SG-2 — Gauge Recovery $SU(3)_c\times SU(2)_L\times U(1)_Y$ (GUT Gate 2): Per-Gate Closure-Attack Dossier

What this is. The closure-attack packet for SG-2 (Gauge recovery — the Standard-Model gauge algebra from internal isometries) on the frozen 13D K₆ branch ONLY. It recaps how our geometry recovers the SM gauge algebra (concise; the full certificate lives on the published site), verifies where the status actually stands, names every open residual, and lays out the attack plan to push the gate further. It is not a rival comparison (that is BATTLE_GATES/), not the one-line status ledger, not a reprint of the manuscript.

Binding discipline (carry verbatim). No status was ever upgraded. Frozen branch dcc66f1b2685 / manifest meta a5b1e6f9d951 READ-ONLY. Honest throughout: the gauge-group OUTCOME is a rival TIE — a filter that string/M/F/NCG/lattice all pass, NOT a framework-discriminating determination; the recovery is computed GIVEN the selected geometry E (the isometry route does not prove this gauge group unique across all geometries, only that this frozen branch yields it); the genuine framework-internal contribution is the carrier-forcedness, not the outcome. SHAPE is selected-not-forced absolutely — forced only inside the declared grammar + MDL metric, with ~9–10 injected reals beyond the 4 anchors. $K_6=SU(3)/T^2$ is the unique clean SU(3) carrier by abelian-isotropy uniqueness (the explicitly-built $\mathbb{CP}^2=SU(3)/U(2)$ route over-produces gauge under the CSDR centralizer rule — its non-abelian $U(2)$ isotropy forces a lose-lose fork: keep $S^2,S^1$ → an extra unwanted $SU(2)\times U(1)$, so Gate-2 fails, or drop them → isotropy-lock, A1.4 violated (the manuscript states this disjunctively, §3.5); separately its family count is a tunable bundle choice killed downstream at Gate-4/chirality. The 11D CP² end-to-end build BREAKS). S² is FORCED by fact F1; $S^1_Y/\mathbb{Z}_2$ is FORCED by fact F2. Closure paths must be non-target-fitted (the κ³/π falsification test: a proposed axiom counts only if it would be written WITHOUT knowing the target). given-E ≠ derivation of E. AXIOM-CLOSED ≠ proven; selection ≠ derivation; dissolved ≠ solved.

0. Header & Verdict

0.1 Abstract — established / open / what would close it

Established (given-E, given the selected & frozen geometry). 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, after the quotients / parities / bundle data act. The witness is the simple-summand multiset $\{\mathfrak{su}(3),\mathfrak{su}(2),\mathfrak{u}(1)\}$, $8+3+1=12$ generators, rank 4, no extra unbroken factor and no missing SM factor (the gate's failure mode), read architecturally with no coupling value consulted. Color is carried by $K_6=SU(3)/T^2$, weak by $S^2$, hypercharge by $S^1_Y/\mathbb{Z}_2$. Beyond the bare recovery, the session establishes three architecture-neutral forcedness results that are the gate's genuine framework-internal content: (C1) $K_6$ is the unique clean SU(3) carrier by abelian-isotropy uniqueness — $T^2$ is the unique purely-abelian SU(3) isotropy, so the CSDR centralizer survives no extra gauge; (C2) $S^2$ is FORCED by fact F1 (no abelian/torus carries non-abelian $SU(2)$); (C3) $S^1_Y/\mathbb{Z}_2$ is FORCED by fact F2 (closed odd-dim → mirror fermions → LEP $Z$-width).

Open (the honest deductions). (1) The gauge-group OUTCOME is a rival TIE — string, M-, F-theory, NCG, and lattice all reproduce $SU(3)\times SU(2)\times U(1)$ by their own routes. Recovering the SM gauge group is a filter every serious framework passes, not a framework-discriminating determiner; on the OUTCOME, SG-2 ties. (2) The recovery is given-E — it certifies this geometry + bundle yields this algebra, not that this algebra is unique across all geometries. The isometry route is an existence-and-equality statement on one frozen branch, not a cross-geometry determination. (3) The forcedness results C1–C3 are stated within the declared grammar ("forces = isometries"; the CSDR centralizer rule; the anti-fitting discipline). Whether they are architecture-neutral enough to bind a reviewer who rejects the grammar is the live attack surface — they are theorems inside the category (R2.5), not category-free. (4) Carrier completeness over ALL SU(3) carriers is established for the clean / purely-abelian-isotropy class (the CP² non-abelian-isotropy route is killed), but a full enumeration-theorem over every homogeneous and non-homogeneous SU(3) carrier is an audit-grade claim, not a delivered theorem. (5) Representation-level recovery (Appendix D charge tables) and the centers / $\mathbb{Z}_6$ global structure are deferred to Gates 3–5; the Gate-2 algebra is blind to the global quotient (Lie-algebra-level only). (6) No coupling-unification numerics are claimed here (that is SG-7); $\alpha_i(M_Z)$ are declared anchors, not Gate-2 outputs.

What would close it (the ladder). The OUTCOME is a TIE and cannot be promoted — closure here does not mean "prove the SM gauge group" (every framework does that). DERIVED-CLOSED for SG-2 means hardening the carrier-forcedness from category-relative theorems to architecture-neutral theorems: a clean abelian-isotropy / CSDR-centralizer uniqueness theorem ("$T^2$ is the unique isotropy whose centralizer in $SU(3)$ adds no gauge"), an F1/F2 generalization (no abelian carrier supplies non-abelian $SU(2)$; no closed odd-dim factor supplies one-sided chirality — both at full generality), and an SU(3)-carrier completeness audit (enumerate every SU(3) carrier and show all non-$T^2$-isotropy ones over-produce gauge or fail chirality). The realistic ceiling per residual is AXIOM-CLOSED: name the grammar posit ("forces = isometries"; "gauge-active = non-trivial centralizer") in plain sight so the category-relativity is a declared axiom, not a hidden assumption. The most likely honest outcome on the OUTCOME-TIE residual is DISCLOSED (it is correct as stated; there is nothing to close).

0.2 Website source-of-truth (link, don't duplicate)

The full construction, the worked algebra recovery, the representation/charge tables, the exotics ledger, and the freeze records are the published manuscript, Paper I:

• GUT.html §6.2 (Gate-2 card, binding status) — https://physics.magflowmeters.com/articles/GUT.html
• §5.1 narrative module (requirement → prospective constraint → what it eliminates → frozen survivor → authority); §3.5 "The Menu of Internal Geometries" (the elimination funnel incl. the CP² verdict); §3.6 the worked anomaly calibration case; Appendix CR module CR2 reader companion (explanatory only, status carried verbatim); Appendix D (formal certificate authority for Gates 2 and 3 — D.1 surviving algebra, D.2 representation assignment, D.3 hypercharge/electric charge, D.4 exotics ledger); Appendix GS (candidate geometry datasheets + GS.2 facts F1/F2/F3 + GS.5 coset shelf + GS.10 funnel + GS.11 one-constraint thought experiment); Appendix R0 (freeze records + frozen parameter manifest).
• Downstream UV / Λ cross-references (do not touch this gate): Paper IV, TOE.html — https://physics.magflowmeters.com/articles/TOE.html.

This dossier recaps only what is needed to attack; the site controls all common material.

1. How OUR geometry recovers the gauge algebra — the closure claim (concise)

Full derivation: GUT.html §6.2 + §5.1 + Appendix D + Appendix GS + CR2. This section is the attack-grade recap, not the derivation.

1.1 The mechanism end-to-end

The gauge sector is recovered by the category-defining translation of the program: gauge forces are the internal isometries of the compact factors (Dictionary row 1; Definition R2.5; §2.9). Gate 2 is the root gate — Gates 3, 4, 5, 7, 8, 9, 10 all consume its output. The pipeline is one architectural read (§5.1):

 K_gauge = K6 × S² × S¹_Y   --isometry algebra-->   g_geom = su(3) ⊕ su(2) ⊕ u(1)
   --quotients/parities/bundle data act-->   g_SM = su(3)_c ⊕ su(2)_L ⊕ u(1)_Y   (EQUALITY)

The load-bearing factors, named so a reader can attack each:

1. Carrier identification (the existence leg). Each compact factor sources one simple summand of the gauge algebra. $K_6=SU(3)/T^2$ is the flag manifold whose isometry algebra is $\mathfrak{su}(3)$ (color); $S^2$ is the homogeneous space of $SU(2)$ (weak); $S^1_Y$ is the parent circle whose translations act as $U(1)_Y$ (hypercharge). Gauge bosons are the KK modes of these isometries (D.1). (Attack handle: this is one factor → one summand by construction of $K_{\rm gauge}$; the existence leg is given-the-selected-geometry.)

2. The equality clause (the no-extra / no-missing leg). The certified claim is equality, not containment: the surviving algebra is exactly $\{\mathfrak{su}(3),\mathfrak{su}(2),\mathfrak{u}(1)\}$ with $8+3+1=12$ generators and rank 4 — no extra unbroken factor (the over-production failure mode) and no missing SM factor (the under-production failure mode). This is what the G02_gauge_recovery certificate's multiset test and no-extra-summand check verify. (Attack handle: equality is the whole force of the gate — containment would be trivial; the over-production check is where CP² dies, §1.3.)

3. The abelian-isotropy / CSDR centralizer rule (the carrier-forcedness leg C1). Under coset dimensional reduction (CSDR), a homogeneous-space carrier $G/H$ gauges $G$ only if the isotropy $H$ does not itself act as gauge. The decisive structural fact: the maximal torus $T^2$ is the unique purely-abelian $SU(3)$ isotropy. Its centralizer in $SU(3)$ adds no surviving gauge — so $K_6=SU(3)/T^2$ is a clean SU(3) carrier. Any non-abelian isotropy (e.g. $U(2)\subset SU(3)$) is gauge-active under the centralizer rule and over-produces gauge. This is the gate's genuine framework-internal contribution — not "the SM has $SU(3)$" (everyone has that) but "this carrier is the clean one, and the cheaper rival is not." (Attack handle: "gauge-active = non-trivial centralizer" is a grammar rule; its architecture-neutrality is R2 below.)

4. Fact F1 forces $S^2$ for weak (carrier-forcedness leg C2). GS.2 F1: flat and abelian geometries cannot supply non-abelian forces — the isometry group of any torus $T^n$ is $U(1)^n$ (abelian), so no torus and no torus orbifold has $SU(2)$ among its isometries. The non-abelian weak force therefore cannot be carried by any abelian factor; it requires a genuinely non-abelian-isometry carrier, and $S^2=SU(2)/U(1)$ is the minimal one. (Attack handle: F1 is a clean classification fact inside the category — string theory routes around it via bundles/branes, a different category; §2.9.)

5. Fact F2 forces the fold $S^1_Y/\mathbb{Z}_2$ for hypercharge (carrier-forcedness leg C3). GS.2 F2: closed odd-dimensional factors produce no net chirality — the chiral index of a Dirac operator on a closed odd-dim manifold vanishes identically, so a bare $S^1_Y$ mirrors every fermion. The mirror sector would show at the LEP $Z$-width (which counts $2.984\pm0.008$ light species, no mirrors). The repair is the $\mathbb{Z}_2$ fold: $S^1_Y/\mathbb{Z}_2$ has fixed-point boundaries, and boundaries re-open the chirality channel (APS index). The hypercharge $U(1)_Y$ is the translation generator along the parent circle, quantized by its topology. (Attack handle: F2 forces the fold for the chirality reason, which is properly SG-3/SG-4; for SG-2 the relevant point is that the hyper carrier is the folded circle, not the bare one.)

6. Representation / charge recovery (deferred to Gates 3–5). Appendix D.2–D.3 assign each surviving multiplet its $(SU(3),SU(2),Y,Q)$ from projector data, with $Q=T_3+Y$ verified componentwise and the global $\mathbb{Z}_6$ identification gluing the three centers. SG-2 itself is blind to the global quotient — the $\mathbb{Z}_6$ is invisible to the Lie algebra and binds only representations (Dictionary row 4). The rep-level recovery and centers are SG-3/SG-4 content and are cited here, not claimed here.

1.2 What "closed" means here, and its conditionality

"Closed" for SG-2 is a rigid algebra-equality recovery + three carrier-forcedness theorems, all category-internal — strictly not a cross-geometry uniqueness proof, not "the SM gauge group is forced by nature," not a coupling-unification result. It is conditional on:

• given-E — the SM chiral content E (gauge group, charges, the family index $-3$) is supplied as the target the recovery is checked against. SG-2 certifies this geometry yields E's gauge algebra; it does not derive E. Cross-geometry uniqueness is explicitly disclaimed.
• given-the-selected-geometry — the frozen 13D K₆ branch with the selected $K_{\rm gauge}$. SHAPE is selected-not-forced absolutely (Kolmogorov/MDL-shortest only over the declared grammar + dimensional ladder + named rivals, ~9–10 injected reals beyond the 4 anchors). The carrier-forcedness theorems C1–C3 are forced within that grammar.
• given-the-grammar — the category "forces = isometries" (R2.5) and the CSDR centralizer rule. A reviewer who rejects the category (e.g. a string theorist who sources gauge from bundles/branes) is outside the scope of F1/F2/C1; the manuscript declares this category-relativity openly (§2.9 honesty note).

The genuine, defensible content (the part that survives the honest accounting):

The conditional / non-genuine content (the rival-TIE and the cross-geometry leg):

So the precise statement of closure: the frozen branch recovers the SM gauge algebra by equality (not containment), and — the framework-internal part — forces its carriers (clean SU(3) by abelian-isotropy uniqueness, weak by F1, hyper-fold by F2) within the declared grammar; the gauge-group OUTCOME itself is a tie every framework passes, and uniqueness across geometries is not claimed.

2. Verify the status — is DERIVED-GIVEN-E real?

This is the verification a skeptic would run. Each witness gets a grade — hand-checkable / symbolic / machine-lane — and an honest reproduces? flag.

2.1 The witness ledger

2.2 What reproduces, plainly

• The algebra recovery reproduces by hand. The isometry algebras of $SU(3)/T^2$, $S^2$, and $S^1$ are textbook; the multiset $\{8,3,1\}$ and rank 4 fall out with no further input. This is the strongest, most defensible leg — it is standard differential geometry on a declared carrier.
• The carrier-forcedness facts F1/F2/C1 reproduce by hand (within the grammar). F1 is the one-line fact that $\mathrm{Isom}(T^n)$ is abelian; F2 is the one-line fact that the odd-dim chiral index vanishes; C1 is the group-theory fact that $T^2$ is the unique maximal torus (unique purely-abelian isotropy) of $SU(3)$. None needs a number; all are deformation-proof structural statements.
• The CP² kill reproduces (the genuine discriminating move). That $U(2)=(SU(2)\times U(1))/\mathbb{Z}_2$ is non-abelian and therefore gauge-active under the centralizer rule is hand-checkable; the 11D CP² build is the corpus's own adversarial witness that the cheaper carrier breaks at Gate 2. This is the part of SG-2 that is not a tie — it is a framework-internal elimination.

2.3 What does NOT yet reproduce (honest gaps in the status)

• The OUTCOME does not "reproduce" as a framework discrimination — it is a TIE. That $SU(3)\times SU(2)\times U(1)$ comes out is true and re-derivable, but it is not evidence for the geometry because every serious framework reproduces it too. The skeptic's correct verdict: the OUTCOME leg of SG-2 is a shared pass, not a framework result. What is framework-specific is the carrier-forcedness, and that is category-relative (R2).
• Cross-geometry uniqueness does NOT reproduce — it is not claimed. There is no theorem "this gauge group is the unique output of all admissible geometries." The recovery is given-E: a single-branch existence+equality certificate. A reviewer asking "why this group, across all geometries?" gets no Gate-2 answer — only "this frozen branch yields it, and the carriers are clean/forced within the grammar."
• The C1 uniqueness is a clean-class result, not an ALL-SU(3)-carriers theorem. Abelian-isotropy uniqueness says: among purely-abelian-isotropy SU(3) carriers, $T^2$ is unique. It does not by itself enumerate every SU(3) carrier (homogeneous and non-homogeneous) and show each non-$T^2$ one fails. The CP² route is killed explicitly; a full SU(3)-carrier completeness audit is an audit-grade promise, not a delivered enumeration theorem (R3).
• The G02 multiset certificate is AUDIT in this packet. The no-extra-summand / multiset check is a finite, fail-closed test but is referenced, not independently re-executed here. Until re-run, "no extra factor survives" is declared-not-independently-verified-here (the same class as other certificate-referenced flags across gates).
• Architecture-neutrality of F1/F2/C1 is asserted, not proven. The manuscript calls these "architecture- neutral" — but they are theorems inside the "forces = isometries" category. Their force against a reviewer who works in a different category (bundles/branes) is declared-conditional (§2.9), not established.

2.4 Why DERIVED-GIVEN-E (not higher, not lower)

• Not FORCED / not an unconditional claim: the OUTCOME is a rival TIE and uniqueness is given-E. There is no cross-geometry forcing of the gauge group; promoting SG-2 to "FORCED" would overclaim the shared OUTCOME as a framework result. The label is held conservatively below any unconditional reading.
• Not PARTIAL / not OPEN: the algebra recovery is rigid — equality (not containment), a deformation-proof multiset, three hand-checkable carrier-forcedness facts, and a genuine elimination (CP² over-produces). There are no fitted reals in this gate (the couplings are anchors handled at SG-7; the scales/normalizations live downstream). The gate is not a "filter-pass with floating inputs" like SG-7/SG-8 — it is a structural recovery.
• DERIVED-GIVEN-E is exactly right: a rigid algebra recovery + carrier-forcedness, conditional on E and on the selected geometry/grammar, with the OUTCOME-TIE and cross-geometry-uniqueness honestly disclaimed. This matches the SCOPED_GUT ledger SG-2 line verbatim and tiers SG-2 with SG-3 (the family-index gate) as the two rigid given-E recoveries.

3. The attack surface — what to attack (ranked by leverage)

Every open residual as a named object with its precise obstruction. Leverage = how much closing it moves the gate (and whether it converts a category-relative theorem into an architecture-neutral one — the only kind of "more closure" available here, since the OUTCOME is a fixed TIE).

3.1 The residual register

3.2 Leverage ranking (attack order)

1. R2 (architecture-neutrality of abelian-isotropy / CSDR uniqueness) — the main framework-internal contribution; hardening it from category-relative to category-neutral is the single highest-value closeable target.
2. R3 (SU(3)-carrier completeness over ALL carriers) — turns "unique clean carrier" into "unique carrier"; the CP² kill is already in hand, so this is finishing an enumeration.
3. R4 / R5 (F1 / F2 generalization) — strengthen C2 / C3 from stated facts to closed theorems.
4. R7 (G02 AUDIT) — cheapest to close (owner artifact); makes the certificate machine-real.
5. R1 / R6 / R8 (OUTCOME-TIE / given-E / grammar) — DISCLOSED honesty residuals; correctly stated, not "closed." The action is to keep them stated correctly, never to dress the OUTCOME tie as a framework win.

The cardinal honest point. R1 is where the gate's integrity lives. The OUTCOME ($SU(3)\times SU(2)\times U(1)$) is a TIE; if a closure path tries to bank "we recover the SM gauge group" as a framework discrimination, it has mis-attributed a shared pass — the analogue of tuning to the known answer at the framing level. The κ³/π falsification test's spirit applies: a Gate-2 closure claim counts only if it is a statement the OTHER frameworks could not also write. The only such statements are the carrier-forcedness ones (R2/R3/R4/R5) — and even those are category-relative until R2/R8 are discharged.

4. THE ATTACK PLAN — closure paths (the core)

For each residual: the technique (closure playbook T1 axiom-floor / T4 no-go / T5 firewall / T6 all-operator-conditional / T7 eliminative / T10 selector), the named axiom it could reduce to (stated so it would be written WITHOUT the target value — the κ³/π falsification test), the specialist target (theorem to hand off) or the owner artifact (certificate/computation) needed, the math to attempt, and the success ladder (DERIVED-CLOSED rare → AXIOM-CLOSED likely → sharper-OPEN → REFUTED).

4.1 R1 — The gauge-group OUTCOME is a rival TIE (the framing spine)

Why first. R1 is not a residual to be closed — it is a residual to be kept correctly stated. Every other SG-2 claim's value depends on not mis-banking the OUTCOME as a framework win. If SG-2 ever reports "we predict/derive the SM gauge group" as a discriminating result, it has committed the cardinal overclaim, because string/M/F/NCG/lattice all recover the same group by their own routes.

Technique: T5 (no tuning to the known answer firewall) at the framing level. The firewall test: for each Gate-2 claim, ask "could a rival framework write this exact sentence?" If yes, it is a TIE leg and must be labeled shared; if no, it is a genuine framework-internal leg.

The firewall, claim by claim:

Named axiom it could reduce to (κ³/π-clean). The honest reduction is not a closure of R1 but a clean statement of the gate's discrimination boundary:

AXIOM-GAUGE-OUTCOME-TIE (framing form): "Recovery of the Standard-Model gauge group is a constraint every serious framework satisfies; SG-2's discriminating content is exclusively the carrier-forcedness within the 'forces = isometries' category — the unique clean SU(3) carrier (abelian isotropy), F1-forced weak, F2-forced hyper-fold — and not the gauge-group outcome itself."

This is writable without any target value — it is a statement about which leg discriminates, not about any number. It passes the falsification test by construction (it explicitly refuses to bank the shared OUTCOME).

Specialist target / owner artifact. None — R1 is a claim-boundary discipline, not a computation. The action is a wording audit: confirm that everywhere SG-2 is cited (the ledger, the dossiers, the public package), the OUTCOME is labeled a shared filter-pass and only the carrier-forcedness is labeled framework-internal.

Success ladder. - DERIVED-CLOSED / AXIOM-CLOSED: not applicable (no axiom to close a tie). - DISCLOSED (the correct endpoint): AXIOM-GAUGE-OUTCOME-TIE named; the OUTCOME labeled shared; the carrier-forcedness labeled the framework-internal contribution. This is essentially already where the corrected ledger stands (the SG-2 line carries "the gauge-group OUTCOME itself is a TIE … a filter every framework passes, not a framework-discriminating determiner"). - REFUTED: would require showing the OUTCOME is not a tie (i.e. some rival cannot recover the SM gauge group) — false; the OUTCOME genuinely is a shared pass.

Honest disposition: DISCLOSED. R1 is correct as stated and is the gate's honesty spine. The closure is to keep it stated, never to promote the tie.

4.2 R2 — Architecture-neutrality of the abelian-isotropy / CSDR-centralizer uniqueness (the main lever)

Why second. This is the gate's principal framework-internal contribution. Currently "$K_6$ is the unique clean SU(3) carrier" is a theorem inside the "forces = isometries" + CSDR grammar. The biggest available gain is to harden it toward architecture-neutrality — a statement a wider class of reviewers must accept.

Technique: T1 (axiom-floor) + T10 (selector). Name the centralizer rule as an explicit posit, then prove the uniqueness given that posit as cleanly and generally as possible.

The structural claim to harden. Under CSDR on a homogeneous carrier $G/H$, the 4D unbroken gauge group is the centralizer $C_G(H)$ (the part of $G$ that commutes with the isotropy embedding), and the isotropy $H$ acts as gauge unless it is "absorbed." For color $G=SU(3)$:

• $H=T^2$ (maximal torus): $C_{SU(3)}(T^2)=T^2$ — no extra non-abelian gauge survives; the carrier is clean. ($T^2$ is the unique connected abelian subgroup of maximal rank — the unique maximal torus up to conjugacy.)
• $H=U(2)$ (the CP² isotropy): $U(2)$ is non-abelian; $C_{SU(3)}(U(2))$ and the gauge-active isotropy together over-produce an unwanted $SU(2)\times U(1)$ (Gate-2 fails) or lock weak/hyper into color (violating the matter-routing rule A1.4). CP² is killed.

Named axiom (κ³/π-clean).

AXIOM-CLEAN-CARRIER (target-blind): "A homogeneous gauge carrier $G/H$ is 'clean' iff its isotropy $H$ is a maximal torus of $G$ (equivalently, $H$ is purely abelian of maximal rank, so its CSDR centralizer adds no non-abelian gauge); for $G=SU(3)$ the unique clean carrier is $SU(3)/T^2=K_6$."

This is writable with no flavor/charge/coupling number in sight — it references only Lie-subgroup structure and the CSDR centralizer. It passes the falsification test.

Specialist target (to hand off). A CSDR-centralizer uniqueness theorem: "Among all homogeneous spaces $SU(3)/H$ with $H$ a closed connected subgroup, the carriers whose CSDR reduction yields exactly $\mathfrak{su}(3)$ as the surviving 4D gauge algebra (no extra factor, no locking) are precisely those with $H$ a maximal torus; up to conjugacy $H=T^2$, so $K_6$ is unique." Hand this to a CSDR / coset-reduction specialist. The architecture-neutral strengthening is to prove it without invoking the anti-fitting rule (the manuscript is explicit that the CP² exclusion "does not rely on the anti-fitting rule alone" — the centralizer argument is the standalone one).

Math to attempt. (1) Enumerate the closed connected subgroups $H\subset SU(3)$ up to conjugacy: $\{1, U(1), T^2, SU(2), U(2), SU(3)\}$ (and the relevant embeddings). (2) For each, compute the CSDR-surviving gauge algebra $=$ centralizer data and check the equality clause ($\mathfrak{su}(3)$ exactly, no extra, no locking). (3) Confirm $H=T^2$ is the unique $H$ passing. This is a bounded, finite Lie-theory computation — the subgroup lattice of $SU(3)$ is small.

κ³/π falsification test. PASSES cleanly: AXIOM-CLEAN-CARRIER references only group structure, never a target value. There is no number to reverse-engineer — the test here is whether the enumeration returns $T^2$-uniqueness, which is a theorem, not a tuned coincidence.

Success ladder. - DERIVED-CLOSED (genuinely reachable): the CSDR-centralizer uniqueness theorem is proven over the full $SU(3)$ subgroup lattice → "$K_6$ is the unique clean SU(3) carrier" becomes a theorem inside CSDR, not an assertion. This is the realistic high outcome — the computation is bounded. - AXIOM-CLOSED (likely floor): AXIOM-CLEAN-CARRIER named; the centralizer rule made an explicit posit; the $T^2$-uniqueness stated relative to it. The category-relativity (it is a CSDR statement) is then a declared axiom, not a hidden one. - sharper-OPEN: the enumeration shows another $H$ also passes the equality clause → C1 weakens (unlikely; $T^2$ is structurally the only abelian maximal-rank isotropy). - REFUTED: $T^2$ itself fails the equality clause under a careful CSDR computation → the whole carrier claim falls (very unlikely; this is standard coset reduction).

Honest disposition: OPEN, but the most tractable physics-internal target in the gate. The subgroup-lattice computation is bounded and could reach DERIVED-CLOSED inside CSDR (architecture-neutral modulo the CSDR grammar). Note the residual architecture-relativity: even a proven CSDR theorem is a statement inside the "forces = isometries" category — full category-neutrality is R8, which is a declared axiom, not a provable theorem.

4.3 R3 — SU(3)-carrier completeness over ALL carriers

The obstruction. C1 (abelian-isotropy uniqueness) covers the purely-abelian-isotropy class and the CP² route is killed explicitly, but there is no delivered enumeration theorem over every SU(3) carrier (homogeneous and non-homogeneous, and product carriers like Witten's $\mathbb{CP}^2\times S^2\times S^1$) showing each non-$T^2$ one over-produces gauge or fails a downstream gate.

Technique: T7 (eliminative) — finish the enumeration. The CP² kill and the wrong-group coset kills ($\mathbb{CP}^{n\ge3}$ gauges $SU(4{+})$; flags of larger groups; $S^{n\ge3}$; lens spaces) are already in GS.5/GS.6/N.4. The task is to assemble these into a single completeness ledger and certify it covers the shelf.

Named axiom (κ³/π-clean).

AXIOM-SU3-CARRIER-SHELF (target-blind): "The complete shelf of internal-isometry SU(3) carriers in the declared category is ${SU(3)/T^2,\ SU(3)/U(2)=\mathbb{CP}^2,\ \text{and products thereof with abelian/}S^2 \text{ factors}}$; every member except $SU(3)/T^2$ either over-produces gauge (non-abelian isotropy, CSDR-active) or delivers the family count only as a tunable bundle choice (anti-fitting fail)."

Writable with no target value — it references only the SU(3) coset shelf and the CSDR/anti-fitting predicates. Passes the falsification test.

Specialist target. A shelf-completeness theorem: enumerate the homogeneous SU(3) carriers (the coset shelf is short and classifiable — GS.5 lists it) plus the relevant products, and certify the elimination row for each (over-production / wrong-group / tunable-family). Hand to a homogeneous-spaces specialist. Extend to a non-homogeneous carrier audit (does any non-homogeneous manifold have $SU(3)$ isometries and survive? — the classification of manifolds with $SU(3)$ isometry is itself bounded).

Math to attempt. (1) Take the GS.5 coset shelf + GS.6 funnel rows for SU(3) carriers. (2) For each, record: surviving gauge algebra (centralizer), the equality verdict, the chirality verdict. (3) Add the product carriers (notably Witten's $\mathbb{CP}^2\times S^2\times S^1$ — passes Gate 2, dies at chirality, GS.11). (4) Assemble the completeness ledger and state the residual scope (homogeneous + classified non-homogeneous).

Success ladder. - DERIVED-CLOSED: full enumeration certified → "$K_6$ is the unique SU(3) carrier (not just the unique clean one)" becomes a theorem over the classified shelf. Reachable for the homogeneous shelf (short, classified). - AXIOM-CLOSED: AXIOM-SU3-CARRIER-SHELF named with the homogeneous shelf certified and the non-homogeneous case flagged as the residual scope. Likely floor. - sharper-OPEN: a non-homogeneous SU(3)-isometry carrier is found that survives the equality clause → the enumeration is incomplete and C1's "unique" downgrades to "unique among classified carriers." - REFUTED: a second clean carrier is found → C1 falls (very unlikely given $T^2$-uniqueness).

Honest disposition: OPEN / AUDIT. The homogeneous shelf is closeable to DERIVED; the full "ALL carriers" (incl. non-homogeneous) is the audit-grade residual. Closing the homogeneous shelf turns "unique clean carrier" into "unique carrier among homogeneous SU(3) carriers" — a genuine, bounded strengthening.

4.4 R4 — F1 generalization (no abelian carrier supplies non-abelian $SU(2)$)

The obstruction. Fact F1 is stated for tori and torus-orbifolds (their isometry group is abelian, so no $SU(2)$). Full generality — no abelian-isometry carrier of any kind supplies a non-abelian gauge factor in this category — is asserted as a class fact, not packaged as a closed theorem with the $S^2$-forcedness as a corollary.

Technique: T1 (axiom-floor) + T4 (no-go). F1 is essentially a no-go: abelian isometry ⇒ abelian gauge.

Named axiom (κ³/π-clean).

AXIOM-NONABELIAN-NEEDS-NONABELIAN-CARRIER (target-blind): "In the 'forces = isometries' category, a non-abelian gauge factor can survive only from a carrier with non-abelian isometry; therefore the non-abelian weak force $SU(2)_L$ requires a non-abelian-isometry carrier, and the minimal such carrier is $S^2=SU(2)/U(1)$."

Writable with no target value — it is a structural statement about abelian vs non-abelian isometry. Passes the falsification test.

Specialist target. A clean no-go corollary: "$\mathrm{Isom}(M)$ abelian $\Rightarrow$ the CSDR-surviving gauge algebra is abelian; hence $SU(2)_L$ requires non-abelian $\mathrm{Isom}$; $S^2$ is the minimal ($\dim=2$, $\mathrm{Isom}=SU(2)$) non-abelian carrier." This is a short Lie-theory statement; hand to the same CSDR specialist as R2.

Math to attempt. Confirm: (1) $\mathrm{Isom}(T^n)=U(1)^n$ abelian (done, textbook). (2) CSDR from an abelian-isometry carrier yields only abelian gauge (centralizer of an abelian group is its own structure → no non-abelian survivor). (3) $S^2$ is the minimal non-abelian-isometry carrier ($S^2=SU(2)/U(1)$, $\dim 2$). Each step is bounded.

Success ladder. AXIOM-CLOSED (likely): AXIOM-NONABELIAN-NEEDS-NONABELIAN-CARRIER named; $S^2$-forcedness a corollary. DERIVED-CLOSED: the no-go ("abelian isometry ⇒ abelian gauge") proven in full generality within CSDR (bounded). sharper-OPEN only if some abelian carrier sneaks a non-abelian factor via bundle data (which is the other category, outside scope). Honest disposition: AXIOM-CLOSED, promotable to DERIVED with the bounded no-go proof; this strengthens C2 (S² forced) from stated-fact to theorem.

4.5 R5 — F2 generalization (closed odd-dim → fold forced for the hyper carrier)

The obstruction. Fact F2 (the chiral index on a closed odd-dim manifold vanishes) forces the fold of the hyper circle (the bare $S^1_Y$ mirrors every fermion → LEP $Z$-width falsifies it). The chirality consequence is properly SG-3/SG-4; for SG-2 the only relevant point is the carrier identity — the hyper carrier is the folded circle $S^1_Y/\mathbb{Z}_2$, not the bare one. The residual: the uniqueness of the $\mathbb{Z}_2$ fold (vs other boundary/orbifold data) is category-internal, not proven minimal.

Technique: T1 (axiom-floor), shared with SG-4. Name the fact; defer the chirality forcedness to SG-4.

Named axiom (κ³/π-clean).

AXIOM-HYPER-CARRIER-FOLD (target-blind): "A closed odd-dimensional carrier produces no net chirality (its Dirac index vanishes); the hypercharge carrier must therefore be the folded circle $S^1_Y/\mathbb{Z}_2$, whose fixed-point boundaries re-open the chirality channel (APS index), rather than the bare $S^1_Y$."

Writable with no target value — index theory + APS. Passes the falsification test.

Specialist target. "Is $\mathbb{Z}_2$ the unique minimal fold of $S^1_Y$ that (a) keeps $U(1)_Y$ as the surviving gauge factor and (b) re-opens a one-sided chirality channel?" Hand to the index-theory / SG-4 specialist. (Note: this is mostly an SG-4 question; for SG-2 the carrier-identity is the deliverable.)

Success ladder. AXIOM-CLOSED: AXIOM-HYPER-CARRIER-FOLD named; the folded circle is the hyper carrier of record. DERIVED-CLOSED (fold-uniqueness) is an SG-4 deliverable, not SG-2's to claim. Honest disposition: AXIOM-CLOSED for the carrier identity; fold-uniqueness deferred to SG-4. For SG-2, $S^1_Y/\mathbb{Z}_2$ is the hyper carrier and $U(1)_Y$ survives — that is all SG-2 needs.

4.6 R6 — The "given-E" conditionality (no cross-geometry uniqueness)

This is a structural-limit / claim-boundary residual, not physics. It cannot be closed without abandoning the given-E posture — and abandoning it would be an overclaim (it would assert the SM gauge group is the unique output of all admissible geometries, which the isometry route does not and cannot show).

Technique: T2-guarded restatement (no relocation). The correct statement is the conditional one.

Named principle (κ³/π-clean).

AXIOM-GIVEN-E-RECOVERY (target-blind): "SG-2 certifies that the selected frozen geometry yields a surviving 4D gauge algebra equal to the Standard-Model algebra; it does not assert that this algebra is the unique output across all admissible geometries. 'given-E' means E is the comparison target, not a derived result."

Writable with no number; it is the scope statement of the gate. Passes the falsification test.

Action (no derivation). Keep the given-E conditional stated wherever SG-2 is cited; never let "the geometry yields the SM gauge group" drift into "the geometry forces the SM gauge group across all geometries."

Success ladder. DISCLOSED-CONSISTENT is the only endpoint. AXIOM-CLOSED/DERIVED-CLOSED are not applicable (there is no theorem of cross-geometry uniqueness to be had from an existence+equality route). Honest disposition: DISCLOSED; correct as stated; not closeable without overclaim.

4.7 R7 — G02 multiset / no-extra-summand certificate (AUDIT)

The obstruction. "No extra unbroken factor survives, no SM factor is missing" is the load-bearing equality claim of the gate, machine-witnessed by the certificates/G02_gauge_recovery/ multiset + no-extra-summand check — but that check is referenced, not independently re-executed in this audit.

Technique: owner artifact (machine-lane), not an axiom. A fail-closed reproducibility task.

Owner artifact needed. (1) Mount the G02_gauge_recovery certificate. (2) Re-run the multiset test on the R1.2/R1.4 isometry data: confirm the surviving simple-summand multiset is exactly $\{8,3,1\}$, rank 4, with the no-extra-summand predicate passing. (3) Re-hash the carrier objects and confirm the branch/manifest hashes (dcc66f1b2685 / a5b1e6f9d951) recompute. (4) Confirm the exotics ledger (D.4) has every candidate marked Absent / Massive.

Math to attempt. None new — execution + verification. The check is mechanical and fail-closed: if any extra summand survives or any SM summand is missing, Gate 2 → Open / not claimed (per §6.12 falsification map).

Success ladder. BLOCKED_INPUTS until the certificate is mounted → then VERIFIED ("claimed certificate pass" becomes machine-real) or REFUTED (an extra/missing factor → downgrade). Highest value-per-effort closeable item — converts an asserted status into a machine-checked one with no new physics.

4.8 R8 — Grammar dependence ("forces = isometries", R2.5)

The obstruction. The entire gate lives inside the category translation forces = internal isometries (R2.5). F1, F2, C1 are theorems inside that grammar; a reviewer who works in a different category (string/brane, where gauge groups arrive through bundle structure groups and branes) is outside the scope of these facts. The manuscript declares this openly (§2.9 category-honesty note).

Technique: T1 (axiom-floor) — name the grammar as the gate's root axiom. The category translation is not provable (it is a modeling choice); the honest move is to make it an explicit, named axiom rather than a hidden assumption.

Named axiom (κ³/π-clean).

AXIOM-FORCES-ARE-ISOMETRIES (target-blind): "In this program's declared category, the 4D gauge group is generated by the continuous isometries of the internal compact factors (Kaluza–Klein); gauge groups are not sourced from bundle structure groups or branes. All Gate-2 forcedness results (F1, F2, abelian-isotropy uniqueness) are theorems relative to this category."

Writable with no number — it is the gate's grammar. Passes the falsification test. It also makes the R1 OUTCOME-TIE sharper: the rivals tie on the outcome precisely because they use a different category to reach it; SG-2's discrimination is inside its own category.

Specialist target / owner artifact. None — this is a declaration, not a computation. The action is to ensure AXIOM-FORCES-ARE-ISOMETRIES is named as the gate's root posit in the axiom ledger (alongside GRANULARITY / SCALE / SHAPE), so the category-relativity of C1–C3 is a declared axiom, not a hidden one.

Success ladder. DISCLOSED / AXIOM-CLOSED. AXIOM-FORCES-ARE-ISOMETRIES named → the category-relativity is explicit. DERIVED-CLOSED is not available (one cannot prove a modeling category is "the right one"). Honest disposition: AXIOM-CLOSED at the named grammar axiom; this is the correct floor and it makes R1 and R2 honest by declaring the category they live in.

4.9 Attack-plan roll-up

REDUCE-vs-RELOCATE verdict on the plan. The plan does not turn one hard problem into three harder ones, and — critically — it does not try to "close" the one thing that cannot be closed (R1: the OUTCOME is a tie; R6: given-E). Each genuine-physics path is a bounded, finite Lie-theory computation on a small object: R2 is the subgroup lattice of $SU(3)$ (six conjugacy classes of closed connected subgroups); R3 is the short, classified coset shelf; R4 is a one-line no-go corollary. None introduces a tunable real — there is nothing to target-fit in SG-2 (no normalization, no scale, no coupling lives in this gate), so the κ³/π blade has little to bite on except the framing (R1), where it correctly forbids banking the OUTCOME tie as a framework win. The remaining residuals are mechanical (R7 owner artifact) or honest declarations (R1, R6, R8). The honest expected outcome of a full campaign: R2 reaches DERIVED-CLOSED-inside-CSDR (the gate's biggest real gain — a uniqueness theorem, not an assertion); R3 reaches DERIVED for the homogeneous shelf; R4 reaches AXIOM-CLOSED→DERIVED; R5/R8 AXIOM-CLOSED; R7 VERIFIED; R1/R6 stay DISCLOSED (correct as stated). No promotion of the gate's label is implied — SG-2 stays DERIVED-GIVEN-E — but its carrier-forcedness moves from category-relative assertions to category-relative theorems, and the OUTCOME-tie stays correctly disclaimed. That is the only kind of "more closure" SG-2 admits: the OUTCOME is, and remains, a tie; the discrimination is, and is hardened to be, the carrier-forcedness.

A. Anchoring & Hardening Map

This is SG-2 run through our internal honesty methodology: for each open residual, decide gap vs wall, hunt the implicit assumption the residual hides, then find the measured invariant truth the residual must ultimately terminate on — and assign the most conservative defensible disposition. The framework, the gap/wall distinction, and the disposition vocabulary used below (DERIVED / DERIVED-GIVEN-E / AXIOM-CLOSED / DISSOLVED / SCHEME-ANCHORED / OPEN / BLOCKED / measured-but-irreducible) are the live ones in the Gaps & Walls Register (https://physics.magflowmeters.com/articles/GAPS_AND_WALLS_REGISTER.html); this section applies that method to the SG-2 residuals named in §3, nothing more.

The decisive structural fact for SG-2. This gate carries no tunable real, no scale, no coupling (§0, §2.3, §4.9 — "no normalization, no scale, no coupling lives in this gate"; $\alpha_i(M_Z)$ are declared anchors handled at SG-7, not Gate-2 outputs). So unlike a quantitative gate, most SG-2 residuals do not terminate on a numeric invariant. They terminate on structural / spectral truths — the observed SM gauge group itself (the content E), and the LEP/SLD light-species count that forbids mirror fermions — or they are claim-boundary / grammar items with no numeric witness, or one computation-debt certificate. SG-7's situation is the opposite kind (Diagnostic-only; threshold rows fitted-not-derived; SCHEME-ANCHORED magnitude) — and Gate-2 by construction consults none of SG-7's numbers (§0). The κ³/π blade therefore has little to bite on in SG-2 except the framing (R1), where it correctly forbids banking the shared OUTCOME as a framework win.

A.1 Per-residual anchoring & hardening table

A.2 Gate-level rollup

• Overall disposition: DERIVED-GIVEN-E — and it stays there. The genuine content (algebra equality $\{8,3,1\}$, rank 4, no extra/missing; carrier-forcedness C1/C2/C3) is rigid within the declared grammar; the OUTCOME is a rival TIE (R1) and cross-geometry uniqueness is not claimed (R6). Hardening moves the carrier-forcedness from category-relative assertions to category-relative theorems — it does not promote the gate's label. Ceiling, as everywhere in the Register: serious candidate, NOT validated.
• Anchors it depends on. SG-2 is structural, so its dependence is mostly on existing structural/spectral anchors, not the numeric four: (1) the observed SM gauge group as the given-E content (R1/R6); (2) the LEP/SLD light-species count $2.984\pm0.008$ (spectrum-E) for the hyper-fold's no-mirror requirement (R5); (3) two grammar-anchors — "forces = isometries" + the CSDR centralizer rule (R2/R8). It depends on none of the numeric anchors {ℏ, M_Pl, α_i(M_Z), y_t, |V_us|}: Gate-2 consults no coupling and no UV number (§0; §2.1 W8). It needs no NEW invariant — the only "new" objects it owes are theorems and one certificate re-run, not measurements.
• Single highest-leverage hardening move: discharge R2 — prove the CSDR-centralizer uniqueness theorem over the (small, six-class) $SU(3)$ subgroup lattice. This converts the gate's principal framework-internal claim ("$K_6$ is the unique clean SU(3) carrier") from an assertion into a DERIVED-CLOSED-inside-CSDR theorem, and it is bounded and hand-off-ready. (Cheapest-by-effort is R7 — mounting and re-running the G02 certificate — which makes the "claimed certificate pass" machine-real with no new physics; it is the best value/effort, but lower leverage on the gate's meaning than R2.)

🎯 Target anchor(s) for this gate

In the terminate-on sense (the internal honesty methodology of the Gaps & Walls Register: every residual must reduce to a measured invariant truth, a named axiom, or an honest OPEN), SG-2 terminates on two measured invariants: the observed particle spectrum E — the SM chiral content, including the observed gauge group $SU(3)_c\times SU(2)_L\times U(1)_Y$ as the given-E content and the LEP/SLD light-species count $2.984\pm0.008$ that forbids mirror fermions — and the measured gauge couplings $\alpha_i(M_Z)$ (the latter are declared anchors, not Gate-2 outputs; SG-2 consults no coupling value — they belong to SG-7). Honest status: the gauge-group OUTCOME is a rival TIE — measured-but-shared, a filter every serious framework passes, so it carries no framework-discriminating anchor and is DISCLOSED (correct as stated, nothing to close), not promotable. The genuine framework-internal content is carrier-forcedness (abelian-isotropy uniqueness, C1/C2/C3), which terminates on E plus the owed SU(3)-carrier completeness theorem — a theorem owed, not an anchor, and therefore OPEN (its full category-neutral form is AXIOM-CLOSED at the named grammar posit "forces = isometries"). No no-witness target is promoted: the cross-geometry-uniqueness target has no witness → OPEN/WALL; the G02 multiset certificate is computation-debt → BLOCKED (input absent on disk) pending re-run. No new anchor is owed by this gate.

5. References & source map

5.1 Website source-of-truth (common material — link, don't duplicate)

• Paper I, GUT.html — https://physics.magflowmeters.com/articles/GUT.html
• §6.2 Gate-2 card (binding status: Claimed certificate pass); §6.12 falsification map (Gate-2 row: extra $U(1)$ / missing $SU(2)$ / wrong rep content → Open / not claimed); §6.13 certificate summary.
• §5.1 narrative module (requirement → prospective constraint → what it eliminates → frozen survivor → layer involvement → authority); §3.5 "The Menu of Internal Geometries" (the elimination funnel; the CP² verdict with abelian-isotropy uniqueness); §3.6 the worked anomaly calibration case (the seven-step template SG-2 follows); §2.9 the category-honesty note (forces = isometries is category-relative).
• Appendix CR module CR2 ("Gauge recovery", reader companion — explanatory only, status carried verbatim).
• Appendix D (formal certificate authority for Gates 2 and 3): D.1 surviving gauge algebra; D.2 representation assignment; D.3 hypercharge / electric charge + D.3.1 explicit charge audit; D.4 exotics and extra modes ledger. (Where any closure language elsewhere conflicts, Appendix D controls.)
• Appendix GS (candidate-geometry datasheets + elimination funnel): GS.2 the three facts F1 (abelian geometries carry no non-abelian force) / F2 (closed odd-dim → no net chirality) / F3 (no isometries → no forces); GS.4 the torus shelf (D2 elimination line); GS.5 coset/homogeneous shelf (the CP² row with the quoted abelian-isotropy verdict); GS.6 the wrong-group cosets; GS.10 the elimination funnel; GS.11 the one-constraint thought experiment (gauge-recovery-alone → Witten's $\mathbb{CP}^2\times S^2 \times S^1$, killed by chirality).
• Appendix R0 / R1 freeze records + frozen parameter manifest (carrier hashes in §0 above).
• Paper IV, TOE.html — https://physics.magflowmeters.com/articles/TOE.html (UV / Λ cross-reference only; does not touch SG-2).

5.2 Corpus locations (authoritative inputs to this dossier)

5.3 Frozen-object hash index (load-bearing; READ-ONLY)

Branch dcc66f1b2685 · manifest meta a5b1e6f9d951 · parity table (fold; SG-3/SG-4 object, co-read for the hyper carrier) ac4d2df3e708 (R1.3). Carrier geometry $K_{\rm gauge}=K_6\times S^2\times S^1_Y$ with $K_6=SU(3)/T^2$ from R1.2 / R1.4 isometry data (registered in R0). UV-package context carried by this branch but set downstream, not consulted by Gate 2: $M_U\sim10^{16}$ GeV, $R_0=1.592\times10^{-17}\,\mathrm{GeV}^{-1}$, threshold $\delta=(+4.8424,-3.1112,-1.7313)$, $\mathbb{Z}_6$, spin-ℂ index $-3$. Machine certificate: certificates/G02_gauge_recovery/ (multiset + no-extra-summand check — AUDIT in this packet, R7). Coupling anchors $\alpha_i(M_Z)$ (R1.8, 6a3b6ef06697) are declared anchors, NOT Gate-2 outputs.

Closing honest statement

SG-2 is DERIVED-GIVEN-E. Its genuine, defensible content is real and rigid: the surviving 4D isometry algebra equals $\mathfrak{su}(3)_c\oplus\mathfrak{su}(2)_L\oplus\mathfrak{u}(1)_Y$ (equality, not containment; the multiset $\{8,3,1\}$, rank 4, no extra/missing factor), and — the framework-internal part — the carriers are forced within the declared grammar: $K_6=SU(3)/T^2$ is the unique clean SU(3) carrier by abelian-isotropy uniqueness (the CSDR centralizer adds no gauge; the cheaper $\mathbb{CP}^2=SU(3)/U(2)$ over-produces gauge via its non-abelian $U(2)$ isotropy and is killed — the 11D CP² build BREAKS), $S^2$ is forced for weak by fact F1, and the folded $S^1_Y/\mathbb{Z}_2$ is forced for hypercharge by fact F2. Its honest open surface is equally clear: the gauge-group OUTCOME is a rival TIE — a filter string/M/F/NCG/lattice all pass, not a framework discrimination; the recovery is given-E (this branch yields the algebra; uniqueness across geometries is not claimed); the carrier-forcedness results are theorems inside the "forces = isometries" category (architecture-neutrality asserted, not proven); SU(3)-carrier completeness over all carriers is established for the clean/homogeneous class but is audit-grade as a full enumeration; and the G02 multiset certificate is AUDIT here. The attack plan reduces these to named, target-blind axioms (AXIOM-CLEAN-CARRIER, AXIOM-SU3-CARRIER-SHELF, AXIOM-NONABELIAN-NEEDS-NONABELIAN-CARRIER, AXIOM-HYPER-CARRIER-FOLD, AXIOM-FORCES-ARE-ISOMETRIES) and bounded finite Lie-theory computations (the $SU(3)$ subgroup lattice; the short coset shelf) — with the κ³/π falsification test guarding the framing (R1: never bank the OUTCOME tie as a framework win), since SG-2 carries no tunable real for the blade to bite elsewhere. The realistic ceiling is a uniqueness theorem inside CSDR for the clean carrier (R2 → DERIVED-CLOSED-inside-CSDR) plus a verified certificate (R7) over a named grammar axiom (R8) — a real strengthening of the carrier-forcedness from assertion to theorem, not a promotion of the gate's label and not a closure of the OUTCOME tie (which is correct as a tie). No status was ever upgraded; frozen branch dcc66f1b2685 / a5b1e6f9d951 READ-ONLY; given-E ≠ derivation of E; the gauge-group OUTCOME is and remains a tie — the discrimination is the carrier-forcedness; nothing applied, nothing deployed.

Dossier built 2026-06-24. Our geometry (13D K₆ branch) only. Common material referenced to the published website source-of-truth, not duplicated.