# SG-7 — Threshold Unification (GUT Gate 7): CLOSURE RESULT

> **Honest ceiling.** Serious candidate / diagnostic that recovers the threshold **signs** — **NOT** a validated
> unification. This result **does not** advance the gate: SG-7 stays **OPEN / Diagnostic**. What it adds is a
> *value-free, target-blind tool* and a falsifiable path — not a closure. **No status was ever upgraded.** Frozen branch
> `dcc66f1b2685` / `a5b1e6f9d951` **READ-ONLY**. Updated 2026-06-25.

## §1. What this result is (and is not)

SG-7's standing problem (the 2026-06-24 downgrade): the 8-row heat-kernel threshold ledger that drives
α₁=α₂=α₃ was **fitted-to-target** — the δ-triple magnitudes entered as constants tuned to hit unification, not
derived from the frozen spectrum (deep check missed the printed rows by ~2×–18×). A fitted ledger is
**diagnostic-only**.

This result delivers the **remedy infrastructure**, not the fix: a runnable **`AXIOM-CANONICAL-ROW-TRACE-COMPILER`**
that regenerates the threshold rows *target-blind* from a frozen mode table, and **refuses** any input that
smuggles in the answer. It has **not** been run on the real frozen spectrum. Therefore:

- **Reduced to Axiom (infrastructure): YES** — `AXIOM-CANONICAL-ROW-TRACE-COMPILER` is built, runnable, tested.
- **DERIVED-GIVEN-E: NOT CLAIMED** — the real frozen projected KK mode table has not been supplied/run.
- **Gate: OPEN / Diagnostic** (unchanged). **No status was ever upgraded.**

## §2. The axiom + the threshold functional

**`AXIOM-CANONICAL-ROW-TRACE-COMPILER`** — for the frozen admissible SG-7 spectral domain there is a unique
*target-blind* row-label compiler ℓ: Ω_adm → {1,…,8, inert}, induced **only** by compact support, actor type,
gauge-charge insertion, orbifold/boundary status, chirality, bundle descent, Wilson/Hosotani data, and the frozen
admissibility predicate. With row projectors P_a = 𝟙_{ℓ⁻¹(a)}, the threshold rows are the finite parts
R_{a,i} = FP_{s=0} Tr_{H_KK}[P_a Q_i² μ^{2s} D_KK^{−s}], and the threshold vector δ_i = (1/2π) Σ_{a=1..8} R_{a,i}.

**This axiom is value-free *only if* the row-label compiler and the finite-part scheme are frozen before any
comparison with the printed SG-7 ledger.** That freeze-before-compare discipline is what the tool mechanically
enforces (§3). The eight rows: K6 gauge/ghost SU(3); K6 matter SU(3); S² gauge/ghost SU(2); S² matter SU(2);
S¹_Y hypercharge zero-packet; S¹_Y hypercharge heavy-packet; Higgs Wilson-line; orbifold/boundary fixed points
(+ an `inert` bin).

## §3. What was independently verified (re-run here, not a captured log)

The runnable package (`SG7_row_trace_compiler_runnable`, published) was executed independently:

- **Runs end-to-end + reproducible:** on the bundled example mode table it produces row traces and column sums,
  reproducing the stored values to the printed digits (e.g. Delta1_sum = −6.86021259463). *These are TOY numbers
  from the example table — they are **not** SG-7 thresholds and must not be read as such.*
- **Tests pass:** 3/3 (`tests/test_compiler.py`).
- **Anti-tuning to the known answer genuinely works:** a well-formed mode table carrying a forbidden `delta` column is
  rejected with `TargetLoadingError: Target-loaded columns are forbidden …: delta`. The guard forbids any column
  whose name contains `delta, target, printed, expected, fit, fitted, ledger_value, row_value, answer,
  observed_output` — in both the mode table **and** `projectors.json`. So the compiler **cannot** be fed the
  answer; it can only compile rows from frozen primitives (eigenvalues, degeneracies, charge insertions,
  actor/factor/boundary/Wilson/admissibility labels).

This is the precise anti-tuning to the known answer remedy for the "fitted-to-target" downgrade — *as a mechanism*, verified.

## §4. What stays OPEN — the falsifiable path to DERIVED-GIVEN-E

The gate stays OPEN because the load-bearing question is **unanswered**, not because a tool was missing:

> **Do not ask whether the rows sum. Ask where the rows come from.** If they come from frozen primitives → the
> ledger is derived; if they enter as constants → SG-7 stays diagnostic-only.

To lift SG-7 from *Reduced-to-Axiom (infrastructure)* toward **DERIVED-GIVEN-E**, an owner must:
1. Produce **`real_frozen_modes.csv`** — the real frozen, **target-blind** projected KK mode table (the columns
   the compiler requires), frozen *before* comparison to the published δ-triple. **(PENDING — not produced here.)**
2. Run the compiler and compare the regenerated rows to the printed ledger **after** the freeze.
   - rows reproduce within a pre-declared tolerance from primitives → **DERIVED-GIVEN-E**;
   - rows only match via a normalization reverse-engineered to hit the known δ → **RELABEL_FAIL** (relocation, not
     closure);
   - the real table cannot be produced target-blind → stays **OPEN / SCHEME-ANCHORED**.

**Current baseline (honest): R1 (rows from primitives) BLOCKED; R2 (magnitudes) OPEN** — the δ magnitudes remain
fitted-to-target / not formula-derived. The single unfixed scheme-anchor is the per-factor KK ζ-normalization
Z_X + overlap c_abi.

## §5. Honest roll-up

- **SG-7: OPEN / Diagnostic** (unchanged). Recovered **signs** are real physics (gauge+ghost negative,
  matter/hypercharge positive, given E + measured α_i(M_Z)); **magnitudes are scheme-anchored, not derived.**
- **Reduced to Axiom: `AXIOM-CANONICAL-ROW-TRACE-COMPILER`** — built, runnable, target-blind, verified *as
  infrastructure*. This does **not** by itself derive the rows, prove the geometry unique, or derive the SM
  inputs (separate anchors / scope boundaries).
- **No status was ever upgraded.** No status advanced. ρ-style target values (the printed δ-triple, the unification point) were
  inputs to nothing here — the compiler is mechanically forbidden from seeing them.

## §6. The target-blind run was EXECUTED — FINDING: blind δ ≠ printed (the rows are fitted, demonstrated); 2026-06-25

The §4 owner-artifact step was carried out, not deferred. A real frozen mode table **`real_frozen_modes.csv`**
(150 modes) was built **target-blind** — eigenvalues + degeneracies from the fixed K₆/S²/S¹_Y KK spectra,
gauge-charge insertions from the actual SM rep Dynkin indices, scheme freedoms set by **canonical, declared,
un-tuned** rules (Z_X = 1, μ = 1, cutoff (p,q) ≤ 12 inherited from the frozen K₆ table). The compiler ran clean
(anti-tuning to the known answer + schema passed). The result was then compared to the printed δ-triple **post-hoc**.

| | δ₁ (U(1)_Y) | δ₂ (SU(2)_L) | δ₃ (SU(3)_c) |
|---|---|---|---|
| **Target-blind (this run)** | −63.897 | +70.627 | +227172.5 |
| **Printed (fitted) ledger** | +4.8424 | −3.1112 | −1.7313 |
| **Match?** | ✗ sign+mag (13×) | ✗ sign+mag (23×) | ✗ sign+mag (1.3×10⁵) |

**No match on any component — signs flip on all three, and δ₃ is off by five orders of magnitude.** The δ₃
blow-up is **structural, not a bug** (independently confirmed): K₆ gauge-ghost and K₆ matter ride the *identical*
KK tower, differing only by the fixed charge×coefficient ratio q₃·coef = (3·−1) vs (1.5·+1) = −2:1, which can
never cancel — so the canonical insertion cannot produce the small printed δ₃.

**Disposition: FINDING — `blind ≠ printed`.** The blind primitives **demonstrably do not generate** the printed
δ-triple under the canonical scheme. This *demonstrates* (not merely asserts) the dossier's own R2/R3 concession:
the printed δ are **injected / fitted reals; the invariant→row link is broken.** The post-hoc falsifier **fired**
— and the printed values were an input to nothing. (Verified SAFE by an independent adversarial pass + an
independent re-run of the column sums.)

**Two honest caveats — this is a demonstration that the obvious reconstruction fails, NOT a clean falsification
of the geometry:**
1. The `log_mu` finite-part is a **self-admitted placeholder** (`finite_part.py`: *"runnable finite-part
   placeholder … should replace or freeze before comparison"*) — **not** the actual ζ-regularized KK finite part.
   The wild magnitudes (especially δ₃) are partly placeholder-scheme artifacts of the missing subtraction.
2. The frozen `spectrum_K6.csv` was not co-located in the run package, so its tower was internal-consistency-
   checked but not byte-traced to source (a flagged UNKNOWN; it does not change the disposition — even granting
   the towers, the triple fails on sign).

**Net:** **SG-7 stays OPEN / Diagnostic** — the gate does **not** move. What changed is the *evidence grade*: the
"fitted-to-target" status is now **demonstrated by a target-blind run**, not just conceded. A genuine
DERIVED-GIVEN-E attempt still requires the **actual** (non-placeholder) ζ-regularized KK finite-part co-packaged
with **byte-anchored** spectra, then a fresh blind run. **No status was ever upgraded.** *(Artifacts published: `real_frozen_modes.csv`,
`outputs/real_run.csv`, the `build_real_modes.py` generator, and `FINDING.md`, inside the runnable package.)*

*The runnable package is published alongside this result so the verification is reproducible (a captured log is
not a derivation). Serious candidate, NOT validated.*