXIOM-OPEN, carried by a regenerated machine certificate).
• KK-mediator channel: DERIVED-on-channel, all-order in KK number (the gauge tower; the unrestricted "no KK mediator" statement is DERIVED-on-channel given R4).

Both legs are genuine results: given the frozen product geometry and the observed spectrum, every dangerous Wilson coefficient in the declared class is an exact zero (not a tuned-small number), the $d\le7$ census closes with no escapee, and a closed-form triality selection rule kills KK leptoquarks at every level. What stays open is everything beyond: the all-order ($d>7$) local-operator channel, the from-scratch internal-sector reconstruction, the formal KK lemma write-up, the seesaw scale $M_R$, and the lifetime (Diagnostic by design). Under the superseded least-closed-residual rubric — a gate's status is its least-closed residual — the mid-audit roll-up read OPEN; on the ratified board SG-9 stands RESOLVED +0 with these residuals shown unchanged.

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

2. Frozen inputs (what SG-9 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 spectrum $E_{\rm frozen}$ is given / charged / inherited — the SM chiral content, the family index $\chi(K_6,E)=-3$, the colour/triality labels. It enters SG-9 as input. SG-9 does not derive $E$. Every "killed" or "vanishes" below is a statement about this $E$ and this selected geometry, not a derivation of either.
• The product gauge geometry $K_{\rm gauge}=K_6\times S^2\times S^1_Y/\mathbb{Z}_2$, with $K_6=SU(3)/T^2$, is selected, not forced (a SHAPE feature inherited OPEN from the geometry gates, R7/R8). SG-9 reads it; SG-9 does not prove product-over-simple-group.

3. Object anchors (given-E / upstream)

SG-9 acts on the active 13D branch $\mathfrak{B}_{\rm active}$, $D=13=4+6+2+1$, whose tenth load-bearing term $\mathcal{E}_{\rm proton}$ supplies the sector machinery. The gauge routing is by isometry:

$$K_6=SU(3)/T^2 \to SU(3)_c,\qquad S^2 \to SU(2)_L,\qquad S^1_Y/\mathbb{Z}_2 \to U(1)_Y,$$

whose zero-mode isometry algebra is the direct sum $\mathfrak{su}(3)_c\oplus\mathfrak{su}(2)_L\oplus\mathfrak{u}(1)_Y$. The matter sectors are the SM Weyl multiplets $Q,u^c,d^c,L,e^c,\nu^c$, the Higgs doublet $H,H^\dagger$, and the covariant derivative $D$ — nothing else is admissible on the frozen branch (that absence is the physics). Status: GIVEN-E / upstream-inherited — not SG-9-derived.

4. Root and master-anchor traceability

Deep roots that are load-bearing for SG-9:

Scale enters only through the (UNKNOWN) seesaw scale $M_R$ on the strictly-separate dim-5 leg; causal order is not a primary load-bearing anchor here.

Master anchors in play: finite invariant ledgers · no unpaid labels · the frozen branch (AUDIT ONLY) · given-$E$ · the declared dangerous operator class · open-residual discipline.

5. The SG-9 anchor ledger (the universal table)

6. The arithmetic — the two strong legs, in full

6.1 The implication chain (the spine)

$$ \underbrace{K_{\rm gauge}=K_6\times S^2\times S^1_Y/\mathbb{Z}_2}_{\text{product, not simple-group}} \;\Rightarrow\; \underbrace{\text{no }X/Y,\ \text{no coloured Higgs}}_{\text{mediators absent}} \;\Rightarrow\; \Pi_q\Pi_\ell=0 \;\Rightarrow\; \Pi_q M\Pi_\ell=0 \;\Rightarrow\; C_{\rm dangerous}\equiv 0 . $$

6.2 The no-mediator identity (Lemma 2, DERIVED-GIVEN-E)

For any sector-respecting mediator $M=\sum_i \Pi_i M_i\Pi_i$, $$\Pi_q M\Pi_\ell=\sum_i(\Pi_q\Pi_i)\,M_i\,(\Pi_i\Pi_\ell)=\sum_i \delta_{qi}\,\delta_{i\ell}\,\Pi_i M_i\Pi_i=0\quad(q\neq\ell).$$ This is a theorem given the projectors, not an axiom. The macro-orthogonality it needs is itself forced by the $SU(3)_c$ colour Casimir: quarks sit in colour $\mathbf 3$ ($C_2=4/3$), leptons in colour $\mathbf 1$ ($C_2=0$), and $V_{\mathbf 3}\otimes V_{\mathbf 1}=0$ as eigenprojectors for distinct Casimir eigenvalues — independent of the $\mathbb{Z}_6/\mathbb{Z}_2$ chamber data and immune to generation mixing (R4′). Consequence: every coefficient in the declared class — $QQQL$, $u^cu^cd^ce^c$, $QLu^cd^c$, $QQu^ce^c$, their dim-7 single-field dressings, and the $\bar d^c\bar d^c\bar u^c$ $n$-side channel — is an exact zero, not suppressed.

6.3 The bounded $d\le7$ census — Lemma 1 and the empty escapee bin

Lemma 1 (parity). A $\Delta B=1$ operator needs $3$ net units of quark triality, hence an odd number of quark fields. A Lorentz scalar needs an even total Weyl count. Odd quarks $+$ even total $\Rightarrow$ an odd number of lepton fields $\Rightarrow$ at least one lepton leg. Therefore every local perturbative $\Delta B=1$ Lorentz-scalar operator at $d\le7$ is a quark$\leftrightarrow$lepton sector crossing, and $\Pi_q M\Pi_\ell=0$ kills it.

The census walks the complete known bases: $d=6$ is the complete Weinberg/Wilczek–Zee set; $d=7$ is the complete Lehman basis; $d=5$ Weinberg $(LH)(LH)$ is $\Delta B=0$ ($\Delta L=2$) and is outside proton safety. The regenerated machine ledger reports:

The ledger committed hash 15f5fc0834e2 regenerates byte-stable; the script reads no frozen file and takes no Super-K / $\tau_p$ input.

6.4 The KK-mediator all-order no-go (STEP4 — DERIVED all-order)

Theorem (STEP4-gauge, all-order in KK number). On $\mathfrak{B}_{\rm active}$ with $K_6=SU(3)/T^2$, every KK excitation of the higher-dimensional gauge field (colour adjoint $\mathbf 8=V(1,1)$) carries $SU(3)_c$ triality 0 at every level $n\ge0$; a leptoquark must carry nonzero triality; hence $\Pi_Q M_{\rm KK}\Pi_L=0$ for every gauge-tower KK mediator, all $n$. It rests on two closed forms:

• (T1) Frobenius/Peter–Weyl selection. $V(p,q)$ appears in $L^2(SU(3)/T^2)$ iff $3\mid(p-q)$ (triality $(p-q)\bmod 3=0$), with multiplicity $m_0(p,q)=\min(p,q)+1$. So $\mathbf 3,\bar{\mathbf 3},\mathbf 6,\bar{\mathbf 6}$ are absent at every level.
• (T2) Triality is the $SU(3)$-centre $\mathbb{Z}_3$ charge → an exact ring homomorphism: $t(A\otimes B)=t(A)+t(B)\bmod 3$.

The leptoquark predicate $(p-q)\bmod 3\neq0$ is period-3 and only residue 0 ever appears, so a single period represents all $n$ — no cutoff is used. The machine certificate (verdict PASS) records 0 leptoquark candidates among 13,467 appearing + 540 product KK modes, the injected-falsifier battery firing on planted $\mathbf 3,\bar{\mathbf 3},\mathbf 6,\bar{\mathbf 6},\mathbf{15}$, and $0$ false positives ($0$ fails / $2009$ checks).

6.5 Diagnostic — the result is specific, not trivial

The cancellation is not vacuous: the spectrum carries a non-zero quark colour Casimir, $C_2(\mathbf 3)=\tfrac43\neq0$, while the cross-sector amplitude vanishes. That a non-trivial invariant is non-zero while every declared dangerous coefficient is exactly zero is what makes the result a real algebraic fact about $E_{\rm frozen}$ rather than an artifact. A second specificity check: the census distinguishes the 9 genuine $\Delta B=\pm1$ scalars (KILLED_PROJECTOR) from the 8 quark-only candidates that fail to form a Lorentz scalar at all (VACUOUS_NO_LORENTZ_SCALAR) — the binner has teeth, and an injected escapee fires the falsifier.

6.6 The obstruction split

$$O_{\rm SG9,\,d\le7}(E)=\big(O_{\rm mediator\text{-}absent}(E),\,O_{\rm projector}(E),\,O_{\rm census}(E)\big),\qquad O_{\rm SG9,\,d\le7}(E_{\rm frozen})=0 .$$ The full gate obstruction additionally carries the all-order ($d>7$) local piece and the formal KK lemma: $$O_{\rm SG9}(E)=\big(O_{d>7}(E),\,O_{\rm KK\text{-}lemma}(E),\,O_{\rm SG9,\,d\le7}(E)\big).$$ We do not assert $O_{\rm SG9}(E_{\rm frozen})=0$: the $d>7$ and lemma-write-up residuals are not closed.

7. Declared structures, split into honest objects

Three single phrases each hide more than one claim:

1. "The operator-safety leg." Splits into (a) the declared-class Wilson zeros — DERIVED-GIVEN-E, an algebraic theorem; and (b) the physical-completeness of the bounded slice — AXIOM-OPEN / Reduced-to-Axiom on $d\le7$, graded below AXIOM-CLOSED on the ground that a finite census is not irreducible and not a measured invariant, and carried by the regenerated machine certificate. The verb is "Reduced," not "solved"; the assumption count does not drop.
2. "No KK mediator." Splits into (a) the gauge-tower statement — DERIVED-on-channel, all-order; and (b) the unrestricted statement, which carries two named conditions inherited from R4 (every propagating zero-mode colour rep is triality-0; matter assignment $t(\text{quark})=1,\,t(\text{lepton})=0$) and is scoped to tree-level single-mediator exchange — not the separate $d>7$ local-operator channel. It is a carried condition pending a written, machine-checked lemma, not a discharged certificate.
3. "The sector projectors." Splits into (a) the macro-orthogonality $\Pi_q\Pi_\ell=0$ — DERIVED-GIVEN-E via the colour Casimir (R4′); and (b) the full internal labeling $\{u,d,e,\nu\}$ reconstructed from scratch — AUDIT (R4). The literal A2.6 phrasing ("four rank-3 idempotents on a 3-dimensional space") is algebraically impossible and must be read as an index/label delta, not four literal $3\times3$ projectors.

8. Open residuals — the full all-order / completion family

The bounded wins above are one face of SG-9. These distinct residuals make up the rest, and none is closed by the $d\le7$ census or the gauge-tower no-go:

• All-order ($d>7$) local-operator completeness — OPEN / computation debt. The dominant residual; the unbounded version of Lemma 1. The plug-able target is a finite structural theorem (parity × triality for all $d$), not an enumeration to a cutoff.
• R4 internal-sector reconstruction — AUDIT. No certificates/R4_bundle_labeling/ on disk (only a VERDICT_STUB); the macro-version is earned (R4′) but the full internal labeling is asserted, not reconstructed target-blind.
• STEP4 formal lemma write-up — AUDIT / BLOCKED. The math is done and machine-checked; the all-order theorem is not yet landed as a discharged lemma in theorems/ADM_MORPHISM_COMPLETENESS/.
• Operator-ledger coverage harness — AUDIT. A bounded hand-census/regenerated ledger; the per-row mechanism witnesses are computed but not yet landed in the published coverage table.
• Seesaw Majorana scale $M_R$ — OPEN / UNKNOWN. No value, formula, or hash; the dim-5 Weinberg leg ($\Delta B=0,\,\Delta L=2$) is kept strictly separate from the $\Delta B=1$ claim.

Two further items are disclosed claim-boundary residuals, not holes to plug: dim-6 baryon-violation pressure is SHARED-OPEN across all GUT-class geometries (the product structure is a filter, not a determiner), and the proton lifetime is Diagnostic only (a permanent observation-pending endpoint).

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

• SG-9 does not derive $E$, and does not prove the proton is absolutely stable. Formally: the declared class has $C_{\rm dangerous}\equiv0$ and the $d\le7$ escapee bin is empty, not $\ker O_{\rm SG9}=\{\text{all admissible operators}\}$ at every dimension.
• The product structure is a filter, not a determiner. "No X/Y" follows from $K_{\rm gauge}$ being a product, but product-vs-simple-group is a selected shape feature (R7); it is not a proof that no rival geometry could be safe — no cross-geometry uniqueness is claimed.
• All-order ($d>7$) local-operator completeness is not computed or closed; it is OPEN/computation-debt.
• The STEP4 KK no-go is not a discharged formal lemma; it is a carried, machine-checked condition pending write-up. "No KK mediator" $\neq$ "no $\Delta B=1$ operator."
• R4's full internal labeling is not reconstructed from scratch (only the macro-orthogonality is DERIVED-GIVEN-E).
• This framework does not predict or bound the proton lifetime; the Super-K bound is never an input. $M_R$ is UNKNOWN and must not be folded into the $\Delta B=1$ claim.
• The frozen-branch hashes are audit anchors; they do not validate the physics. The cited certificates/G10_proton_safety/ does not exist — the real backing is the R5 ledger.
• Bounded ($d\le7$) closure is not all-order closure. $O_{\rm SG9,\,d\le7}=0$ does not imply $O_{\rm SG9}=0$.

10. Specialist closure plan

Each open residual is a concrete, finite, target-blind work-package. Shared traps named so they are not repeated: never reverse-engineer a posit to "no dangerous operators above $d=7$" (the script must run identically in a proton-decay world); the fully general "no operator at any dimension" form is an unbounded universal negative (a unicorn, not a target); do not cite the phantom G10_proton_safety/; never let bounded-census confidence inflate the SHARED-OPEN verdict; do not relocate the residual by enlarging the declared class to "whatever vanishes."

1. Hole C — R4 internal reconstruction (the shared blocker; do first). Reconstruct $\mathcal E_{\rm matter}$ target-blind from the A2.3 factorisation + R1.4 projectors + global $\mathbb{Z}_6$ centre (a68ee92a75be) + $\mathbb{Z}_2$ orbifold (ac4d2df3e708); confirm $\Pi_i\Pi_j=\delta_{ij}\Pi_i$ and $\Pi_q\Pi_\ell=0$ to machine precision and re-hash 3b8d68559f5e. Bounded linear-algebra-over-representations. Falsifier: a non-zero overlap $\langle\Pi_q\cdot\Pi_\ell\rangle$ downgrades the gate. Unblocks Holes A and B.
2. Hole E — $M_R$ inversion (half-day; pays two debts). Compute the required $M_R^{\rm req}=N_\nu^2/|\Delta m^2|$ and test whether it lands on a frozen scale already committed ($M_U\approx10^{16}$ GeV, $R_0^{-1}$, or a chamber-flux scale). DERIVED-CLOSED if a chamber computation yields $M_R$ parameter-free; REFUTED if a definite chamber $M_R$ misses $M_R^{\rm req}$. Run the inversion before adopting any identity (the corpus $M_R=\kappa M_U$ candidate is ~231× off — do not bank a tuned $\kappa$-power). Keep strictly separate from $\Delta B=1$.
3. Hole B — STEP4 formal lemma. Write the Frobenius/Peter–Weyl triality selection rule as a formal all-order-in-KK lemma with its three conditions explicit, and land the certificate in theorems/ADM_MORPHISM_COMPLETENESS/. Falsifier: a propagating zero-mode field in a nonzero-triality colour rep on the branch.
4. Hole A — all-order $d>7$ (the dominant residual). Generalize Lemma 1's parity × triality argument to all $d$, with Holes B/C discharged. Success = a closed-form proof that quark-count parity ⇒ lepton-leg at every $d$ with a dimension-independent dressing argument; a refuting admissible $d>7$ operator (neither sector-crossing nor mediator-required) is a valid structure-first close.
5. Hole D — ledger coverage harness. Regenerate the CSV (bc1e4e84840a) target-blind, confirm byte-equality, attach a machine-checked mechanism witness to every "Covered" row; point at the real R5 ledger, not the phantom G10 folder.
6. Hole F — corpus-editorial fixes. Relabel the "QQQL HH (dim-7)" row to single-H $d=7$ (it is actually $d=8$); footnote the $\bar d^c\bar d^c\bar u^c$ row's mechanism (absent triplet, not the projector); demote the phantom G10_proton_safety/ citation. Owner edits; none opens an escapee.

Closing Holes A–D upgrades SG-9 from "two strong legs, gate open" toward whole-gate closure — and even then, only given $E$ and the selected geometry. Holes E and F are reinforcing. The SHARED-OPEN pressure, the lifetime, and the inherited R7/R8 selection are claim-boundary endpoints, not plug targets.

11. Completion tests for this page

Required presence (all met): gate roll-up RESOLVED +0 (frozen mid-audit reading: OPEN (hardened)) · $O_{\rm SG9,\,d\le7}(E_{\rm frozen})=0$ with empty escapee bin · operator-safety leg Reduced-to-Axiom on $d\le7$ · KK channel DERIVED-on-channel all-order · $E$ not derived · frozen hashes (AUDIT ONLY) · product-geometry SELECTED (R7) · no X/Y · no coloured Higgs · macro-orthogonality $\Pi_q\Pi_\ell=0$ (R4′) · no-mediator identity $\Pi_q M\Pi_\ell=0$ · Lemma 1 · the $d\le7$ census table (escapee 0, hash 15f5fc0834e2) · STEP4 T1/T2 (0 candidates / 13,467 + 540 modes) · diagnostic $C_2(\mathbf 3)=\tfrac43\neq0$ · all-order $d>7$ residual · R4 AUDIT residual · STEP4 lemma residual · $M_R$ UNKNOWN residual · SHARED-OPEN disclosure · lifetime Diagnostic · filter-not-determiner anti-claim · hashes-don't-validate anti-claim · phantom-cert anti-claim.

Required absence (all held): no claim SG-9 is physics-closed · $E$ derived · the proton proven absolutely stable · a lifetime predicted/bounded · cross-geometry uniqueness · the product structure sold as forced · all-order $d>7$ computed · the STEP4 lemma discharged · R4 full internal reconstructed · "all KK are BRST-exact" · hashes validate physics · the G10 certificate cited as real · bounded closure = all-order closure · any reader-visible build-process vocabulary.

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

See also: the anchoring method · Layer 2 — no unpaid exact labels (why a structural zero is not a tuned-small number) · Layer 4 — carrier-forcing & the given-E wall · the shape-minimality challenge (why "product-not-simple" is selected, not forced) · the full SG-9 dossier.