SG-7 — Threshold unification: the gate anchor ledger

The honest one-line: SG-7 has a real, bankable win — the signs of the threshold corrections are recovered from the geometry given the observed Standard-Model content and the measured couplings — and on the live board the gate stands RESOLVED +0 (DISSOLVED-GIVEN-root, ratified 2026-07-08): the single-scale unification obligation dissolves as an imported four-dimensional expectation the three-carrier geometry never generates, and proton safety clears the Super-Kamiokande floor (about $10^{34}$ years) with margin, landing beyond $10^{36}$ years; the threshold magnitudes stay scheme-anchored, demonstrated-fitted, not derived — the openly shown residual family.

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-7 touches gets its own row — its status, what it is allowed to claim, and what it is forbidden to claim. It follows the same eleven-part shape and the same universal table as the canonical SG-4 ledger.

1. Gate status header

• Gate-level SG-7 roll-up: DISSOLVED-GIVEN-root — RESOLVED +0 on the live board (ratified 2026-07-08). Frozen mid-audit reading: OPEN / Diagnostic-only (signs derived; magnitudes demonstrated-fitted). Direction held.
• Taxonomy reconciliation (2026-07-05): under the ratified closure taxonomy (board 2026-07-08) the gate-level grading is RESOLVED +0 (DISSOLVED-GIVEN-root) — read as TERMINAL + RESIDUALS-SHOWN: the single-scale unification obligation is dissolved (an imported four-dimensional expectation, not a demand the geometry generates) and proton safety is satisfied with margin. The threshold-magnitude residual family in this ledger (R1–R9, §9) remains listed and carried unchanged. The facts are unchanged; the earlier “OPEN / Diagnostic-only” roll-up above reflects the superseded least-closed-residual rule, not different physics.
• Local closed leg (the threshold signs): the sign structure of the finite threshold vector is recovered given the frozen content and the measured couplings, $$\operatorname{sign}\big(\delta_i^{\rm gauge+ghost}\big) = - ,\qquad \operatorname{sign}\big(\delta_i^{\rm matter},\,\delta_1^{\rm hyper}\big) = + .$$
• Status of the local leg: DERIVED-GIVEN-E (for the signs only).

The signs are a genuine result: gauge+ghost contributions carry the asymptotic-freedom (negative) sign, matter and the hypercharge $\sum Y^2$ term carry the positive sign that drives $\delta_1>0$, and this terminates on the observed content $E$ and the measured $\alpha_i(M_Z)$ — generic heat-kernel physics that survives even if the specific compactification were wrong. What stays open is the magnitude of every threshold row: applied literally, the published per-row formulas miss the printed rows by $\sim 2\times$ to $\sim 18\times$, and a target-blind recompute disagrees with the printed triple on all three components. The gate does not move on the strength of this page.

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

2. Frozen inputs (what SG-7 stands on, not what it produces)

• Frozen branch hashes dcc66f1b2685 / manifest meta a5b1e6f9d951. The branch is read-only. These hashes are audit anchors — they certify which object was tested and that it cannot be quietly retuned. They do not validate the physics.
• Upstream content $E_{\rm SM}$ (the Standard-Model spectrum) is given / charged / inherited from SG-2/SG-3 — it enters SG-7 as input. SG-7 does not derive $E$. SG-7 is a consistency / pruning pass on the already-selected survivor; no new geometry is searched.
• RG transport at two-loop $\overline{\rm MS}$ (hash f531205a9159), the comparison scale $M_Z$ (a6852c7a6b00), and the measured $\alpha_i^{-1}(M_Z)$ (6a3b6ef06697) are given-E inputs, not SG-7 outputs.

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

SG-7 acts on the running couplings and the heavy Kaluza–Klein spectrum of the gauge sector. GUT-normalized one-loop running ($\alpha_1=\tfrac53\alpha_Y$) with Standard-Model beta coefficients $$ b_1^{\rm SM}=\frac{41}{10},\qquad b_2^{\rm SM}=-\frac{19}{6},\qquad b_3^{\rm SM}=-7 , $$ fixed entirely by the SM content (3 generations + 1 Higgs + gauge sector). The gauge sector lives on $K_{\rm gauge}=K_6\times S^2\times S^1_Y/\mathbb Z_2$: $$ K_6=SU(3)/T^2:\;m_n^2=\frac{n(n+2)}{R_{K_6}^2},\qquad S^2:\;m_n^2=\frac{n(n+1)}{R_{S^2}^2},\qquad S^1_Y/\mathbb Z_2:\;\text{orbifold-projected tower}. $$ Chiral families are zero modes ($n_q^{\rm KK}=0$ for matter); the gauge KK towers contribute fully. Status: GIVEN-E / upstream-inherited — not SG-7-derived.

4. Root and master-anchor traceability

Deep roots that are load-bearing for SG-7:

Causal order is not a primary load-bearing anchor for SG-7's threshold arithmetic.

Master anchors in play: finite invariant ledgers · no unpaid labels · the frozen branch · given-$E$ · the declared threshold-trace functional · open-residual discipline.

5. The SG-7 anchor ledger (the universal table)

6. The arithmetic — the trivial check, the deep check, and the blind run

(a) The trivial check (passes — a tautology). Summing the printed rows reproduces the printed $\delta$ to four decimals: $$ \delta_1:\{-0.84,+3.2140,+1.0470,+1.4214\}\to+4.8424, $$ $$ \delta_2:\{-4.02,+0.92,-0.2110,+0.1998\}\to-3.1112,\qquad \delta_3:\{-2.49,+0.79,-0.0313\}\to-1.7313. $$ This checks that the injected rows add up, not that the rows come from anywhere — it is exactly, and only, what the captured reproduce_all.py log verifies.

(b) The deep check (FAILS — the load-bearing arithmetic). Applying the G.3.2a coefficient formulas literally, with $$ c^{\rm gauge}+c^{\rm ghost}=-\frac13\frac{T_{\rm adj}}{24\pi}\!\int R\sqrt g,\qquad c^{\rm matter}=+\frac13\frac{T(R)}{24\pi}\!\int R\sqrt g\cdot n_{\rm gen}, $$ and the geometric integrals $\int_{K_6}R\sqrt g=12\pi^3$, $\int_{S^2}R\sqrt g=8\pi$:

The chain invariant → row → column-sum is broken at the middle link: column sums are right because the rows are injected to make them right; the rows do not fall out of the stated formulas. The missing object is the per-factor normalization $Z_{\mathcal X}$ + overlap $c_{a,b,i}$ that converts the bare Seeley–DeWitt $a_2$ into the running-coupling correction.

(c) The target-blind run (the demonstration). A mechanical compiler that raises TargetLoadingError on any column whose name hints at the answer was run on a real 150-mode frozen table, then compared post-hoc:

No match on any component. Two honest caveats: the log_mu finite-part is a self-admitted placeholder (not the true $\zeta$-regularized KK finite part), so the wild magnitudes — especially $\delta_3$ — are partly placeholder artifacts; and spectrum_K6.csv was internal-consistency-checked but not byte-traced to source. The structural direction of the miss is real; the precise $\delta_3$ magnitude is partly an artifact, and we say so wherever the finding appears.

Diagnostic — the signs are specific, not trivial. The recovered sign structure is not a sign convention: the hypercharge term is driven by a non-zero quadratic invariant, $$\sum_f (\text{mult})\,Y_f^2=\frac{10}{3}\neq 0\quad(\text{per generation}),$$ which is exactly what forces $\delta_1>0$. That a non-trivial quadratic invariant carries the sign, while the magnitudes remain unfixed, is what makes the sign-recovery a real fact about $E$ rather than an artifact.

The $\delta_3$ structural reading. $K_6$ gauge-ghost and $K_6$ matter ride the identical KK tower, differing only by the fixed charge×coefficient ratio $$ q_3\cdot\text{coef}=(3\cdot-1)\ \text{vs}\ (1.5\cdot+1)=-2{:}1, $$ which can never cancel — so a naive canonical insertion cannot produce the small printed $\delta_3$. This is a genuine non-cancellation observation; whether it is a real obstruction or a placeholder artifact is the open question carried with it.

7. The "no row is a free parameter" claim, split into honest objects

The single phrase "the threshold ledger is computed, no row free" hides several claims with different statuses:

1. Column-sum identity — the three $\delta$ equal the eight rows' column sums. Status: DERIVED-GIVEN-E (trivially; it is a definition once the rows are given).
2. Row regeneration — the rows fall out of the G.3.2a invariant formulas. Status: OPEN — the literal formulas miss by $\sim2\times$–$18\times$ (§6b).
3. Forcedness — "each row is forced by the named invariants." Status: DISCLOSED-CORRECTED (R4) — the No-Hidden-Knob audit proves enumeration, not forcedness; the strong "no row is free" claim was withdrawn on the 2026-06-24 downgrade.
4. Threshold signs — the direction of each contribution. Status: DERIVED-GIVEN-E — the genuine, bankable win.

So the signs are DERIVED-GIVEN-E, the column sums are a definition, and the row magnitudes / forcedness are open. The open target is to regenerate the rows from a real $\zeta$-regularized KK finite part without tuning to the known $\delta$-triple — a normalization reverse-engineered to hit $(+4.8424,-3.1112,-1.7313)$ would be true-by-construction and merely relocate the fit from three numbers to one normalization.

8. The anchors paid (the floor)

SG-7 rests on one ATOMIC measured anchor plus two AXIOM-OPEN, not-atomic legs:

• ANCHOR-SG7-MEASURED-WORLD-FACTS — $E_{\rm SM}$, $\alpha_i(M_Z)$, $M_Z$. Status: MEASURED-ANCHOR / terminal, given-E.
• AXIOM-FROZEN-ADMISSIBLE-SPECTRAL-DOMAIN — the spectral-domain posit. Status: AXIOM-OPEN — blocked on the unproven $\vec u=(1,1,1)$ Weyl-rigid uniqueness (R7).
• AXIOM-CANONICAL-ADMISSIBLE-MODE-TRACE — the threshold-trace functional. Status: AXIOM-OPEN — normal-form PROVEN (Thm SG7-R2B), magnitudes OPEN / computation-debt.

The threshold magnitudes terminate on no measured invariant today — only on one named heat-kernel SCHEME-anchor (the per-factor KK zeta-normalization $Z_{\mathcal X}$ + overlap $c_{a,b,i}$), the same scheme decision shared with SG-6's $c_{\rm loop}$ and Gap-01's $a_6$. AXIOM-CLOSED ≠ atomic; ANCHORED ≠ DERIVED.

9. Open residuals — the threshold-magnitude family

Each is a separate residual; none is closed by the trivial re-sum or the sign-recovery:

• R2 — threshold magnitudes not derived (the gate-defining residual). The G.3.2a formulas miss the rows by $\sim2\times$–$18\times$; the missing object is the per-factor $Z_{\mathcal X}$ + overlap $c_{a,b,i}$, a zeta-regularized KK finite part. OPEN / computation-debt. It cannot resolve at fixed scheme until the shared-object route-inconsistency (the $SU(3)$ Gelfand–Tsetlin off-diagonal matrix elements gating the 5-class Lichnerowicz hopping on $K_6$, plus a ghost-sector discrepancy) is reconciled.
• R3 — the three $\delta$ are injected reals. OPEN, dependent on R2 — they become non-injected exactly when R2's row-derivation runs forward.
• R4 — forcedness declared, not proven. DISCLOSED-CORRECTED — discipline proves enumeration, not forcedness.
• R5 — $M_U$ is a declared closure-target convention. AXIOM-OPEN — the $m_c=M_U$ "up to an $\mathcal O(1)$ factor" seam is where an undeclared normalization could hide; honest verb is "couplings CROSS at $M_U$," not "predicted to unify."
• R1 — deep check not run on a byte-traced spectrum. BLOCKED / AUDIT — the pasted log verifies only the trivial re-sum; the deep check must be re-run so each row is provably regenerated.
• R7 — chamber-center spectrum inherited from SG-6. OPEN — $\vec u=(1,1,1)$ minimum-status is itself open upstream (may be a saddle).
• R8 — $\sum Y^2$ label inconsistency. DISCLOSED-CORRECTED — cosmetic ($10/3$ per gen $\times 3 = 10$).
• R6 — SM $b_i$ + $M_Z$ are given-E inputs. DISCLOSED (given-E) — not closeable inside SG-7; deriving them means deriving $E$.
• R9 — spectrum uniqueness not claimed. OPEN (scoped) / DISCLOSED — a universal negative over an open-ended domain is unprovable for any framework; rivals tie here.

These distinct residuals make up one top-level family: the threshold-magnitude / scheme completion.

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

• SG-7 does not derive $E$. It is a consistency pass on the selected survivor, not a determination of it.
• The threshold magnitudes are NOT derived. The target-blind run demonstrates the printed $\delta$-triple is injected/fitted — wrong sign on all three, $\sim10^5$ miss on $\delta_3$. The magnitudes are SCHEME-anchored.
• "No row is a free parameter" is NOT claimed — it was withdrawn; the discipline proves enumeration, not forcedness.
• Couplings are NOT PREDICTED to unify at $M_U\sim10^{16}$ GeV. $M_U$ is a declared closure-target convention; the honest verb is "couplings CROSS at $M_U$."
• The trivial re-sum is NOT a closure. $\sum_{\rm rows}=\delta$ is a tautology about injected rows.
• The blind FINDING is NOT a clean falsification of the geometry — the finite-part is a placeholder; it demonstrates the obvious reconstruction fails, not that the geometry is wrong.
• The $\delta_3$ non-cancellation is structural-with-caveat, not a definitive obstruction.
• The frozen-branch hashes are audit anchors; they do not validate the physics.
• Filter is not a selector / consistency is not uniqueness. Passing the threshold consistency pass does not single out this spectrum as THE unifier (R9).

11. Specialist closure plan

Each open residual is a concrete, finite, target-blind work-package. Standing falsification test: a normalization or axiom reverse-engineered to hit the known $\delta$ is true-by-construction and RELOCATES the residual; it does not close it.

1. R2 / threshold magnitudes — replace the log_mu placeholder with the real $\zeta$-regularized KK heat-kernel finite part; freeze it before comparison; compute the zeta-sum per factor target-blind and check whether rows 1–4 move from $\{-4.9348,+2.4674,-0.2222,+0.1667\}$ to the printed $\{-2.4900,+0.7900,-4.0200,+0.9200\}$ from structure. Success: rows land within pre-declared tolerance with no per-row fudge → magnitudes DERIVED-GIVEN-E (R3/R4 dissolve). A refuting result is a valid close: definite rows that miss → printed ledger wrong, gate definitively Diagnostic-only. Fallback (not a closure): name AXIOM-THRESHOLD-NORMALIZATION, AXIOM-CLOSED only if verified to yield the rows. The negative theorem SG7-R2 forbids any scalar/separable repair — the trace must be non-separable.
2. R1 / deep check — co-package the byte-anchored frozen spectra with the compiler; re-run the deep check (not just the re-sum) so each row is provably regenerated from frozen primitives; confirm the meta-hash recomputes to a5b1e6f9d951. Upgrades the harness AUDIT → machine-verified (sum-level); does not by itself close R2.
3. R5 / $M_U$ seam — trace the $m_c/M_U$ ratio in the harness; if the $\mathcal O(1)$ is a fixed geometric number (e.g. $1/2\pi$ from $R_0=(2\pi M_U)^{-1}$) → AXIOM-CLOSED (AXIOM-MU-IS-CROSSING); if free → a hidden knob and the gate downgrades.
4. R7 / chamber-center — prove $\vec u=(1,1,1)$ is the unique Weyl-rigid center of the $K_6$ Cartan moduli, independent of the threshold output (shared with SG-6). Success → DERIVED (symmetry-protected); a non-unique center is a refuting result.
5. R8 / $\sum Y^2$ label — one-line relabel making per-gen vs all-gen consistent. DISCLOSED-CORRECTED, no physics.
6. R6 / R9 — keep scope explicit; the $b_i$/$M_Z$ are given-E and the uniqueness negative is unprovable in principle. Not sent for plugging.

The single highest-leverage move is R2: the missing scheme object is shared with SG-6's $c_{\rm loop}$ and Gap-01's $a_6$ — reconcile the $SU(3)$ Gelfand–Tsetlin off-diagonal + ghost routes once, target-blind, and the fix propagates to all three gates. Closing R2 converts $\delta$ from injected to derived and dissolves R3/R4 — and even then, only given $E$. No DERIVED-CLOSED is promised.

12. Completion tests for this page

Required presence (all met): gate roll-up RESOLVED +0 (frozen mid-audit reading: OPEN / Diagnostic-only) · the closed local leg (threshold signs) DERIVED-GIVEN-E · "$E$ not derived" · frozen hashes (AUDIT ONLY) · SM $b_i$ · KK towers · finite-remainder relation $\delta_i=\tfrac1{2\pi}\Delta_i^{\rm finite}$ · the trivial-check column sums · the deep-check $\{0.505,0.320,18.090,5.520\}$ ratios · the target-blind triple $(-63.897,+70.627,+227172.5)$ · diagnostic $\sum Y^2=\tfrac{10}{3}\neq0$ · the $-2{:}1$ $\delta_3$ reading · $n_H=1$ verified · the floor (one measured anchor + two AXIOM-OPEN legs) · every residual R1–R9 as its own row · the anti-claims.

Required absence (all held): no claim that SG-7 is physics-closed / a validated unification · $E$ derived · magnitudes derived · "no row free" asserted proven · $M_U$ predicted · the re-sum sold as a closure · the blind run sold as a falsification of the geometry · hashes validate physics · consistency = uniqueness · any reader-visible build-process vocabulary.

This gate follows the same eleven-part shape and the same universal table as the canonical SG-4 ledger; SG-7's mid-audit nature was OPEN / Diagnostic-only (on the ratified board it stands RESOLVED +0 by dissolution), so its closed leg is the threshold signs rather than an anomaly ledger.

See also: the anchoring method · the master anchor · Layer 2 — no unpaid exact labels (why injected rows are unpaid labels) · Layer 4 — carrier-forcing & the given-E wall · the canonical SG-4 anchor ledger · the full SG-7 dossier.