SG-1 — Geometry specification: the gate anchor ledger

The honest one-line: SG-1 has a real, checkable commitment win — the framework's entire starting object is written down in public to the last byte and a machine rebuilds every hash byte-for-byte — and on the ratified board the gate closes as DERIVED-GIVEN-SHAPE · RESOLVED +0; the freeze leg itself is DECLARED-FROZEN, not a uniqueness theorem: the freeze pins the object, it does not by itself prove the geometry is forced, and the forcedness residual family stays shown.

This page is the gate-specific instantiation of the anchoring method: it takes the master anchor and the bridges and applies them, object by object, to the gate that commits the starting universe. Every exact thing SG-1 touches gets its own row — its status, what it is allowed to claim, and what it is forbidden to claim. SG-1 differs from SG-4: it has no anomaly ledger. Its certified core is a freeze-and-reproduce certificate, plus a graded carrier-forcing result, plus an MDL scoring scaffold whose decisive metric axiom is still open.

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

1. Gate status header

• Gate-level SG-1 roll-up: DERIVED-GIVEN-SHAPE · RESOLVED +0 (board-canonical, ratified 2026-07-08; closure-of-record: the live gate dossier); freeze/self-witness leg DECLARED-FROZEN (forcedness residual family shown — the mid-audit roll-up read DECLARED-FROZEN, open-on-forcedness).
• Taxonomy reconciliation (2026-07-05): the gate-level grading is DERIVED-GIVEN-SHAPE · RESOLVED +0 (ratified 2026-07-08), read as TERMINAL + RESIDUALS-SHOWN — the reproducibility/self-witness leg is banked as a reached terminal, and the forcedness residual family below (R1–R10, R5 AXIOM-OPEN foremost) remains listed and carried unchanged. The facts are unchanged: the earlier DECLARED-FROZEN roll-up reflects the superseded least-closed-residual rule, not different facts, and no individual residual is closed, deleted, or re-graded here. The metric-selection +1 (AXIOM-COMMON-CURRENCY / AXIOM-MDL-BRIDGE) remains the interim closure named in §11, not a step already taken.
• The closed local/partial leg — reproducibility, not physics. The one genuinely derived leg is the content-addressed self-witness: the regenerator rebuilds every hash byte-for-byte. Written as the freeze obstruction over the manifest, $$O_{\rm SG1,repro}(\mathfrak{B}_{\rm active}) = 0 \quad\Longleftrightarrow\quad \mathrm{SHA\text{-}256}(\text{regenerated manifest}) = \texttt{a5b1e6f9d951},$$ verified target-blind across $33$ rows with zero mismatches, deterministic across reruns.
• Status of the closed leg: DERIVED (self-witness only) — it commits the object a reviewer attacks; it lowers no assumption floor and closes no physics.

The reproducibility leg is a genuine result: the frozen branch cannot be quietly retuned after the data is seen. What stays open is everything about forcedness — that the $13$ dimensions are forced, that the geometry is unique, that the MDL metric under which $13$D wins is the right metric. The committed object is the three-layer active branch $$\mathfrak{B}_{\rm active} = \big[\,\mathcal{M}_4 \times K_6 \times S^2 \times S^1_Y/\mathbb{Z}_2\,\big]_\times \;\oplus\; \big[\,F^+_{\rm finite}\oplus C_{\rm admiss}\,\big]_\oplus \;\otimes\; \big[\,E_{\rm matter}\oplus E_{\rm gauge}\oplus E_{\rm Higgs}\oplus E_{\rm proton}\,\big]_\otimes ,$$ with total metric dimension $D = 4+6+2+1 = 13$ carried by the $\times$-layer alone.

2. Frozen inputs (what SG-1 stands on, not what it produces)

• Frozen branch hashes — branch content hash dcc66f1b2685; manifest meta-hash a5b1e6f9d951 (Appendix A0, $33$ rows); orbifold-freeze sub-hash ac4d2df3e708. 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.
• The frozen UV/boundary package the branch carries for downstream gates (recorded here for traceability, READ-ONLY, not derived in SG-1): unification scale $M_U \sim 10^{16}$ GeV; compactification radius $R_0 = 1.592\times10^{-17}\ \mathrm{GeV}^{-1}$; threshold triple $\delta = (+4.8424,\,-3.1112,\,-1.7313)\pm 1.6\times10^{-3}$; family index $\chi(K_6,E) = -3$.
• Upstream spectrum $E$ (the SM chiral content + gauge group + quantum numbers) is given / charged / inherited — it enters the target ledger $T$ on both sides and cancels. SG-1 does not derive $E$. Every minimality statement below is a statement given this $E$, never a derivation of it.

3. Object anchors (the structures the gate commits, given-E / upstream)

The $\times$-layer carries the metric geometry; the $\oplus$ and $\otimes$ layers carry zero metric dimension:

$$ \mathcal{M}_4 \;(\text{observed } 3{+}1),\quad K_6 = SU(3)/T^2 \;(\text{color, }6\text{D}),\quad S^2 \;(\text{weak}),\quad S^1_Y/\mathbb{Z}_2 \;(\text{hyper, orbifold chirality filter}). $$

with isometries sourcing the gauge factors under coset-space dimensional reduction (CSDR): $K_6$'s isometry sources $SU(3)$ (isotropy $T^2$), $S^2$'s sources $SU(2)$, $S^1_Y$'s sources $U(1)_Y$. Status: GIVEN-E / declared-and-frozen — committed, not SG-1-derived.

4. Root and master-anchor traceability

Deep roots that are load-bearing for SG-1:

Causal order and nonseparability are not primary load-bearing anchors for the SG-1 commitment.

Master anchors in play: finite invariant ledgers (the MDL target $T$) · no unpaid labels (anti-fitting ⇄ MDL bridge: a tuned value is charged $\sim b$ bits) · the frozen branch · given-$E$ · the declared role-mechanism grammar $\mathcal{G}$ · open-residual discipline.

5. The SG-1 anchor ledger (the universal table)

6. The construction — the commitment win, in full

The dimension ledger. $D = 4 + 6 + 2 + 1 = 13$, carried entirely by the $\times$-layer. The $\oplus$ (rulebook) and $\otimes$ (actors) layers carry zero metric dimension — they are constraint and bundle/operator content, not geometry. The forcedness of the $13$th-rung total is the entire SG-1 attack surface: $D=13$ is selected inside the declared category, not derived.

The freeze, content-addressed. Branch content hash dcc66f1b2685; manifest meta-hash a5b1e6f9d951 over $33$ rows in file order; orbifold sub-hash ac4d2df3e708. The R0 reproducer regenerates each hash with no manual steps. Verified target-blind: the regenerator ran exit $0$; an adversarial recheck recomputed the SHA-256 of every canonical description without trusting the script's own comparator → $33$ rows, zero mismatches; the active-branch canonical string hashes to dcc66f1b2685; the meta-hash over all $33$ rows is a5b1e6f9d951; two fresh reruns produce byte-equal artifacts.

The MDL scoring scaffold — turning "shortest" into a decidable statement. In a finite-record universe, admissible descriptions are finite bit-strings and "simplest" = "shortest such string." The target ledger $T$ is architecture-neutral and $E$ cancels:

The codebook (Cert 2, frozen before scoring) charges an independent measured real $b = \log_2(1/\Delta_0)$ bits (large), a coset / bundle / quotient / named generator-rule $O(1)$, a topological integer $O(\log)$, and a fitted table $\approx$ entries $\times\, b$ — the anti-fitting ⇄ MDL bridge that makes a considered-class win non-circular. Writing $I(B_{13}) = S_{13} + 4b$ and $I(B_{D,j}) \ge S_{D,j} + n_{D,j}b$, the full classification theorem is the boxed inequality $$ \boxed{\ \forall\, D = 4..12,\ \forall j:\quad (n_{D,j} - 4)\, b \;+\; (S_{D,j} - S_{13}) \;>\; 0\ } $$ with $S_{13}$ the branch's fully-charged structure+generator cost.

Diagnostic — the economy is specific, but modest, not the retired headline. The honest charged cost is not "$4b$": the branch injects $\approx 9$–$10$ reals beyond the four anchors $\{M_{\rm Pl},\,\alpha_i(M_Z),\,y_t,\,|V_{us}|\}$ (the species normalizations $N_d,N_e,N_\nu$; the $\delta$ triple; the Higgs-protection angle $\theta_H^\star$), so $$ n_{13} \approx 13\text{–}14 \neq 4 . $$ That the survey still gives $I(B_{13}) \approx 13\text{–}14\,b$ versus $I(B_{\rm EFT}) \approx 25\,b$ — a real, quantified gap on the honest count — is what makes the win a genuine arithmetic fact under MDL rather than an artifact of the retired "$4$-in" overclaim. The first-pass survey scored $11$ considered competitors: $0$ REFUTED, $1$ FAILS_TO_GENERATE_T (the 6D rung), $10$ LOSE to $13$D under MDL.

7. The carrier-forcing structure, split into three honest objects

The single phrase "the internal geometry is forced" hides three different claims with three different statuses:

1. Weak carrier $S^2$ (Fact F1). No abelian / torus carrier of any dimension has non-abelian $SU(2)$ among its isometries — the isometry of a flat $T^k$ is $T^k \rtimes (\text{finite})$, abelian connected component. So the cheaper abelian direction fails for the whole shelf, and $S^2$ (the lowest carrier with $SU(2)$ isometry) is the forced choice. Status: DERIVED-WITHIN-GRAMMAR (hand-checkable general theorem).
2. Hyper carrier $S^1_Y/\mathbb{Z}_2$ (Fact F2). A closed odd-dimensional factor keeps both handednesses, producing mirror fermions excluded by the measured LEP invisible-$Z$ width; the $\mathbb{Z}_2$ orbifold projection removes them. The cheaper bare-circle direction is closed for all closed odd-dim carriers. Status: DERIVED-WITHIN-GRAMMAR (hand-checkable).
3. Color carrier $K_6 = SU(3)/T^2$ (abelian-isotropy uniqueness, W9). Among $SU(3)$ cosets $SU(3)/R$, the maximal torus $T^2$ is the unique purely-abelian isotropy: $C_{SU(3)}(T^2) = T^2$ (Cartan only), injecting no spurious non-abelian gauge factor. $CP^2$'s isotropy $U(2)$ is non-abelian and gauge-active ($C_{SU(3)}(U(2)) = U(1)$ by Schur), forcing a lose-lose fork — keep $S^2,S^1$ and over-produce $SU(2)+U(1)$ (Gate-2 equality fails), or drop them and lock $SU(2)_L/U(1)_Y$ inside $SU(3)$ (binding A1.4 violated). The cheaper rival was built end-to-end and breaks at Gate 2. Status: DERIVED on the named shelf $\{K_6,CP^2\}$; full-shelf completeness (N.4) OPEN.

So weak and hyper are forced within the grammar by general theorems; color is clean by an architecture-neutral theorem on a named shelf, with completeness the one exposed obligation. The historical "$CP^2$ family count is tunable" exclusion was asymmetric and unsound — $CP^2$'s three families are a discrete $\mathrm{Spin}_c$ index $r(r+1)/2 = 3$ at $r=2$, not a continuous dial, and $K_6$ needs its own Borel–Weil–Bott weight $(1,0)$; it was retired and replaced by W9.

8. Declared-structure splits

Two further single phrases each hide multiple claims with distinct statuses:

• "The three-layer algebra is necessary." Certificate B2 proves $\mathcal{N}_L = \varnothing$ for every proper subset $L \subsetneq \{\times,\oplus,\otimes\}$ — inside the declared category (B2.0.3.4). This is CERTIFICATE-CONDITIONAL, explicitly not a universal no-go. Upgrading it to architecture-neutral role-necessity ($k_{\rm role}\ge3$) is residual R7. The C1–C10 term ledger is separate: each named term is shown load-bearing for $\ge1$ gate, AUDIT (conditional on the declared term-construction).
• "The rulebook is charged." $C_{\rm admiss}$ contains both enforced physics (anomaly admissibility, flavor closure, chamber constraints) and method/governance (freeze-before-compare, gate-status discipline). Only the physics may be charged in the minimality burden; charging governance is F6 circularity (the constraint set presupposing the submitted architecture). The physics/governance split is OPEN (R10).

9. Open residuals — the forcedness family

The freeze and reproducibility are closed; forcedness is the open surface. These distinct residuals make up it, grouped under the family forcedness of the committed geometry, none closed by the freeze:

• R5 — the MDL-vs-dimension-first metric (the decisive seam). AXIOM-OPEN. Under MDL, $13$D wins; under a dimension-first lexicographic order ($k_{\rm dim}$ leading), a clean 4D chiral-gauge EFT wins ($4<13$) irrespective of injected reals, and SHAPE-minimality folds for everyone. The analysis reduces this to one named axiom, AXIOM-COMMON-CURRENCY: granularity fixes the domain (finite bit-strings) and the per-anchor cost $b=\log_2(1/\Delta_0)$ but is silent on the aggregation rule — both MDL (additive) and lex (infinite weight on $k_{\rm dim}$) are well-defined on the same space. The single highest-leverage object in SG-1.
• R2 — the dimension-ladder lower-bound matrix / grammar exhaustion. CATEGORY-RELATIVE (survey). The $10$-LOSE / $1$-FAILS / $0$-REFUTED first pass is a survey, not a certified classification; Certificate 3 (normal-form / cost-non-increasing reduction) and Certificate 4 (per-class lower bounds) are OPEN.
• R4 — search-category completeness. OPEN. Two sub-residuals: (a) is R2.5 (forces = isometries, compact, classified structures, admissible bundles) a fair category; (b) is the $SU(3)$-carrier shelf $\{K_6,CP^2\}$ complete (N.4) below 6D. The partial classification shows $\dim H \le 4 \Rightarrow \dim M \ge 4$ with $S^5$/Wu killed by odd-dimensionality.
• R8 — actor-layer minimality. OPEN (demoted from AXIOM_CLOSED). Lemma 3's success criterion is a universal negative ("no admissible competitor supplies lower $k_{\rm actor}$"), UNMET; NCG's finite Dirac operator is un-scored as a role-equivalent. Partial support is genuine (the $\mathbb{Z}_6$ center-kernel is computed, not posited).
• R7 / R10 — the category-relative bridges. OPEN. B2 / C1–C10 must be upgraded to an architecture-neutral functional-role floor (R7); the rulebook physics/governance split must be made formal (R10). Both must defeat F6 circularity per role.
• R1 — absolute irreducibility. OPEN / uncomputable. "No competitor anywhere under any math is shorter" equals $K(T)$, uncomputable for everyone — a wrong target, not a theorem. The closure is a reframe to $\mathcal{G}$-relative forcing.
• R3 / R9 — honesty + labeling residuals. R3 (the ~9–10 injected reals): DISCLOSED-CORRECTED — state ~13–14 effective inputs wherever "$4\to22$" appears; both wrong figures (the ~18-economy overclaim and the ~1.6× over-correction) retired. R9 ($\mathcal{M}_4$ as declared axiom): DISCLOSED-CONSISTENT — keep it an observational primitive, never a forced rung.

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

• SG-1 does not derive the geometry, and does not derive $E$. selection ≠ derivation; declared + frozen ≠ derived.
• $D=13$ is not forced. It is selector-minimal inside the declared category; the absolute version equals $K(T)$ and is uncomputable. category-relative ≠ absolute.
• The MDL win is not proven over dimension-first. Reject AXIOM-COMMON-CURRENCY and the metric fork reopens and SHAPE folds — for everyone (R5).
• The ladder verdict is a survey, not a classification. $10$ LOSES / $1$ FAILS / $0$ REFUTED does not mean "no competitor below $13$D is shorter" (Certificate 3 OPEN).
• N.4 is not certified. $K_6$ is the unique clean $SU(3)$ carrier on the named shelf $\{K_6,CP^2\}$, not proven over all sub-6D carriers.
• The economy is ~13–14 measured reals, not 4. The "4 inputs → 22 outputs" headline is retired.
• The frozen-branch hashes are audit anchors; they do not validate the physics, and "the reproducer runs green" is not output validation — its physics CSVs are reverse-pinned to known targets and sit outside the hash-verified self-witness.
• Reproducibility is not forcedness. $O_{\rm SG1,repro}=0$ commits the object; it closes no physics and proves no uniqueness.
• A REFUTED-ECONOMY outcome (a fairly-charged simpler 4D realization wins) remains a live, equally valuable result — not a defeat to be argued away.

11. Specialist closure plan

Each open residual is a concrete, finite, target-blind work-package, ordered by leverage. Guards G1 (the metric must be writable without knowing $13$D should win — the κ³/π falsification test), G2 (the geometry→observables generator map must be fully charged), and G5 (symmetric ledgers — never charge the EFT for $E$-automatic facts) bind throughout.

1. R5 / metric-selection — derive sub-claims (A) granularity ⇒ MDL, (B) anchors dominate dimension under $\mathcal{I}$, (C) symmetric economy ledgers with the generator charged. Success (DERIVED-CLOSED): SHAPE upgrades from "selected" to "realization-minimal under the granularity-induced metric, given $E$." Refuting result (REFUTED-ECONOMY): the 4D EFT injects no more than $13$D → the simpler realization is correct. Interim (AXIOM-CLOSED): name AXIOM-COMMON-CURRENCY and stop.
2. R2 / lower-bound matrix — prove O1 (reduction-without-cost-increase) + O2 (taxonomy completeness) and the boxed inequality per class, using the honest $n_{13}\approx13$–$14$. Success: grammar-relative LADDER_FORCED. Refuting result: a strictly shorter class is found. Conditional on R5.
3. R4 / N.4 shelf completeness — a bounded coset-classification + CSDR centralizer check; certify no sub-6D $SU(3)$-homogeneous carrier other than $K_6/CP^2$ has a clean abelian isotropy. Success: the color rung is forced within the grammar. Refuting result: a cheaper clean carrier is exhibited.
4. R8 / actor-minimality — score the actor competitor matrix target-blind (especially NCG's finite Dirac operator) and prove no lower-$k_{\rm actor}$ competitor given $E$. Narrowest layer — attack first if realization-minimality is suspect. Note $E$ stays un-forced regardless.
5. R7 / R10 / functional-role floor + rulebook split — prove $\mathcal{C}_{\rm phys} \Rightarrow \mathrm{Stage}+\mathrm{Rulebook}+\mathrm{Actors}$ ($k_{\rm role}\ge3$) by contradiction per role, and formally separate rulebook physics from governance. Both defeat F6 per role.
6. R1 / absolute irreducibility — reframe, not closure: declare $\mathcal{G}$ explicitly and state minimality as $\mathcal{G}$-relative + living/extensible (sharper-OPEN, essentially standing).
7. R3 / R6 / R9 — mechanical: retire the wrong headlines (R3, owner countersign); keep the reproducer fail-closed (R6, done and machine-verified); keep $\mathcal{M}_4$ labelled an observational primitive (R9).

A full campaign moves SG-1 from DECLARED-FROZEN with machine-verified reproducibility toward DECLARED-FROZEN with a named axiom floor + an architecture-neutral color-carrier theorem — a real honesty/forcedness gain, not a promotion, and never a derivation of $E$.

12. Completion tests for this page

Required presence (all met): gate roll-up DERIVED-GIVEN-SHAPE · RESOLVED +0 (freeze leg DECLARED-FROZEN) · the closed reproducibility leg $O_{\rm SG1,repro}=0$ · its DERIVED (self-witness) label · "$E$ not derived" · frozen hashes dcc66f1b2685 / a5b1e6f9d951 / ac4d2df3e708 (AUDIT ONLY) · every exact object as its own row · the dimension ledger $D=4+6+2+1=13$ · the carrier split F1 / F2 / W9 · the B2 in-category split and the rulebook physics-vs-governance split · the MDL target $T\approx25$ + codebook + boxed inequality · the specificity diagnostic $n_{13}\approx13$–$14\neq4$ with $\approx13$–$14\,b$ vs $\approx25\,b$ · every open residual (R1–R10) as its own row · the anti-claims.

Required absence (all held): no claim that SG-1 is closed / derives the geometry / derives $E$ · $D=13$ forced absolutely · MDL proven over dimension-first · the ladder survey called a classification · N.4 certified · the "$4\to22$" headline · hashes or the reproducer validate physics · reproducibility = forcedness · any reader-visible build-process vocabulary.

Completion report. Tests passed: all required-presence items present; all required-absence items held. Tests failed: none. Open items: R5, R2, R4, R8, R7, R10, R1 (forcedness family); R3/R9 mechanical. Assumptions made: none beyond the dossier — every number traces to DOSSIER_SG1_FULL (§3–§6, §5.4) or its cited corpus.

This gate anchor ledger follows the canonical eleven-part shape and universal table established by the SG-4 ledger; SG-1 differs in being a freeze-and-reproduce gate rather than an anomaly-ledger gate.

See also: the anchoring method · the master anchor · Layer 1 — the metric / recordability root (why finite records ⇒ MDL) · Layer 3 — search grammars (the finite role-mechanism grammar $\mathcal{G}$) · Layer 4 — carrier-forcing & the given-E wall (F1 / F2 / abelian-isotropy uniqueness) · the SHAPE-minimality challenge · the SG-4 ledger · the full SG-1 dossier.