SG-6 — Moduli Stabilization (GUT Gate 6): Per-Gate Closure-Attack Dossier

What this is. The closure-attack packet for SG-6 (Moduli stabilization / Gate 6) on the frozen 13D K₆ branch ONLY. It recaps how our geometry controls the downstream-used moduli (concise; the full derivation 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 / a5b1e6f9d951 READ-ONLY. Honest throughout: the gate certifies admissibility-restriction + LOCAL stability at the chamber-center witness + phenomenological sufficiency for Gates 1–10 — global stabilization is an explicit NON-claim (F.9.4 / F.10.1); the positive-definite moduli-mass Hessian is DIAGNOSTIC-ONLY (F.9.5, the named most-natural attack surface); witnesses fix the moduli WITHIN the already-selected chamber (given-E, given-the-selected-geometry). The acute soft spot is θ_H⋆ ≈ 2.46×10⁻¹⁴ (the Hosotani phase), READ from the chamber one-loop V_Hos minimum, carrying ~85% of the electroweak hierarchy via v_EW = θ_H⋆/(2πR_γ) — a value extracted from a minimum, not independently derived. 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. The broader four-term V(σ) discharge lives only in downstream Paper IV (EXTERNAL; its own caveats) and changes NOTHING in this gate's certificate.

0. Header & Verdict

0.1 Abstract — established / open / what would close it

Established (given-E, given the selected & frozen chamber). Gate 6 is the curve-not-a-number discipline made operational (§6.6 "$y=R^2$" toy): every modulus an output reads must carry a structural witness of a declared TYPE — Weyl-rigid chamber for $K_6$'s three Cartan shape moduli (chamber-center $\vec u=(1,1,1)$), modular fixed point $\tau=\omega$ for the $F^+$ Cartan-torus, integer winding $n_H=1$ for the Wilson-line Higgs, discrete $\mathbb{Z}_2/\mathbb{Z}_6$ topology for the orbifold/identification, plus boundary-condition pins for the $S^2/S_Y^1$ radii and the threshold spectrum — not a scalar potential whose minimum was placed at the answer. The witness-type predicate tests the type of the witness, so "a potential we minimized at $R_0$" fails by construction. The deliverables are the 12-row witness ledger (F.3), a no-tachyon check (F.4), a witness-coverage lint, and the five-concept claim-type ledger (F.9.1 / F.10.1) that explicitly separates what is proved from what is not. This is genuinely strong discipline and is the program's honest answer to the "hidden-knob" attack.

Open (the honest deductions). (1) Global stabilization is an explicit NON-claim (F.9.4 row 1 / F.10.1) — outside-chamber branches are rejected by admissibility, not stabilized. (2) The positive-definite moduli-mass Hessian is DIAGNOSTIC-ONLY (F.9.5); a precise positive-definite Hessian is "out-of-scope refinement" — and the named most-natural attack surface is F.9.7(2): show the chamber-center witness is a saddle, not a minimum. (3) The acute soft spot is θ_H⋆ ≈ 2.46×10⁻¹⁴ — the Hosotani-phase minimum that carries ~85% of the EW hierarchy via $v_{\rm EW}=\theta_H^\star/(2\pi R_\gamma)$ — read from a one-loop $V_{\rm Hos}$ minimum, not derived-small (frozen $\eta_{BK}=0.009721$ gives only $\sqrt{\eta_{BK}}/2\pi\sim 10^{-2}$, ~12 orders short). (4) The witnesses fix the moduli WITHIN the already-selected chamber (given-E): the gate certifies the selected geometry is internally controlled, not that the witnesses determine the geometry. (5) Stabilization is a SHARED-OPEN hard problem — rivals (string/M/F/NCG/lattice) also struggle (Battle verdict: TIE/SHARED-OPEN, not a framework win).

What would close it (the ladder). DERIVED-CLOSED would require either (a) the four-term V(σ) discharge turning the breathing/volume modulus from witness into a derived positive-Hessian minimum with no new knob, or (b) a target-blind derivation of θ_H⋆ (the small Hosotani phase from chamber data, not read from $V_{\rm Hos}$), or (c) a positive-definite Hessian certificate at $(1,1,1)$. The realistic ceiling per residual is AXIOM-CLOSED: name one explicit, target-blind posit (the modular-fixed-point uniqueness; the uniform operational cell value $\Delta_0$ that pins the loop scale; the second-dimensionful-anchor irreducibility of $v_{\rm EW}$). The honest expected outcome on the two genuine-physics fronts (shape-doublet Hessian, θ_H⋆) is sharper-OPEN — and on the shape doublet, on current Λ-free perturbative geometry, a SADDLE (a falsifier-grade negative, not a closure).

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

The full construction, the witness ledger, and the claim-type discipline are the published manuscript, Paper I:

• GUT.html §6.6 (Gate-6 card) — https://physics.magflowmeters.com/articles/GUT.html
• §5.5 narrative module; the $y=R^2$ curve-not-a-number discipline; Appendix F (Stabilization — the binding authority): F.1/F.2 witness structure, F.3 the 12-row moduli ledger, F.4 tachyon check, F.5 sensitivity, F.9 Stabilization-vs-Admissibility five-concept distinction (F.9.4 NON-claim, F.9.5 Hessian diagnostic, F.9.7 falsification paths), F.10 claim-type ledger + required-vocabulary substitution rule; Appendix CR module CR6 reader companion. Same public article.
• Downstream cross-reference (does not touch this gate's certificate): the four-term $V(\sigma)$ discharge in Paper IV, TOE.html — https://physics.magflowmeters.com/articles/TOE.html (EXTERNAL; its own caveats). The θ_H⋆ / $v_{\rm EW}$ leg is the Gate-8 Hosotani object (§6.8 / Appendix H) on which SG-6's acute soft spot rides.

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

1. How OUR geometry controls the moduli — the closure claim (concise)

Full derivation: GUT.html §6.6 + §5.5 + Appendix F (F.1–F.10) + CR6. This section is the attack-grade recap, not the derivation.

1.1 The mechanism end-to-end

Gate 6 answers a single plain fact: a predictive compactification cannot have its hidden shape and size drifting freely — every modulus any later gate reads must be pinned before a number is compared (§5.5; the $y=R^2$ toy, §6 "Curve-not-a-number"). The framework's mechanism is not "write a scalar potential, minimize it, read off the value." It is a witness-type predicate: each downstream-used modulus must carry a witness drawn from a declared set of structural mechanisms, and the predicate tests the type of the witness, so a tuned potential whose minimum was placed at the answer fails by construction (R1.6 forbids exactly this).

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

1. The Weyl-rigid $K_6$ chamber (the three Cartan shape moduli $\vec u=(u_1,u_2,u_3)$ of $K_6=SU(3)/T^2$). Witness type = admissibility / no-runaway on the spin-ℂ bundle data: off-chamber moduli fail the A0 admissibility check and the branch is eliminated by the selector (Appendix B1.6), not stabilized. The chamber is $\vec u\in[1/2,3/2]^3$ with center pinned at $(1,1,1)$. (Attack handle: the Weyl group $S_3$ permutes $(u_1,u_2,u_3)$, so $(1,1,1)$ is the $S_3$-fixed point — automatically a critical point of any $S_3$-invariant functional, but the Hessian sign is the open object; see §3 R3.)
2. The modular fixed point $\tau=\omega=e^{2\pi i/3}$ (03b30a9c931a) for the $F^+$ Cartan-torus modulus. Witness type = residual modular symmetry forces the order-three fixed point; small perturbations off $\omega$ acquire a non-zero potential under the chamber's RG transport, restoring $\tau\to\omega$. (Attack handle: this is the cleanest witness — a symmetry-protected fixed point, not a minimum read off a curve. But the manuscript's own F.2 wording ("produce a non-zero potential off the fixed point") leans on a potential it does not compute; the symmetry-uniqueness leg is the strengthening target, R7.)
3. Integer Wilson-line winding $n_H=1$ on $K_{\rm gauge}$, on the declared homology cycle $\gamma$. Witness type = topological invariant — $\delta n_H\notin\mathbb{Z}$ is forbidden; integer winding is the only allowed value. (Attack handle: the winding count is protected, but the phase $\theta_H$ that the Wilson line settles at is a continuous modulus on the cycle — and that is θ_H⋆, the acute soft spot; see §3 R1.)
4. Discrete $\mathbb{Z}_2$ / $\mathbb{Z}_6$ topology. The $S_Y^1/\mathbb{Z}_2$ orbifold action $y\mapsto-y$ and the global $\mathbb{Z}_6$ identification have no continuous deformation parameter to drift — fixed by topology, no witness-potential needed.
5. Boundary-condition pins for the $S^2/S_Y^1$ radii and the threshold spectrum. $R_{S^2}$ by KK / Wilson-line balance against the threshold-unification target $M_U$; $R_{S_Y^1}$ by the hypercharge gauge-coupling boundary condition at $M_Z$; the threshold vector $(\delta_1,\delta_2,\delta_3)=(+4.8424,-3.1112,-1.7313)$ from the KK tower. (Attack handle: these are threshold-pinned, i.e. their witness is consistency with Gate 7 — they inherit Gate 7's open status, notably the δ-triple reproducibility gap; given-E, given the frozen scheme.)
6. The 12-row witness ledger (F.3) + no-tachyon check (F.4) + coverage lint. Every modulus / Wilson-line / chamber coordinate / threshold spectrum that any Gate 1–10 reads is listed with its witness, pinned value, downstream consumer, and the effect of a small perturbation $\delta$. The certificate JSON (F.6) reports stabilized_object_count: 6 "potential/witness" classes covering the 12 ledger rows, tachyon_check: no tachyon on the active branch.

The pipeline is one discipline, not one computation:

 every downstream-used modulus
   --witness-type predicate-->  {Weyl-rigid | modular fixed pt | integer winding | discrete topology | BC pin}
     --type test (NOT minimize-at-answer)-->  PASS (witnessed) / FAIL (tuned-or-floating)
       --12-row ledger + no-tachyon + coverage lint-->  Claimed certificate pass (given-E, in-chamber)

1.2 What "closed" means here, and its conditionality

"Closed" for SG-6 is moduli-control by structural witness under declared assumptions — strictly not global stabilization, not a proven positive-definite Hessian, not a derivation of the chamber. It is the conjunction of exactly three claim types (F.9.3 / F.10.3):

• (i) Admissibility restriction — non-chamber configurations rejected by $\mathcal{C}_{\rm admiss}$ (Appendix C6). Proven (it is an $\oplus$-layer rulebook rule).
• (ii) Local stability at the chamber-center witness $\vec u=(1,1,1)$ — small in-chamber perturbations return to the center. Claimed at tree-plus-one-loop within the declared scheme.
• (iii) Phenomenological sufficiency — the remaining moduli do not move any Gate 1–10 output beyond its published tolerance. Claimed under the Weyl-rigid chamber + freeze discipline.

It is conditional on:

• given-E — the SM chiral content and the family index $\chi=-3$ are supplied as input from SG-2/SG-3; the witnesses operate on the selected bundle, they do not derive it.
• given-the-selected-geometry — the witnesses fix the moduli inside the already-selected chamber. SG-6 certifies the selected geometry is internally controlled; it does not prove the witnesses determine the geometry. (This is the load-bearing conditionality of the whole gate.)

What the gate explicitly does NOT claim (F.9.4 / F.10.1, carried verbatim):

So the precise statement of closure: the gate certifies that every downstream-used modulus carries a structural witness (not a tuned potential), that small in-chamber perturbations are controlled at the chamber-center, and that the residual moduli do not spoil Gates 1–10 — given the SM content and the selected geometry. It does NOT prove a positive-definite Hessian, does NOT claim global stabilization, and the one modulus carrying the electroweak hierarchy (θ_H⋆) is READ from a minimum, not derived.

2. Verify the status — is PARTIAL 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 witness-type discipline reproduces by inspection. The predicate is a type test; a reader can walk F.3 row by row and confirm each witness is structural (admissibility / symmetry fixed-point / integer winding / discrete topology / BC pin) and not "a number we minimized to." This is the genuine, strong leg of the gate, and it is exactly the $y=R^2$ curve-not-a-number toy applied to every dial the certificates touch.
• The discrete/topological witnesses reproduce by hand. $n_H=1$, the $\mathbb{Z}_2$ orbifold, the $\mathbb{Z}_6$ identification, the cycle $\gamma$ — all are topological invariants with no continuous knob; nothing to reproduce numerically and nothing to drift.
• The $S_3$-fixed-point structure reproduces by hand. Because the Weyl group $S_3$ permutes $(u_1,u_2,u_3)$, the center $(1,1,1)$ is automatically a critical point of any $S_3$-invariant functional ($\partial V=0$ for free). This upgrades the manuscript's "fixed by fiat" wording to "fixed by symmetry as a critical point" — a genuine, free, target-blind structural fact (TEST 2).

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

• The positive-definite Hessian does NOT reproduce — it is diagnostic-only, and the one Λ-free piece that CAN be computed gives a SADDLE. F.9.5 explicitly declines the Hessian computation as "out-of-scope refinement." When the chamber Hessian is decomposed by symmetry into $3=\mathbf{1}\oplus\mathbf{2}$ (TEST 2), the shape doublet ($S_3$-doublet, traceless) is computable Λ-free from the flag-manifold scalar curvature — and the result is $\partial^2 R/\partial\varepsilon^2|_{(1,1,1)}=-1<0$, a saddle (the structure-constant triangle term $-5/2$ overwhelms the diagonal $+2$). The breathing/volume singlet rides the uncomputed $c_{\rm loop}$ (the 13D FRG-2 wall). So the F.9.7(2) saddle attack surface is not merely open — on current perturbative geometry it leans toward saddle, rescuable only by a Casimir term whose net sign needs the absent admissible-rep multiplicity table (TEST_CASIMIR_RESCUE).
• θ_H⋆ does NOT reproduce as a derivation — it is read from a minimum. The Hosotani phase $\theta_H^\star\approx2.46\times10^{-14}$, which carries ~85% of the EW hierarchy via $v_{\rm EW}=\theta_H^\star/(2\pi R_\gamma)$, is extracted from the one-loop $V_{\rm Hos}$ minimum, not computed-small from chamber data. The frozen $\eta_{BK}=0.009721$ gives only $\sqrt{\eta_{BK}}/2\pi\sim 10^{-2}$ — ~12 orders short. The lightness-of-$v$ leg leans on a read modulus.
• Global stabilization does NOT reproduce — it is not claimed. F.9.4 row 1 says so explicitly; outside-chamber branches are rejected by admissibility, not stabilized. There is no all-loop positive-definite Hessian and no claim of "no flat directions anywhere."
• Phenomenological sufficiency inherits downstream AUDIT. The "residual moduli don't spoil Gates 1–10" claim is only as machine-real as the downstream harnesses it relies on — and the Gate-7 threshold pipeline (which consumes the SG-6 radii/spectrum) is itself AUDIT (δ-triple injected reals, reproduce_all.py absent). SG-6's sufficiency claim is therefore declared-not-independently-verified at the numerical level.

2.4 Why PARTIAL (not higher, not lower)

• Not DERIVED-GIVEN-E (the SG-2/SG-3 tier): SG-6 is not a rigid integer/representation recovery. It has a diagnostic-only Hessian (with a Λ-free saddle in the shape sector), a read-from-minimum θ_H⋆ carrying the hierarchy, an explicit global-stabilization non-claim, and downstream-inherited AUDIT — too many open residuals for the DERIVED tier.
• Not OPEN/DECLARED-FROZEN: the witness-type discipline is real, strong, and non-trivial; the discrete/topological witnesses and the $S_3$-critical-point structure are genuine target-blind facts; the claim-type ledger honestly fences what is and is not proved. The gate is well clear of "unfixed knobs."
• PARTIAL is exactly right: genuine structural moduli-control under declared assumptions, a diagnostic-only Hessian named as the most-natural attack surface, an explicit non-claim on global stabilization, and a read-from-minimum acute soft spot. This matches the SCOPED_GUT ledger SG-6 line verbatim.

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, for SG-6, how much of the electroweak hierarchy / how much of the certificate it touches).

3.1 The residual register

3.2 Leverage ranking (attack order)

1. R1 (θ_H⋆ read from a minimum) — carries ~85% of the hierarchy; the single highest-value physics target.
2. R3 (the diagnostic Hessian — the named saddle attack surface) — already computed to a saddle in the shape sector; the explicit F.9.7 falsification path.
3. R5 (the $c_{\rm loop}$ / breathing-mode singlet wall) — decides the singlet Hessian sign; the shared deep 13D FRG-2 wall.
4. R6 (the Casimir rescue) — the only route that could flip R3; gated on the missing multiplicity table.
5. R4 (given-E / within-chamber) — conditions everything; honesty residual shared with SG-1/SG-3.
6. R2 / R7 / R8 / R9 — disclosed non-claim / cleanest-witness strengthening / conditional tachyon / inherited AUDIT.

The cardinal honest point. R1 and R5 are where the "85% hierarchy" and the "singlet Hessian" live, and both reduce to the same single object: $\mu_{\rm cell}$, the spectral value of the uniform operational cell $\Delta_0$ (the log-scheme / cell-scale factor of the 6-D $a_6$ determinant). The decisive, already-run firewall verdict (CLOSURE_CAMPAIGN §3) is that $\mu_{\rm cell}$ has no $v$-independent readout — its only available anchor is $\partial_\sigma V=0$, which IS the hierarchy — so anchoring $\mu_{\rm cell}$ there to "predict" $v$ is circular by construction (the κ³/π signature). Any closure path that pins $\mu_{\rm cell}$ from the EW hierarchy has RELOCATED, not removed, the input. The κ³/π falsification test applies in full force here. R2/R7/R8/R9 are the non-physics-flip residuals (claim-boundary / symmetry-strengthening / conditional-check / inherited reproducibility); closing them improves honesty and machine-reality but does not derive the hierarchy or flip the Hessian.

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 (CSV/ruling/computation) needed, the math to attempt, and the success ladder (DERIVED-CLOSED rare → AXIOM-CLOSED likely → sharper-OPEN → REFUTED).

4.1 R1 — θ_H⋆ ≈ 2.46×10⁻¹⁴ read from the $V_{\rm Hos}$ minimum (the acute soft spot)

Why first. This single modulus carries ~85% of the electroweak hierarchy. If θ_H⋆ could be derived target-blind from chamber data instead of read off the one-loop Hosotani potential, the lightness-of-$v$ leg would stop leaning on a read number and SG-6 would deliver the hierarchy's dominant factor. If it provably cannot, the gate's honest standing tightens to "the hierarchy is a second irreducible anchor."

Technique: T5 (no tuning to the known answer firewall) + T7 (eliminative) — is θ_H⋆ a derivation or a relocation? This is exactly the SCALE firewall, pointed at the Hosotani phase.

The two routes, and which the firewall kills:

Named axiom it could reduce to (κ³/π-clean). The honest reduction is not a closure but a precise statement of what R1 reduces to once the two routes are killed:

AXIOM-VEW-SECOND-ANCHOR (target-blind form): "the electroweak scale $v_{\rm EW}$ is a second independent dimensionful anchor on the same footing as $M_{\rm Pl}$, irreducible in this geometry because no second mass can be built from $\{M_{\rm Pl}, \hbar, \text{dimensionless frozen geometry}\}$ without a σ-carrying length (Buckingham-π), and because the carrier θ_H⋆ is a periodic-minimum location, not a transmutation exponent."

This is writable with no value of $v$ or θ_H⋆ in sight — it is a dimensional + mechanistic statement about what kind of object the hierarchy is. It PASSES the κ³/π falsification test (it carries no reverse-engineered coefficient). It does not derive θ_H⋆; it names why θ_H⋆ must be read, not computed, in this structure.

Specialist target (to hand off). A target-blind θ_H⋆ derivation theorem: "Given only the frozen chamber data (cycle $\gamma$, $n_H=1$, $\eta_{BK}$, $R_\gamma$) and a UV reference frozen before and independently of $v_{\rm obs}$, compute $\theta_H^\star$ and check whether it returns $\approx2.46\times10^{-14}$ with no $v$ in the inputs." If a specialist can produce such a derivation, R1 closes (DERIVED) and the hierarchy's dominant factor is delivered; if the derivation provably requires $\partial_\sigma V=0$ (the hierarchy itself), R1 is confirmed RELOCATION and AXIOM-VEW-SECOND-ANCHOR stands.

Math to attempt. (1) Write $V_{\rm Hos}(\theta_H)$ from the frozen cycle/winding/$\eta_{BK}$ data. (2) Locate its minimum θ_H⋆ without consulting $v_{\rm obs}$. (3) Check whether the minimum lands at $\sim10^{-14}$ from chamber data alone, or only after a scale is pinned by $v$. (4) Run the $v$-independence audit: does the threshold-weight swing across the σ-band ($\sim34\times$, the documented forbidden count-only "35" stationarity) confirm $\mu_{\rm cell}$ is the knob?

Success ladder. - DERIVED-CLOSED: θ_H⋆ computed target-blind from chamber data, returning $\sim10^{-14}$ with no $v$ input → the hierarchy's dominant factor delivered. Unlikely — the dimensional near-no-go (Buckingham-π) blocks a second mass from $\{M_{\rm Pl},\hbar,\text{shape}\}$. - AXIOM-CLOSED (likely): AXIOM-VEW-SECOND-ANCHOR named; θ_H⋆/$v_{\rm EW}$ conceded as a second irreducible dimensionful anchor with a principled (dimensional + mechanistic) reason, not "we failed to reduce it." This is the honest ceiling and is essentially where the SCALE frontier already stands. - sharper-OPEN: the derivation attempt returns fail-closed on $\mu_{\rm cell}$ (the FRG-2 wall) without deciding circularity. - REFUTED: a chamber-only computation gives a θ_H⋆ that misses $10^{-14}$ → the lightness-of-$v$ leg is a genuine prediction-vs-data failure.

Honest disposition: AXIOM-CLOSED at AXIOM-VEW-SECOND-ANCHOR; θ_H⋆ as "derived-small" stays OPEN/RELOCATION. The firewall has already confirmed Path A circular and Path B wrong-mechanism; the genuine banked sub-result is the Higgs-mass quadratic protection (proven, finite, 12-orders-short). Saying precisely "the hierarchy is a second anchor, here is the structural reason it must be" is the closure — not a higher claim.

4.2 R3 — The diagnostic Hessian: the named saddle attack surface (F.9.7-2)

Why second. F.9.5 grades the positive-definite moduli-mass Hessian DIAGNOSTIC-ONLY and F.9.7(2) names the single most-natural falsifier: show the chamber-center witness is a saddle, not a minimum. This is the explicitly-named attack surface, and it is already partially executed against the gate.

Technique: T7 (eliminative) by symmetry decomposition — split the Hessian, compute the computable piece. The Weyl group $S_3$ permutes $(u_1,u_2,u_3)$, so the 3-dim perturbation space splits exactly into $3=\mathbf{1}\oplus\mathbf{2}$ (TEST 2):

$(1,1,1)$ is a minimum iff both eigenvalues > 0, and they decouple by symmetry.

The Λ-free computation, already run (TEST_SHAPE_DOUBLET_STABILITY). The flag-manifold scalar curvature $R(u)=\sum_i 1/u_i - \tfrac12 T(u)$, $T(u)=\sum_k u_k/(u_iu_j)$, along the doublet ray $u=(1+\varepsilon,1-\varepsilon,1)$ gives: - diagonal $\sum 1/u_i$: $\to +2$ to the $\varepsilon^2$ coefficient (convexity of $1/u$), - structure-constant triangle $T$: $\to -5/2$, - net $\partial^2 R/\partial\varepsilon^2|_{(1,1,1)} = +2 - 5/2 = \mathbf{-1} < 0$ → SADDLE.

The boundary (Gibbons–Hawking) term is small and mildly destabilizing; the Wilson-line term is exactly zero in the doublet direction. On purely perturbative geometry, the shape doublet is a SADDLE — the structure-constant coupling between the three root planes makes the equal-scale flag metric unstable to shape deformation. (This is the well-documented fact that $SU(3)/T^2$ carries three invariant Einstein metrics and the normal one is not the shape-sector extremizer.)

The only rescue, and its falsification test (TEST_CASIMIR_RESCUE). A graded Casimir / KK vacuum-energy term can flip the doublet sign. The per-sector sign is LOCKED (bosons stabilize, fermions destabilize, by convexity $g''(0)=2(w_1^2+w_2^2)>0$ and $f'(m^2)>0$, scheme-independent). The net sign is $\mathrm{sgn}(\text{boson-surplus}-\text{fermion-surplus})$ over admissible $(p,q)$ at $(1,1,1)$ — which needs the admissible-rep multiplicity table (ABSENT from the corpus) and a magnitude $\mu_{\rm cell}\cdot q\gtrsim 0.5$. The frozen heat-kernel ledger (G.3.2a) weakly favors boson dominance (adjoint gauge+ghost $-2.49/-4.02$ vs matter $+0.79/+0.92$, ~3:1) → conjectured rescue — but the opposing frozen fact is $\chi=-3$ (a low-lying fermionic chiral surplus that destabilizes), and banking boson-dominance without the multiplicity table is NEEDS-TUNING.

Named axiom it could reduce to (κ³/π-clean). Not a closure, but the precise reduction of the doublet question:

AXIOM-SHAPE-DOUBLET-CASIMIR (target-blind form): "the shape-doublet Hessian at the chamber center is $-1_{\rm curvature} + q\,\mu_{\rm cell}\,(+2_{\rm convexity})$, with the convexity factor $+2$ and the per-sector sign locked by geometry; the net sign is fixed by the graded boson−fermion surplus of the admissible-rep spectrum." Writable with no Λ and no target value — it references only the root-system curvature and the graded multiplicity. PASSES the falsification test as a statement; the test is then whether the multiplicity table yields a net positive.

Specialist target / owner artifact. (1) Owner artifact: the admissible-rep multiplicity table at $(1,1,1)$ + the regularized zeta (from certificates/ + reproduce_all.py / the KK eigenvalue file R0.7) — this is the single missing input that decides the net Casimir sign. (2) Specialist target: "Given the multiplicity table, compute $\mathrm{sgn}(\text{boson-surplus}-\text{fermion-surplus})$ and the magnitude $\mu_{\rm cell}\cdot q$, and report whether the doublet net Hessian is $>0$ (rescue) or $\le 0$ (saddle persists)."

Math to attempt. (1) The doublet curvature $-1$ is done (banked, target-blind). (2) Mount the multiplicity table; form the graded doublet-weighted sum $\sum (-1)^F d\,n\,(w_1^2+w_2^2)$. (3) Decide the net sign. (4) Combine the singlet (R5) and doublet results for the full Hessian signature.

Success ladder. - DERIVED-CLOSED: both eigenvalues (singlet via R5, doublet via the multiplicity table) proven $>0$ target-blind → the Hessian is positive-definite, F.9.7(2) defeated, the diagnostic upgrades to a derived minimum. Gated on both the multiplicity table AND the FRG-2 $c_{\rm loop}$ — unlikely in one step. - AXIOM-CLOSED: AXIOM-SHAPE-DOUBLET-CASIMIR named and the multiplicity table yields a net positive doublet → the shape sector reduced to one symmetry/spectral posit. Best realistic outcome for the doublet. - sharper-OPEN / saddle-leaning (current standing): doublet curvature is a saddle ($-1$); the rescue needs the absent table; the singlet rides the FRG-2 wall. The honest current state is "saddle on perturbative geometry, conditional rescue pending owner artifacts." - REFUTED: multiplicity table computed and the net doublet stays negative → the chamber-center is a saddle, F.9.7(2) succeeds, and the moduli-mass-generation row of F.9.2 downgrades (per F.10.4-2, to Open / not claimed).

Honest disposition: sharper-OPEN, saddle-leaning. The named falsification path is the closest to firing of any SG-6 residual: the one Λ-free piece that can be computed is a saddle, and the rescue is gated on a missing multiplicity table and the $\mu_{\rm cell}$ value (shared with R1/R5). Do not bank the boson-dominance rescue without the table.

4.3 R5 — $c_{\rm loop}$ / breathing-mode singlet = the 13D FRG-2 β-vector wall

The obstruction. The breathing/volume singlet eigenvalue rides $V''(\sigma)$ at the $e^{-6\sigma}$ term — i.e. $c_{\rm loop}$, the coefficient of the σ-fluctuation one-loop determinant, equivalently the $a_6$ Seeley–DeWitt heat-kernel coefficient on the 6-manifold $K_6$. $c_{\rm loop}$ is BLOCKED_MISSING_FRG_MATCHING — the uncomputed 13D FRG-2 β-vector, the deep core shared with Gap-04 and a6.

Technique: T7 (eliminative) — is the wall the continuum or the scale? Replace the continuum mode integral by a finite cell-sum under the uniform operational cell law $N\le B/\Delta_0$ (TEST_CLOOP_CELLSUM_ATTACK).

What the cell-sum genuinely buys, and what it does not. The cell-sum does kill the UV ($a\to0$) divergence — the continuum-as-source-of-divergence assumption is genuinely false, and dropping it is banked (ROOT-1). But finiteness does not imply uniqueness: a finite supertrace of a log-running 6-D determinant still carries a log-scheme degree of freedom $\mu_{\rm cell}$. Three residual ambiguities survive: (3a) the cutoff is a mode-COUNT, but $a_6$ needs a mode-SCALE; (3b) the η-dependence the ledger flagged IS the scheme knob; (3c) supertrace ordering / sign is not fixed by a count. So:

$$ c_{\rm loop} = f(\text{frozen }K_6\text{ spectrum},\ \chi=-3,\ e^{-6\sigma}\text{ scaling}) \times [\text{one log-scheme factor fixed by }\mu_{\rm cell}]. $$

Named axiom it could reduce to (κ³/π-clean). This is exactly the GRANULARITY root's residue:

AXIOM-UNIFORM-CELL-VALUE (target-blind form): "there exists a single uniform operational cell $\Delta_0>0$ on the operational distinguishability metric (existence is the GRANULARITY root, AXIOM-COSTFLOOR; the spectral VALUE of $\Delta_0$ is a FLOOR-VALUES-RESIDUE on the same footing as $\hbar, k_B$), and $\mu_{\rm cell}$ is its spectral value at the $K_6$ scale." Writable with no Λ and no hierarchy value. PASSES the falsification test as an existence + value-residue statement.

The κ³/π pivot (the firewall, already run). The single anchor that would pin $\mu_{\rm cell}$ is one measurement fixing $\Delta_0$ in spectral units at the $K_6$ scale — and the only such anchor available is the moduli-stabilization point $\partial_\sigma V=0$. But $\partial_\sigma V=0$ IS the electroweak hierarchy, so anchoring $\mu_{\rm cell}$ there and then "predicting" the hierarchy is tuning to the known answer — explicitly FORBIDDEN (the CLOSURE_CAMPAIGN firewall designated $\partial_\sigma V=0$ the forbidden anchor; the earlier TEST_CLOOP suggestion to use it is superseded). The honest anchor is $\Delta_0$'s spectral value as an independent floor-value input, on the footing of $\hbar$ — which the corpus does not supply.

Specialist target / owner artifact. (1) Owner artifact: the KK eigenvalue file / reproduce_all.py (R0.7) that would pin $\mu_{\rm cell}$ if an independent floor-value for $\Delta_0$ existed. (2) Specialist target (the deep one): "Compute the 13D FRG-2 β-vector and hence $c_{\rm loop}=\mathrm{tr}[a_6]$ of the σ-fluctuation determinant on the frozen $K_6$ branch, target-blind." This is the shared Gap-04 / hierarchy / a6 wall.

Success ladder. - DERIVED-CLOSED: the FRG-2 β-vector computed → $c_{\rm loop}$ sign decided → the singlet Hessian eigenvalue fixed. Blocked on the deep 13D FRG wall; not reachable here. - AXIOM-CLOSED: AXIOM-UNIFORM-CELL-VALUE named, with $\mu_{\rm cell}$ conceded as a floor-value residue (no new knob, no target). Honest ceiling; already the GRANULARITY root's standing. - sharper-OPEN / BLOCKED_INPUTS (current standing): the cell-sum lowers the wall (finite, single-parameter) but does not remove it; the one residual $\mu_{\rm cell}$ has no $v$-independent readout. - REFUTED: the FRG computation gives a definite $c_{\rm loop}$ with the wrong sign → the breathing singlet is a saddle → chamber-center is not a minimum.

Honest disposition: BLOCKED_INPUTS, reducible to AXIOM-UNIFORM-CELL-VALUE. The breathing-singlet Hessian is the same FRG-2 wall as the hierarchy; the cell-sum converts "uncomputed continuum determinant" into "finite cell-sum + one named floor-value anchor," which closes nothing but names the bottom precisely. The $\partial_\sigma V=0$ anchor is forbidden (it is the hierarchy).

4.4 R6 — The Casimir net sign (the only route that could flip R3)

This is the dependent of R3, broken out because it is the single decision that flips saddle→minimum. Fully covered as the rescue analysis under §4.2: per-sector sign LOCKED (bosons +, fermions −, scheme-independent); net sign = boson-surplus − fermion-surplus over the absent admissible-rep multiplicity table; magnitude threshold $\mu_{\rm cell}\cdot q\gtrsim0.5$.

Technique: owner artifact (machine-lane) + T7. Mount the multiplicity table + regularized zeta; form the graded doublet-weighted sum; decide the net sign and magnitude.

The κ³/π guard. The frozen heat-kernel ledger weakly favors boson dominance (~3:1), but the opposing frozen fact is $\chi=-3$. Banking "boson-dominated rescue" without the table is NEEDS-TUNING — the conjecture $q>0$ is exactly the kind of plausible-but-unverified sign that the falsification test forbids until the table is computed. The magnitude carries $\mu_{\rm cell}$ (shared with R1/R5), which has no $v$-independent readout.

Success ladder. BLOCKED_INPUTS until the multiplicity table is mounted → then VERIFIED-RESCUE (net $>0$, $\mu_{\rm cell}\cdot q\gtrsim0.5$ → minimum) or SADDLE-CONFIRMED (net $\le0$ → F.9.7(2) fires). Highest value-per-effort physics item if the table exists — it converts the named saddle attack surface from "leaning saddle" into a decided verdict with no new tuning.

Honest disposition: BLOCKED_INPUTS / NEEDS-TUNING. The rescue is conditionally available but unbankable without the absent multiplicity table; do not assert boson dominance as a result.

4.5 R2 — Global stabilization is an explicit NON-claim

This is a claim-boundary residual, not a physics flip. Global stabilization (no flat directions anywhere in the active-branch moduli space) is explicitly NOT claimed (F.9.4 row 1 / F.10.1) — outside-chamber configurations are rejected by admissibility, not stabilized. The task is not to "solve" it but to keep the boundary honest and to state what would close it.

Technique: T6 (all-operator-conditional) via the manuscript's own F.10.2 vocabulary rule. The required-vocabulary substitution rule already bans "full stabilization" / "no flat directions" / "all moduli stabilized" in body text without the chamber-restriction qualifier. Compliance is mechanical.

Named principle (κ³/π-clean): AXIOM-CHAMBER-RESTRICTION — "outside-chamber configurations are rejected by the admissibility rulebook $\mathcal{C}_{\rm admiss}$ (an $\oplus$-layer rule), not dynamically stabilized; the stabilization claim is scoped to the moduli used downstream by Gates 1–10." This is the F.9.4/F.10.3 statement; it carries no target value and is the honest scope wall.

What would close it (and why it is SHARED-OPEN). Global stabilization is the textbook hard problem on which all rivals struggle (Battle verdict: TIE/SHARED-OPEN — not a framework win). The downstream four-term $V(\sigma)$ discharge (Paper IV) is the only object that even attempts the volume modulus globally, and it is EXTERNAL with its own caveats and "promotions: zero" discipline; it changes nothing in this gate's certificate (the F.8 downstream-corpus pointer says so explicitly).

Success ladder. AXIOM-CLOSED is not applicable as a flip (no axiom closes global stabilization here). The endpoint is DISCLOSED-CONSISTENT: the non-claim is explicit, the vocabulary rule is enforced, and the SHARED-OPEN status is stated honestly. The cleanest, lowest-risk honesty item.

Honest disposition: DISCLOSED non-claim; keep it disclosed. Do not let any downstream cross-reference (Paper IV) be cited to upgrade this gate's certificate.

4.6 R4 — Witnesses fix moduli WITHIN the selected chamber (given-E)

The obstruction. The witnesses certify that the selected geometry is internally controlled; they do not prove the witnesses determine the geometry. The chamber, the bundle $E$, and the family index $\chi=-3$ are inputs (from SG-1/SG-3). Selection ≠ derivation; given-E ≠ derivation of E.

Technique: T1 (axiom-floor), shared with SG-1/SG-3. The honest reduction names the inheritance:

AXIOM-GIVEN-E-CHAMBER (κ³/π-clean): "the stabilization witnesses operate on the selected 13D K₆ branch and the supplied SM chiral content E; SG-6 certifies internal moduli-control of that selected geometry, not a cross-geometry determination." Writable with no target value — it is a scope statement inherited from SG-1 (DECLARED-FROZEN, SHAPE selected-not-forced-absolute, ~9–10 injected reals beyond the 4 anchors) and SG-3 (DERIVED-GIVEN-E).

Specialist target. None unique to SG-6 — this is the SG-1 SHAPE-minimality / SG-3 given-E conditionality flowing downstream. The contained SG-6 strengthening is R7 (the modular-fixed-point uniqueness), which is the one witness that could be promoted from "selected" to "symmetry-forced."

Success ladder. AXIOM-CLOSED (AXIOM-GIVEN-E-CHAMBER named) is essentially already standing via the SG-1/SG-3 labels. DERIVED-CLOSED is not available at SG-6 (it would require closing SG-1's SHAPE-forcedness, which is SELECTED-not-forced-absolute). Honest disposition: AXIOM-CLOSED at the inherited scope axiom; the within-chamber conditionality is the load-bearing caveat of the gate.

4.7 R7 — $\tau=\omega$: strengthen "non-zero potential" to symmetry-uniqueness

The obstruction. F.2 asserts that perturbations off $\tau=\omega$ "produce a non-zero potential" restoring $\tau\to\omega$, but the potential is not computed. The leg should rest on modular-symmetry uniqueness of $\omega$ — a fixed point of an order-three modular symmetry is symmetry-protected, not tuned — which is stronger than "a potential we did not compute."

Technique: T1 (axiom-floor) — name the symmetry, prove the uniqueness.

AXIOM-MODULAR-FIXED-POINT (κ³/π-clean): "the $F^+$ Cartan-torus modulus sits at the order-three modular fixed point $\tau=\omega=e^{2\pi i/3}$, the unique modular-symmetric / Weyl-rigid point of the chamber." Writable with no flavor or hierarchy number — a symmetry statement. Stronger than "read from a minimum" because an order-three fixed point is symmetry-protected.

Specialist target. Hand the moduli specialist: "Show $\tau=\omega$ is the unique fixed point of the $F^+$ Cartan-torus modular group, independent of any one-loop potential." A clean, bounded group-theory claim.

Math to attempt. Enumerate the fixed points of the chamber's modular group on the upper half-plane; confirm $\omega$ is the order-three fixed point and characterize uniqueness within the chamber.

Success ladder. AXIOM-CLOSED (AXIOM-MODULAR-FIXED-POINT named; symmetry-protected) is the realistic endpoint; DERIVED-CLOSED (uniqueness theorem proven) closes R7 outright and converts the $\tau$-witness from "potential we did not compute" to "symmetry-forced point." sharper-OPEN if the fixed point is non-unique. Honest disposition: AXIOM-CLOSED likely, DERIVED if uniqueness is proven; this is the one witness that can be genuinely strengthened by a contained computation. (Note: this also strengthens SG-8's inherited $\tau=\omega$ soft spot.)

4.8 R8 — No-tachyon check is conditional on the declared spectrum

The obstruction. F.4 asserts no tachyonic mode "on the active branch" at the declared witnesses, but a deformation away from a witness could produce one (and then the certificate fails). The check is for the frozen spectrum, not proven stable under the deformation classes the gate exposes it to.

Technique: T6 (all-operator-conditional) within the admissibility-bounded deformation class. The deformations the gate must survive are bounded by admissibility (off-chamber ⇒ branch eliminated). The honest statement is conditional, and the F.5 sensitivity analysis already bounds the in-chamber responses.

Named principle (κ³/π-clean): AXIOM-NO-TACHYON-IN-CHAMBER — "on the active branch at the declared witnesses, and under admissibility-bounded in-chamber deformations, no tachyonic mode appears." Carries no target value; it is the F.4 statement scoped to the admissible deformation class.

Success ladder. DISCLOSED-CONSISTENT (already substantially in place via F.4 + F.5). The residual is a consistency sweep over the admissibility-bounded deformations, not a closure. (Note: R3's saddle finding is the spatial (Hessian) analog of the tachyon check — a saddle direction is a flat/negative mass direction in field space; R8 and R3 are linked, and a confirmed shape-doublet saddle would put pressure on R8 in that direction.)

Honest disposition: DISCLOSED-CONSISTENT, conditional on the declared spectrum; dependent on R3.

4.9 R9 — Phenomenological sufficiency inherits the downstream AUDIT (Gate-7 radii/spectrum)

The obstruction. The "residual moduli don't spoil Gates 1–10" claim relies on the downstream harnesses that consume the SG-6 radii and threshold spectrum — chiefly the Gate-7 threshold pipeline, which is itself AUDIT (the three δ are injected reals quoted to five sig figs but not regenerated; the G.3.2a row formulas do not reproduce the decimals; reproduce_all.py + the ledger CSVs are ABSENT). SG-6's sufficiency leg is only as machine-real as those harnesses.

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

Owner artifact needed. (1) Mount the Gate-7 ledger CSVs (appendix_F_heat_kernel_ledger.csv, appendix_F_threshold_outputs.csv). (2) Run reproduce_all.py against the frozen radii + spectrum, target-blind. (3) Confirm the threshold vector + unification residual regenerate. (4) Confirm a small $\delta$ on the SG-6 radii moves the Gate-7 output within the F.3 $\delta$-response column.

Math to attempt. None new — execution + verification. Mechanical and fail-closed: if a perturbation of an SG-6 radius moves a Gate 1–10 output beyond tolerance and is not bounded by F.3, phenomenological sufficiency fails (F.10.4-3) and the offending downstream gate downgrades.

Success ladder. BLOCKED_INPUTS until the Gate-7 CSVs/script are mounted → then VERIFIED (sufficiency machine-real) or REFUTED (a perturbation breaches tolerance → downgrade). Value-per-effort high, but the artifact is owned by SG-7 — closing SG-7's δ-harness AUDIT closes this too.

Honest disposition: AUDIT, inherited from SG-7; closes when SG-7's reproduction harness is mounted.

4.10 Attack-plan roll-up

REDUCE-vs-RELOCATE verdict on the plan. The plan does not turn one hard problem into three harder ones. Crucially, the two genuine-physics fronts collapse onto one shared object: $\mu_{\rm cell}$, the spectral value of the uniform operational cell $\Delta_0$. R1 (the hierarchy via θ_H⋆), R3/R6 (the doublet Casimir magnitude), and R5 (the breathing-singlet $c_{\rm loop}$) all reduce to "what is $\mu_{\rm cell}$, and can it be read $v$-independently?" — and the already-run firewall answers: no $v$-independent readout exists; the only anchor ($\partial_\sigma V=0$) IS the hierarchy. So the plan reduces the gate to a single named floor-value residue plus a SADDLE-or-rescue verdict gated on one absent multiplicity table — it does not multiply the difficulty. Each path either (a) names a single target-blind axiom that pays a debt in plain sight (R1, R2, R4, R5, R7, R8), (b) is a bounded owner-artifact computation that decides a sign (R3/R6 the multiplicity table; R9 the Gate-7 CSVs), and every $\mu_{\rm cell}$-touching path carries the explicit κ³/π falsification test (the $\partial_\sigma V=0$ anchor is forbidden) so a circular closure cannot be banked. The honest expected outcome of a full campaign: ~4 AXIOM-CLOSED (R1 as 2nd-anchor, R4, R5 to the cell residue, R7), ~2 DISCLOSED-CONSISTENT (R2, R8), 1 sharper-OPEN/saddle-leaning (R3), and 2 BLOCKED_INPUTS pending owner artifacts (R6 multiplicity table, R9 Gate-7 CSVs). No DERIVED-CLOSED is promised; the gate would move from PARTIAL-with-diagnostic-Hessian to PARTIAL-with-a-named-axiom-floor + a decided-or-saddle Hessian verdict — a real honesty/structural gain, not a promotion. The single most likely negative — and the most valuable one — is R3 firing: the shape-doublet saddle, if the Casimir rescue fails, REFUTES the moduli-mass-generation row and downgrades it per F.10.4-2. That is the gate working as designed.

A. Anchoring & Hardening Map

This section runs SG-6 through our internal honesty methodology — the same hardening method that produced the live Gaps & Walls Register: classify each residual as a gap (a route exists; what is missing is a finished computation or a measured input) or a wall (the route itself is the problem) → hunt the implicit assumption it hides → name the measured-invariant truth it must terminate on → assign the most conservative defensible disposition — exactly that method (the framework, the gap-vs-wall split, and the disposition vocabulary are defined in the linked Register). The discipline is honest bookkeeping, not closure: no theorem breaks a wall, and the ceiling on every line below is serious candidate / NOT validated. The only existing measured anchors a residual may terminate on are {ℏ, M_Pl, spectrum-E, α_i(M_Z), y_t, |V_us|}; anything else must name a NEW invariant, hinge on a SCHEME-anchor, be a COMPUTATION-DEBT, or have NO witness → OPEN. Disposition words are used exactly as the Register defines them: DERIVED / DERIVED-GIVEN-E / AXIOM-CLOSED / DISSOLVED / SCHEME-ANCHORED / OPEN / BLOCKED / measured-but-irreducible.

A.1 Per-residual anchoring map (one row per §3.1 residual)

A.2 Gate-level rollup

• Overall disposition. SG-6 is PARTIAL → conservatively OPEN (matching the live Register's SG-6 line). The genuine, strong content is the witness-type predicate (a type test that makes "a potential we minimized at the answer" fail by construction) plus the discrete/topological witnesses and the S₃-critical-point structure — these are real, target-blind facts, not promotions. The open surface terminates as 3 AXIOM-CLOSED (R1 as a second irreducible anchor, R4 the given-E scope, R7 the modular fixed point — likely DERIVED), 1 reducible-to- AXIOM-CLOSED but BLOCKED (R5 → AXIOM-UNIFORM-CELL-VALUE, gated on the FRG-2 wall), 2 DISCLOSED-CONSISTENT (R2 global-stabilization non-claim, R8 conditional no-tachyon), 1 OPEN/saddle-leaning (R3 — the named F.9.7(2) falsifier, already −1 on Λ-free geometry), and 2 BLOCKED_INPUTS pending owner artifacts (R6 the multiplicity table, R9 the SG-7 reproduction harness). No DERIVED-CLOSED is promised; No status was ever upgraded.
• Anchors it depends on. spectrum-E (R4: the gate certifies internal moduli-control given the supplied chiral content + χ=−3, never derives it) and ℏ-footing only (R5: μ_cell is a floor-value residue on the footing of ℏ, but its spectral value is not supplied — it is a NEW invariant, not the measured ℏ itself). It depends on no clean termination on M_Pl / α_i(M_Z) / y_t / |V_us|; the one residual that touches a number (R1, the hierarchy) terminates on measured-but-irreducible v_EW, not on an existing anchor. R9 inherits SG-7's SCHEME-anchor, not a measured one.
• Single highest-leverage hardening move. Mount the admissible-rep multiplicity table (currently ABSENT). It is the one owner artifact that decides the net Casimir sign and thereby flips the named saddle attack surface (R3/R6) from "leaning saddle" to a decided verdict — VERIFIED-RESCUE (minimum) or SADDLE-CONFIRMED (F.9.7(2) fires, the moduli-mass-generation row downgrades per F.10.4-2). It is the highest physics value-per-effort step and the only one that converts a diagnostic into a falsifiable result without new tuning. The deeper twin wall (R1/R5, the hierarchy and the breathing singlet) collapses onto a single object — μ_cell — which the already-run firewall confirms has no v-independent readout (its only anchor ∂_σV=0 IS the hierarchy; the κ³/π falsification test fires), so that front hardens only to a named axiom floor, never to a derivation.

🎯 Target anchor(s) for this gate

The body of SG-6 carries no new measured anchor and terminates on nothing of its own: the chamber witnesses fix the moduli WITHIN the already-selected chamber, given-E (witness-type test, not a tuned potential minimized at the answer) — so the witness ledger terminates on spectrum-E (R4, AXIOM-CLOSED at AXIOM-GIVEN-E-CHAMBER) and on the ℏ-footing only (R5: μ_cell is a NEW invariant on the footing of ℏ, not the measured ℏ — SCHEME-ANCHORED / BLOCKED), and global stabilization is an explicit NON-claim (R2, DISCLOSED-CONSISTENT — it terminates on no number). The one acute open leg is θ_H⋆ ≈ 2.46×10⁻¹⁴, read from the chamber one-loop V_Hos minimum and carrying ~85% of the EW hierarchy via v_EW = θ_H⋆/(2πR_γ): its target anchor is v_EW = 246 GeV, which here is measured-but-irreducible (a second dimensionful anchor, not deducible from {M_Pl, ℏ, dimensionless frozen geometry}) — θ_H⋆ as "derived-small" stays OPEN/RELOCATION, AXIOM-CLOSED at AXIOM-VEW-SECOND-ANCHOR. No new anchor is introduced; No status was ever upgraded. The hardening move is not a new datum but the θ_H⋆ / Hessian derivation — a target-blind θ_H⋆ from {γ, n_H=1, η_BK, R_γ} + a pre-v_obs UV reference, plus the positive-definite Hessian decision at (1,1,1) gated on the ABSENT admissible-rep multiplicity table (R3/R6).

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
• §6.6 Gate-6 card (binding status: Claimed certificate pass under declared admissibility and moduli-control assumptions); §6.12 falsification map (Gate 6 row: Diagnostic only if Hessian falsified / Open if phenomenological sufficiency falsified); §6.13 certificate summary.
• §5.5 narrative module; the $y=R^2$ curve-not-a-number discipline (§6 opening of the gate index).
• Appendix F (Stabilization — the binding authority for Gate 6): F.1 stabilization claim; F.2 potential/witness structure; F.3 the 12-row moduli ledger; F.4 spectrum + tachyon check; F.5 sensitivity; F.6 certificate JSON; F.9 Stabilization-vs-Admissibility (F.9.1 five concepts, F.9.2 what is proved, F.9.4 the NON-claim, F.9.5 the Hessian diagnostic, F.9.6 required vocabulary, F.9.7 falsification paths); F.10 claim-type ledger (F.10.1 ledger, F.10.2 vocabulary substitution rule, F.10.3 binding statement, F.10.4 reviewer falsification path).
• Appendix CR module CR6 — reader companion (explanatory only; mints no status).
• §6.8 / Appendix H — the Gate-8 Hosotani object on which R1 (θ_H⋆ / $v_{\rm EW}$) rides; Appendix G — the Gate-7 threshold pipeline on which R9 (sufficiency) depends.
• Paper IV, TOE.html — https://physics.magflowmeters.com/articles/TOE.html (the four-term $V(\sigma)$ discharge; EXTERNAL; cross-reference only — changes nothing in this gate's certificate per F.8).

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

5.2 Corpus locations (authoritative inputs to this dossier)

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

Branch dcc66f1b2685 · manifest meta a5b1e6f9d951 · three Cartan radii 634438ce0776 / 2381d472c62e / 0e8b8dba2cf0 · chamber-center $\vec u=(1,1,1)$ (A1.2) · $\tau=\omega$ 03b30a9c931a · $\eta_{BK}$ 84e94518d3f5 · RG transport f531205a9159 · $M_Z$ a6852c7a6b00 · $N_{u,d,e}$ 20dc4e0b8220 · $\theta_F$ 1ff57f48d45a · spin-ℂ index / $\chi=-3$ (R1.4) 0fd19c9ae0c1 · threshold vector $(\delta_1,\delta_2,\delta_3)=(+4.8424,-3.1112,-1.7313)$. θ_H⋆ (Hosotani minimum), $\mu_{\rm cell}$ / $\Delta_0$ spectral value, $c_{\rm loop}$, the admissible-rep multiplicity table, and $M_R$: no SG-6 hash — read / uncomputed / ABSENT.

Closing honest statement

SG-6 is PARTIAL. Its genuine, defensible content is real and strong: the witness-type predicate — every downstream-used modulus carries a structural witness of declared TYPE (Weyl-rigid chamber / modular fixed point / integer winding / discrete topology / boundary-condition pin), with "a potential we minimized at the answer" failing by construction — backed by the 12-row witness ledger, a no-tachyon check, a coverage lint, and the honest claim-type ledger that fences exactly what is proved. Its open surface is equally clear: global stabilization is an explicit NON-claim (F.9.4/F.10.1); the positive-definite Hessian is DIAGNOSTIC-ONLY (F.9.5) — and on current Λ-free perturbative geometry the shape-doublet Hessian is a SADDLE ($-1$, the named F.9.7(2) attack surface leaning toward firing, rescuable only by a Casimir term whose net sign needs an ABSENT multiplicity table); the acute soft spot is θ_H⋆ ≈ 2.46×10⁻¹⁴, carrying ~85% of the EW hierarchy, read from a one-loop minimum, not derived-small; and the witnesses fix the moduli within the selected chamber (given-E). The attack plan reduces these to named, target-blind axioms and bounded owner-artifact computations — and crucially collapses the two genuine-physics fronts (the hierarchy and the Hessian sign) onto one shared object $\mu_{\rm cell}$, which the already-run firewall confirms has no $v$-independent readout ($\partial_\sigma V=0$, its only anchor, IS the hierarchy — the forbidden, circular anchor; the κ³/π falsification test fires). The realistic ceiling is a named axiom floor (AXIOM-VEW-SECOND-ANCHOR, AXIOM-UNIFORM-CELL-VALUE, AXIOM-MODULAR-FIXED-POINT) plus a decided-or-saddle Hessian verdict — not a promotion; and the most valuable likely outcome is a negative: the shape-doublet saddle firing F.9.7(2) if the Casimir rescue fails. No status was ever upgraded; frozen branch dcc66f1b2685 / a5b1e6f9d951 READ-ONLY; given-E ≠ derivation of E; selection ≠ derivation; dissolved ≠ solved; 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.