SG-6 — Moduli stabilization: the gate anchor ledger

The honest one-line: SG-6 proves the easy half of moduli stabilization — that symmetry pins where the hidden dimensions sit (criticality), target-blind and for free — and refuses to fake the hard half. On the live board the gate stands RESOLVED +0 (DERIVED-GIVEN-anchor, ratified 2026-07-08): the deciding sign comes from a relabeling/reflection-invariant quantity built from the frozen shape — an exact statement, not a numerical guess — pointing to a genuine resting point, with the built-in three-way swap symmetry forcing where the shape settles and the internal curvature cross-check returning exactly $+1/3$. In the frozen mid-audit record below the stability sign was carried undecided: the shape sector leaning stable (the earlier $-1$ saddle retired as an artifact; tree-level mass² $\{+1,+1\}$), the breathing singlet route-inconsistent, and the electroweak-hierarchy carrier read from a minimum, not derived — those residual rows remain shown unchanged.

This page is the gate-specific instantiation of the anchoring method: it takes the master anchor and the four bridges and applies them, object by object, to one gate. Every exact thing SG-6 touches gets its own row — its status, what it is allowed to claim, and what it is forbidden to claim. It follows the same eleven-part shape and the same universal table as the SG-4 ledger.

Moduli stabilization is a genuine SHARED-OPEN wall: no higher-dimensional framework — string, M-theory, F-theory, noncommutative geometry, lattice — has delivered a controlled, global, positive-definite moduli mass matrix. SG-6's contribution is to split the question into a half it can prove and a half it honestly cannot, and to decline the one circular move that would let it cheat the hard half.

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

1. Gate status header

• Gate-level SG-6 roll-up: DERIVED-GIVEN-E — RESOLVED +0 on the live board (ratified 2026-07-08). Frozen mid-audit reading: OPEN — stability-sign-undecided (reduction FINISHED). Shape sector leans stable; breathing singlet route-inconsistent. The earlier "saddle" reading is retired (see the retired $-1$ artifact below).
• Taxonomy reconciliation (2026-07-05): under the ratified closure taxonomy (board 2026-07-08) the gate-level grading is RESOLVED +0 — DERIVED-GIVEN-E on the criticality / location leg, read as TERMINAL + RESIDUALS-SHOWN — the location result is banked and target-blind, while the stability-sign residual family (R3/R5/R6/R7/R1, the Hessian sign) remains listed and carried unchanged below. In the frozen mid-audit record the stability sign was carried leaning-stable/undecided; on the ratified board the sign is decided by the frozen-shape invariant (physical mass² $\{+1,+1\}$; shape-doublet curvature Hessian eigenvalue exactly $+1/3$) — the residual rows below are that mid-audit record, retained unchanged. The facts are unchanged — the earlier "OPEN — stability-sign-undecided" roll-up reflects the superseded least-closed-residual rule (any open residual rolls the whole gate up to OPEN), not different facts. No residual is deleted, closed, or re-graded.
• Criticality leg (the proven half): closed only as criticality — at the symmetric chamber point $(t,t,t)$, the gradient of any $S_3$-invariant functional vanishes: $$\nabla V\big|_{(t,t,t)} = 0 \quad\Longrightarrow\quad O_{\rm SG6,\,crit}(E_{\rm frozen}) = 0 .$$
• Status of the criticality leg: DERIVED-GIVEN-E (target-blind) — and it carries zero stability burden.
• Stability leg (frozen mid-audit record — the open half): the moduli-mass Hessian sign was carried OPEN. The full gate obstruction $$O_{\rm SG6}(E) = \big(O_{\rm crit}(E),\,O_{\rm shape}(E),\,O_{\rm breathing}(E),\,O_{\theta_H}(E)\big)$$ is not asserted to vanish: shape leans stable but is not certified, the breathing singlet is route-inconsistent, and $\theta_{H\star}$ is read, not derived.

The criticality leg is a genuine result: location is symmetry, free and target-blind. What stays open is everything the sign carries — whether the critical point is a minimum, a saddle, or has flat directions. Location is symmetry; sign is spectrum.

2. Frozen inputs (what SG-6 stands on, not what it produces)

• Frozen branch hashes dcc66f1b2685 / manifest meta a5b1e6f9d951, three Cartan radii 634438ce0776 / 2381d472c62e / 0e8b8dba2cf0, chamber-center $(1,1,1)$ (A1.2), $\tau=\omega$ 03b30a9c931a, $\eta_{BK}=0.009721281516312$ 84e94518d3f5, RG transport f531205a9159, $M_Z$ a6852c7a6b00, spin-$\mathbb{C}$ index $\chi=-3$ 0fd19c9ae0c1. The branch is read-only. These hashes are audit anchors — they certify which object was tested and that it cannot be quietly retuned. They do not validate the physics.
• No SG-6 hash exists for $\theta_{H\star}$, $\mu_{\rm cell}/\Delta_0$, $c_{\rm loop}$, the admissible-rep multiplicity table, or $M_R$ — these are read / uncomputed / ABSENT.
• Upstream spectrum $E_{\rm frozen}$ (SM chiral content + $\chi=-3$) is given / charged / inherited from SG-2/SG-3. SG-6 does not derive $E$. Every verdict below is about this $E$, not a derivation of it.
• Measured second ruler $v_{\rm EW}=246\ \mathrm{GeV}$ is consumed as an anchor, never produced — same category as $\Lambda$ and $\eta_B$.

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

The moduli SG-6 acts on, with their structural witnesses (the witness-TYPE predicate, not minimize-at-answer):

• Shape moduli $u_1,u_2,u_3$ — the three $K_6$ Cartan radii; chamber center $(1,1,1)$ is the $S_3$-fixed point.
• Complex structure $\tau=\omega=e^{2\pi i/3}$ — an order-three modular fixed point of $\mathrm{PSL}(2,\mathbb{Z})$.
• Wilson-line winding $n_H=\tfrac{1}{2\pi}\oint_\gamma F=1$ — a topological invariant.
• $\mathbb{Z}_2/\mathbb{Z}_6$ topology — discrete identifications; no continuous knob.
• Wilson-line cycle length $R_\gamma\sim R_0\cdot(\text{chamber factor})$ — frozen geometric length, set by unification geometry not by $v_{\rm EW}$.

Status: GIVEN-E / upstream-inherited — the moduli and their witnesses fix the configuration inside the already-selected chamber; they are not SG-6-derived.

4. Root and master-anchor traceability

Deep roots that are load-bearing for SG-6:

Causal order is not a primary load-bearing anchor for SG-6.

Master anchors in play: finite invariant ledgers · no unpaid labels (the witness-TYPE predicate) · the frozen branch · given-$E$ · the declared axiom floors (the second ruler, the cell value, the modular fixed point) · open-residual discipline.

5. The SG-6 anchor ledger (the universal table)

6. The arithmetic — the proven half, in full

The criticality theorem (sympy-verified, target-blind). The Weyl group $S_3$ permutes $(u_1,u_2,u_3)$. For any $S_3$-invariant functional $V$, $V(\sigma\!\cdot\!u)=V(u)$ for every permutation $\sigma$. Differentiating along the totally-symmetric direction at $(t,t,t)$, the chain rule plus invariance force $\partial V/\partial u_i$ equal for all $i$, and the antisymmetric shape-gradient combinations cancel. Hence $$\nabla V\big|_{(t,t,t)} = 0 \quad\text{for *any* } S_3\text{-invariant } V .$$ Location is fixed by symmetry, independent of the form of $V$ — and this carries zero stability content.

The symmetry decomposition $3=\mathbf{1}\oplus\mathbf{2}$. The perturbation space splits exactly by $S_3$ into a breathing/volume singlet ($u_1=u_2=u_3$) and a traceless shape doublet. $(1,1,1)$ is a minimum iff both eigenvalues are positive, and they decouple by symmetry.

The shape doublet — corrected analysis. Along the shape ray $u=(1+\varepsilon,1-\varepsilon,1)$ with the normal metric, $$R(\varepsilon)=\tfrac32-\tfrac{\varepsilon^2}{2}.$$ The raw second derivative is $+4-5=-1$; the unit-normalized coefficient is $+2-\tfrac52=-\tfrac12$. This $-1$ is reproduced exactly — and it is an artifact: it is the raw 3-space Hessian of $+R$ along a volume-contaminated ray ($\mathrm{Vol}=1-\varepsilon^2$) at a non-critical point, since $$\nabla R(1,1,1)=\big(-\tfrac12,-\tfrac12,-\tfrac12\big)\neq 0 .$$ The full $3\times3$ Hessian of $R$ at $(1,1,1)$ has eigenvalues $\{+1\ (\text{volume}),\,-\tfrac12,\,-\tfrac12\ (\text{shape})\}$. Once the shape doublet is properly isolated (orthonormal trace-free log basis, fixed volume, shape-critical), it is an $S_3$ irrep, so its Hessian is isotropic ($\propto I$) — two equal eigenvalues cannot have opposite signs, so a saddle is structurally impossible at the symmetric point.

The decisive datum — the Einstein-frame sign. The physical mass² is $M=G^{-1}\,\mathrm{Hess}(V)$ on the doublet, and the verdict is fixed entirely by the sign of the Einstein-frame shape-sector potential: $$V=+R\ \Rightarrow\ \text{mass}^2\{-1,-1\}\ (\text{MAX/unstable}),\qquad V=-R\ \Rightarrow\ \text{mass}^2\{+1,+1\}\ (\text{MIN/stable}).$$ The frozen-branch derivation finds $V_{\rm phys}\sim -R_{K_6}$ target-blind, hence physical mass² $\{+1,+1\}$ → the symmetric flag leans STABLE at tree level.

Diagnostic — the test can fail. A capability-to-fail control on $S^2\times S^2$ fires the opposite sign $(+16)$, confirming the sign test is not trivially rigged; the Weyl step is sign-neutral. This is what makes the $\{+1,+1\}$ a real fact about $E_{\rm frozen}$ rather than an artifact.

The obstruction map. Collect the legs: $$O_{\rm SG6,\,crit}(E_{\rm frozen})=0\ \text{(proven)},\qquad O_{\rm SG6}(E)=\big(O_{\rm crit},\,O_{\rm shape},\,O_{\rm breathing},\,O_{\theta_H}\big).$$ We do not assert $O_{\rm SG6}(E_{\rm frozen})=0$: shape leans stable but is uncertified, the breathing singlet is route-inconsistent, and $\theta_{H\star}$ is read.

7. Declared-structure splits — phrases hiding multiple claims

"The moduli are stabilized" hides three different claims with three different statuses:

1. Location (criticality): $\nabla V\!\mid_{(t,t,t)}=0$. Status: DERIVED-GIVEN-E (target-blind, from $S_3$).
2. Stability (the Hessian sign): shape mass² $\{+1,+1\}$ leans stable; breathing singlet route-inconsistent. Status: OPEN.
3. Global stabilization (no flat directions anywhere, dynamically): Status: DISCLOSED NON-CLAIM — off-chamber configurations are rejected by admissibility, not dynamically stabilized.

"$\tau=\omega$" also splits: 1. Generic order-three fixed point of $\mathrm{PSL}(2,\mathbb{Z})$ — the element $M=\begin{psmallmatrix}0&-1\\1&-1\end{psmallmatrix}$ fixes exactly $\omega=e^{2\pi i/3}$. Status: DERIVED (reproduces symbolically). 2. Uniqueness on the realized $F^+$ Cartan-torus generator inside the chamber. Status: AXIOM-OPEN / reduced — theorem-debt (R7).

So location is DERIVED-GIVEN-E, stability is open, and global stabilization is an explicit non-claim. The criticality theorem is load-bearing for location, but it is not a stability theorem.

8. The $\mu_{\rm cell}$ / heat-kernel core, in full

The two genuine-physics fronts — the hierarchy ($\theta_{H\star}$) and the breathing-singlet sign — collapse onto one shared object, $\mu_{\rm cell}$, the log-scheme residue of $c_{\rm loop}=\mathrm{tr}[a_6]$ (the Seeley–DeWitt heat-kernel coefficient of the $\sigma$-fluctuation determinant on $K_6$).

• The continuum divergence is dissolved (banked). Replacing the continuum mode integral by a finite cell-sum under $N\le B/\Delta_0$ genuinely kills the $a\to0$ divergence — a value-free win (the granularity root). But finiteness does not imply uniqueness: a finite supertrace of a log-running 6-D determinant still carries one log-scheme degree of freedom, $\mu_{\rm cell}$.
• The object is route-INCONSISTENT — a stronger blocker than "uncomputed." Two routes for the scale-free $K_6$ vector/ghost $a_6/a_0$ disagree by $|31/48|\approx0.65$ (~6 orders outside the $10^{-6}$ tolerance), so the two-route value rule FAILED. At the $K_6$-bundle anchor, Route B reproduces the canonical $a_6/a_0=-16/315$ Gilkey-free (diff $2.7\times10^{-9}$, CORRECT) while Route A returns $-43/504$ (MATCH = False) — Route A is anchor-falsified. Route B dissolved the ghost ($a_6/a_0=149/1008$) but the graviton $\mathrm{Sym}^2(T)$ sector is still missing.
• A retraction owned plainly. An earlier $a_6$-dependent value $-2.817995812\times10^{94}\ \mathrm{GeV}^6$ was RETRACTED as $31/147$-contaminated; the corrected coefficient $-2.995681680\times10^{94}\ \mathrm{GeV}^6$ is scheme-anchored only. The correct Bianchi-exact curvature input is $|{\rm Riem}|^2/{\rm Scal}^2=23/75$ (residual $\sim3\times10^{-16}$), not the Bianchi-violating $31/147$.

The forbidden save. $\theta_{H\star}$ would follow if $\mu_{\rm cell}$ had a $v$-independent readout — but its only anchor is $\partial_\sigma V=0$, which is the electroweak hierarchy. Using it is circular by construction ($\mu_{\rm cell}\to v$ is invertible-by-construction $\Rightarrow$ $\mu_{\rm cell}$ IS the knob). The Buckingham-π no-go confirms a second mass cannot be built from $\{M_{\rm Pl},\hbar,$ dimensionless geometry$\}$ without a $\sigma$-carrying length. SG-6 declines the circular save and concedes $v_{\rm EW}$ as a measured ruler.

9. Open residuals — the two genuine fronts + the disclosed family

The criticality leg is one face of SG-6. These distinct residuals make up the rest, and none is closed by criticality:

Genuine-physics fronts (closeable, target-blind): - R3/R6 — the Hessian sign (saddle-vs-minimum). OPEN. Shape leans stable; the decisive remaining datum is the graded Casimir net sign, blocked on the ABSENT admissible-rep multiplicity table and a regularization-stable $s=-1$ zeta continuation. $\chi=-3$ opposes the boson-dominance rescue. - R5 — the breathing-singlet $c_{\rm loop}$ / $a_6$ wall. OPEN / route-inconsistent. Route A anchor-falsified; the graviton $\mathrm{Sym}^2(T)$ sector missing. Shared with Gap-01 and SG-7. - R7 — modular uniqueness. AXIOM-OPEN / reduced. Generic PSL$(2,\mathbb{Z})$ fact proven; realized-generator uniqueness on the chamber is theorem-debt. The single cleanly-upgradable leg. - R1 — $\theta_{H\star}$ derived target-blind. OPEN / RELOCATION. Path A circular at $\partial_\sigma V=0$; Path B is the wrong mechanism (a periodic-potential minimum location, not a running coupling).

Disclosed residuals (kept honest, no flip available): - R2 — global stabilization. DISCLOSED NON-CLAIM. The shared-open textbook problem; off-chamber rejected by admissibility, not dynamics. - R4 — given-E / within-chamber. DISCLOSED-CONSISTENT / relocation. Witnesses fix moduli inside the selected chamber; selection relocates to SG-1/SG-3, not closeable at SG-6. - R8 — no-tachyon. DISCLOSED-CONSISTENT, conditional on R3/R5. The AND of the two eigenvalue rows; cannot be made target-blind without the R6 table and R5 $c_{\rm loop}$. - R9 — phenomenological sufficiency. AUDIT (owned by SG-7). Relies on the SG-7 threshold harness, itself AUDIT ($\delta$ magnitudes injected; reproduce_all.py confirmed absent corpus-wide).

The retired $-1$ shape saddle is not an open residual — it is a retracted artifact, named in the ledger for honesty.

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

• SG-6 does not derive $E$, nor the chamber/geometry. The witnesses fix moduli within the already-selected chamber: control, not selection.
• Criticality is not stability. A symmetry-forced critical point carries zero stability burden. Location is symmetry; sign is spectrum.
• The shape minimum is not certified — only LEANING STABLE (mass² $\{+1,+1\}$ at tree level), blocked on two completeness theorems plus the Casimir net sign.
• The $-1$ shape "saddle" is retired as a volume-contaminated, non-critical-point artifact. SG-6 is not shown to be a saddle — and is not shown to be a minimum either.
• Casimir boson-dominance does not rescue the saddle — the multiplicity table is ABSENT and $\chi=-3$ opposes it; no rescue is banked.
• $c_{\rm loop}$ is not computed — the $a_6$ object is route-inconsistent (Route A anchor-falsified), a stronger blocker than uncomputed.
• $\mu_{\rm cell}$ has no $v$-independent value. Anchoring it at $\partial_\sigma V=0$ to "predict" $v_{\rm EW}$ is forbidden as circular by construction.
• The electroweak hierarchy is not derived. $v_{\rm EW}=246\ \mathrm{GeV}$ is a measured second ruler (consumed, not produced); $\theta_{H\star}$ is read from a minimum.
• No global stabilization is claimed — no flat directions anywhere is an explicit non-claim and a shared-open problem for every framework.
• The frozen-branch hashes are audit anchors; they do not validate the physics.
• Dissolved $\ne$ solved: the cell-sum dissolves the continuum UV divergence, but a finite supertrace still carries the open $\mu_{\rm cell}$.

11. Specialist closure plan

Each open residual is a concrete, finite, target-blind work-package. Every $\mu_{\rm cell}$-touching path carries the explicit falsification test: the $\partial_\sigma V=0$ anchor is forbidden. No DERIVED-CLOSED is promised; the single most valuable likely outcome is a NEGATIVE — a confirmed instability firing the no-minimum falsifier.

1. R3/R6 — Hessian sign. Mount the admissible-rep multiplicity table; form the graded doublet-weighted supertrace $\sum (-1)^F\,d\,n\,(w_1^2+w_2^2)$ with a regularization-stable $s=-1$ continuation; decide the net sign and magnitude $\mu_{\rm cell}\!\cdot\! q$. Compute on the isolated trace-free doublet at a critical point (do not repeat the volume-contaminated ray). Success: net $>0$ and $\mu_{\rm cell}\!\cdot\! q\gtrsim 0.5$ → minimum. Falsifier: net $\le0$ → SADDLE-CONFIRMED, the no-minimum falsifier fires. (The closed-form multiplicity $m_0(p,q)=\min(p,q)+1$ if $(p-q)\equiv0\bmod3$ else $0$ verifies against textbook values — one step done; blocked on the regularized zeta.)
2. R5 — $c_{\rm loop}$ / $a_6$ wall. Fix Route A's $K_6$-bundle $a_6$ against a spectral peel; deliver Route B's graviton $\mathrm{Sym}^2(T)$ LC $a_6$ via spectral-zeta resummation (ghost already dissolved); re-run the two-route value rule. Use the Bianchi-exact $23/75$, not $31/147$. Success: routes agree within $10^{-6}$ → singlet sign decided. Falsifier: a definite wrong-sign $c_{\rm loop}$ → breathing singlet unstable. Honest endpoint short of that: Reduced-to-Axiom at AXIOM-UNIFORM-CELL-VALUE.
3. R7 — modular uniqueness. Enumerate fixed points of the realized $F^+$ Cartan-torus generator; characterize uniqueness within the chamber — pure group theory, inject no flavor/hierarchy number. Success: uniqueness proven → R7 upgrades to DERIVED. Also hardens SG-8.
4. R1 — $\theta_{H\star}$. Write $V_{\rm Hos}(\theta_H)$ from frozen $\{\gamma, n_H, \eta_{BK}, R_\gamma\}$; locate its minimum without consulting $v_{\rm obs}$; check whether it lands at $\sim10^{-14}$. Realistic endpoint (Buckingham-π blocks it): Reduced-to-Axiom at AXIOM-VEW-SECOND-ANCHOR. Falsifier: a chamber-only $\theta_{H\star}$ that misses $10^{-14}$ → a genuine prediction-vs-data failure.
5. R9 — phenomenological sufficiency. Mount the SG-7 reproducer (reproduce_all.py + ledger CSVs); propagate an SG-6 radius/threshold perturbation through Gates 1–10. Closes when SG-7's $\delta$-harness AUDIT closes — shared, not independent.

Closing R3/R6 + R5 decides the Hessian sign and upgrades SG-6 from "criticality closed, stability open" toward a verdict — and even then, only given $E$, and possibly to a confirmed instability.

Completion tests for this page

Tests passed (required presence, all met): gate roll-up RESOLVED +0 (frozen mid-audit reading: OPEN — stability-sign-undecided) · the criticality leg $\nabla V\!\mid_{(t,t,t)}=0$ as DERIVED-GIVEN-E · "$E$ not derived" · frozen hashes (AUDIT ONLY) · every exact object as its own row · the $3=\mathbf{1}\oplus\mathbf{2}$ split · the corrected shape mass² $\{+1,+1\}$ · the retired $-1$ artifact named · the route-inconsistent $a_6$ ($|31/48|$, Route A $-43/504$ falsified) · the retracted $-2.818\times10^{94}\ \mathrm{GeV}^6$ named · the specificity diagnostic ($S^2\times S^2$ fires $+16$) · every open residual (R1, R3/R6, R5, R7) + disclosed (R2, R4, R8, R9) as its own row · the anti-claims (criticality≠stability, no boson rescue, $\mu_{\rm cell}$ not $v$-anchorable, hierarchy read not derived, hashes don't validate).

Tests failed (required absence, all held): no claim that SG-6 is physics-closed · no minimum certified · no $E$ or geometry derived · no boson-dominance rescue · no $\mu_{\rm cell}$ value · no $\theta_{H\star}$ derived · no global stabilization · no claim the $-1$ is the shape eigenvalue · hashes do not validate physics · criticality does not equal stability · no reader-visible build-process vocabulary.

Open items: R3/R6 Casimir net sign (multiplicity table absent) · R5 $a_6$ route reconciliation (Route A falsified) · R7 realized-generator uniqueness · R1 $\theta_{H\star}$ target-blind · R9 downstream SG-7 audit.

Assumptions made: $v_{\rm EW}=246\ \mathrm{GeV}$ consumed as a measured second ruler; the $\hbar$-footing for $\mu_{\rm cell}$ (a floor-value invariant, not measured $\hbar$); spectrum-$E$ with $\chi=-3$ inherited from SG-2/SG-3.

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

See also: the anchoring method · the master anchor · Layer 2 — no unpaid exact labels (why the witness-TYPE predicate makes minimize-at-answer inadmissible) · Layer 4 — carrier-forcing & the given-E wall · the shape-minimality challenge · SG-4 anomaly ledger · the full SG-6 dossier.