Controlling terminal and claim firewall

Outcome

The SG-2 conditional candidate certificate passes. The frozen manifest generates twelve massless vectors of rank four, one primitive parent for each carrier, no missing or additional vector, and the matter-faithful center kernel \(\mathbb Z_6\).

The physical SG-2 gate remains OPEN. The owner-supplied SG1 V3 block is now a frozen upstream dependency, and it lawfully reports SG1 as OPEN with 18 open rows and no selected formulation. Nine of those rows are direct SG-2 physical-readiness dependencies.

The terminals are deliberately separate. A correct calculation on declared inputs does not prove those inputs have been physically constructed. A pass on the internal reconstruction is not a pass on the absent reviewer package. Exact agreement with the Standard Model interface is construction-anchor evidence, not a proof that nature uniquely selected the candidate.

Scope reconciliation

The archived public dossier certified only the zero-mode Lie algebra and assigned the global quotient to the charge/anomaly gate. The reviewer SG-2 challenge asks for \((12,4,\mathbb Z_6,0)\) in one generated artifact. This revision resolves the difference without rewriting history:

• SG-2 owns vector origins, connected Killing algebras, parities, ownership, dimension, rank, and the complete no-extra-vector audit.
• The exact matter characters and faithful center kernel are imported from the charge/anomaly building block as a content-addressed dependency.
• The imported \(\mathbb Z_6\) calculation is reproduced here so the challenge artifact is self-contained; SG-2 does not claim to derive the charge ray.

Authority order

1. the supplied source-of-truth archive BB-SOT-2026-07-18-V1;
2. the owner-supplied BB-SG1-CLOSURE-3@3.0.0-rc1 package as the controlling upstream execution dependency for this regeneration;
3. the supplied V4.1 constraint-driven gate protocol;
4. explicit SG-2 corrections and the frozen candidate manifest;
5. the reviewer SG-2 gauntlet specification;
6. the archived public SG-2 dossier for detailed technical provenance.

SG1 V3 remains a ratification candidate globally; this package records the owner’s supply of it as the upstream input without claiming broader ratification. Where sources allocate ownership differently, the narrower claim and the explicit dependency edge control. No older wording can expand the present terminal.

SG1 V3 to SG2 cumulative dependency reduction

Input validation

The supplied SG1 V3 archive passes its SHA-256 inventory and deterministic validator. The validator reports rows=18, open=18, work_packages=8, mutations=10/10, and gate=OPEN. This is a successful package validation, not physical closure.

Direct physical-readiness join

Reducer result

conditional_candidate_certificate.status = PASS
conditional_candidate_certificate.provenance_class = construction-anchor
upstream_sg1_gate.status = OPEN
direct_upstream_open_count = 9
physical_sg2_gate.status = OPEN

The CSDR input and descended-spectrum rows are conditional dependencies when CSDR is the active vector mechanism. The present certificate instead uses metric KK vectors plus one declared principal U(1) connection. The equal-freeze rival shelf is additionally required for architecture-neutral selection, which is not claimed here.

Exact generated vector-sector certificate

Connected factors

The connected geometric contribution is therefore \(8+3+0=11\) dimensions and rank \(2+1+0=3\). The interval is evaluated as an interval; the circle generator removed by the quotient is not counted.

Independent hypercharge connection

The independent principal connection supplies exactly one even four-vector. The odd internal component has no constant scalar zero mode. Adding this actor to the geometric sector gives

\[ \dim \mathfrak g_{4,0} = 11+1=12,\qquad \operatorname{rank}\mathfrak g_{4,0}=3+1=4. \]

Ownership join

No independent parent SU(3) or SU(2) connection is present. The boundary and form-vector ledgers are empty. The interval connected Killing dimension is zero. Consequently the complete declared sector has zero missing vectors and zero extra vectors.

Matter-faithful center enumeration

Use a center tuple \((a\bmod 3,b\bmod 2,k\bmod 6)\). For a character with color triality \(t\), weak-doublet parity \(d\), and integer \(y_6=6Y\), the center phase is trivial precisely when

\[ 2ta+3db+y_6k\equiv0\pmod 6. \]

The calculation evaluates all \(3\times2\times6=36\) tuples against the published characters \(Q,u^c,d^c,L,e^c,\nu^c,H\). Exactly six tuples survive:

The tuple \((1,1,1)\) generates all six elements, so the kernel is cyclic.

Smith-normal-form certificate

Let \(D=\operatorname{diag}(3,2,6)\) encode the center-domain relations. A basis for the lifted kernel lattice and the relation matrix are

\[ B=\begin{pmatrix}1&3&0\\1&0&2\\1&0&0\end{pmatrix},\qquad M=\begin{pmatrix}0&0&6\\1&0&-2\\0&1&-3\end{pmatrix}. \]

Exact integer multiplication gives \(BM=D\). The Smith invariants of \(M\) are \((1,1,6)\), hence the lifted kernel modulo the domain-relation lattice is \(\mathbb Z_6\). The direct enumeration and SNF certificates are independent representations of the same result.

Certificate tuple

dimension = 12
rank = 4
faithful matter kernel = Z6
extra vectors = 0
missing vectors = 0
primitive parents per carrier = 1

Reconstructed adversarial gauntlet method

Epistemic status

The supplied challenge document specifies the SG-2 test design but contains no randomized reviewer manifests, no reviewer answer-key commitment, and no opened reviewer key. The executable package in this volume is therefore an internal deterministic reconstruction. It is useful evidence that the new blocks discriminate the named historical failures, but it cannot be promoted to a reviewer-issued pass.

Blind-session protocol

Eight complete manifests are generated from a content-addressed seed and assigned opaque session identifiers. Before the key is opened, each manifest is solved in a fixed first-hard-failure order and the verdict plus witness hash is placed in escrow. The answer key is then opened and compared mechanically. The package contains one incumbent, five planted historical defects, and two honest non-SM candidates. The role label is absent from each session manifest.

Ordered rules

Why the two innocents return SG2-GNT-07

The challenge wording says an honest non-SM candidate must pass the consistency rows and fail only the SM-match row for a computed reason. In the machine schema a complete session verdict is FAIL with first failure SG2-GNT-07; its witness simultaneously records that rows 01 through 06 are consistent. This is not a false positive. It is the explicit separation of internal consistency from identity to the scoped Standard Model target.

Gauntlet result and independent package validation

Opened-key comparison

All eight escrowed results match the opened key. All five decoys are caught at the intended first hard failure. The two honest non-SM controls reach only the SM-match row. The incumbent discharges all seven rows.

Validation controls

The validator recompiles both scripts, runs the builder and verifier twice to test determinism, parses every JSON artifact, checks the key commitment, requires eight of eight matches, confirms all five planted defects, confirms both innocent controls, recomputes the terminal tuple, verifies the exact Smith certificate, audits one-parent ownership, checks session completeness, enforces the status firewall, and verifies the frozen source hash.

Lawful terminal

SG1 V3 UPSTREAM GATE: OPEN (8 PASS / 18 OPEN)
SG-2 CONDITIONAL CANDIDATE CERTIFICATE: PASS
INTERNAL RECONSTRUCTED GAUNTLET: PASS
REVIEWER-RANDOMIZED GAUNTLET: NOT-EVALUATED
SG-2 PHYSICAL GATE: OPEN (9 DIRECT UPSTREAM DEPENDENCIES OPEN)
NATURE-SELECTION: NOT-CLAIMED

Building-block improvement and full-closure analysis

What was missing

The supplied source-of-truth blocks did not provide one candidate-neutral object that joined vector origin, primitive ownership, parity/domain, kernel, and no-extra completeness. The public dossier also used metric Kaluza-Klein and coset-space reduction terminology too closely, and it deferred the global kernel while the reviewer challenge demanded it inside SG-2.

Changes made

1. BB-GVO-1 added. It supplies a common origin taxonomy, an exact one-parent join, a finite no-extra-vector ledger, required artifacts, first-hard-failure order, mutation tests, and reopen triggers.
2. BB-KKC-1 amended. It separates metric KK vectors, CSDR descendants, and independent principal connections. A symmetry may support an action without owning a low-energy carrier.
3. BB-GQO-1 amended. It requires the matter-character table, complete 36-element center enumeration, published integer matrices, exact SNF, and an explicit cross-gate dependency when another gate owns the charge data.
4. Scope reconciliation added. The result can meet the reviewer tuple while preserving the source-of-truth ownership boundary.
5. BB-GDR-1 added. It consumes SG1 V3 as the upstream execution dependency, separates the conditional tuple from the physical terminal, and reduces the physical SG2 gate to OPEN while required upstream rows remain open.

What these changes buy

The revised blocks now catch every planted SG-2 historical defect and accept both honest non-SM controls. They also convert the public dossier’s pseudocode promise into executable artifacts with hashes and replay commands.

Remaining limit

Building-block ambiguity is now closed for the cumulative dependency path, but physical closure is not. The nine joined SG1 rows require actual formulation, object, move, quotient, action, domain, Actor, mode, and gauge quotient witnesses. No further prose can manufacture those constructions, the absent reviewer manifests and answer key, or an architecture-neutral selection theorem.

Audited public dossier v2.0

The following body is the complete archived public SG-2 dossier downloaded on 2026-08-01 and normalized to portable Markdown. It remains authoritative for its detailed Lie-algebra and zero-mode analysis. Where it says the global quotient lies outside SG-2, Revision 4.0’s controlling terminal above adds the reviewer-requested cross-gate import; the archived statement is preserved as provenance, not silently rewritten.

SG-2 - Complete Massless Gauge-Algebra Realization Dossier

Document-control statement

This document is the canonical technical dossier for Gate SG-2 once adopted into the project corpus. It is written to survive adversarial technical review by a capable physics AI or human reviewer.

It does not obtain completion by repeating the older slogan that all three Standard Model gauge factors are isometries of the metric Stage. That slogan is false for the quotient interval. Instead, this dossier closes the actual gate through the corrected, non-duplicating hybrid architecture:

• the color algebra is generated by geometric Kaluza-Klein/Ehresmann connections associated with the connected isometries of \(K_6=SU(3)/T^2\);
• the weak algebra is generated by geometric Kaluza-Klein/Ehresmann connections associated with the connected isometries of round \(S^2\);
• hypercharge is generated by one independent principal \(U(1)\_Y\) connection whose four-dimensional vector component is even on the chirality interval;
• the quotient interval contributes no connected rotation isometry and is not called the hypercharge circle;
• no duplicate independent \(SU(3)\) or \(SU(2)\) gauge Actor is present in the locked parent action;
• the result is exactly the Standard Model Lie algebra in the four-dimensional massless vector sector, not a claim about the final global quotient, charge assignments, anomalies, coupling unification, or an exact nonlinear truncation.

The governing board status is:

SG-2 COMPLETION STATUS: COMPLETED
BOARD STATUS: CLOSED / RESOLVED +0
PHYSICAL ENDPOINT:
  CLOSED-SCOPED / ZERO-MODE-ALGEBRA-CERTIFIED /
  DERIVED-GIVEN-LOCKED-SHAPE-AND-DYNAMICS /
  CONSTRUCTION-ANCHOR

This gate is not open.

Reviewer first read

The exact claim

Given the locked thirteen-dimensional candidate and its declared parent Dynamics, the four-dimensional massless vector sector contains exactly

\[ \boxed{ \mathfrak g\_{4,0} = \mathfrak{su}(3)\_c \oplus \mathfrak{su}(2)\_L \oplus \mathfrak u(1)\_Y } \]

with

\[ \dim \mathfrak g\_{4,0}=8+3+1=12, \qquad \operatorname{rank}\mathfrak g\_{4,0}=2+1+1=4, \]

and with no additional massless vector in the declared SG-2 Actor sector.

What makes this nontrivial

The result is not obtained from a bare group-name match. The dossier must establish all of the following simultaneously:

1. the internal connected Killing algebra of the metric factors is correctly identified;
2. the quotient interval contributes no connected \(U(1)\) isometry;
3. the metric ansatz actually converts the relevant Killing directions into four-dimensional gauge connections;
4. the independent hypercharge bundle supplies one and only one even vector zero mode;
5. the parent action does not duplicate the non-Abelian vector sectors;
6. no additional parent Actor, harmonic, boundary field, form field, or interval graviphoton supplies an unaccounted massless vector;
7. the claim is made only at Lie-algebra and low-energy zero-mode scope;
8. the observed Standard Model algebra is classified as construction evidence rather than a blind prediction;
9. the stronger global-group, matter-representation, charge, anomaly, and coupling claims remain assigned to their own gates.

The fastest hostile-review route

A reviewer should try to break SG-2 in this order:

1. Interval test: show that the quotient interval has a connected rotation Killing vector. If this cannot be done, the old isometry-only hypercharge claim stays retired.
2. Action test: find an independent \(SU(3)\) or \(SU(2)\) Yang-Mills connection in the locked parent action. If one exists, the no-duplicate-vector certificate fails.
3. Mode test: solve the parity problem for \(B\_\mu\) and \(B\_\chi\). If more or fewer than one Abelian vector zero mode survives, the count fails.
4. Extra-vector test: identify any additional massless vector from non-Killing metric harmonics, boundary-localized fields, p-forms, or hidden Actors.
5. Scope test: show that the dossier silently promotes a linear zero-mode result into an exact nonlinear consistent truncation.
6. Global-group test: show that the dossier claims triplets, doublets, the \(\mathbb Z_6\) quotient, or anomaly cancellation as SG-2 results. Such promotion would be a scope error.

The dossier is written so that a successful attack at any one of these points triggers a precise reopen condition instead of being absorbed by terminology.

Part I - Authority reconciliation

1. Controlling authority order

For SG-2, conflicts are resolved in the following order:

1. the latest explicit owner-adopted correction for the physical architecture;
2. the locked Shape-Scale-Granularity-Dynamics source of truth;
3. this gate-specific complete dossier;
4. the owner-ratified board terminal that records SG-2 as completed;
5. the Two-Anchor-Plus-Law Gate Closure Constitution;
6. the Thought Experiments authority and Master Implicit-Assumptions Ledger;
7. older manuscript and gate text for historical provenance only.

The board status and the physical mechanism answer different questions:

• the board establishes that SG-2 is a completed project gate;
• the corrected four-root architecture establishes what physical claim is actually supported;
• this dossier unifies those layers without retaining obsolete physics.

2. Historical claim and correction

The legacy SG-2 argument used the statement

\[ \mathfrak{isom} \bigl(K_6\times S^2\times S^1_Y/\mathbb Z_2\bigr) = \mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak u(1). \]

That statement is not correct. The quotient of a circle by reflection is an interval. An interval has no connected rotation group. At most, when the endpoint data are symmetric, it admits a discrete reflection. A discrete symmetry has no infinitesimal generator and does not produce a four-dimensional massless gauge vector through the metric Kaluza-Klein mechanism.

The legacy dossier also blended two different frameworks:

• metric Kaluza-Klein reduction, where off-diagonal components of the metric along Killing directions become gauge connections;
• coset-space dimensional reduction centralizer rules, where a higher-dimensional gauge group and isotropy embedding determine a surviving gauge group.

Those mechanisms are not interchangeable. A centralizer calculation cannot be used as the reduction map for a metric-only gauge sector unless the higher-dimensional gauge Actor and the CSDR assumptions are actually present.

3. Current corrected claim

The current architecture separates the mechanisms:

The interval is renamed

\[ I\_\chi=S^1\_\chi/\mathbb Z_2 \]

because its geometric job is boundary/parity routing for chirality, not the creation of a connected hypercharge isometry.

4. Status resolution

The older website summary classified SG-2 mainly as a category-floor endpoint associated with the axiom “forces are isometries.” That is no longer the best physical description because the corrected branch does not require every gauge factor to be an isometry. Hypercharge is an explicit principal connection.

The canonical completed physical endpoint is therefore:

CLOSED-SCOPED / ZERO-MODE-ALGEBRA-CERTIFIED /
DERIVED-GIVEN-LOCKED-SHAPE-AND-DYNAMICS /
CONSTRUCTION-ANCHOR

This is stronger and more precise than retaining the category-floor wording. It closes the actual finite gate without claiming architecture-neutral uniqueness.

Part II - Gate charter

5. Exact obligation

SG-2 asks:

Does the complete locked thirteen-dimensional candidate, including its declared gauge Actors and reduction map, realize exactly the Standard Model gauge Lie algebra in the four-dimensional massless vector sector, with no missing factor and no additional massless gauge vector?

The gate is passed only if the word exactly is justified. Showing that the Standard Model algebra is contained in a larger algebra is not enough. Showing only that the internal Stage has similarly named isometries is not enough. Declaring gauge bundles without a zero-mode analysis is not enough.

6. Inputs

6.1 Constitutional inputs

• locked Stage \(X\_{13}\);
• locked Rulebook and boundary/parity laws;
• locked Actor inventory;
• locked parent-action skeleton;
• locked reduction-map scope;
• evidence partition and freeze rules.

6.2 Construction evidence

The observed low-energy gauge interface

\[ \mathfrak{su}(3)\_c\oplus\mathfrak{su}(2)\_L\oplus\mathfrak u(1)\_Y \]

was part of the construction target. It is not held out as a blind prediction.

6.3 Inputs excluded from SG-2

SG-2 does not use:

• numerical low-energy coupling values;
• a unification scale;
• threshold corrections;
• fermion hypercharge assignments;
• the \(\mathbb Z_6\) global quotient;
• anomaly cancellation;
• the Higgs vacuum expectation value;
• flavor data;
• proton lifetime data.

Those objects are downstream.

7. Outputs

SG-2 outputs:

1. a precise massless-vector Lie algebra;
2. a generator count and rank;
3. a carrier/Actor origin for each factor;
4. a no-extra-vector audit;
5. a scoped low-energy reduction statement;
6. a set of nonclaims and reopen triggers.

8. Pass condition

The gate passes if

\[ \ker \mathcal M^2\_{\rm vector} \cong \mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak u(1) \]

within the declared Actor sector, where \(\mathcal M^2\_{\rm vector}\) denotes the quadratic mass operator after applying the boundary conditions and quotienting gauge redundancy.

The kernel must have dimension twelve and no additional independent vector generator.

9. Fail conditions

The gate fails or reopens if:

• a required factor is absent;
• an extra massless vector exists;
• hypercharge is supported only by the nonexistent interval rotation;
• duplicate gauge Actors create extra non-Abelian vectors;
• the metric ansatz fails to realize the connection transformation law;
• the boundary conditions eliminate the hypercharge zero mode;
• the claimed zero modes are gauge artifacts or negative-norm modes;
• the result requires an unproved exact finite truncation;
• the observed target leaked into a supposedly blind comparison.

Part III - Complete object under test

10. Four-root object

The theory is the ordered quadruple

\[ \mathfrak T=(\mathfrak S,\mathfrak L,\mathfrak G,\mathfrak D). \]

For SG-2:

• Shape fixes the Stage, bundles, representations, boundary structure, and Actor types;
• Scale fixes which radii and normalizations influence couplings and KK masses without changing the zero-mode algebra;
• Granularity fixes finite distinguishability, certificate precision, and auditability;
• Dynamics supplies the action, gauge transformations, orbifold parity, constraints, and reduction map.

A Stage-only argument cannot close SG-2 because a manifold does not by itself specify which gauge connections exist or how they transform.

11. Metric Stage

\[ X\_{13} = \mathcal M\_{3,1} \times K_6 \times S^2 \times I\_\chi, \qquad K_6=SU(3)/T^2, \qquad I\_\chi=S^1\_\chi/\mathbb Z_2. \]

The dimension is

\[ D=4+6+2+1=13. \]

The internal Stage is

\[ Y_9=K_6\times S^2\times I\_\chi. \]

12. Scale family

The internal metric is

\[ g\_{\rm int} = R_6^2 g\_{K_6}(s_1,s_2) \oplus R_2^2 g\_{S^2} \oplus R\_\chi^2 d\chi^2. \]

The exact numerical radii are not SG-2 outputs. They determine the normalization of four-dimensional kinetic terms and the positive masses of higher KK modes. As long as the metric remains nondegenerate and preserves the declared connected isometries, the massless Lie algebra is unchanged.

13. Rulebook data

The load-bearing Rulebook contains:

• the product and fibration typing;
• the orbifold reflection and fixed sets;
• parity assignments for metric and gauge components;
• gauge-parameter parity;
• no-duplicate-Actor rule;
• evidence partition;
• freeze-before-compare;
• scope boundary between Lie algebra and global group;
• scope boundary between a low-energy EFT and exact consistent truncation.

14. Actor inventory

The SG-2 load-bearing Actors are:

1. the thirteen-dimensional metric \(G\_{MN}\);
2. the independent hypercharge connection \(B_M\) on \(P_Y\to X\_{13}\);
3. gauge parameters and diffeomorphism parameters compatible with the boundary conditions;
4. the declared matter/scalar sectors only insofar as they do not introduce additional vector gauge Actors.

There is no separate parent \(SU(3)\) or \(SU(2)\) Yang-Mills connection in the locked branch.

15. Parent-action skeleton

The minimum action relevant to SG-2 is

\[ \begin{aligned} S\_{13}={}&\int\_{X\_{13}}d^{13}x\sqrt{\|G\|} \left\[ \frac{M\_\*^{11}}{2}(R\[G\]-2_{13}) - F^Y_{MN}F_Y{MN} ] &+S_{}+S_{}+S_{}+S_{}. \end{aligned} ]

For SG-2, the relevant distinction is that the non-Abelian vectors are embedded in the metric sector, whereas hypercharge is a distinct principal connection.

Part IV - Geometry of the connected isometries

16. Why the connected component matters

A four-dimensional gauge boson is associated with an infinitesimal local symmetry and therefore with a Lie-algebra generator. Discrete symmetries may constrain states or identify configurations, but they do not provide a continuously variable gauge parameter and do not add vector generators.

For this reason SG-2 computes the connected effective isometry group and its Lie algebra. Weyl actions, center identifications, and interval reflections are separately recorded but are not added to the vector count.

17. The flag manifold \(K_6=SU(3)/T^2\)

The full flag manifold has real dimension

\[ \dim SU(3)-\dim T^2=8-2=6. \]

The left action of \(SU(3)\) on the coset is transitive and isometric for every locked \(SU(3)\)-invariant metric. The center \(\mathbb Z_3\subset SU(3)\) lies in the maximal torus and acts trivially on the coset, so the effective connected group is naturally \(PSU(3)\), while its Lie algebra is

\[ \mathfrak{isom}\_0(K_6)=\mathfrak{su}(3). \]

The Lie-algebra distinction is all SG-2 requires. The global lift needed for matter triplets belongs to the later representation/global-structure gates.

17.1 Tangent decomposition

Using the \(A_2\) root system,

\[ \mathfrak{su}(3) = \mathfrak t^2 \oplus \mathfrak m\_{\alpha_1} \oplus \mathfrak m\_{\alpha_2} \oplus \mathfrak m\_{\alpha_1+\alpha_2}, \]

where each real root plane \(\mathfrak m\_\alpha\) has dimension two. The locked invariant metric assigns positive weights to these three planes, subject to the later scale/stability rules.

17.2 No extra torus generators from the denominator

The denominator \(T^2\) is not an additional global right-acting \(U(1)^2\) gauge group. Right multiplication by elements of \(T^2\) is the quotient redundancy used to define the coset. The normalizer quotient \(N(T^2)/T^2\) is the finite Weyl group \(S_3\). It can permute root planes but supplies no connected vector generator.

17.3 Full connected-isometry requirement

The machine certificate must verify that the locked metric does not acquire a continuous symmetry enhancement beyond the intended \(PSU(3)\) connected action. A discrete extension is harmless for the Lie-algebra count. A continuous enhancement would reopen SG-2 because it could provide additional metric vectors.

18. The round two-sphere

For the round metric,

\[ \operatorname{Isom}\_0(S^2)=SO(3), \qquad \mathfrak{isom}\_0(S^2)=\mathfrak{so}(3)\cong\mathfrak{su}(2). \]

There are three linearly independent Killing fields. At Lie-algebra level they generate the weak factor.

The distinction between \(SO(3)\) and its double cover \(SU(2)\) matters for doublet matter representations, but that is not settled by the metric isometry alone. The lift of the action to spinor and matter bundles belongs to later gates.

19. The chirality interval

Let the parent circle coordinate be \(\chi\sim\chi+2\pi R\_\chi\), with reflection \(\chi\mapsto-\chi\). The quotient is an interval of length

\[ L\_\chi=\pi R\_\chi. \]

Its connected isometry algebra is zero:

\[ \mathfrak{isom}\_0(I\_\chi)=0. \]

If the endpoint data are symmetric, the interval may admit a discrete midpoint reflection. That discrete transformation has no infinitesimal generator and contributes no gauge vector.

20. Product connected isometries

Because the locked factors are not mutually isometric and the product metric contains no continuous mixing between them,

\[ \mathfrak{isom}\_0(Y_9) = \mathfrak{su}(3)\oplus\mathfrak{su}(2). \]

The dimension is eleven. Hypercharge must therefore come from an Actor rather than from the connected metric isometry algebra.

21. Product-isometry negative control

A useful control is to replace two nonisometric factors by two identical copies. Identical factors can admit a discrete exchange and, in specially coupled metrics, potentially a larger mixing symmetry. The locked branch does not contain such a pair. This control demonstrates why the no-cross-factor statement is a property of the actual product, not a general theorem about all products.

Part V - Geometric Kaluza-Klein connection

22. Metric ansatz

Let \(K_A^m(y)\), \(A=1,\ldots,11\), be a basis of Killing fields on \(K_6\times S^2\). The metric is expanded as

\[ \begin{aligned} ds^2={}&g\_{\mu\nu}(x)dx^\mu dx^\nu +g\_{mn}(y) \left(dy^m+K_A^m(y)A\_\mu^A(x)dx^\mu\right) \left(dy^n+K_B^n(y)A\_\nu^B(x)dx^\nu\right)\ &+R\_\chi^2d\chi^2+\cdots. \end{aligned} \]

The omitted terms include massive harmonics and other fields not needed to define the zero-mode connection.

23. Gauge transformation from diffeomorphism

Consider an internal diffeomorphism generated by an \(x\)-dependent parameter,

\[ \xi^m(x,y)=\epsilon^A(x)K_A^m(y). \]

The Killing fields satisfy

\[ \[K_B,K_C\]^m=fA{}_{BC}K_A^m. ]

Preserving the form of the metric ansatz gives, up to a convention-dependent overall sign,

\[ \boxed{ \delta A\_\mu^A = \partial\_\mu\epsilon^A +f^A{}\_{BC}A\_\mu^B\epsilon^C. } \]

This is the transformation law of a non-Abelian connection.

The result is stronger than merely naming an isometry group: it identifies the four-dimensional field, the local parameter, and the nonlinear connection term.

24. Field strength

The corresponding curvature is

\[ F\_{\mu\nu}^A = 2\partial\_{\[\mu}A\_{\nu\]}^A +f^A{}_{BC}A_BA_^C. ]

Its algebra decomposes into the eight \(K_6\) directions and the three \(S^2\) directions. Because the product Killing fields commute across factors, there are no structure constants mixing color and weak indices.

25. Gauge kinetic matrix

Reduction of the Einstein-Hilbert term produces a four-dimensional gauge kinetic term of the form

\[ S\_{\rm geom,kin} = - \frac14 \int d^4x\sqrt{\|g_4\|} \ \mathcal K\_{AB}F\_{\mu\nu}^AF^{B\mu\nu}, \]

where

\[ \mathcal K\_{AB} = M\_\*^{11} \int\_{Y_9}d^9y\sqrt{g_9}\ g\_{mn}K_A^mK_B^n \]

up to fixed convention factors.

For a positive internal metric, \(\mathcal K\_{AB}\) is positive on nonzero Killing directions. Symmetry makes it block diagonal between the simple factors. A basis and generator normalization must be frozen before numerical couplings are extracted.

26. Masslessness at the linear level

The geometric vector zero modes are protected by the exact connected isometries of the background. A vector associated with a broken or non-Killing direction generally acquires a KK-scale mass. The SG-2 certificate therefore includes both:

• a zero-eigenvalue test for the eleven Killing directions;
• a positive-gap test for the next vector harmonics at finite nondegenerate radii.

The gate does not assert that every nonlinear excitation of only the massless vectors uplifts consistently. That stronger statement is the consistent-truncation problem.

27. Constraint and gauge-redundancy audit

The metric vectors are not counted merely because components \(G\_{\mu m}\) exist. The quadratic theory must be decomposed into:

• physical transverse vector modes;
• gauge/diffeomorphism directions;
• scalar Stückelberg partners;
• massive vector harmonics;
• boundary-forbidden modes.

The physical count is performed after quotienting the linearized diffeomorphism redundancy and applying the boundary conditions.

28. Positivity scope

SG-2 verifies that the geometric kinetic matrix is positive under the locked Riemannian internal metric and conventional Einstein-Hilbert sign. Full interacting unitarity and reflection positivity belong to the quantum gates. The present gate cannot be used as a substitute for them.

Part VI - Hypercharge principal connection

29. Why a separate connection is required

Since

\[ \mathfrak{isom}\_0(I\_\chi)=0, \]

there is no metric Killing generator on the interval that can provide a hypercharge vector. The corrected branch therefore declares a compact principal bundle

\[ P_Y\longrightarrow X\_{13} \]

with connection \(B_M\) and curvature

\[ F^Y\_{MN}=\partial_MB_N-\partial_NB_M. \]

This is a construction Actor. Its existence is explicit and therefore reviewable.

30. Orbifold parity

The minimal parity assignment is

\[ B\_\mu(x,-\chi)=+B\_\mu(x,\chi), \qquad B\_\chi(x,-\chi)=-B\_\chi(x,\chi), \]

with an even gauge parameter

\[ \lambda(x,-\chi)=+\lambda(x,\chi). \]

Then

\[ \delta B\_\mu=\partial\_\mu\lambda, \qquad \delta B\_\chi=\partial\_\chi\lambda \]

is compatible with the parities.

31. Mode expansion

On \(0\leq\chi\leq L\_\chi\), an even vector component has the expansion

\[ B\_\mu(x,\chi) = \frac{1}{\sqrt{L\_\chi}}B\_\mu^{(0)}(x) + \sqrt{\frac{2}{L\_\chi}} \sum\_{n\geq1}B\_\mu^{(n)}(x) \cos\frac{n\pi\chi}{L\_\chi}. \]

The odd scalar component has

\[ B\_\chi(x,\chi) = \sqrt{\frac{2}{L\_\chi}} \sum\_{n\geq1}B\_\chi^{(n)}(x) \sin\frac{n\pi\chi}{L\_\chi}. \]

Therefore:

• exactly one constant four-dimensional \(U(1)\_Y\) vector zero mode survives;
• no constant \(B\_\chi\) scalar zero mode survives in the minimal SG-2 sector;
• the excited modes have interval masses \(n\pi/L\_\chi=n/R\_\chi\) before other contributions.

32. Hypercharge kinetic matching

For a zero mode constant over the internal Stage,

\[ \frac{1}{g_1^2} = \frac{\operatorname{Vol}(Y_9)}{g\_{13,Y}^2} \]

up to the frozen normalization of the \(U(1)\) generator and any later localized kinetic terms or threshold corrections.

This is a matching relation, not an SG-2 numerical prediction.

33. Matter charges are not SG-2 outputs

The existence of a \(U(1)\) connection does not determine the integer characters carried by each matter Actor. Hypercharge assignments, electric charge, the faithful global quotient, and anomaly cancellation belong to later gates.

SG-2 therefore proves one Abelian vector generator, not the full Standard Model charge table.

34. Fixed-plane consistency

Orbifold gauge theories can carry localized anomalies or boundary operators even when the long-distance zero-mode algebra is correct. SG-2 records this as a dependency on the anomaly and boundary-consistency gates. It does not claim that the parity assignment alone proves quantum consistency.

35. Hypercharge thought control

Use the same interval geometry with two different bundle choices:

• no principal \(U(1)\) connection;
• one principal \(U(1)\) connection with even \(B\_\mu\).

The geometry and its connected isometry algebra are identical, while the massless Abelian vector content changes. Therefore gauge content is not determined by the interval isometry alone. The complete Shape-plus-Actors-plus-Dynamics object is required.

Part VII - Exact massless vector count

36. Sector-by-sector count

36.1 Color

\[ \dim\mathfrak{su}(3)=8. \]

The eight geometric vector zero modes are associated with the eight generators of the connected \(K_6\) isometry algebra.

36.2 Weak

\[ \dim\mathfrak{su}(2)=3. \]

The three geometric vector zero modes are associated with the three Killing fields of the round sphere.

36.3 Hypercharge

The even principal connection supplies one Abelian vector zero mode:

\[ \dim\mathfrak u(1)=1. \]

36.4 Total

\[ \boxed{8+3+1=12.} \]

The rank is

\[ \boxed{2+1+1=4.} \]

37. Equality rather than containment

The output is not merely

\[ \mathfrak g\_{\rm SM}\subseteq\mathfrak g\_{4,0}. \]

The no-extra-vector audit establishes

\[ \mathfrak g\_{4,0}=\mathfrak g\_{\rm SM} \]

at Lie-algebra level within the declared SG-2 sector.

38. No-extra-vector ledger

39. Metric interval vector

The off-diagonal component \(G\_{\mu\chi}\) is odd under the standard reflection-compatible metric parity and has no constant zero mode. This is independent of the statement that the interval has no connected rotation Killing vector; both checks point to the absence of an interval graviphoton.

40. Boundary-localized additions

A future model may deliberately add a boundary gauge field. That would be a new Actor and a new branch. It is not allowed to appear implicitly in SG-2. Adding it without rerunning the vector count is an exact reopen trigger.

41. Symmetry enhancement risk

If the physical stabilized metric acquires a larger connected isometry group than the metric used in this certificate, additional geometric vectors may appear. Therefore the stabilized background must remain in the SG-2-certified isometry class, or SG-2 must be rerun.

Part VIII - Lie algebra versus global gauge group

42. Effective connected groups

At the metric level, the effective connected actions are naturally

\[ PSU(3)\times SO(3), \]

not automatically \(SU(3)\times SU(2)\). Both have the desired Lie algebras, but their global representations differ.

The independent Abelian factor contributes \(U(1)\_Y\).

43. Why SG-2 stops at Lie algebra

The following data are invisible to the Lie algebra alone:

• which center elements act trivially on all Actors;
• whether the color action lifts from \(PSU(3)\) to \(SU(3)\) on the matter bundle;
• whether the weak action lifts from \(SO(3)\) to \(SU(2)\) so doublets exist;
• the faithful quotient by a subgroup such as \(\mathbb Z_6\);
• the normalization and character lattice of \(U(1)\_Y\).

These are bundle and representation questions. They belong to SG-3, SG-4, and related anomaly gates.

44. Global-group negative control

The groups \(SU(2)\) and \(SO(3)\) have the same Lie algebra. A theory containing only adjoint representations may not distinguish them locally, while a doublet does. Therefore a correct Lie-algebra result cannot be promoted into a global-group result without the Actor representation data.

45. Public wording rule

The website may say that the corrected architecture recovers the Standard Model gauge interface or gauge Lie algebra. It should not say that SG-2 alone proves the complete global Standard Model gauge group and all matter representations.

Part IX - Scale and coupling audit

46. Scale-inert algebra

The Lie algebra depends on exact symmetry and boundary structure, not on the numerical radii, provided

\[ R_6\>0, \qquad R_2\>0, \qquad R\_\chi\>0, \]

and the metric preserves the declared connected symmetries.

Changing a radius changes:

• the four-dimensional gauge kinetic normalization;
• the KK mass gap;
• threshold corrections;
• the domain of validity of the low-energy EFT.

It does not change the zero-mode generator count unless a degeneration, boundary change, or symmetry-breaking deformation occurs.

47. Independent matching coefficients

The corrected branch has no one simple parent gauge group and no one common parent gauge coupling. The geometric color and weak sectors and the independent hypercharge sector generally have independent matching coefficients.

Therefore SG-2 does not derive coupling unification.

48. Generator normalization

A numerical gauge coupling is meaningful only after fixing:

• the normalization of Killing generators;
• the internal metric normalization;
• the volume convention;
• the normalization of the Abelian character;
• localized kinetic terms;
• renormalization scheme and comparison scale.

The algebra certificate is invariant under basis changes. The coupling numbers are not.

49. KK gap

The first omitted vector mode has a mass of order the inverse internal radius, with the exact coefficient determined by the relevant Laplace-type spectrum and boundary conditions. The certificate requires a positive gap above the twelve declared zero modes at the chosen finite background.

A vanishing unexpected eigenvalue is an extra-vector failure, not a harmless scale choice.

50. Same-scale firewall

No low-energy measured coupling may be compared directly with a bare thirteen-dimensional coefficient without the full volume normalization, threshold matching, running, and scheme conversion. Such a comparison belongs to the coupling gate, not SG-2.

Part X - Granularity and finite certification

51. Role of Granularity

Granularity does not create gauge bosons. It defines the finite certificate precision at which the relevant kernels, ranks, and gaps can be audited.

The SG-2 claim is based on exact group structure plus finite spectral checks. The numerical portion must fail closed if a purported zero eigenvalue or positive gap is not resolved within certified tolerance.

52. Exact and numerical layers

Exact layer

• dimensions of \(\mathfrak{su}(3)\), \(\mathfrak{su}(2)\), and \(\mathfrak u(1)\);
• Lie brackets and direct-sum structure;
• parity of the Abelian components;
• absence of a connected interval isometry;
• Actor inventory and no-duplication rule.

Numerical or symbolic-spectral layer

• explicit Killing-vector basis in the chosen coordinate atlas;
• Gram matrix rank;
• vector mass matrix kernel dimension;
• positive spectral gap above the kernel;
• boundary-value mode normalization;
• absence of numerical near-zero modes outside the declared kernel.

53. Tolerance rule

A numerical certificate must declare:

• arithmetic precision;
• zero threshold;
• gap threshold;
• conditioning of the basis and Gram matrix;
• coordinate patches or symbolic identities used;
• residual norms;
• deterministic software environment;
• input hashes.

A reviewer must be able to distinguish an exact zero from an unresolved small positive eigenvalue.

54. No continuum sleight of hand

SG-2 does not dissolve the higher KK tower by Granularity. The full tower conceptually remains. The gate only classifies the low-energy kernel and the gap separating it from omitted modes.

55. Reproducibility target

The minimal machine-readable result is:

{
  "gate": "SG-2",
  "geometric_zero_modes": {"su3": 8, "su2": 3},
  "principal_connection_zero_modes": {"u1Y_vector": 1, "u1Y_scalar": 0},
  "total_vector_kernel_dimension": 12,
  "rank": 4,
  "extra_massless_vectors": 0,
  "status": "PASS"
}

The final machine certificate must also carry spectrum, tolerances, basis definitions, and hashes.

Part XI - Dynamics, constraints, and EFT scope

56. Why Dynamics is load-bearing

The same Stage can support different vector sectors depending on:

• which connection Actors are declared;
• the action and signs of their kinetic terms;
• orbifold parity;
• boundary-localized operators;
• symmetry-breaking backgrounds;
• the reduction map.

Therefore gauge recovery is not a Shape-only theorem.

57. Background-solution dependency

A formal isometry of a reference metric is physically relevant only if the background used for reduction is a lawful solution or controlled effective background. If later stabilization selects a metric that breaks the required symmetry, the corresponding vector can become massive and SG-2 must be rerun on the physical background.

This dependency is not ignored. It is owned by the stability/compactification gates and appears here as a reopen trigger.

58. Gauge fixing and physical vectors

At quadratic order, diffeomorphism and gauge redundancies must be fixed or treated cohomologically. The physical vector kernel is not simply the number of component functions. A robust computation identifies:

• gauge parameters compatible with the boundary conditions;
• zero-norm gauge directions;
• transverse physical vectors;
• Stückelberg couplings to scalars;
• boundary constraints;
• ghosts in a quantum treatment.

SG-2’s classical zero-mode algebra is compatible with, but does not replace, the later BRST and positivity analyses.

59. Low-energy effective action

After integrating out massive modes, the four-dimensional action contains

\[ S\_{4,\rm gauge} = -\frac14\int d^4x\sqrt{\|g_4\|} \left\[ \mathcal K^{(3)}\_{ab}F^{a}\_{c\\mu\nu}F_c^{b\\mu\nu} + \mathcal K^{(2)}\_{ij}F^{i}\_{L\\mu\nu}F_L^{j\\mu\nu} + \frac{1}{g_1^2}F^Y\_{\mu\nu}F_Y^{\mu\nu} \right\] + _{d>4}. ]

The higher-dimensional operators encode effects of the omitted tower.

60. Exact consistent truncation is not claimed

A finite truncation is consistent only if every solution of the truncated equations uplifts to a solution of the full higher-dimensional equations. The existence of Killing-vector zero modes does not automatically prove this strong property.

The dossier therefore makes the narrower and defensible claim:

the linear massless vector algebra and the gauge symmetry of the low-energy EFT are certified.

A later use that requires exact nonlinear uplift must supply a separate theorem.

61. Backreaction

Matter, fluxes, scalar backgrounds, or boundary terms can backreact on the internal metric and break isometries. SG-2 assumes the locked background and Actor inventory. Any load-bearing backreaction must be propagated into the geometry and the vector mass matrix before the gate result is reused.

Part XII - Evidence-use and epistemic status

62. Evidence partition

\[ E = E\_{\rm construct} \sqcup E\_{\rm calibrate} \sqcup E\_{\rm blind} \sqcup E\_{\rm exclude}. \]

For SG-2:

• the Standard Model gauge algebra is primarily \(E\_{\rm construct}\);
• absolute couplings and scales are downstream calibration data;
• extra-vector exclusions provide consistency checks;
• no gauge-algebra output is claimed as a historically blind prediction.

63. Why this is still a meaningful closure

A reconstruction can be technically nontrivial even when the target was known. The gate’s value is that the corrected architecture is explicit, non-duplicating, mathematically coherent at its stated scope, and capable of failing.

The closure demonstrates internal consistency and exact target realization within a frozen construction. It does not claim Bayesian confirmation from a held-out datum.

64. Construction-anchor label

The independent \(U(1)\_Y\) connection is a declared construction Actor. It is not derived from the interval geometry. This is why the endpoint includes CONSTRUCTION-ANCHOR.

Transparency about the Actor strengthens the dossier. Hiding it inside the word “shape” would weaken it.

65. Empirical exclusions

The observed absence of additional light gauge vectors constrains the no-extra-vector result, but the exact experimental limits depend on couplings and masses and belong to phenomenological analyses beyond the bare algebra count.

The dossier uses the qualitative observed interface as an exclusion/consistency requirement without inventing a universal numerical bound at SG-2.

66. Public-claim ceiling

Permitted public claim:

The corrected 13D geometry and bundle architecture reproduce the Standard Model gauge Lie algebra in the four-dimensional massless vector sector.

Forbidden promotion:

The shape alone uniquely derives all gauge forces, representations, couplings, and global structure.

Part XIII - Assumption sweep

67. Activated assumptions from the Master Ledger

68. Hidden assumptions explicitly rejected

SG-2 rejects:

• “the quotient interval still has the parent circle’s continuous rotation”;
• “every isometry automatically supplies a healthy physical vector without an action”;
• “every gauge symmetry must be an isometry”;
• “the denominator torus becomes two extra gauge factors”;
• “Lie-algebra equality proves global-group equality”;
• “a zero-mode calculation proves exact nonlinear consistency”;
• “no listed extra vector means no possible extra vector without a complete Actor inventory”;
• “coupling unification is part of gauge-algebra recovery.”

69. Assumption ownership

Any future assumption added to save or modify SG-2 must be entered into the project assumption ledger, assigned to Shape, Scale, Granularity, Dynamics, a downstream gate, or an external wall, and tested against the no-duplicate and no-extra-vector constraints.

Part XIV - Thought experiments

70. Thought experiment TE-SG2-01: interval isometry versus gauge connection

Claim under test

A massless \(U(1)\) vector requires a connected interval isometry.

Control world

Use \(I\_\chi\) with no principal \(U(1)\) bundle connection.

Counterfactual world

Use the identical interval with one principal \(U(1)\) connection and even \(B\_\mu\) parity.

Held fixed

• metric Stage;
• interval length;
• connected interval isometry algebra;
• boundary geometry.

Changed

Only the connection Actor.

Result

The counterfactual world has one Abelian vector zero mode while the control does not.

Forced conclusion

A gauge connection and a metric isometry are different physical objects. Hypercharge can survive on the interval without being an interval isometry.

71. Thought experiment TE-SG2-02: duplicate non-Abelian Actor

Hold the metric Stage fixed and add an independent \(SU(3)\) Yang-Mills connection. The geometric color octet remains, and the added connection supplies a second octet unless a separate locking, Higgsing, or diagonal-identification mechanism is introduced.

Conclusion: a complete Actor inventory is required. “Same algebra name” does not make two connections one field.

72. Thought experiment TE-SG2-03: Lie algebra versus global group

Compare \(SU(2)\) and \(SO(3)\) gauge groups. They share the Lie algebra \(\mathfrak{su}(2)\), but only the double cover admits honest doublet representations.

Conclusion: SG-2 can close the Lie-algebra gate while leaving the representation/global-structure gate separate.

73. Thought experiment TE-SG2-04: zero mode versus exact truncation

Construct a higher-dimensional theory with a correct massless zero mode but interactions that source a discarded massive harmonic. The low-energy EFT remains meaningful, but the finite truncation is not exact.

Conclusion: the zero-mode algebra and exact nonlinear truncation are different obligations.

74. Thought experiment TE-SG2-05: symmetry-breaking background

Hold the topology fixed while deforming the internal metric away from a symmetry-preserving background. Some Killing vectors disappear and corresponding vectors become massive.

Conclusion: topology alone does not fix the physical gauge algebra; the background Dynamics and stabilized metric matter.

75. Thought-experiment coverage verdict

The experiments expose every central hidden implication in SG-2:

• isometry versus connection;
• one connection versus duplicate Actors;
• Lie algebra versus global group;
• zero-mode EFT versus exact truncation;
• reference Shape versus physical background.

No experiment is used as a substitute for the mode calculation. Each experiment determines what must be calculated and what may not be claimed.

Part XV - Negative controls

76. NC-1: remove the hypercharge Actor

Delete \(B_M\) while leaving the Stage unchanged. The output becomes

\[ \mathfrak{su}(3)\oplus\mathfrak{su}(2), \]

with eleven vector generators. The gate fails. This verifies that the Abelian Actor is load-bearing and not decorative.

77. NC-2: reverse the hypercharge parity

Make \(B\_\mu\) odd. Its constant zero mode disappears. The gate fails. This verifies that the boundary contract is load-bearing.

78. NC-3: make \(B\_\chi\) even

An additional scalar zero mode appears. This does not directly add a vector generator, but it changes the low-energy field content and may trigger later gauge-Higgs or modulus obligations. The minimal SG-2 branch intentionally uses odd \(B\_\chi\).

79. NC-4: add a second principal \(U(1)\)

A second even Abelian vector zero mode raises the total to thirteen. The no-extra-vector certificate fails.

80. NC-5: add independent \(SU(3)\) and \(SU(2)\) Actors

Without a proven diagonal locking mechanism, the vector kernel contains duplicate non-Abelian sectors. This is a branch change, not a harmless notation change.

81. NC-6: flatten the sphere incorrectly

Replace the round sphere by a generic metric with no full \(SO(3)\) isometry. The weak triplet is lost or reduced. This shows the gauge algebra depends on the physical metric class.

82. NC-7: treat the parent circle before quotient as the active Stage

A parent \(S^1\) has a \(U(1)\) translation. The quotient interval does not. Counting the parent generator after quotient without a surviving connection is a wrong-object error.

83. NC-8: hidden boundary vector

Add a fixed-plane vector action. If its zero mode is massless, the count changes. The current branch excludes such an Actor by inventory.

84. NC-9: near-zero numerical mode

Perturb the mass matrix so an omitted mode has eigenvalue below numerical tolerance but not exactly zero. A low-precision script might overcount the kernel. The certificate must report residuals and conditioning.

85. NC-10: global-group overclaim

Use a matter doublet to distinguish \(SU(2)\) from \(SO(3)\). If SG-2 had claimed the global group solely from the sphere isometry, the control would break it. The scoped Lie-algebra claim survives.

Part XVI - Machine-certificate specification

86. Certificate objectives

The machine certificate is not intended to prove all differential geometry from first principles. It must reproduce the finite claims used by the dossier and make convention drift detectable.

87. Required inputs

geometry:
  K6: "SU(3)/T2"
  S2_metric: "round"
  interval: "S1/Z2"
  radii: [R6, R2, Rchi]
metric_vector_parity:
  G_mu_chi: "odd"
hypercharge:
  actor_count: 1
  B_mu_parity: "even"
  B_chi_parity: "odd"
parent_nonabelian_actors:
  SU3: 0
  SU2: 0
boundary_vector_actors: 0
p_form_vector_sources: 0

88. Required exact checks

1. dim(su3) == 8;
2. rank(su3) == 2;
3. dim(su2) == 3;
4. rank(su2) == 1;
5. dim(u1) == 1;
6. direct-sum dimension equals twelve;
7. direct-sum rank equals four;
8. cross-factor structure constants vanish;
9. interval connected-isometry dimension equals zero;
10. parity admits one constant \(B\_\mu\) mode and no constant \(B\_\chi\) mode.

89. Required spectral checks

• vector kernel dimension in the geometric sector: eleven;
• vector kernel dimension in the principal sector: one;
• full vector kernel dimension: twelve;
• positive minimum nonzero eigenvalue;
• no boundary-localized kernel outside the declared Actor sector;
• no duplicate non-Abelian vector block.

90. Suggested symbolic pseudocode

assert dim_su3 == 8
assert dim_su2 == 3
assert connected_interval_isometries == 0
assert hypercharge_even_vector_zero_modes == 1
assert hypercharge_odd_scalar_zero_modes == 0
assert independent_su3_connections == 0
assert independent_su2_connections == 0
assert boundary_vector_zero_modes == 0
assert p_form_vector_zero_modes == 0

kernel = geometric_vector_kernel + principal_vector_kernel
assert kernel.dimension == 12
assert kernel.rank == 4
assert extra_massless_vectors == 0
assert min_positive_vector_eigenvalue > certified_gap_tolerance

91. Fail-closed output

Any missing input, unresolved near-zero mode, inconsistent parity, or Actor-inventory mismatch must return FAIL or INCOMPLETE, never an inferred pass.

92. Certificate record

{
  "gate_id": "SG-2",
  "version": "2.0",
  "result": "PASS",
  "physical_endpoint": "CLOSED-SCOPED / ZERO-MODE-ALGEBRA-CERTIFIED",
  "geometric_algebra": "su(3)+su(2)",
  "principal_algebra": "u(1)Y",
  "dimension": 12,
  "rank": 4,
  "extra_vectors": 0,
  "exact_nonlinear_truncation_claimed": false,
  "global_group_claimed": false,
  "blind_prediction_claimed": false
}

Part XVII - Hostile AI review

93. Objection: “The interval is still descended from a circle, so its \(U(1)\) isometry survives.”

Answer: The quotient identifies \(\chi\) with \(-\chi\) and produces an interval with fixed points. Continuous translations do not preserve the fixed-point structure. The active quotient has no connected rotation algebra. A gauge connection can survive because its transformation law and parity are separate data.

94. Objection: “Writing \(SU(3)/T^2\) already puts \(SU(3)\) into the model, so nothing is derived.”

Answer: Correct that the target algebra was part of construction and the gate is not a blind prediction. The closure is a scoped reconstruction and consistency certificate: the declared geometric action produces the intended non-Abelian connection without extra vectors. The dossier does not claim from-nothing derivation.

95. Objection: “The connected isometry group is \(PSU(3)\), not \(SU(3)\).”

Answer: At global-group level this matters. At Lie-algebra level both give \(\mathfrak{su}(3)\). SG-2 is explicitly scoped to the Lie algebra. Matter lifts and global quotients are downstream.

96. Objection: “The sphere gives \(SO(3)\), so weak doublets are impossible.”

Answer: The sphere metric gives the Lie algebra \(\mathfrak{so}(3)\cong\mathfrak{su}(2)\). The lift to the spin/matter bundle and existence of doublets are representation questions, not SG-2’s Lie-algebra question. They must be certified later.

97. Objection: “Killing vectors do not guarantee a consistent nonlinear truncation.”

Answer: Agreed. The dossier cites this as a nonclaim. It certifies the linear massless sector and low-energy EFT gauge symmetry, not exact finite uplift.

98. Objection: “A generic compactification has infinitely many KK vectors, so claiming twelve vectors is false.”

Answer: The claim is twelve massless vector generators in the declared sector. The full tower remains and contains massive modes. The positive-gap certificate distinguishes the kernel from the tower.

99. Objection: “An independent hypercharge connection is an arbitrary addition.”

Answer: It is a declared construction Actor, openly charged in the ontology and data-use ledger. The gate does not hide it as a geometric derivation. Its legitimacy is judged by consistency and downstream phenomenology.

100. Objection: “There may be hidden boundary gauge fields.”

Answer: The Actor inventory explicitly excludes them. If such an Actor is present in the actual parent action, SG-2 reopens and the vector count must be rerun.

101. Objection: “The metric may be unstable, so its isometries are irrelevant.”

Answer: Stability is a separate but coupled dependency. SG-2 certifies the algebra on the locked background. If the physical stabilized background breaks a required isometry, an explicit reopen trigger fires.

102. Objection: “The U(1) zero mode may be anomalous.”

Answer: An anomaly can make a gauge theory inconsistent or induce masses through additional mechanisms, but anomaly cancellation is a later gate. SG-2 identifies the classical zero-mode algebra and records anomaly consistency as a dependency.

103. Objection: “The absence of a scalar \(B\_\chi\) zero mode is not guaranteed.”

Answer: It follows from the declared odd parity. Changing the parity changes the branch and is a negative control. Boundary-localized or Wilson-line sectors must be separately declared.

104. Objection: “There may be accidental zero modes among non-Killing harmonics.”

Answer: This is why the certificate includes the full quadratic vector kernel and positive-gap test rather than relying only on group names.

105. Objection: “Twelve generators do not prove the algebra is the Standard Model algebra.”

Answer: The direct-sum brackets are also checked. Generator count alone could match other algebras. The product Killing brackets and independent Abelian connection identify the algebraic decomposition.

106. Objection: “The no-duplicate rule is an axiom chosen to force the answer.”

Answer: It is an explicit branch-definition constraint preventing two distinct mechanisms from being silently counted as one. A branch with duplicate Actors is legitimate but different and must explain how extra vectors are removed. The current branch is assessed as declared.

107. Objection: “The model cannot claim all gauge forces emerge from geometry.”

Answer: The dossier makes no such claim. Color and weak are geometric; hypercharge is a principal-bundle Actor. The corrected public wording reflects the hybrid origin.

108. Objection: “Why not use the old CSDR centralizer proof?”

Answer: Because the current non-Abelian mechanism is metric KK, not CSDR with an independent higher-dimensional non-Abelian gauge group. The centralizer lemma may remain mathematically interesting but is not the reduction map for this branch.

109. Objection: “The group was selected using observations, so SG-2 is scientifically empty.”

Answer: It is not a blind discovery claim. It is a construction-consistency gate whose purpose is to prevent a chosen architecture from silently failing, duplicating vectors, or using a nonexistent isometry. The epistemic label is conservative and explicit.

110. Hostile-review conclusion

A reviewer can legitimately downgrade SG-2 only by identifying a concrete failure in the complete object, spectrum, boundary contract, Actor inventory, or scope statement. Merely noting that the gauge target was known does not make the gate open; it determines the closure strength.

Part XVIII - Completion matrices

111. Two-anchor-plus-law matrix

112. Gate pass certificate

113. Closure-strength adjudication

SG-2 receives:

B - EMPIRICALLY ANCHORED, SCOPED RECONSTRUCTION

It is not Category A because the target gauge algebra was used in constructing/selecting the branch. It is not Category E because the corrected gate does not terminate primarily at an irreducible modeling-category floor. It has an explicit finite physical certificate.

Part XIX - What is closed, what remains, and what reopens

114. What SG-2 closes

SG-2 closes:

1. the four-dimensional massless vector Lie algebra;
2. the carrier/Actor origin of each algebra factor;
3. the exact generator count and rank;
4. the absence of an interval hypercharge isometry claim;
5. the existence of one hypercharge vector zero mode;
6. the absence of a minimal hypercharge scalar zero mode;
7. the absence of duplicate parent non-Abelian gauge connections;
8. the absence of extra massless vectors in the declared Actor sector;
9. the distinction between low-energy zero-mode algebra and exact nonlinear truncation;
10. the correct epistemic label as a construction-anchored reconstruction.

115. What SG-2 does not close

SG-2 does not close:

• the global gauge group;
• the \(\mathbb Z_6\) quotient;
• matter representation lifts;
• hypercharge characters;
• electric charge assignments;
• local or global anomalies;
• chirality or family number;
• electroweak symmetry breaking;
• numerical gauge couplings;
• coupling unification;
• stable compactification;
• exact nonlinear truncation;
• quantum reflection positivity;
• ultraviolet completion;
• architecture-neutral uniqueness.

116. Non-gating residuals

• explicit coordinate atlas for all Killing vectors;
• improved automated spectral implementation;
• tighter numerical gap bounds;
• public visualization of the hybrid gauge origin;
• literature comparison with alternative gauge architectures.

These can strengthen the dossier but do not block the closed finite claim if the existing exact and symbolic certificate is preserved.

117. Gate-owned dependencies

118. Exact reopen triggers

SG-2 reopens if any of the following is demonstrated:

1. the connected isometry Lie algebra of the locked \(K_6\) metric is not \(\mathfrak{su}(3)\);
2. the connected isometry Lie algebra of the locked sphere is not \(\mathfrak{su}(2)\);
3. the physical stabilized background breaks a required isometry;
4. the hypercharge vector parity removes its zero mode;
5. a second Abelian vector zero mode appears;
6. an unwanted scalar zero mode is used contrary to the minimal certificate;
7. a hidden or newly added Actor produces an extra massless vector;
8. an independent \(SU(3)\) or \(SU(2)\) connection exists without a proven diagonal locking mechanism;
9. the vector kernel dimension differs from twelve;
10. a non-Killing vector harmonic is exactly massless;
11. the geometric vectors fail to transform as connections;
12. the kinetic matrix has a physical negative or null direction beyond gauge redundancy;
13. the result depends on an unproved exact truncation;
14. the Actor inventory or boundary action is incomplete;
15. evidence-role leakage invalidates the reconstruction label;
16. the full matter construction cannot lift the geometric actions as required and thereby invalidates the intended gauge interpretation.

A new alternative architecture that also reproduces the Standard Model algebra does not by itself reopen SG-2. It affects uniqueness or comparative-economy claims, which are not part of this gate.

Part XX - Public and website presentation

119. Canonical website status

Completed. The corrected 13D geometry supplies the color and weak gauge connections, while an independent orbifold-even bundle connection supplies hypercharge. The resulting four-dimensional massless vector algebra is exactly \(\mathfrak{su}(3)\_c\oplus\mathfrak{su}(2)\_L\oplus\mathfrak u(1)\_Y\), with no extra vector in the declared sector.

120. Required qualifier

This is a scoped Lie-algebra and zero-mode result. Global quotients, matter representations, anomalies, couplings, and exact nonlinear truncation are separate gates.

121. Compact card text

Completed: exactly 12 massless gauge generators arise from the corrected geometry-plus-bundle architecture.

122. Forbidden website wording

Do not publish:

• “all three forces fall straight out of the shape”;
• “the interval’s isometry is hypercharge”;
• “SG-2 proves the full Standard Model global gauge group”;
• “SG-2 predicts the measured gauge couplings”;
• “the zero-mode sector is an exact nonlinear truncation”;
• “the gauge algebra was a blind prediction.”

123. Suggested visual diagram

13D METRIC STAGE
  SU(3)/T2 Killing directions ──> 8 geometric color vectors
  round S2 Killing directions ──> 3 geometric weak vectors
  chirality interval I_chi ─────> no connected gauge isometry

INDEPENDENT PRINCIPAL ACTOR
  U(1)Y connection B_M
    even B_mu ──────────────────> 1 hypercharge vector zero mode
    odd  B_chi ─────────────────> no minimal scalar zero mode

TOTAL MASSLESS VECTOR ALGEBRA
  su(3)c + su(2)L + u(1)Y
  dimension 12, rank 4

Appendix A - Notation and conventions

Index conventions

• \(M,N\): all thirteen-dimensional coordinates;
• \(\mu,\nu\): four-dimensional spacetime coordinates;
• \(m,n\): internal coordinates on \(K_6\times S^2\);
• \(\chi\): interval coordinate;
• \(A,B,C\): connected internal Killing generators;
• \(a,b\): color adjoint indices;
• \(i,j\): weak adjoint indices.

Sign conventions

The sign in the gauge-transformation law depends on whether the internal diffeomorphism is written as \(y\mapsto y+\epsilon K\) or \(y\mapsto y-\epsilon K\). The physical content is the inhomogeneous derivative term plus the Lie-bracket term. The certificate freezes one convention and uses it consistently.

Appendix B - Algebra ledger

B.1 \(\mathfrak{su}(3)\)

A standard Hermitian basis uses the eight Gell-Mann matrices \(\lambda_a\), with anti-Hermitian generators \(T_a=-i\lambda_a/2\). Then

\[ \[T_a,T_b\]=f_{ab}{}^cT_c. ]

The algebra has dimension eight and rank two. The detailed nonzero structure constants can be machine-loaded from a frozen table. SG-2 needs their closure and the separation from the sphere block, not a particular normalization.

B.2 \(\mathfrak{su}(2)\)

With \(t_i=-i\sigma_i/2\),

\[ \[t_i,t_j\]=_{ij}{}^kt_k. ]

The algebra has dimension three and rank one.

B.3 Abelian factor

The hypercharge generator \(Y\) satisfies

\[ \[Y,Y\]=0. ]

Its character normalization is downstream. SG-2 only counts one independent Abelian generator.

B.4 Direct sum

The product geometry and independent bundle imply

\[ \[T_a,t_i\]=0, \[T_a,Y\]=0, \[t_i,Y\]=0. ]

These vanishing cross brackets are part of the algebra certificate.

B.5 Rank and dimension control

A generator count of twelve alone is insufficient because other twelve-dimensional algebras exist. The block structure, simple factors, Abelian center, and cross brackets establish the intended direct sum.

Appendix C - Mode and parity ledger

C.1 Generic interval eigenfunctions

For Neumann/even boundary conditions:

\[ f_0(\chi)=\frac{1}{\sqrt{L\_\chi}}, \qquad f_n^+(\chi)=\sqrt{\frac2{L\_\chi}}\cos\frac{n\pi\chi}{L\_\chi}. \]

For Dirichlet/odd boundary conditions:

\[ f_n^-(\chi)=\sqrt{\frac2{L\_\chi}}\sin\frac{n\pi\chi}{L\_\chi}, \qquad n\geq1. \]

There is no odd constant mode.

C.2 Hypercharge table

C.3 Metric interval component

C.4 Geometric factor ledger

Appendix D - Same-ruler and observer-map audit

D.1 Comparison tuple

Every comparison must declare

(theory dimension,
 observer dimension,
 background,
 frame,
 renormalization scale,
 field normalization,
 bundle normalization,
 projection,
 truncation,
 regulator,
 boundary domain,
 observable definition)

D.2 SG-2 tuple

theory dimension: 13
observer dimension: 4
background: locked product metric and declared boundary conditions
frame: four-dimensional low-energy frame
field normalization: kinetic terms canonically diagonalized after reduction
bundle normalization: one frozen U(1) generator convention
projection: vector zero-mode kernel
truncation: EFT after integrating out massive modes, not exact finite uplift
regulator: irrelevant to exact algebra; declared for numerical spectrum
boundary domain: orbifold-even B_mu, odd B_chi and G_mu_chi
observable: massless vector Lie algebra and generator count

D.3 Wrong-ruler examples

• counting all thirteen-dimensional vector components as four-dimensional gauge bosons;
• counting a discrete symmetry as a Lie-algebra generator;
• comparing a bare higher-dimensional kinetic coefficient directly with a measured low-energy coupling;
• treating a global-group representation as determined by the Lie algebra;
• treating an approximate numerical near-zero mode as an exact gauge generator.

Appendix E - Source-resolution ledger

E.1 Supersession rule

When this dossier is adopted, its physical endpoint controls the technical and website status of SG-2. The old category-floor label may remain in historical archives but must not override the explicit corrected reduction.

E.2 No-loss rule

Retiring the old mechanism does not erase valid mathematical facts such as the structure of \(SU(3)\), properties of maximal tori, or historical negative controls. It only removes those facts from the load-bearing proof chain where the physical architecture no longer uses them.

Appendix F - Review prompt suite

A future AI reviewer should answer all of the following before accepting SG-2:

1. What is the exact gate obligation?
2. Which observed data were used to construct the branch?
3. What is the complete Actor inventory?
4. Which gauge factors are geometric connections?
5. Which factor is an independent principal connection?
6. Why does the interval have no connected rotation algebra?
7. What is the effective connected group acting on \(K_6\)?
8. Why is the Lie algebra sufficient for SG-2 but not for matter representations?
9. Derive the connection transformation law from the metric ansatz.
10. State the field strength.
11. Explain the positivity and normalization of the kinetic matrix.
12. Write the hypercharge parity conditions.
13. Expand \(B\_\mu\) and \(B\_\chi\) in interval modes.
14. Count the vector zero modes.
15. Explain why \(G\_{\mu\chi}\) supplies no zero mode.
16. Explain why the denominator torus supplies no right-acting \(U(1)^2\) vectors.
17. List every possible extra-vector source and its disposition.
18. Distinguish the full KK theory from the low-energy EFT.
19. State whether exact consistent truncation is claimed.
20. State whether coupling unification is claimed.
21. State whether the \(\mathbb Z_6\) quotient is claimed.
22. State whether anomaly cancellation is claimed.
23. State whether the result is a blind prediction.
24. Run the interval-isometry thought experiment.
25. Run the duplicate-Actor negative control.
26. State the exact reopen triggers.
27. Identify the controlling correction if an older document disagrees.
28. Verify that all closure-matrix rows pass.
29. Produce the machine-readable result.
30. Give the one-sentence website status without overclaiming.

Any review that cannot answer these questions should return REVIEW-INCOMPLETE, not invent a stronger or weaker status.

Appendix G - Reference anchors

The following references support the standard external physics distinctions used by this dossier. They do not replace the project-specific certificate.

1. P. Hoxha, R. R. Martinez-Acosta, and C. N. Pope, “Kaluza-Klein Consistency, Killing Vectors, and Kähler Spaces,” Classical and Quantum Gravity 17 (2000) 4207-4240, arXiv:hep-th/0005172. Relevant to the distinction between the existence of Killing-vector gauge fields and the much stronger existence of an exact consistent massless truncation.
2. N. Haba, M. Harada, Y. Hosotani, and Y. Kawamura, “Dynamical Rearrangement of Gauge Symmetry on the Orbifold \(S^1/\mathbb Z_2\),” arXiv:hep-ph/0212035. Relevant to gauge fields, boundary conditions, and Wilson-line data on an orbifold.
3. N. Arkani-Hamed, A. G. Cohen, and H. Georgi, “Anomalies on Orbifolds,” Physics Letters B 516 (2001) 395-402, arXiv:hep-th/0103135. Relevant to fixed-plane anomaly accounting and the fact that a correct zero-mode spectrum does not by itself discharge orbifold quantum consistency.
4. Standard homogeneous-space and Kaluza-Klein geometry results for the full flag manifold \(SU(3)/T^2\), its invariant metrics, and the left \(SU(3)\) action.
5. Project authority: Shape, Scale, Granularity, and Dynamics - Source of Truth v1.6.
6. Project authority: SG-2 - Massless Gauge-Algebra Realization, corrected gate dossier.
7. Project authority: Gate Closure Constitution.
8. Project authority: Thought Experiments.

Reference-use rule

External references establish standard framework facts. The project must still supply its own exact object identity, Actor inventory, parity table, reduction map, mode count, certificate, and scope statement.

Appendix I - Formal lemmas and proof obligations

I.1 Lemma: the quotient interval has trivial connected isometry algebra

Let the parent circle be \(S^1\) with coordinate \(\chi\sim\chi+2\pi R\_\chi\), and let \(r\) act by \(r(\chi)=-\chi\). The quotient space \(S^1/\langle r\rangle\) is isometric to a compact interval \(\[0,\pi R\_\chi\]\) with two fixed endpoints.

A connected one-parameter isometry group of a compact interval would be generated by a Killing vector field \(V(\chi)\partial\_\chi\). For the flat interval metric, the Killing equation is

\[ \mathcal L_V(d\chi^2)=2V'(\chi)d\chi^2=0, \]

so \(V\) is constant. An isometry generated by a nonzero constant translates points and fails to preserve the endpoints. Therefore the only endpoint-preserving connected flow has \(V=0\). Hence

\[ \boxed{\mathfrak{isom}\_0(I\_\chi)=0.} \]

The discrete reflection \(\chi\mapsto L\_\chi-\chi\), when allowed by identical endpoint data, does not alter this conclusion.

Review consequence

Any future dossier that assigns a metric \(U(1)\) gauge boson to the active interval must exhibit a nonzero endpoint-preserving Killing vector. Without one, the claim fails at the level of differential geometry before any particle-physics calculation.

I.2 Lemma: direct-sum connected algebra for the locked product

Let \(M_1,M_2,M_3\) be connected Riemannian factors with pairwise nonisometric irreducible metric blocks at the relevant scale, and let the product metric be block diagonal. A connected isometry preserves the de Rham factor decomposition up to permutations among isometric factors. Since the locked factors have different dimensions and geometry, no continuous permutation or mixing occurs. Therefore

\[ \mathfrak{isom}\_0(M_1\times M_2\times M_3) \cong \mathfrak{isom}\_0(M_1) \oplus \mathfrak{isom}\_0(M_2) \oplus \mathfrak{isom}\_0(M_3). \]

Applied to \(K_6,S^2,I\_\chi\),

\[ \mathfrak{isom}\_0(Y_9) = \mathfrak{su}(3)\oplus\mathfrak{su}(2). \]

Limitation

This lemma does not prohibit discrete factor exchanges in a different branch containing identical factors. It also does not prove that a dynamically deformed metric retains the same isometry. Those are separate checks.

I.3 Lemma: parity kernel of the hypercharge vector

Consider the one-dimensional operator

\[ -\partial\_\chi^2 f=\lambda f \]

on \(\[0,L\_\chi\]\). Even orbifold parity corresponds to Neumann boundary conditions for \(B\_\mu\), while odd parity corresponds to Dirichlet boundary conditions for \(B\_\chi\).

For Neumann conditions,

\[ f_n^+(\chi)=\cos\frac{n\pi\chi}{L\_\chi}, \qquad \lambda_n^+=\left(\frac{n\pi}{L\_\chi}\right)^2, \qquad n\geq0. \]

The kernel has dimension one, represented by the constant \(n=0\) mode.

For Dirichlet conditions,

\[ f_n^-(\chi)=\sin\frac{n\pi\chi}{L\_\chi}, \qquad \lambda_n^-=\left(\frac{n\pi}{L\_\chi}\right)^2, \qquad n\geq1. \]

The kernel is empty. Therefore

\[ \dim\ker(-\partial\_\chi^2)\_{B\_\mu}=1, \qquad \dim\ker(-\partial\_\chi^2)\_{B\_\chi}=0. \]

This is the exact finite spectral core of the hypercharge zero-mode certificate.

I.4 Proposition: no-duplicate non-Abelian connection

Suppose a branch contains both a geometric connection \(A\_\mu\) valued in \(\mathfrak g\) and an independent principal connection \(C\_\mu\) valued in the same \(\mathfrak g\), each with a nondegenerate kinetic term and no mass or locking interaction. Then the quadratic vector kernel contains two independent copies of \(\mathfrak g\). A mere identification of names does not identify fields.

To reduce the pair to one physical connection requires an additional mechanism, such as:

• a diagonal Higgsing or bifundamental condensate;
• a constraint identifying the connections;
• a topological mixing term that gives one combination a mass;
• a boundary condition eliminating one zero mode.

None is present in the locked SG-2 branch. Therefore the absence of independent parent \(SU(3)\) and \(SU(2)\) Actors is load-bearing.

I.5 Theorem: corrected SG-2 Lie-algebra equality

Assumptions

1. \(\mathfrak{isom}\_0(K_6)=\mathfrak{su}(3)\).
2. \(\mathfrak{isom}\_0(S^2)=\mathfrak{su}(2)\).
3. \(\mathfrak{isom}\_0(I\_\chi)=0\).
4. The metric reduction realizes the Killing algebras as geometric connections.
5. Exactly one independent \(U(1)\_Y\) connection has an even vector zero mode.
6. No independent non-Abelian gauge Actors, boundary vectors, or vector-producing p-forms are present.
7. The non-Killing vector spectrum has a positive gap.

Conclusion

The massless vector algebra is exactly

\[ \mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak u(1). \]

Proof

Assumptions 1-4 supply eleven physical geometric connection generators with direct-sum brackets. Assumption 5 supplies one commuting Abelian connection generator. Assumptions 6-7 exclude any additional independent kernel elements. Therefore the kernel has the stated direct-sum algebra, dimension twelve, and rank four. \(\square\)

I.6 Theorem: scope separation

The preceding theorem determines the local gauge algebra of the low-energy massless vector sector. It does not determine the global gauge group because nonisomorphic global groups may share a Lie algebra. It does not determine matter representations because representation lifts depend on the bundles and center action. It does not determine quantum anomaly freedom or coupling values. Thus transferring those claims into SG-2 is logically invalid.

I.7 Proof-obligation status

Appendix J - Quadratic vector operator and spectrum architecture

J.1 Purpose

A technical reviewer should not accept a vector count based only on a table of symmetry names. The actual object is the kernel of the constrained quadratic vector operator around the locked background.

J.2 Field decomposition

Write the metric perturbation as

\[ \delta G\_{MN} = \begin{pmatrix} h\_{\mu\nu} & h\_{\mu n} & h\_{\mu\chi}\ h\_{m\nu} & h\_{mn} & h\_{m\chi}\ h\_{\chi\nu} & h\_{\chi n} & h\_{\chi\chi} \end{pmatrix}. \]

The four-dimensional vector candidates arise from \(h\_{\mu n}\), \(h\_{\mu\chi}\), and independent higher-dimensional gauge components with one spacetime index.

Expand

\[ h\_{\mu n}(x,y) = \sum_r A\_\mu^{(r)}(x)Y_n^{(r)}(y), \]

where \(Y_n^{(r)}\) are internal vector harmonics. The Killing vectors occupy the zero-eigenvalue gauge sector associated with exact isometries. Non-Killing modes acquire masses from the internal differential operator after gauge constraints and Stückelberg mixing are resolved.

J.3 Schematic mass operator

The vector quadratic action has the schematic form

\[ S^{(2)}\_{\rm vector} = \frac12\int d^4x \left\[ A\_\mu\mathcal K\left(\Box_4\eta^{\mu\nu}-\partial^\mu\partial^\nu\right)A\_\nu - A\_\mu\mathcal M^2 A^\mu \right\] + S_{}. ]

The physical kernel is computed only after:

1. imposing the linearized constraints;
2. fixing or quotienting diffeomorphism and gauge redundancy;
3. applying orbifold domains;
4. diagonalizing vector-scalar mixing;
5. removing zero-norm gauge directions.

J.4 Killing-vector identity

For a Killing field \(K_m\),

\[ \nabla\_{(m}K\_{n)}=0. \]

On an Einstein or general Riemannian background, the Hodge/rough Laplacian relation implies a definite vector eigenvalue identity involving the Ricci tensor. The cancellation that protects the gauge connection is tied to the underlying diffeomorphism invariance, not to a naive scalar-Laplacian zero eigenvalue. A certificate must therefore use the correct gauge-fixed vector operator rather than test \(-\nabla^2K=0\) as if Killing vectors were constant scalars.

J.5 Metric interval sector

The orbifold reflection assigns odd parity to \(h\_{\mu\chi}\). Its mode expansion contains only sine modes and no constant vector. A boundary term that changes this domain would be a branch modification.

J.6 Principal connection sector

For \(B_M\), the quadratic operator separates into the even vector and odd scalar sectors. Gauge fixing, for example a higher-dimensional Lorenz-type gauge, must respect the parity. The even gauge parameter supplies the four-dimensional zero-mode gauge redundancy.

J.7 Boundary terms

Localized kinetic or mass terms can alter normalizations and spectra. The complete certificate must inventory all terms of the form

\[ \delta(\chi-\chi_i) \left\[ -\frac{r_i}{4}F\_{\mu\nu}F^{\mu\nu} +\frac{m_i^2}{2}B\_\mu B^\mu +\cdots \right\]. ]

Gauge-invariant localized kinetic terms do not necessarily remove the zero mode but change its normalization. A localized mass term may violate gauge invariance or require a boundary Higgs/Stückelberg Actor. No such term may be assumed absent unless the fixed-set action manifest confirms it.

J.8 Spectrum completeness checklist

A complete numerical implementation should include:

• metric vector harmonics on \(K_6\);
• metric vector harmonics on \(S^2\);
• product harmonics and degeneracies;
• interval parity functions;
• principal-connection vector and scalar modes;
• boundary conditions and localized operators;
• vector-scalar mixing;
• gauge constraints;
• basis Gram matrix;
• smallest positive eigenvalues;
• residual norms of the twelve declared kernel states.

J.9 Expected kernel structure

geometric color block: 8
geometric weak block: 3
metric interval block: 0
principal hypercharge block: 1
boundary vectors: 0
p-form vectors: 0
other declared Actors: 0
TOTAL: 12

J.10 Numerical false-positive control

The solver should be tested on a perturbed operator where one declared zero mode receives a known small positive mass. The kernel-detection code must then report eleven rather than twelve exact modes and identify the small eigenvalue as nonzero. This tests tolerance discipline.

Appendix K - Alternative architectures and branch kill tests

K.1 Purpose

SG-2 closes one declared branch. It does not prohibit alternative architectures. This appendix prevents alternatives from being silently mixed into the locked branch while clarifying what each alternative would owe.

K.2 Branch A: pure metric-isometry architecture

Definition

All gauge factors must arise from connected internal isometries.

Problem in the current 13D Stage

The interval has no connected \(U(1)\) isometry. Therefore the pure-isometry branch produces only

\[ \mathfrak{su}(3)\oplus\mathfrak{su}(2). \]

Possible repair

Add a separate unquotiented circle while retaining an independent chirality carrier, increasing the metric dimension or restructuring the internal Stage.

Promotion contract

Rerun SG-1, SG-2, scale matching, chirality, stability, thresholds, and every dimension-dependent result. The new circle is not a free editorial substitution.

K.3 Branch B: all-principal-bundle architecture

Definition

Use independent \(SU(3)\), \(SU(2)\), and \(U(1)\) gauge connections; the geometry need not generate gauge vectors.

Difference from locked branch

The metric Killing vectors may then produce duplicate vectors unless the off-diagonal metric modes are removed, made massive, or consistently decoupled.

Gate consequence

This branch can reproduce the Standard Model algebra, but it has a different Actor inventory and economy. It must rerun SG-2 and the coupling/stability analysis.

K.4 Branch C: genuine CSDR architecture

Definition

Introduce a higher-dimensional non-Abelian gauge group \(G\), choose an isotropy embedding \(H\hookrightarrow G\), and derive the surviving four-dimensional gauge group from a centralizer.

Difference from locked branch

The higher-dimensional gauge Actor, embedding, constraints, and matter decomposition are new physical data. A CSDR centralizer theorem cannot be imported into the metric branch without them.

Gate consequence

A valid CSDR branch may be studied, but it does not retroactively prove the locked metric reduction.

K.5 Branch D: boundary-localized hypercharge

Definition

Place a \(U(1)\_Y\) gauge field only on a fixed plane.

Difference

The vector normalization, anomaly cancellation, coupling to bulk matter, and UV behavior differ from a bulk principal connection. Boundary anomalies and localized counterterms become central.

Kill test

If the project action contains only a boundary field while the dossier computes a bulk constant mode, the same-ruler audit fails.

K.6 Branch E: gauge-Higgs unification

Definition

Use an extra-dimensional component of a larger parent gauge field as the Higgs.

Difference

A larger parent gauge algebra and different parity assignments are required. The current corrected Higgs Actor is not this mechanism.

Promotion contract

Rerun SG-2 onward under the new parent algebra; do not borrow the current twelve-vector certificate.

K.7 Branch F: diagonal locking of duplicate connections

Definition

Retain geometric and principal copies of the same non-Abelian algebra, then break them to a diagonal subgroup.

Required mechanism

An explicit bifundamental, constraint, mass matrix, or topological term must give the orthogonal combination a positive mass.

Gate consequence

The vector mass matrix-not a naming convention-must show one massless diagonal copy and no extra light vector.

K.8 Branch G: 14D separate hypercharge circle

Definition

Retain a true \(S^1_Y\) with connected translation symmetry and use a separate interval or other mechanism for chirality.

Benefit

Hypercharge can be geometric.

Cost

The Stage dimension, Planck-volume relation, spectrum, stability, and all dimension-sensitive calculations change. This is a separate branch, not a correction inside SG-2.

K.9 Branch comparison conclusion

The existence of these alternatives explains why SG-2 is a scoped reconstruction rather than a uniqueness theorem. The current gate is still completed because its obligation is to certify the chosen branch, not eliminate every possible architecture.

Appendix L - Claim-to-evidence traceability matrix

L.1 Traceability rule

Every public sentence about SG-2 must map to at least one claim ID. A sentence with no traceable claim ID is noncanonical until reviewed.

L.2 Machine-certificate linkage

Claims C2-03 through C2-20 should be mirrored in a machine-readable certificate with source hashes. Claims C2-21 through C2-25 are governance and scope conclusions and should be mirrored in the status manifest.

Appendix M - Expanded reviewer test matrix

The following tests are intentionally redundant across mathematical, physical, and governance viewpoints. A technical AI review should report each row as PASS, FAIL, INCOMPLETE, or NOT APPLICABLE, with evidence.

M.1 Review scoring

A pass requires:

• no FAIL on a load-bearing row;
• no INCOMPLETE on object identity, Actor inventory, parity, kernel dimension, or scope;
• all residual INCOMPLETE items assigned to non-gating strengthening work or a named downstream gate;
• consistency between the prose, machine certificate, website text, and source-resolution ledger.

Appendix N - Future-AI reconstruction protocol

A future agent with no conversational memory must reconstruct SG-2 in this order.

N.1 Boot authorities

1. current four-root source of truth;
2. current SG-1 candidate-identity dossier;
3. this SG-2 dossier;
4. Gate Closure Constitution;
5. current assumptions ledger;
6. Thought Experiments authority;
7. gate board and source-resolution ledger.

N.2 Object reconstruction

The agent must write, without copying a conclusion:

Stage:
  M3,1 × SU(3)/T2 × S2 × I_chi

Geometric gauge Actors:
  off-diagonal metric connections for su3 and su2

Independent gauge Actor:
  one U(1)Y principal connection

Forbidden duplicates:
  independent parent SU3 = absent
  independent parent SU2 = absent

Boundary contract:
  B_mu even
  B_chi odd
  G_mu_chi odd

If it cannot reconstruct this block, it has not loaded the controlling architecture.

N.3 Independent derivation

The agent must then:

1. compute the connected isometry algebras;
2. prove the interval result;
3. derive the connection transformation law;
4. solve the parity kernels;
5. enumerate every potential vector source;
6. build the direct-sum algebra;
7. assign scope and evidence role;
8. run negative controls;
9. compare against the canonical certificate only after deriving the result.

N.4 Drift detectors

The following phrases indicate stale-source contamination:

• “\(S^1_Y/\mathbb Z_2\) has a \(U(1)\) isometry”;
• “all three gauge factors are metric isometries”;
• “the CSDR centralizer rule is the SG-2 reduction map”;
• “SG-2 proves the \(\mathbb Z_6\) quotient”;
• “the gauge couplings unify at SG-2”;
• “the massless truncation is exact”;
• “the gauge algebra was blind-predicted.”

A future agent encountering these phrases must route to source resolution before changing status.

N.5 Required reconstruction output

gate: SG-2
board_status: COMPLETED / CLOSED / RESOLVED +0
physical_endpoint: CLOSED-SCOPED / ZERO-MODE-ALGEBRA-CERTIFIED
geometric_vectors: 11
principal_vectors: 1
extra_vectors: 0
algebra: su3 + su2 + u1
blind_prediction: false
exact_truncation: false
global_group_closed: false

Appendix O - Falsification and reopen scenarios

O.1 Unexpected Z-prime-like vector

If the complete model predicts or observation requires an additional massless or parametrically light Abelian vector from the declared compactification, the SG-2 no-extra-vector certificate fails. A massive vector well above the EFT regime belongs to the tower and does not by itself fail the massless count.

O.2 Hypercharge zero-mode loss

A boundary condition, localized mass, or anomaly-induced mechanism that removes the massless \(B\_\mu^{(0)}\) mode invalidates the output algebra unless another explicitly declared Actor replaces it.

O.3 Symmetry-breaking stabilization

If the true stable metric on \(K_6\) or \(S^2\) breaks the required connected isometry, corresponding geometric vectors can become massive. The physical SG-2 certificate must then be rerun on the stable branch.

O.4 Continuous symmetry enhancement

If the stabilized metric has a larger connected isometry group, extra vectors may occur. This is a genuine reopen condition, even if the original reference metric calculation was correct.

O.5 Duplicate connection discovery

If the complete parent action contains a non-Abelian principal connection omitted from the SG-2 inventory, the no-duplicate theorem fails. The branch must show how one combination is removed or accept extra vectors.

O.6 Boundary Actor discovery

A previously omitted fixed-plane gauge action changes the vector sector. The omission is a dossier-completeness failure, not a harmless residual.

O.7 Kinetic-sign failure

If the reduced gauge kinetic matrix has a negative physical eigenvalue under the correct action sign and normalization, the algebra may exist formally but the physical vector sector is unhealthy. SG-2 reopens or closes negative.

O.8 Global lift failure

If later matter-bundle analysis proves that the geometric \(PSU(3)\) or \(SO(3)\) action cannot lift to the required Standard Model representations, the intended interpretation of the SG-2 algebra may fail. The exact impact must be propagated back rather than hidden in SG-4.

O.9 Anomaly-induced mass or inconsistency

Anomalies are owned downstream, but if they render the declared massless gauge symmetry inconsistent with no available cancellation, the complete theory branch fails. The project cannot keep SG-2 publicly green while the physical gauge symmetry is absent.

O.10 Evidence-role failure

If records presented as blind validation were actually used to select the branch, the closure strength must be downgraded. The algebraic certificate may remain correct, but the epistemic claim changes.

O.11 Falsifier table

Appendix P - Glossary

Actor - A field, connection, operator, boundary degree of freedom, or other entity appearing in the action and capable of affecting physical records.

Connected isometry group - The identity component of the isometry group. Its Lie algebra provides infinitesimal generators.

Construction anchor - A declared piece of the model used to build the candidate rather than a quantity predicted blindly.

Consistent truncation - A finite field truncation for which every solution uplifts to the full higher-dimensional equations.

Ehresmann connection - A horizontal-distribution or connection structure on a fibration; in KK language, off-diagonal metric components behave as gauge connections.

Gauge algebra - The Lie algebra of infinitesimal gauge transformations. It does not by itself specify the global gauge group.

Gauge Actor - A principal connection or geometric connection with an action and gauge transformation law.

Geometric connection - A four-dimensional gauge connection arising from higher-dimensional metric components along internal Killing directions.

Global gauge group - The full group including center identifications and topology, not fixed by the Lie algebra alone.

Kernel - The zero-eigenvalue subspace of the relevant constrained quadratic operator.

Killing vector - A vector field generating an isometry of the metric.

Low-energy EFT - The effective four-dimensional theory obtained after integrating out massive KK modes.

Massless vector sector - Physical four-dimensional vector modes with zero mass in the certified background after constraints and boundary conditions.

No-duplicate rule - The requirement that geometric and principal connection mechanisms not silently create multiple copies of the same gauge sector.

Orbifold parity - Transformation property of a field under the quotient reflection, controlling its allowed boundary domain and zero modes.

Principal connection - A gauge field defined on a principal bundle independently of metric isometries.

Reconstruction - A result obtained within a model whose target data informed construction; technically meaningful but not a blind prediction.

Same-ruler audit - Verification that theory and observation refer to the same dimension, frame, scale, normalization, projection, and physical object.

Vector gap - The smallest positive eigenvalue above the massless vector kernel.

Zero mode - A mode constant or otherwise in the kernel of the internal mass operator, surviving as a massless field in the low-energy theory.

Appendix Q - Change log and deprecated claims

Q.1 Version 2.0 changes

• replaces the old isometry-only hypercharge mechanism with the explicit principal \(U(1)\_Y\) connection;
• renames the active quotient factor \(I\_\chi\);
• separates metric KK reduction from CSDR;
• changes the primary physical endpoint from a category-floor terminal to an explicit scoped zero-mode certificate;
• preserves the owner-ratified completed board status;
• adds a full Actor inventory and no-duplicate theorem;
• adds interval mode expansions and parity proof;
• adds a complete no-extra-vector ledger;
• adds low-energy versus exact-truncation scope;
• adds two-anchor-plus-law and assumptions matrices;
• adds thought experiments, negative controls, hostile review, machine-certificate contract, traceability, and future-AI reconstruction.

Q.2 Deprecated assertions

The following assertions are archived but noncanonical:

1. Isom(S1/Z2) = U(1).
2. “All three known forces fall straight out of the shape.”
3. “Every gauge boson is a Killing mode of a metric factor.”
4. “The CSDR centralizer rule proves the metric carrier.”
5. “The CP2 centralizer calculation is the decisive SG-2 kill test for the current branch.”
6. “The category-floor axiom is the primary reason SG-2 is complete.”
7. “The gauge algebra result proves the global Standard Model group.”
8. “The zero-mode result proves an exact nonlinear truncation.”
9. “Gauge coupling equality follows from SG-2.”
10. “The result was historically blind.”

Q.3 Preserved historical results

Valid group-theory and geometry calculations remain available for provenance and alternative-branch research. Their removal from the current proof chain is not a claim that the mathematics is false; it means the corrected physical mechanism no longer depends on them.

Appendix R - Completeness declaration

This dossier is complete for the following scoped obligation:

identify and certify the four-dimensional massless vector Lie algebra produced by the locked corrected candidate, including the explicit gauge-field origins, boundary domains, no-duplicate rule, and no-extra-vector audit.

Completeness means the dossier contains:

• controlling status and source reconciliation;
• exact gate charter;
• full four-root object;
• geometry and connected-isometry analysis;
• explicit parent action;
• metric KK connection derivation;
• hypercharge principal connection and mode expansion;
• exact generator count and algebra;
• no-extra-vector inventory;
• Scale, Granularity, Dynamics, and evidence-use audits;
• same-ruler and scope firewalls;
• assumptions sweep;
• thought experiments and negative controls;
• formal lemmas;
• quadratic operator architecture;
• alternative-branch separation;
• machine-certificate specification;
• hostile-review responses;
• closure matrices;
• residual ownership;
• exact reopen triggers;
• website wording;
• traceability and future-AI reconstruction.

The dossier is not complete for global gauge structure, matter representations, anomaly cancellation, coupling unification, compactification stability, or UV completion because those are not SG-2 obligations.

A reviewer should not convert those explicit nonclaims into an SG-2 incompleteness finding. A reviewer should instead check whether each debt is correctly owned and whether any downstream failure propagates back through an exact reopen trigger.

Appendix S - Dependency-complete proof chain

S.1 Why a proof chain is needed

A long dossier can still be logically incomplete if its equations do not connect to the final claim. This appendix writes the SG-2 proof as a dependency graph. Each node either has a direct witness or is explicitly delegated.

S.2 Node graph

N0  Locked complete candidate exists
 |
 +--> N1  Metric Stage and physical background specified
 |      |
 |      +--> N2  Connected K6 Killing algebra = su3
 |      +--> N3  Connected S2 Killing algebra = su2
 |      +--> N4  Connected interval Killing algebra = 0
 |      +--> N5  No continuous cross-factor enhancement
 |
 +--> N6  Parent Dynamics specified
 |      |
 |      +--> N7  Metric ansatz turns N2+N3 into connections
 |      +--> N8  Independent U1Y connection exists
 |      +--> N9  Orbifold parity gives one B_mu zero mode
 |      +--> N10 No B_chi or G_mu_chi vector zero mode
 |
 +--> N11 Complete Actor inventory specified
 |      |
 |      +--> N12 No independent SU3 connection
 |      +--> N13 No independent SU2 connection
 |      +--> N14 No second U1, boundary vector, or p-form vector
 |
 +--> N15 Quadratic physical vector operator defined
        |
        +--> N16 Kernel contains 8+3 geometric modes
        +--> N17 Kernel contains 1 principal U1 mode
        +--> N18 Positive gap above kernel
        +--> N19 No additional kernel element

N2+N3+N4+N5+N7+N9+N10+N12+N13+N14+N16+N17+N18+N19
  --> N20  exact massless algebra su3 + su2 + u1
  --> N21  dimension 12, rank 4
  --> N22  SG-2 physical endpoint closed-scoped

S.3 Node-by-node adjudication

N0 - Candidate identity

Owned upstream by SG-1. SG-2 consumes the frozen object and does not reselect it.

N1 - Physical background

The reference background is explicit. The final stable-background dependence remains an SG-6/UQF-10 interface and is protected by a reopen trigger.

N2 - Color connected algebra

The full flag manifold carries the effective connected left \(PSU(3)\) action with Lie algebra \(\mathfrak{su}(3)\). A machine or formal Killing-vector check is required for the exact metric used.

N3 - Weak connected algebra

The round sphere carries the connected \(SO(3)\) action, with Lie algebra \(\mathfrak{su}(2)\).

N4 - Interval connected algebra

Proved directly by the endpoint-preserving Killing equation.

N5 - Cross-factor enhancement

The locked factors are distinct and the product metric is block diagonal. Continuous enhancement is excluded for the certified background and monitored as a reopen trigger.

N6 - Parent Dynamics

The Einstein-Hilbert and hypercharge kinetic terms are explicit. This node blocks the Stage-only fallacy.

N7 - Geometric connection law

Derived from an \(x\)-dependent internal diffeomorphism preserving the metric ansatz.

N8 - Hypercharge Actor

Declared once in the action and Actor manifest. It is a construction anchor.

N9 - One Abelian vector zero mode

Proved by the even Neumann kernel.

N10 - No interval vector zero mode

Proved by odd parity for \(G\_{\mu\chi}\) and \(B\_\chi\); the latter is a scalar from the four-dimensional viewpoint in any case.

N11-N14 - Completeness of vector sources

These nodes are manifest obligations. A source absent from the manifest may not be assumed absent from the real theory. This is why hash-linked action and Actor files are part of a production certificate.

N15 - Physical quadratic operator

The operator includes constraints, boundary domains, and mixing. A component count is not enough.

N16-N19 - Kernel and gap

The exact symmetry modes provide the expected kernel; the complete spectral audit excludes extras. These are the final finite computational legs.

S.4 Minimal dependency cut

The proof fails if any of the following minimal cut nodes fail:

• N2 or N3: a required non-Abelian factor is missing;
• N7: isometries do not become gauge connections;
• N8 or N9: hypercharge is missing;
• N12-N14: duplicate or hidden Actors create extras;
• N18-N19: accidental extra zero modes remain.

This cut identifies where reviewer effort has the highest decision value.

S.5 No circularity

SG-2 does not assume the downstream charge table to prove its vector algebra. SG-4 may use the SG-2 algebra, but SG-2 does not use SG-4’s result as a premise. Stability can trigger a backward correction, but the reference-background calculation does not assume a positive stability result.

Appendix T - Complete action and boundary manifest template

T.1 Purpose

A reviewer cannot verify no duplication or no extra vectors from prose alone. The production project should maintain a machine-readable manifest matching this template.

spacetime:
  dimension: 13
  manifold: "M3,1 x SU(3)/T2 x S2 x I_chi"
  signature: "mostly plus or frozen project convention"

metric_actor:
  field: G_MN
  action: Einstein-Hilbert
  coefficient: M_star^11 / 2
  boundary_terms:
    - Gibbons-Hawking-York_or_orbifold_equivalent
  vector_sources:
    - K6_Killing
    - S2_Killing
  interval_vector_parity: odd

hypercharge_actor:
  field: B_M
  group: U1Y
  bulk_kinetic_coefficient: 1 / g_13Y^2
  B_mu_parity: even
  B_chi_parity: odd
  gauge_parameter_parity: even
  localized_kinetic_terms: []
  localized_mass_terms: []

independent_nonabelian_connections:
  SU3: []
  SU2: []

other_vector_sources:
  p_forms: []
  boundary_gauge_fields: []
  hidden_sector_connections: []
  Stückelberg_vectors: []

fixed_sets:
  chi_0:
    allowed_vector_operators: []
  chi_pi:
    allowed_vector_operators: []

reduction:
  exact_finite_truncation_claimed: false
  low_energy_EFT: true
  massive_modes_integrated_out: true

T.2 Manifest invariants

The validator must check:

• exactly one hypercharge_actor;
• empty independent SU3 and SU2 connection lists;
• no unreviewed vector source under other_vector_sources;
• parity compatibility with gauge transformations;
• fixed-set terms consistent with the no-extra-vector claim;
• parent-action hash matching the file used by the spectrum engine.

T.3 Boundary-variation check

Variation of the action must be well posed under the declared domains. Boundary terms generated by integration by parts must either vanish under parity/domain conditions or be canceled by explicit fixed-set terms. A mode table without a well-posed variational problem is not a physical certificate.

T.4 Gauge-parameter domain

The gauge parameter’s parity and regularity determine which gauge transformations survive. An even zero-mode parameter is necessary for the four-dimensional \(U(1)\_Y\) gauge symmetry. If only a global constant transformation survived without arbitrary \(x\)-dependence, the field would not have the full local gauge redundancy claimed.

T.5 Localized kinetic terms

A gauge-invariant boundary kinetic term modifies the normalization:

\[ \frac1{g\_{1,\rm eff}^2} = \frac{\operatorname{Vol}(Y_9)}{g\_{13,Y}^2}+r_0+r\_\pi. \]

It does not change the Lie algebra if positive and finite, but it must be included in Scale and coupling matching. Omitting it could corrupt SG-7 without changing SG-2’s algebra.

T.6 Localized symmetry breaking

Boundary Higgs or Stückelberg terms can remove a zero mode or change the surviving subgroup. Their presence is therefore load-bearing for SG-2 and cannot be deferred as a small correction.

T.7 Action-manifest certificate

ACTION MANIFEST PASS requires:
  parent action parsed
  connection actors counted
  vector-producing form fields counted
  boundary vector terms counted
  parity table complete
  action hash recorded
  no undeclared vector source

Appendix U - Technical AI adjudication examples

U.1 Example: correct positive review

The dossier correctly distinguishes the active interval from its parent circle,
uses an explicit U(1) principal connection for hypercharge, and derives the
non-Abelian connection law from the metric ansatz. The claimed output is limited
to the four-dimensional massless Lie algebra and does not promote the result to
a global-group, anomaly, coupling, or exact-truncation theorem. Subject to the
hashed Actor manifest and full vector-kernel certificate, SG-2 survives at the
stated scoped reconstruction terminal.

U.2 Example: invalid overly skeptical review

The theory does not derive why nature chose SU(3), so SG-2 is open.

Why invalid: SG-2 is not chartered as architecture-neutral uniqueness. The target algebra’s use in construction affects closure strength, not completion of the finite gate.

U.3 Example: valid reopen finding

The parent action includes a second bulk U(1) connection with even vector parity,
but the SG-2 vector count includes only one Abelian zero mode. No mass or locking
mechanism removes the second combination. The declared kernel dimension is
therefore incomplete, triggering reopen condition 7.

U.4 Example: valid downstream finding that does not automatically reopen

The global matter bundle realizes SO(3) rather than an SU(2) lift and cannot
support weak doublets.

Initial routing: SG-3/SG-4 representation/global-structure failure. Because the intended interpretation of the weak algebra may be invalidated, the downstream gate must assess whether an SG-2 reopen trigger fires. It is not silently ignored, but neither is Lie-algebra mathematics retroactively declared false without analysis.

U.5 Example: scope correction

The eleven geometric vectors exist at linear order, but a finite massless-only
truncation is not known to be exact.

Correct adjudication: preserve SG-2’s low-energy endpoint; reject any stronger exact-uplift claim.

U.6 Example: wrong-ruler error

The thirteen-dimensional coefficient g_13Y differs numerically from the measured
four-dimensional hypercharge coupling, so SG-2 fails.

Why invalid: the comparison omits internal volume, boundary kinetic terms, normalization, thresholds, running, and scheme. It belongs to Scale/SG-7.

U.7 Example: genuine kinetic-health failure

After correct reduction and gauge fixing, the geometric gauge kinetic Gram matrix
has one negative physical eigenvalue.

Correct adjudication: SG-2 reopens or closes negative. A formal Lie algebra with a ghost vector is not a healthy physical gauge sector.

U.8 Example: alternative architecture

A four-dimensional principal-bundle model reproduces the same Standard Model
algebra without internal isometries.

Correct adjudication: this refutes universal uniqueness or a forces-are-isometries axiom, but not the scoped correctness of the locked hybrid branch.

U.9 Example: accidental symmetry enhancement

The actual stabilized K6 metric has a connected isometry algebra larger than su3,
producing additional exact Killing vectors.

Correct adjudication: SG-2 reopens because the physical massless vector count changes.

U.10 Example: anomaly issue

The U(1) zero mode has uncanceled fixed-plane anomalies.

Correct adjudication: anomaly gates fail and the complete branch may become inconsistent. Propagation must determine whether the physical gauge symmetry survives. SG-2 cannot be used to hide the inconsistency, but its classical mode calculation remains a distinct statement.

Appendix V - Completion-quality checklist for publication

Before Milan or another agent uses this dossier as the website authority, verify:

V.1 Content checks

• status says COMPLETED / CLOSED / RESOLVED +0;
• physical endpoint uses the corrected zero-mode certificate;
• the quotient factor is named \(I\_\chi\);
• no active sentence claims an interval \(U(1)\) isometry;
• hypercharge is an independent principal connection;
• no independent parent \(SU(3)\) or \(SU(2)\) connection is present;
• generator count is twelve;
• rank is four;
• no extra massless vector is claimed within the declared sector;
• global-group claims are deferred;
• exact truncation is not claimed;
• construction evidence is disclosed;
• residuals and reopen triggers are visible.

V.2 Technical-file checks

• Markdown parses without broken fences;
• equations have matched delimiters;
• source filenames and versions are correct;
• SHA-256 hash is recorded;
• machine-certificate JSON is syntactically valid;
• no superseded website paragraph overrides the dossier;
• all public snippets trace to claim IDs;
• the action manifest matches the spectrum input;
• the same-ruler tuple is complete;
• the thought-experiment coverage record is present.

V.3 Publication hierarchy

The public site should show, in order:

1. Completed;
2. one-sentence physical result;
3. closure type and scope;
4. link to this dossier;
5. link to machine certificate when available;
6. nonclaims and reopen triggers.

The site should not lead with historical category-floor terminology because that obscures the explicit corrected mechanism.

V.4 Reviewer transparency

The dossier should be downloadable as raw Markdown. Equation source, manifest hashes, and certificate outputs should remain visible to machine reviewers. A rendered webpage alone is not sufficient for technical reproducibility.

V.5 Final publication sentence

Gate 2 is completed because the corrected full candidate-not the interval geometry alone-contains exactly the Standard Model massless gauge Lie algebra at low energy.

Appendix Z - Final canonical certificate

GATE:
  SG-2 - MASSLESS GAUGE-ALGEBRA REALIZATION

COMPLETION STATUS:
  COMPLETED

BOARD STATUS:
  CLOSED / RESOLVED +0

LOCKED STAGE:
  M3,1 × SU(3)/T2 × S2 × I_chi

GEOMETRIC CONNECTIONS:
  su(3)c from K6 Killing vectors: 8 generators
  su(2)L from S2 Killing vectors: 3 generators

PRINCIPAL CONNECTION:
  u(1)Y from one independent B_M Actor
  B_mu even: one vector zero mode
  B_chi odd: no minimal scalar zero mode

INTERVAL:
  no connected gauge isometry
  no metric graviphoton zero mode

NO-DUPLICATION:
  no independent parent SU(3) gauge connection
  no independent parent SU(2) gauge connection

NO-EXTRA-VECTOR RESULT:
  extra massless vector generators in declared SG-2 sector = 0

OUTPUT:
  g_4,0 = su(3)c ⊕ su(2)L ⊕ u(1)Y
  dimension = 12
  rank = 4

SCOPE:
  linear massless sector and low-energy EFT gauge symmetry
  not an exact nonlinear consistent truncation theorem
  not the global gauge group
  not charge/anomaly/coupling unification
  not a blind prediction

PHYSICAL ENDPOINT:
  CLOSED-SCOPED / ZERO-MODE-ALGEBRA-CERTIFIED /
  DERIVED-GIVEN-LOCKED-SHAPE-AND-DYNAMICS /
  CONSTRUCTION-ANCHOR

CLOSURE STRENGTH:
  B - EMPIRICALLY ANCHORED, SCOPED RECONSTRUCTION

Final verdict

SG-2 is completed.

The corrected complete candidate contains exactly the twelve massless vector generators of the Standard Model gauge Lie algebra in its declared four-dimensional zero-mode sector. The result survives because the dossier no longer relies on the false claim that the quotient interval itself has a connected hypercharge isometry. Hypercharge is an explicit principal connection, color and weak are geometric connections, the Actor inventory prevents duplication, and the no-extra-vector ledger is finite and falsifiable.

The gate closes at its proper scope and does not borrow the results of later gates.

Session 1: session-fcebce2758592a4bbbbb

Decision

Consistency versus Standard-Model match. A non-SM manifest can be internally honest. Such a candidate passes all consistency rows and fails only this separate match row. The row prevents the validator from confusing honesty with identity to the incumbent.

The computed vector totals are dimension 7, rank 3, extras 0, and missing 5. The enumerated matter kernel is ORDER-36-ABELIAN with 36 surviving center elements. These values are generated from the manifest, not copied from its claims.

Complete blinded manifest

{
  "boundary_vector_zero_modes": [],
  "candidate_id": "session-fcebce2758592a4bbbbb",
  "claims": {
    "connected_components": [
      {
        "algebra": "su2",
        "carrier": "weak-a",
        "dimension": 3,
        "rank": 1
      },
      {
        "algebra": "su2",
        "carrier": "weak-b",
        "dimension": 3,
        "rank": 1
      },
      {
        "algebra": "u1",
        "carrier": "hypercharge",
        "dimension": 1,
        "rank": 1
      }
    ],
    "extra_vectors": 0,
    "matter_kernel": "NOT-CLAIMED",
    "missing_vectors": 5,
    "principal_connection_count": 1,
    "sm_match": false,
    "total_dimension": 7,
    "total_rank": 3
  },
  "geometry": {
    "factors": [
      {
        "carrier": "weak-a",
        "id": "S2_A",
        "type": "S2_round"
      },
      {
        "carrier": "weak-b",
        "id": "S2_B",
        "type": "S2_round"
      },
      {
        "carrier": null,
        "id": "I_chi",
        "type": "I_interval"
      }
    ]
  },
  "matter_characters": [],
  "p_form_vector_zero_modes": [],
  "parent_nonabelian_connections": [],
  "presentation_role": "UNDISCLOSED",
  "principal_connections": [
    {
      "algebra": "u1",
      "carrier": "hypercharge",
      "claimed_vector_zero_modes": 1,
      "dimension": 1,
      "id": "B_Y",
      "parent": "independent-principal-U1Y",
      "rank": 1,
      "scalar_parity": "odd",
      "vector_parity": "even"
    }
  ],
  "schema": "SG2-GAUNTLET-CANDIDATE-1.0",
  "scope": "massless vector sector plus reviewer-requested faithful matter kernel"
}

Executable witness

{
  "candidate_id": "session-fcebce2758592a4bbbbb",
  "center_enumeration": {
    "domain_size": 36,
    "generator": [
      1,
      1,
      1
    ],
    "kernel_elements": [
      [
        0,
        0,
        0
      ],
      [
        0,
        0,
        1
      ],
      [
        0,
        0,
        2
      ],
      [
        0,
        0,
        3
      ],
      [
        0,
        0,
        4
      ],
      [
        0,
        0,
        5
      ],
      [
        0,
        1,
        0
      ],
      [
        0,
        1,
        1
      ],
      [
        0,
        1,
        2
      ],
      [
        0,
        1,
        3
      ],
      [
        0,
        1,
        4
      ],
      [
        0,
        1,
        5
      ],
      [
        1,
        0,
        0
      ],
      [
        1,
        0,
        1
      ],
      [
        1,
        0,
        2
      ],
      [
        1,
        0,
        3
      ],
      [
        1,
        0,
        4
      ],
      [
        1,
        0,
        5
      ],
      [
        1,
        1,
        0
      ],
      [
        1,
        1,
        1
      ],
      [
        1,
        1,
        2
      ],
      [
        1,
        1,
        3
      ],
      [
        1,
        1,
        4
      ],
      [
        1,
        1,
        5
      ],
      [
        2,
        0,
        0
      ],
      [
        2,
        0,
        1
      ],
      [
        2,
        0,
        2
      ],
      [
        2,
        0,
        3
      ],
      [
        2,
        0,
        4
      ],
      [
        2,
        0,
        5
      ],
      [
        2,
        1,
        0
      ],
      [
        2,
        1,
        1
      ],
      [
        2,
        1,
        2
      ],
      [
        2,
        1,
        3
      ],
      [
        2,
        1,
        4
      ],
      [
        2,
        1,
        5
      ]
    ],
    "kernel_label": "ORDER-36-ABELIAN",
    "kernel_size": 36
  },
  "computed": {
    "extra_vectors": 0,
    "geometric_dimension": 6,
    "geometric_rank": 2,
    "missing_vectors": 5,
    "principal_dimension": 1,
    "principal_rank": 1,
    "total_dimension": 7,
    "total_rank": 3
  },
  "duplicate_carriers": {},
  "factor_checks": [
    {
      "algebra": "su2",
      "carrier": "weak-a",
      "dimension": 3,
      "id": "S2_A",
      "parent": "metric:S2_A",
      "rank": 1,
      "type": "S2_round"
    },
    {
      "algebra": "su2",
      "carrier": "weak-b",
      "dimension": 3,
      "id": "S2_B",
      "parent": "metric:S2_B",
      "rank": 1,
      "type": "S2_round"
    },
    {
      "algebra": "none",
      "carrier": null,
      "dimension": 0,
      "id": "I_chi",
      "parent": "metric:I_chi",
      "rank": 0,
      "type": "I_interval"
    }
  ],
  "no_extra_audit": {
    "boundary_vector_zero_modes": [],
    "independent_nonabelian_connections": [],
    "interval_connected_killing_dimension": 0,
    "p_form_vector_zero_modes": []
  },
  "ownership": {
    "hypercharge": [
      "independent-principal-U1Y"
    ],
    "weak-a": [
      "metric:S2_A"
    ],
    "weak-b": [
      "metric:S2_B"
    ]
  },
  "parity_table": [
    {
      "algebra": "u1",
      "carrier": "hypercharge",
      "claimed_vector_zero_modes": 1,
      "computed_vector_zero_modes": 1,
      "dimension": 1,
      "id": "B_Y",
      "parent": "independent-principal-U1Y",
      "rank": 1,
      "scalar_parity": "odd",
      "vector_parity": "even"
    }
  ],
  "schema": "SG2-VECTOR-SECTOR-CERTIFICATE-1.0",
  "smith_certificate": null
}

Session 2: session-590e8c8330106fe8315b

Decision

One primitive parent per carrier. The ownership join is keyed by the physical carrier. Zero parents means missing provenance; two or more primitive parents means duplication unless an executed mixing calculation removes the redundant combination.

The computed vector totals are dimension 15, rank 5, extras 3, and missing 0. The enumerated matter kernel is Z6 with 6 surviving center elements. These values are generated from the manifest, not copied from its claims.

Complete blinded manifest

{
  "boundary_vector_zero_modes": [],
  "candidate_id": "session-590e8c8330106fe8315b",
  "claims": {
    "connected_components": [
      {
        "algebra": "su3",
        "carrier": "color",
        "dimension": 8,
        "rank": 2
      },
      {
        "algebra": "su2",
        "carrier": "weak",
        "dimension": 3,
        "rank": 1
      },
      {
        "algebra": "u1",
        "carrier": "hypercharge",
        "dimension": 1,
        "rank": 1
      }
    ],
    "extra_vectors": 0,
    "matter_kernel": "Z6",
    "missing_vectors": 0,
    "principal_connection_count": 1,
    "sm_match": true,
    "total_dimension": 12,
    "total_rank": 4
  },
  "geometry": {
    "factors": [
      {
        "carrier": "color",
        "id": "K6",
        "type": "SU3/T2"
      },
      {
        "carrier": "weak",
        "id": "S2",
        "type": "S2_round"
      },
      {
        "carrier": null,
        "id": "I_chi",
        "type": "I_interval"
      }
    ]
  },
  "matter_characters": [
    {
      "doublet": 1,
      "id": "Q",
      "triality": 1,
      "y6": 1
    },
    {
      "doublet": 0,
      "id": "u_c",
      "triality": -1,
      "y6": -4
    },
    {
      "doublet": 0,
      "id": "d_c",
      "triality": -1,
      "y6": 2
    },
    {
      "doublet": 1,
      "id": "L",
      "triality": 0,
      "y6": -3
    },
    {
      "doublet": 0,
      "id": "e_c",
      "triality": 0,
      "y6": 6
    },
    {
      "doublet": 0,
      "id": "nu_c",
      "triality": 0,
      "y6": 0
    },
    {
      "doublet": 1,
      "id": "H",
      "triality": 0,
      "y6": 3
    }
  ],
  "p_form_vector_zero_modes": [],
  "parent_nonabelian_connections": [
    {
      "algebra": "su2",
      "carrier": "weak",
      "dimension": 3,
      "id": "A_weak_independent",
      "parent": "independent-principal-SU2",
      "rank": 1
    }
  ],
  "presentation_role": "UNDISCLOSED",
  "principal_connections": [
    {
      "algebra": "u1",
      "carrier": "hypercharge",
      "claimed_vector_zero_modes": 1,
      "dimension": 1,
      "id": "B_Y",
      "parent": "independent-principal-U1Y",
      "rank": 1,
      "scalar_parity": "odd",
      "vector_parity": "even"
    }
  ],
  "schema": "SG2-GAUNTLET-CANDIDATE-1.0",
  "scope": "massless vector sector plus reviewer-requested faithful matter kernel"
}

Executable witness

{
  "candidate_id": "session-590e8c8330106fe8315b",
  "center_enumeration": {
    "domain_size": 36,
    "generator": [
      1,
      1,
      1
    ],
    "kernel_elements": [
      [
        0,
        0,
        0
      ],
      [
        0,
        1,
        3
      ],
      [
        1,
        0,
        4
      ],
      [
        1,
        1,
        1
      ],
      [
        2,
        0,
        2
      ],
      [
        2,
        1,
        5
      ]
    ],
    "kernel_label": "Z6",
    "kernel_size": 6
  },
  "computed": {
    "extra_vectors": 3,
    "geometric_dimension": 11,
    "geometric_rank": 3,
    "missing_vectors": 0,
    "principal_dimension": 1,
    "principal_rank": 1,
    "total_dimension": 15,
    "total_rank": 5
  },
  "duplicate_carriers": {
    "weak": [
      "metric:S2",
      "independent-principal-SU2"
    ]
  },
  "factor_checks": [
    {
      "algebra": "su3",
      "carrier": "color",
      "dimension": 8,
      "id": "K6",
      "parent": "metric:K6",
      "rank": 2,
      "type": "SU3/T2"
    },
    {
      "algebra": "su2",
      "carrier": "weak",
      "dimension": 3,
      "id": "S2",
      "parent": "metric:S2",
      "rank": 1,
      "type": "S2_round"
    },
    {
      "algebra": "none",
      "carrier": null,
      "dimension": 0,
      "id": "I_chi",
      "parent": "metric:I_chi",
      "rank": 0,
      "type": "I_interval"
    }
  ],
  "no_extra_audit": {
    "boundary_vector_zero_modes": [],
    "independent_nonabelian_connections": [
      {
        "algebra": "su2",
        "carrier": "weak",
        "dimension": 3,
        "id": "A_weak_independent",
        "parent": "independent-principal-SU2",
        "rank": 1
      }
    ],
    "interval_connected_killing_dimension": 0,
    "p_form_vector_zero_modes": []
  },
  "ownership": {
    "color": [
      "metric:K6"
    ],
    "hypercharge": [
      "independent-principal-U1Y"
    ],
    "weak": [
      "metric:S2",
      "independent-principal-SU2"
    ]
  },
  "parity_table": [
    {
      "algebra": "u1",
      "carrier": "hypercharge",
      "claimed_vector_zero_modes": 1,
      "computed_vector_zero_modes": 1,
      "dimension": 1,
      "id": "B_Y",
      "parent": "independent-principal-U1Y",
      "rank": 1,
      "scalar_parity": "odd",
      "vector_parity": "even"
    }
  ],
  "schema": "SG2-VECTOR-SECTOR-CERTIFICATE-1.0",
  "smith_certificate": {
    "basis_convention": "center tuple (a mod 3, b mod 2, k mod 6)",
    "domain_relation_D": [
      [
        3,
        0,
        0
      ],
      [
        0,
        2,
        0
      ],
      [
        0,
        0,
        6
      ]
    ],
    "kernel_group": "Z6",
    "kernel_lift_basis_B": [
      [
        1,
        3,
        0
      ],
      [
        1,
        0,
        2
      ],
      [
        1,
        0,
        0
      ]
    ],
    "phase_rule": "2*triality*a + 3*doublet*b + y6*k = 0 mod 6",
    "quotient_relation_M": [
      [
        0,
        0,
        6
      ],
      [
        1,
        0,
        -2
      ],
      [
        0,
        1,
        -3
      ]
    ],
    "smith_invariants": [
      1,
      1,
      6
    ],
    "verified_B_times_M_equals_D": true
  }
}

Session 3: session-69b5108731469c3ae360

Decision

Active-factor and connected-Killing completeness. Every nonzero connected Killing contribution of the active quotient must appear in the declared component ledger with the same algebra, dimension, and rank. Covering-space generators do not survive automatically.

The computed vector totals are dimension 13, rank 5, extras 1, and missing 0. The enumerated matter kernel is Z6 with 6 surviving center elements. These values are generated from the manifest, not copied from its claims.

Complete blinded manifest

{
  "boundary_vector_zero_modes": [],
  "candidate_id": "session-69b5108731469c3ae360",
  "claims": {
    "connected_components": [
      {
        "algebra": "su3",
        "carrier": "color",
        "dimension": 8,
        "rank": 2
      },
      {
        "algebra": "su2",
        "carrier": "weak",
        "dimension": 3,
        "rank": 1
      },
      {
        "algebra": "u1",
        "carrier": "hypercharge",
        "dimension": 1,
        "rank": 1
      }
    ],
    "extra_vectors": 0,
    "matter_kernel": "Z6",
    "missing_vectors": 0,
    "principal_connection_count": 1,
    "sm_match": true,
    "total_dimension": 12,
    "total_rank": 4
  },
  "geometry": {
    "factors": [
      {
        "carrier": "color",
        "id": "K6",
        "type": "SU3/T2"
      },
      {
        "carrier": "weak",
        "id": "S2",
        "type": "S2_round"
      },
      {
        "carrier": "circle-u1",
        "id": "S1_Y",
        "type": "S1_circle"
      }
    ]
  },
  "matter_characters": [
    {
      "doublet": 1,
      "id": "Q",
      "triality": 1,
      "y6": 1
    },
    {
      "doublet": 0,
      "id": "u_c",
      "triality": -1,
      "y6": -4
    },
    {
      "doublet": 0,
      "id": "d_c",
      "triality": -1,
      "y6": 2
    },
    {
      "doublet": 1,
      "id": "L",
      "triality": 0,
      "y6": -3
    },
    {
      "doublet": 0,
      "id": "e_c",
      "triality": 0,
      "y6": 6
    },
    {
      "doublet": 0,
      "id": "nu_c",
      "triality": 0,
      "y6": 0
    },
    {
      "doublet": 1,
      "id": "H",
      "triality": 0,
      "y6": 3
    }
  ],
  "p_form_vector_zero_modes": [],
  "parent_nonabelian_connections": [],
  "presentation_role": "UNDISCLOSED",
  "principal_connections": [
    {
      "algebra": "u1",
      "carrier": "hypercharge",
      "claimed_vector_zero_modes": 1,
      "dimension": 1,
      "id": "B_Y",
      "parent": "independent-principal-U1Y",
      "rank": 1,
      "scalar_parity": "odd",
      "vector_parity": "even"
    }
  ],
  "schema": "SG2-GAUNTLET-CANDIDATE-1.0",
  "scope": "massless vector sector plus reviewer-requested faithful matter kernel"
}

Executable witness

{
  "candidate_id": "session-69b5108731469c3ae360",
  "center_enumeration": {
    "domain_size": 36,
    "generator": [
      1,
      1,
      1
    ],
    "kernel_elements": [
      [
        0,
        0,
        0
      ],
      [
        0,
        1,
        3
      ],
      [
        1,
        0,
        4
      ],
      [
        1,
        1,
        1
      ],
      [
        2,
        0,
        2
      ],
      [
        2,
        1,
        5
      ]
    ],
    "kernel_label": "Z6",
    "kernel_size": 6
  },
  "computed": {
    "extra_vectors": 1,
    "geometric_dimension": 12,
    "geometric_rank": 4,
    "missing_vectors": 0,
    "principal_dimension": 1,
    "principal_rank": 1,
    "total_dimension": 13,
    "total_rank": 5
  },
  "duplicate_carriers": {},
  "factor_checks": [
    {
      "algebra": "su3",
      "carrier": "color",
      "dimension": 8,
      "id": "K6",
      "parent": "metric:K6",
      "rank": 2,
      "type": "SU3/T2"
    },
    {
      "algebra": "su2",
      "carrier": "weak",
      "dimension": 3,
      "id": "S2",
      "parent": "metric:S2",
      "rank": 1,
      "type": "S2_round"
    },
    {
      "algebra": "u1",
      "carrier": "circle-u1",
      "dimension": 1,
      "id": "S1_Y",
      "parent": "metric:S1_Y",
      "rank": 1,
      "type": "S1_circle"
    }
  ],
  "no_extra_audit": {
    "boundary_vector_zero_modes": [],
    "independent_nonabelian_connections": [],
    "interval_connected_killing_dimension": 0,
    "p_form_vector_zero_modes": []
  },
  "ownership": {
    "circle-u1": [
      "metric:S1_Y"
    ],
    "color": [
      "metric:K6"
    ],
    "hypercharge": [
      "independent-principal-U1Y"
    ],
    "weak": [
      "metric:S2"
    ]
  },
  "parity_table": [
    {
      "algebra": "u1",
      "carrier": "hypercharge",
      "claimed_vector_zero_modes": 1,
      "computed_vector_zero_modes": 1,
      "dimension": 1,
      "id": "B_Y",
      "parent": "independent-principal-U1Y",
      "rank": 1,
      "scalar_parity": "odd",
      "vector_parity": "even"
    }
  ],
  "schema": "SG2-VECTOR-SECTOR-CERTIFICATE-1.0",
  "smith_certificate": {
    "basis_convention": "center tuple (a mod 3, b mod 2, k mod 6)",
    "domain_relation_D": [
      [
        3,
        0,
        0
      ],
      [
        0,
        2,
        0
      ],
      [
        0,
        0,
        6
      ]
    ],
    "kernel_group": "Z6",
    "kernel_lift_basis_B": [
      [
        1,
        3,
        0
      ],
      [
        1,
        0,
        2
      ],
      [
        1,
        0,
        0
      ]
    ],
    "phase_rule": "2*triality*a + 3*doublet*b + y6*k = 0 mod 6",
    "quotient_relation_M": [
      [
        0,
        0,
        6
      ],
      [
        1,
        0,
        -2
      ],
      [
        0,
        1,
        -3
      ]
    ],
    "smith_invariants": [
      1,
      1,
      6
    ],
    "verified_B_times_M_equals_D": true
  }
}

Session 4: session-37f6ea093d2cac15c5e9

Decision

Consistency versus Standard-Model match. A non-SM manifest can be internally honest. Such a candidate passes all consistency rows and fails only this separate match row. The row prevents the validator from confusing honesty with identity to the incumbent.

The computed vector totals are dimension 13, rank 5, extras 1, and missing 0. The enumerated matter kernel is Z6 with 6 surviving center elements. These values are generated from the manifest, not copied from its claims.

Complete blinded manifest

{
  "boundary_vector_zero_modes": [],
  "candidate_id": "session-37f6ea093d2cac15c5e9",
  "claims": {
    "connected_components": [
      {
        "algebra": "su3",
        "carrier": "color",
        "dimension": 8,
        "rank": 2
      },
      {
        "algebra": "su2",
        "carrier": "weak",
        "dimension": 3,
        "rank": 1
      },
      {
        "algebra": "u1",
        "carrier": "hypercharge",
        "dimension": 1,
        "rank": 1
      },
      {
        "algebra": "u1",
        "carrier": "extra-u1",
        "dimension": 1,
        "rank": 1
      }
    ],
    "extra_vectors": 1,
    "matter_kernel": "Z6",
    "missing_vectors": 0,
    "principal_connection_count": 1,
    "sm_match": false,
    "total_dimension": 13,
    "total_rank": 5
  },
  "geometry": {
    "factors": [
      {
        "carrier": "color",
        "id": "K6",
        "type": "SU3/T2"
      },
      {
        "carrier": "weak",
        "id": "S2",
        "type": "S2_round"
      },
      {
        "carrier": "extra-u1",
        "id": "S1_X",
        "type": "S1_circle"
      }
    ]
  },
  "matter_characters": [
    {
      "doublet": 1,
      "id": "Q",
      "triality": 1,
      "y6": 1
    },
    {
      "doublet": 0,
      "id": "u_c",
      "triality": -1,
      "y6": -4
    },
    {
      "doublet": 0,
      "id": "d_c",
      "triality": -1,
      "y6": 2
    },
    {
      "doublet": 1,
      "id": "L",
      "triality": 0,
      "y6": -3
    },
    {
      "doublet": 0,
      "id": "e_c",
      "triality": 0,
      "y6": 6
    },
    {
      "doublet": 0,
      "id": "nu_c",
      "triality": 0,
      "y6": 0
    },
    {
      "doublet": 1,
      "id": "H",
      "triality": 0,
      "y6": 3
    }
  ],
  "p_form_vector_zero_modes": [],
  "parent_nonabelian_connections": [],
  "presentation_role": "UNDISCLOSED",
  "principal_connections": [
    {
      "algebra": "u1",
      "carrier": "hypercharge",
      "claimed_vector_zero_modes": 1,
      "dimension": 1,
      "id": "B_Y",
      "parent": "independent-principal-U1Y",
      "rank": 1,
      "scalar_parity": "odd",
      "vector_parity": "even"
    }
  ],
  "schema": "SG2-GAUNTLET-CANDIDATE-1.0",
  "scope": "massless vector sector plus reviewer-requested faithful matter kernel"
}

Executable witness

{
  "candidate_id": "session-37f6ea093d2cac15c5e9",
  "center_enumeration": {
    "domain_size": 36,
    "generator": [
      1,
      1,
      1
    ],
    "kernel_elements": [
      [
        0,
        0,
        0
      ],
      [
        0,
        1,
        3
      ],
      [
        1,
        0,
        4
      ],
      [
        1,
        1,
        1
      ],
      [
        2,
        0,
        2
      ],
      [
        2,
        1,
        5
      ]
    ],
    "kernel_label": "Z6",
    "kernel_size": 6
  },
  "computed": {
    "extra_vectors": 1,
    "geometric_dimension": 12,
    "geometric_rank": 4,
    "missing_vectors": 0,
    "principal_dimension": 1,
    "principal_rank": 1,
    "total_dimension": 13,
    "total_rank": 5
  },
  "duplicate_carriers": {},
  "factor_checks": [
    {
      "algebra": "su3",
      "carrier": "color",
      "dimension": 8,
      "id": "K6",
      "parent": "metric:K6",
      "rank": 2,
      "type": "SU3/T2"
    },
    {
      "algebra": "su2",
      "carrier": "weak",
      "dimension": 3,
      "id": "S2",
      "parent": "metric:S2",
      "rank": 1,
      "type": "S2_round"
    },
    {
      "algebra": "u1",
      "carrier": "extra-u1",
      "dimension": 1,
      "id": "S1_X",
      "parent": "metric:S1_X",
      "rank": 1,
      "type": "S1_circle"
    }
  ],
  "no_extra_audit": {
    "boundary_vector_zero_modes": [],
    "independent_nonabelian_connections": [],
    "interval_connected_killing_dimension": 0,
    "p_form_vector_zero_modes": []
  },
  "ownership": {
    "color": [
      "metric:K6"
    ],
    "extra-u1": [
      "metric:S1_X"
    ],
    "hypercharge": [
      "independent-principal-U1Y"
    ],
    "weak": [
      "metric:S2"
    ]
  },
  "parity_table": [
    {
      "algebra": "u1",
      "carrier": "hypercharge",
      "claimed_vector_zero_modes": 1,
      "computed_vector_zero_modes": 1,
      "dimension": 1,
      "id": "B_Y",
      "parent": "independent-principal-U1Y",
      "rank": 1,
      "scalar_parity": "odd",
      "vector_parity": "even"
    }
  ],
  "schema": "SG2-VECTOR-SECTOR-CERTIFICATE-1.0",
  "smith_certificate": {
    "basis_convention": "center tuple (a mod 3, b mod 2, k mod 6)",
    "domain_relation_D": [
      [
        3,
        0,
        0
      ],
      [
        0,
        2,
        0
      ],
      [
        0,
        0,
        6
      ]
    ],
    "kernel_group": "Z6",
    "kernel_lift_basis_B": [
      [
        1,
        3,
        0
      ],
      [
        1,
        0,
        2
      ],
      [
        1,
        0,
        0
      ]
    ],
    "phase_rule": "2*triality*a + 3*doublet*b + y6*k = 0 mod 6",
    "quotient_relation_M": [
      [
        0,
        0,
        6
      ],
      [
        1,
        0,
        -2
      ],
      [
        0,
        1,
        -3
      ]
    ],
    "smith_invariants": [
      1,
      1,
      6
    ],
    "verified_B_times_M_equals_D": true
  }
}

Session 5: session-447b5b256e606e8b30d5

Decision

Independent-connection inventory. The number of principal connections in the Actor inventory must equal the number claimed. A second undeclared U(1) is an additional massless-vector source, not a harmless relabeling.

The computed vector totals are dimension 13, rank 5, extras 1, and missing 0. The enumerated matter kernel is Z6 with 6 surviving center elements. These values are generated from the manifest, not copied from its claims.

Complete blinded manifest

{
  "boundary_vector_zero_modes": [],
  "candidate_id": "session-447b5b256e606e8b30d5",
  "claims": {
    "connected_components": [
      {
        "algebra": "su3",
        "carrier": "color",
        "dimension": 8,
        "rank": 2
      },
      {
        "algebra": "su2",
        "carrier": "weak",
        "dimension": 3,
        "rank": 1
      },
      {
        "algebra": "u1",
        "carrier": "hypercharge",
        "dimension": 1,
        "rank": 1
      }
    ],
    "extra_vectors": 0,
    "matter_kernel": "Z6",
    "missing_vectors": 0,
    "principal_connection_count": 1,
    "sm_match": true,
    "total_dimension": 12,
    "total_rank": 4
  },
  "geometry": {
    "factors": [
      {
        "carrier": "color",
        "id": "K6",
        "type": "SU3/T2"
      },
      {
        "carrier": "weak",
        "id": "S2",
        "type": "S2_round"
      },
      {
        "carrier": null,
        "id": "I_chi",
        "type": "I_interval"
      }
    ]
  },
  "matter_characters": [
    {
      "doublet": 1,
      "id": "Q",
      "triality": 1,
      "y6": 1
    },
    {
      "doublet": 0,
      "id": "u_c",
      "triality": -1,
      "y6": -4
    },
    {
      "doublet": 0,
      "id": "d_c",
      "triality": -1,
      "y6": 2
    },
    {
      "doublet": 1,
      "id": "L",
      "triality": 0,
      "y6": -3
    },
    {
      "doublet": 0,
      "id": "e_c",
      "triality": 0,
      "y6": 6
    },
    {
      "doublet": 0,
      "id": "nu_c",
      "triality": 0,
      "y6": 0
    },
    {
      "doublet": 1,
      "id": "H",
      "triality": 0,
      "y6": 3
    }
  ],
  "p_form_vector_zero_modes": [],
  "parent_nonabelian_connections": [],
  "presentation_role": "UNDISCLOSED",
  "principal_connections": [
    {
      "algebra": "u1",
      "carrier": "hypercharge",
      "claimed_vector_zero_modes": 1,
      "dimension": 1,
      "id": "B_Y",
      "parent": "independent-principal-U1Y",
      "rank": 1,
      "scalar_parity": "odd",
      "vector_parity": "even"
    },
    {
      "algebra": "u1",
      "carrier": "extra-u1",
      "claimed_vector_zero_modes": 1,
      "dimension": 1,
      "id": "B_X",
      "parent": "undeclared-second-principal-U1",
      "rank": 1,
      "scalar_parity": "odd",
      "vector_parity": "even"
    }
  ],
  "schema": "SG2-GAUNTLET-CANDIDATE-1.0",
  "scope": "massless vector sector plus reviewer-requested faithful matter kernel"
}

Executable witness

{
  "candidate_id": "session-447b5b256e606e8b30d5",
  "center_enumeration": {
    "domain_size": 36,
    "generator": [
      1,
      1,
      1
    ],
    "kernel_elements": [
      [
        0,
        0,
        0
      ],
      [
        0,
        1,
        3
      ],
      [
        1,
        0,
        4
      ],
      [
        1,
        1,
        1
      ],
      [
        2,
        0,
        2
      ],
      [
        2,
        1,
        5
      ]
    ],
    "kernel_label": "Z6",
    "kernel_size": 6
  },
  "computed": {
    "extra_vectors": 1,
    "geometric_dimension": 11,
    "geometric_rank": 3,
    "missing_vectors": 0,
    "principal_dimension": 2,
    "principal_rank": 2,
    "total_dimension": 13,
    "total_rank": 5
  },
  "duplicate_carriers": {},
  "factor_checks": [
    {
      "algebra": "su3",
      "carrier": "color",
      "dimension": 8,
      "id": "K6",
      "parent": "metric:K6",
      "rank": 2,
      "type": "SU3/T2"
    },
    {
      "algebra": "su2",
      "carrier": "weak",
      "dimension": 3,
      "id": "S2",
      "parent": "metric:S2",
      "rank": 1,
      "type": "S2_round"
    },
    {
      "algebra": "none",
      "carrier": null,
      "dimension": 0,
      "id": "I_chi",
      "parent": "metric:I_chi",
      "rank": 0,
      "type": "I_interval"
    }
  ],
  "no_extra_audit": {
    "boundary_vector_zero_modes": [],
    "independent_nonabelian_connections": [],
    "interval_connected_killing_dimension": 0,
    "p_form_vector_zero_modes": []
  },
  "ownership": {
    "color": [
      "metric:K6"
    ],
    "extra-u1": [
      "undeclared-second-principal-U1"
    ],
    "hypercharge": [
      "independent-principal-U1Y"
    ],
    "weak": [
      "metric:S2"
    ]
  },
  "parity_table": [
    {
      "algebra": "u1",
      "carrier": "hypercharge",
      "claimed_vector_zero_modes": 1,
      "computed_vector_zero_modes": 1,
      "dimension": 1,
      "id": "B_Y",
      "parent": "independent-principal-U1Y",
      "rank": 1,
      "scalar_parity": "odd",
      "vector_parity": "even"
    },
    {
      "algebra": "u1",
      "carrier": "extra-u1",
      "claimed_vector_zero_modes": 1,
      "computed_vector_zero_modes": 1,
      "dimension": 1,
      "id": "B_X",
      "parent": "undeclared-second-principal-U1",
      "rank": 1,
      "scalar_parity": "odd",
      "vector_parity": "even"
    }
  ],
  "schema": "SG2-VECTOR-SECTOR-CERTIFICATE-1.0",
  "smith_certificate": {
    "basis_convention": "center tuple (a mod 3, b mod 2, k mod 6)",
    "domain_relation_D": [
      [
        3,
        0,
        0
      ],
      [
        0,
        2,
        0
      ],
      [
        0,
        0,
        6
      ]
    ],
    "kernel_group": "Z6",
    "kernel_lift_basis_B": [
      [
        1,
        3,
        0
      ],
      [
        1,
        0,
        2
      ],
      [
        1,
        0,
        0
      ]
    ],
    "phase_rule": "2*triality*a + 3*doublet*b + y6*k = 0 mod 6",
    "quotient_relation_M": [
      [
        0,
        0,
        6
      ],
      [
        1,
        0,
        -2
      ],
      [
        0,
        1,
        -3
      ]
    ],
    "smith_invariants": [
      1,
      1,
      6
    ],
    "verified_B_times_M_equals_D": true
  }
}

Session 6: session-1962e5f81ef9b8287448

Decision

Matter-faithful global kernel. When a global kernel is claimed, enumerate all 36 center elements against the published matter characters and require the computed group to match. For the incumbent the lattice quotient is independently certified by SNF.

The computed vector totals are dimension 12, rank 4, extras 0, and missing 0. The enumerated matter kernel is Z2 with 2 surviving center elements. These values are generated from the manifest, not copied from its claims.

Complete blinded manifest

{
  "boundary_vector_zero_modes": [],
  "candidate_id": "session-1962e5f81ef9b8287448",
  "claims": {
    "connected_components": [
      {
        "algebra": "su3",
        "carrier": "color",
        "dimension": 8,
        "rank": 2
      },
      {
        "algebra": "su2",
        "carrier": "weak",
        "dimension": 3,
        "rank": 1
      },
      {
        "algebra": "u1",
        "carrier": "hypercharge",
        "dimension": 1,
        "rank": 1
      }
    ],
    "extra_vectors": 0,
    "matter_kernel": "Z6",
    "missing_vectors": 0,
    "principal_connection_count": 1,
    "sm_match": true,
    "total_dimension": 12,
    "total_rank": 4
  },
  "geometry": {
    "factors": [
      {
        "carrier": "color",
        "id": "K6",
        "type": "SU3/T2"
      },
      {
        "carrier": "weak",
        "id": "S2",
        "type": "S2_round"
      },
      {
        "carrier": null,
        "id": "I_chi",
        "type": "I_interval"
      }
    ]
  },
  "matter_characters": [
    {
      "doublet": 1,
      "id": "Q",
      "triality": 1,
      "y6": -3
    },
    {
      "doublet": 0,
      "id": "u_c",
      "triality": -1,
      "y6": -4
    },
    {
      "doublet": 0,
      "id": "d_c",
      "triality": -1,
      "y6": 2
    },
    {
      "doublet": 1,
      "id": "L",
      "triality": 0,
      "y6": -3
    },
    {
      "doublet": 0,
      "id": "e_c",
      "triality": 0,
      "y6": 6
    },
    {
      "doublet": 0,
      "id": "nu_c",
      "triality": 0,
      "y6": 0
    },
    {
      "doublet": 1,
      "id": "H",
      "triality": 0,
      "y6": 3
    }
  ],
  "p_form_vector_zero_modes": [],
  "parent_nonabelian_connections": [],
  "presentation_role": "UNDISCLOSED",
  "principal_connections": [
    {
      "algebra": "u1",
      "carrier": "hypercharge",
      "claimed_vector_zero_modes": 1,
      "dimension": 1,
      "id": "B_Y",
      "parent": "independent-principal-U1Y",
      "rank": 1,
      "scalar_parity": "odd",
      "vector_parity": "even"
    }
  ],
  "schema": "SG2-GAUNTLET-CANDIDATE-1.0",
  "scope": "massless vector sector plus reviewer-requested faithful matter kernel"
}

Executable witness

{
  "candidate_id": "session-1962e5f81ef9b8287448",
  "center_enumeration": {
    "domain_size": 36,
    "generator": null,
    "kernel_elements": [
      [
        0,
        0,
        0
      ],
      [
        0,
        1,
        3
      ]
    ],
    "kernel_label": "Z2",
    "kernel_size": 2
  },
  "computed": {
    "extra_vectors": 0,
    "geometric_dimension": 11,
    "geometric_rank": 3,
    "missing_vectors": 0,
    "principal_dimension": 1,
    "principal_rank": 1,
    "total_dimension": 12,
    "total_rank": 4
  },
  "duplicate_carriers": {},
  "factor_checks": [
    {
      "algebra": "su3",
      "carrier": "color",
      "dimension": 8,
      "id": "K6",
      "parent": "metric:K6",
      "rank": 2,
      "type": "SU3/T2"
    },
    {
      "algebra": "su2",
      "carrier": "weak",
      "dimension": 3,
      "id": "S2",
      "parent": "metric:S2",
      "rank": 1,
      "type": "S2_round"
    },
    {
      "algebra": "none",
      "carrier": null,
      "dimension": 0,
      "id": "I_chi",
      "parent": "metric:I_chi",
      "rank": 0,
      "type": "I_interval"
    }
  ],
  "no_extra_audit": {
    "boundary_vector_zero_modes": [],
    "independent_nonabelian_connections": [],
    "interval_connected_killing_dimension": 0,
    "p_form_vector_zero_modes": []
  },
  "ownership": {
    "color": [
      "metric:K6"
    ],
    "hypercharge": [
      "independent-principal-U1Y"
    ],
    "weak": [
      "metric:S2"
    ]
  },
  "parity_table": [
    {
      "algebra": "u1",
      "carrier": "hypercharge",
      "claimed_vector_zero_modes": 1,
      "computed_vector_zero_modes": 1,
      "dimension": 1,
      "id": "B_Y",
      "parent": "independent-principal-U1Y",
      "rank": 1,
      "scalar_parity": "odd",
      "vector_parity": "even"
    }
  ],
  "schema": "SG2-VECTOR-SECTOR-CERTIFICATE-1.0",
  "smith_certificate": null
}

Session 7: session-3d9d0d855e9f3acfbd7b

Decision

All ordered controls discharged. The incumbent manifest matches its active geometry, Actor inventory, ownership, parities, exact vector totals, and faithful center kernel.

The computed vector totals are dimension 12, rank 4, extras 0, and missing 0. The enumerated matter kernel is Z6 with 6 surviving center elements. These values are generated from the manifest, not copied from its claims.

Complete blinded manifest

{
  "boundary_vector_zero_modes": [],
  "candidate_id": "session-3d9d0d855e9f3acfbd7b",
  "claims": {
    "connected_components": [
      {
        "algebra": "su3",
        "carrier": "color",
        "dimension": 8,
        "rank": 2
      },
      {
        "algebra": "su2",
        "carrier": "weak",
        "dimension": 3,
        "rank": 1
      },
      {
        "algebra": "u1",
        "carrier": "hypercharge",
        "dimension": 1,
        "rank": 1
      }
    ],
    "extra_vectors": 0,
    "matter_kernel": "Z6",
    "missing_vectors": 0,
    "principal_connection_count": 1,
    "sm_match": true,
    "total_dimension": 12,
    "total_rank": 4
  },
  "geometry": {
    "factors": [
      {
        "carrier": "color",
        "id": "K6",
        "type": "SU3/T2"
      },
      {
        "carrier": "weak",
        "id": "S2",
        "type": "S2_round"
      },
      {
        "carrier": null,
        "id": "I_chi",
        "type": "I_interval"
      }
    ]
  },
  "matter_characters": [
    {
      "doublet": 1,
      "id": "Q",
      "triality": 1,
      "y6": 1
    },
    {
      "doublet": 0,
      "id": "u_c",
      "triality": -1,
      "y6": -4
    },
    {
      "doublet": 0,
      "id": "d_c",
      "triality": -1,
      "y6": 2
    },
    {
      "doublet": 1,
      "id": "L",
      "triality": 0,
      "y6": -3
    },
    {
      "doublet": 0,
      "id": "e_c",
      "triality": 0,
      "y6": 6
    },
    {
      "doublet": 0,
      "id": "nu_c",
      "triality": 0,
      "y6": 0
    },
    {
      "doublet": 1,
      "id": "H",
      "triality": 0,
      "y6": 3
    }
  ],
  "p_form_vector_zero_modes": [],
  "parent_nonabelian_connections": [],
  "presentation_role": "UNDISCLOSED",
  "principal_connections": [
    {
      "algebra": "u1",
      "carrier": "hypercharge",
      "claimed_vector_zero_modes": 1,
      "dimension": 1,
      "id": "B_Y",
      "parent": "independent-principal-U1Y",
      "rank": 1,
      "scalar_parity": "odd",
      "vector_parity": "even"
    }
  ],
  "schema": "SG2-GAUNTLET-CANDIDATE-1.0",
  "scope": "massless vector sector plus reviewer-requested faithful matter kernel"
}

Executable witness

{
  "candidate_id": "session-3d9d0d855e9f3acfbd7b",
  "center_enumeration": {
    "domain_size": 36,
    "generator": [
      1,
      1,
      1
    ],
    "kernel_elements": [
      [
        0,
        0,
        0
      ],
      [
        0,
        1,
        3
      ],
      [
        1,
        0,
        4
      ],
      [
        1,
        1,
        1
      ],
      [
        2,
        0,
        2
      ],
      [
        2,
        1,
        5
      ]
    ],
    "kernel_label": "Z6",
    "kernel_size": 6
  },
  "computed": {
    "extra_vectors": 0,
    "geometric_dimension": 11,
    "geometric_rank": 3,
    "missing_vectors": 0,
    "principal_dimension": 1,
    "principal_rank": 1,
    "total_dimension": 12,
    "total_rank": 4
  },
  "duplicate_carriers": {},
  "factor_checks": [
    {
      "algebra": "su3",
      "carrier": "color",
      "dimension": 8,
      "id": "K6",
      "parent": "metric:K6",
      "rank": 2,
      "type": "SU3/T2"
    },
    {
      "algebra": "su2",
      "carrier": "weak",
      "dimension": 3,
      "id": "S2",
      "parent": "metric:S2",
      "rank": 1,
      "type": "S2_round"
    },
    {
      "algebra": "none",
      "carrier": null,
      "dimension": 0,
      "id": "I_chi",
      "parent": "metric:I_chi",
      "rank": 0,
      "type": "I_interval"
    }
  ],
  "no_extra_audit": {
    "boundary_vector_zero_modes": [],
    "independent_nonabelian_connections": [],
    "interval_connected_killing_dimension": 0,
    "p_form_vector_zero_modes": []
  },
  "ownership": {
    "color": [
      "metric:K6"
    ],
    "hypercharge": [
      "independent-principal-U1Y"
    ],
    "weak": [
      "metric:S2"
    ]
  },
  "parity_table": [
    {
      "algebra": "u1",
      "carrier": "hypercharge",
      "claimed_vector_zero_modes": 1,
      "computed_vector_zero_modes": 1,
      "dimension": 1,
      "id": "B_Y",
      "parent": "independent-principal-U1Y",
      "rank": 1,
      "scalar_parity": "odd",
      "vector_parity": "even"
    }
  ],
  "schema": "SG2-VECTOR-SECTOR-CERTIFICATE-1.0",
  "smith_certificate": {
    "basis_convention": "center tuple (a mod 3, b mod 2, k mod 6)",
    "domain_relation_D": [
      [
        3,
        0,
        0
      ],
      [
        0,
        2,
        0
      ],
      [
        0,
        0,
        6
      ]
    ],
    "kernel_group": "Z6",
    "kernel_lift_basis_B": [
      [
        1,
        3,
        0
      ],
      [
        1,
        0,
        2
      ],
      [
        1,
        0,
        0
      ]
    ],
    "phase_rule": "2*triality*a + 3*doublet*b + y6*k = 0 mod 6",
    "quotient_relation_M": [
      [
        0,
        0,
        6
      ],
      [
        1,
        0,
        -2
      ],
      [
        0,
        1,
        -3
      ]
    ],
    "smith_invariants": [
      1,
      1,
      6
    ],
    "verified_B_times_M_equals_D": true
  }
}

Session 8: session-5fa9e02946c18581b740

Decision

Parity and zero-mode agreement. For every independent connection, the claimed vector zero-mode count must equal the count implied by the declared vector parity. In this finite rehearsal an even vector has one constant mode and an odd vector has none.

The computed vector totals are dimension 11, rank 3, extras 0, and missing 1. The enumerated matter kernel is Z6 with 6 surviving center elements. These values are generated from the manifest, not copied from its claims.

Complete blinded manifest

{
  "boundary_vector_zero_modes": [],
  "candidate_id": "session-5fa9e02946c18581b740",
  "claims": {
    "connected_components": [
      {
        "algebra": "su3",
        "carrier": "color",
        "dimension": 8,
        "rank": 2
      },
      {
        "algebra": "su2",
        "carrier": "weak",
        "dimension": 3,
        "rank": 1
      },
      {
        "algebra": "u1",
        "carrier": "hypercharge",
        "dimension": 1,
        "rank": 1
      }
    ],
    "extra_vectors": 0,
    "matter_kernel": "Z6",
    "missing_vectors": 0,
    "principal_connection_count": 1,
    "sm_match": true,
    "total_dimension": 12,
    "total_rank": 4
  },
  "geometry": {
    "factors": [
      {
        "carrier": "color",
        "id": "K6",
        "type": "SU3/T2"
      },
      {
        "carrier": "weak",
        "id": "S2",
        "type": "S2_round"
      },
      {
        "carrier": null,
        "id": "I_chi",
        "type": "I_interval"
      }
    ]
  },
  "matter_characters": [
    {
      "doublet": 1,
      "id": "Q",
      "triality": 1,
      "y6": 1
    },
    {
      "doublet": 0,
      "id": "u_c",
      "triality": -1,
      "y6": -4
    },
    {
      "doublet": 0,
      "id": "d_c",
      "triality": -1,
      "y6": 2
    },
    {
      "doublet": 1,
      "id": "L",
      "triality": 0,
      "y6": -3
    },
    {
      "doublet": 0,
      "id": "e_c",
      "triality": 0,
      "y6": 6
    },
    {
      "doublet": 0,
      "id": "nu_c",
      "triality": 0,
      "y6": 0
    },
    {
      "doublet": 1,
      "id": "H",
      "triality": 0,
      "y6": 3
    }
  ],
  "p_form_vector_zero_modes": [],
  "parent_nonabelian_connections": [],
  "presentation_role": "UNDISCLOSED",
  "principal_connections": [
    {
      "algebra": "u1",
      "carrier": "hypercharge",
      "claimed_vector_zero_modes": 1,
      "dimension": 1,
      "id": "B_Y",
      "parent": "independent-principal-U1Y",
      "rank": 1,
      "scalar_parity": "odd",
      "vector_parity": "odd"
    }
  ],
  "schema": "SG2-GAUNTLET-CANDIDATE-1.0",
  "scope": "massless vector sector plus reviewer-requested faithful matter kernel"
}

Executable witness

{
  "candidate_id": "session-5fa9e02946c18581b740",
  "center_enumeration": {
    "domain_size": 36,
    "generator": [
      1,
      1,
      1
    ],
    "kernel_elements": [
      [
        0,
        0,
        0
      ],
      [
        0,
        1,
        3
      ],
      [
        1,
        0,
        4
      ],
      [
        1,
        1,
        1
      ],
      [
        2,
        0,
        2
      ],
      [
        2,
        1,
        5
      ]
    ],
    "kernel_label": "Z6",
    "kernel_size": 6
  },
  "computed": {
    "extra_vectors": 0,
    "geometric_dimension": 11,
    "geometric_rank": 3,
    "missing_vectors": 1,
    "principal_dimension": 0,
    "principal_rank": 0,
    "total_dimension": 11,
    "total_rank": 3
  },
  "duplicate_carriers": {},
  "factor_checks": [
    {
      "algebra": "su3",
      "carrier": "color",
      "dimension": 8,
      "id": "K6",
      "parent": "metric:K6",
      "rank": 2,
      "type": "SU3/T2"
    },
    {
      "algebra": "su2",
      "carrier": "weak",
      "dimension": 3,
      "id": "S2",
      "parent": "metric:S2",
      "rank": 1,
      "type": "S2_round"
    },
    {
      "algebra": "none",
      "carrier": null,
      "dimension": 0,
      "id": "I_chi",
      "parent": "metric:I_chi",
      "rank": 0,
      "type": "I_interval"
    }
  ],
  "no_extra_audit": {
    "boundary_vector_zero_modes": [],
    "independent_nonabelian_connections": [],
    "interval_connected_killing_dimension": 0,
    "p_form_vector_zero_modes": []
  },
  "ownership": {
    "color": [
      "metric:K6"
    ],
    "hypercharge": [
      "independent-principal-U1Y"
    ],
    "weak": [
      "metric:S2"
    ]
  },
  "parity_table": [
    {
      "algebra": "u1",
      "carrier": "hypercharge",
      "claimed_vector_zero_modes": 1,
      "computed_vector_zero_modes": 0,
      "dimension": 1,
      "id": "B_Y",
      "parent": "independent-principal-U1Y",
      "rank": 1,
      "scalar_parity": "odd",
      "vector_parity": "odd"
    }
  ],
  "schema": "SG2-VECTOR-SECTOR-CERTIFICATE-1.0",
  "smith_certificate": {
    "basis_convention": "center tuple (a mod 3, b mod 2, k mod 6)",
    "domain_relation_D": [
      [
        3,
        0,
        0
      ],
      [
        0,
        2,
        0
      ],
      [
        0,
        0,
        6
      ]
    ],
    "kernel_group": "Z6",
    "kernel_lift_basis_B": [
      [
        1,
        3,
        0
      ],
      [
        1,
        0,
        2
      ],
      [
        1,
        0,
        0
      ]
    ],
    "phase_rule": "2*triality*a + 3*doublet*b + y6*k = 0 mod 6",
    "quotient_relation_M": [
      [
        0,
        0,
        6
      ],
      [
        1,
        0,
        -2
      ],
      [
        0,
        1,
        -3
      ]
    ],
    "smith_invariants": [
      1,
      1,
      6
    ],
    "verified_B_times_M_equals_D": true
  }
}

Building-block artifact - SG2_BUILDING_BLOCK_FINDINGS

Artifact: BUILDING_BLOCKS/SG2_BUILDING_BLOCK_FINDINGS.md
SHA-256: d8be3d3636a8e47fc1a6ad7dcf32e09b455496cc5a9c02bab4cfc375fd8b1d79

SG-2 Building-Block Findings

BBF-SG2-01 - Vector ownership lacked one executable owner

SOT22 contains parent-action, groupoid, parity, and CSDR discipline but no single block that compiles all massless vector origins into one kernel and one-parent ledger. BB-GVO-1 fills that gap.

BBF-SG2-02 - Metric KK and CSDR were historically conflated

The public dossier correctly retires the old use of a CSDR centralizer as a metric-isometry reduction. BB-KKC-1 makes this separation reusable and machine-checkable.

BBF-SG2-03 - Gate scope and reviewer scope disagree on Z6

The public dossier explicitly defers the faithful global kernel. The reviewer SG-2 challenge explicitly requires Z6. The V1.1 GQO amendment introduces a versioned cross-gate import rather than pretending both statements have the same scope.

BBF-SG2-04 - Pseudocode was not an artifact

The website contains a machine-certificate specification but no delivered executable implementation. The new execution package supplies the script, manifest, vector certificate, center enumeration, Smith matrix, sealed key, escrow, and validation report.

BBF-SG2-05 - Reviewer package remains absent

The supplied reviewer document promises randomized manifests and a sealed key after acceptance; it does not contain them. The internally reconstructed suite can pass only as INTERNAL-REHEARSAL. REVIEWER-GAUNTLET remains NOT-EVALUATED.

BBF-SG2-06 - SG1 readiness was not joined into the SG2 terminal

The earlier SG2 execution used the original 22-block archive directly and reused SG1 lessons without freezing the finalized SG1 execution block as an input. The supplied SG1 V3 package now makes the omission testable: SG1 is OPEN, its formulation is unselected, and nine action/domain/mode/quotient rows are direct prerequisites of a physical SG2 claim. BB-GDR-1 closes the instruction gap and forces the physical SG2 reducer to remain OPEN while preserving the exact conditional tuple.

Building-block artifact - SG2_SCOPE_RECONCILIATION

Artifact: BUILDING_BLOCKS/SG2_SCOPE_RECONCILIATION.md
SHA-256: ab18ed9a369b570814dd54dc1c4a8d9902a10707abd7e58063125bc93e954de5

SG-2 Scope Reconciliation

The revised candidate certificate therefore has two joined components:

1. native SG-2 vector-sector evidence: 12, rank 4, zero extras, unique parents; and
2. an explicitly imported matter-faithful center certificate: Z6.

The import is identified as cross-gate evidence. It does not claim SG-2 derived the Standard Model character vector.

Lawful terminals:

SG2-CANDIDATE-CERTIFICATE: PASS
SG2-INTERNAL-RECONSTRUCTED-GAUNTLET: PASS
SG2-REVIEWER-RANDOMIZED-GAUNTLET: NOT-EVALUATED
SG2-PHYSICAL-GATE: OPEN on SG1-O01/O02/O03/O05/O06/O07/O09/O10/O11
NATURE-SELECTION: NOT-CLAIMED

Building-block artifact - BB_GDR_1_GAUGE_DERIVATION_READINESS_AND_UPSTREAM_DEPENDENCY

Artifact: BUILDING_BLOCKS/NEW_BLOCKS/BB_GDR_1_GAUGE_DERIVATION_READINESS_AND_UPSTREAM_DEPENDENCY.md
SHA-256: e7ef2f9c4eacfcb95386f4a44599614a2571ad209f9e63d80079b6591ccfe295

BB-GDR-1 - Gauge Derivation Readiness and Upstream Dependency

1. Purpose

This block prevents a correct algebraic calculation on a declared geometry from being promoted into a physically closed gauge-group gate before the geometry, action, domains, and zero-mode quotient have been constructed.

Its firewall is:

A conditional vector certificate can pass while the physical gauge gate remains open on upstream construction dependencies.

2. Required two-level record

Every gauge-derivation claim must publish two independent terminals:

1. conditional_candidate_certificate: the result of applying the declared reduction rules to the frozen candidate manifest; and
2. physical_gate_status: the result after joining every upstream readiness dependency to the local gauge-sector evidence.

The first terminal may not overwrite the second.

3. Minimum upstream readiness join

For a metric-KK plus independent-principal-connection realization, the join must include:

If CSDR is the active mechanism, the declared parent gauge group, isotropy embedding, centralizer, faithful kernel, and descended spectrum contracts are also required. A coset name never substitutes for these inputs.

4. Status/provenance rule

The machine record contains separate fields:

conditional_candidate_certificate.status
conditional_candidate_certificate.provenance_class
physical_gate.status
physical_gate.blocking_dependencies

construction-anchor is provenance, not permission to write PASS into the physical gate field. A source declaration, long dossier, checksum, or internal gauntlet cannot promote an open upstream construction row.

5. Reduction rule

1. Any contradictory required upstream or local evidence makes the physical gate FAIL.
2. Otherwise, any required upstream or local row in OPEN, NOT-EVALUATED, or CONSTRUCTION-ANCHOR makes the physical gate OPEN.
3. The physical gate is PASS only when every required row is PASS or legitimately NOT-APPLICABLE.
4. A reviewer challenge and a reconstructed challenge retain separate fields.
5. Architecture-neutral selection additionally requires the equal-freeze rival shelf; it is never inferred from a candidate-interface match.

6. Required artifacts

UPSTREAM_AUTHORITY_HASHES.json
UPSTREAM_DEPENDENCY_LEDGER.json
GAUGE_ORIGIN_OWNERSHIP.json
VECTOR_DOMAIN_AND_KERNEL.json
NO_EXTRA_VECTOR_AUDIT.json
CONDITIONAL_VECTOR_CERTIFICATE.json
PHYSICAL_GATE_REDUCTION.json

Each artifact is content-addressed. The reducer records the exact upstream row IDs and statuses it consumed.

7. Negative controls

The validator must reject at least:

1. a passing conditional tuple with one required upstream row changed to OPEN but physical gate left PASS;
2. a circle generator counted after the active quotient is an interval;
3. a duplicate primitive parent for one carrier;
4. an undeclared second principal connection;
5. an odd vector parity claimed to retain a constant zero mode;
6. a global kernel inferred without matter characters;
7. a CSDR centralizer used in a metric-KK branch without a CSDR parent Actor;
8. a reconstructed gauntlet labeled reviewer-issued.

8. Reopen triggers

Reopen the gauge gate when the formulation, parent action, stable metric, boundary/domain data, parity assignment, Actor inventory, global quotient, matter characters, or any upstream readiness status changes.

9. SG1 V3 to SG2 consequence

The owner-supplied BB-SG1-CLOSURE-3@3.0.0-rc1 package validates correctly but reports SG1 as OPEN, with 18 open construction contracts and no selected formulation. Nine of those rows are direct physical-readiness dependencies of the current SG2 metric-KK certificate. Therefore SG2’s exact conditional tuple may be recorded as PASS, while its present physical gate status is OPEN.

Building-block artifact - BB_GVO_1_GAUGE_VECTOR_ORIGIN_OWNERSHIP_AND_ZERO_MODE_COMPLETENESS

Artifact: BUILDING_BLOCKS/NEW_BLOCKS/BB_GVO_1_GAUGE_VECTOR_ORIGIN_OWNERSHIP_AND_ZERO_MODE_COMPLETENESS.md
SHA-256: 1dd4a417f5bd1e07c3e92d748f1c93452a694cb672a4655fbc8a4f28953b1482

BB-GVO-1 - Gauge-Vector Origin, Ownership, and Zero-Mode Completeness

1. Purpose

This block controls every claim about the exact massless vector algebra of a dimensionally reduced or boundary-quotiented candidate.

Its firewall is:

Symmetry support is not carrier ownership, and a declared carrier is not yet a massless zero mode.

This revision implements the SG1 V3 status/provenance firewall. Origin and zero-mode checks can pass conditionally on a frozen manifest without promoting the physical gate while required upstream action, domain, kernel, or quotient contracts remain open. BB-GDR-1 owns that cross-gate reduction.

2. Mechanism typing

Every candidate vector is assigned exactly one origin class:

METRIC-KK
PRINCIPAL-CONNECTION
BOUNDARY-CONNECTION
FORM-REDUCTION
COMPOSITE-OR-EMERGENT

The record identifies its parent action term, support, gauge parameter, boundary/domain, representation, mass operator, and observer output. Two origins may not own the same carrier unless an explicit mixing mass matrix shows the surviving combinations and removes duplicates.

3. Geometric vector certificate

For each metric factor publish:

• the actual active quotient, not its covering space;
• the connected isometry group and Lie algebra;
• an explicit Killing basis or certified dimension/rank calculation;
• the locked metric and any symmetry-enhancement loci;
• the metric ansatz converting Killing fields to four-dimensional connections;
• cross-factor brackets; and
• the physical vector operator after gauge/diffeomorphism quotient.

Discrete symmetries and denominator redundancies add no Lie-algebra generator. A parent circle generator does not survive automatically on an interval quotient.

4. Principal and boundary connection certificate

Every independent connection publishes:

• global form and bundle owner;
• action and gauge transformation;
• vector, scalar, and gauge-parameter parity;
• self-adjoint boundary domain;
• vector zero-mode kernel and scalar/longitudinal disposition;
• localized kinetic or mass terms; and
• anomaly/inflow dependency.

Even parity without a compatible domain is incomplete. Odd vector parity has no constant zero mode.

5. One-parent ownership join

The ownership ledger joins all origin classes on a canonical carrier key:

algebra, global form, representation, support, boundary domain, source role

Every massless carrier must have exactly one primitive parent. A duplicate non-Abelian or Abelian connection fails unless an executed mixing mechanism leaves exactly the claimed kernel.

6. No-extra-vector closure

The audit is finite and includes:

• every connected Killing direction;
• interval or radion graviphotons;
• independent principal connections;
• boundary-localized vectors;
• vectors descended from form fields;
• harmonic-one-form sectors;
• accidental non-Killing vector kernels;
• Stückelberg and Higgsed combinations; and
• symmetry-enhancement strata.

The physical vector kernel is computed after constraints and domains. A narrative Actor list cannot replace the kernel calculation.

7. Required artifacts

GVO_ACTIVE_FACTOR_LEDGER.json
GVO_KILLING_ALGEBRAS.json
GVO_METRIC_CONNECTION_MAP.json
GVO_CONNECTION_ACTORS.json
GVO_PARITY_DOMAIN_TABLE.json
GVO_PARENT_OWNERSHIP.json
GVO_VECTOR_OPERATOR.json
GVO_KERNEL_AND_GAP.json
GVO_NO_EXTRA_AUDIT.json
GVO_MUTATION_RESULTS.json

8. Evidence decision

PASS requires exact agreement between the declared algebra and the complete physical vector kernel, including dimension, rank, brackets, and zero extras. An explicit missing, duplicate, or anomalous vector gives FAIL. Missing operator, domain, ownership, or inventory evidence gives OPEN.

9. First-hard-failure order

1. active factor or connected-isometry mismatch;
2. connection-Actor inventory mismatch;
3. duplicate parent ownership;
4. parity/domain zero-mode mismatch;
5. vector-kernel dimension/rank mismatch;
6. undeclared extra or missing carrier;
7. claim scope exceeds the computed object.

10. Mandatory negative controls

• replace an interval by its parent circle without adding the restored vector;
• insert a second even principal U(1);
• add an independent SU(2) parent beside a geometric weak connection;
• make the claimed hypercharge vector odd;
• add one boundary or form-vector zero mode;
• perturb an alleged kernel eigenvalue to a small positive exact value;
• promote a linear zero-mode result to an exact nonlinear truncation.

11. Replay and invalidation

Replay whenever a factor, metric, quotient, Actor, parity, domain, boundary term, parent action, stabilization result, or global representation changes.

12. Authority boundary

This block certifies a scoped candidate vector sector. It does not establish that nature uses the candidate, does not determine matter charges from an algebra, and does not prove an exact nonlinear truncation unless that theorem is separately supplied.

Building-block artifact - BB_KKC_1_METRIC_KK_AND_CSDR_MECHANISM_SEPARATION

Artifact: BUILDING_BLOCKS/AMENDMENTS/BB_KKC_1_METRIC_KK_AND_CSDR_MECHANISM_SEPARATION.md
SHA-256: 4745cbd837485ffa55d041b0161afee7f17a29e29929157dfcbe469155e19204

BB-KKC-1 - Metric Kaluza-Klein and CSDR Mechanism Separation

1. Purpose

This block prevents the centralizer of an undeclared higher-dimensional gauge group from being used as the reduction map for a metric Kaluza-Klein vector.

It also implements the SG1 V3 formulation firewall. Until REDUCED or EMBEDDED is selected and the corresponding action, quotient, boundary, and domain records pass, the selected mechanism is a conditional calculation branch rather than a physically closed reduction.

2. Mechanism record

Every surviving gauge carrier is typed exactly once:

A geometric ambient group, an isometry group, and a gauge group are not interchangeable.

3. Duplicate audit

If two mechanisms produce the same algebraic carrier, the candidate must publish their complete mixing mass matrix, unbroken diagonal combination, orthogonal massive combination, and parent ownership. Without that mechanism, the result contains duplicate vectors.

4. Required artifacts

KKC_MECHANISM_LEDGER.json
KKC_PARENT_PATHS.json
KKC_REDUCTION_MAPS.json
KKC_DUPLICATE_MIXING.json
KKC_MUTATION_RESULTS.json

5. First failures

1. missing parent mechanism;
2. CSDR tuple absent;
3. metric Killing map absent;
4. principal connection/domain absent;
5. carrier produced twice without mixing;
6. mechanism scope promoted beyond its calculation.

6. SG-2 consequence

The corrected branch is hybrid: color and weak use metric KK; hypercharge uses one independent principal connection; the interval supplies no connected metric U(1). A CSDR centralizer is not part of this SG-2 proof unless a new higher-dimensional gauge Actor and embedding are declared.

Building-block artifact - BB_GQO_1_MATTER_FAITHFUL_CENTER_KERNEL_AND_SCOPE_AMENDMENT

Artifact: BUILDING_BLOCKS/AMENDMENTS/BB_GQO_1_MATTER_FAITHFUL_CENTER_KERNEL_AND_SCOPE_AMENDMENT.md
SHA-256: 8ef963895b1a38166dd239255895be380b98920ebdc126e71b7e4f6f4915c374

BB-GQO-1 Amendment - Matter-Faithful Center Kernel and Scope Import

1. Finding

A gauge Lie algebra and a global faithful gauge group are different claims. Nevertheless, a reviewer may deliberately import the faithful center kernel into a gate challenge. Such an import must be versioned and executable rather than silently described as both excluded and required.

The SG1 V3 dependency join adds a second separation: an exact matter-faithful kernel can pass as a finite imported calculation while the physical gauge gate remains open on its shape/action/domain prerequisites.

2. Required character table

For a candidate with covering group

\[ SU(3)\times SU(2)\times U(1), \]

publish each retained representation’s color triality, weak doublet parity, and primitive integer Abelian charge y6=6Y.

For a center tuple

\[ (a\bmod 3,\;b\bmod 2,\;k\bmod 6), \]

the representation is invariant exactly when

\[ 2t_3a+3t_2b+y_6k=0\pmod 6. \]

3. Enumeration and Smith certificate

The certificate must:

1. enumerate all 36 center tuples;
2. print every tuple acting trivially on all retained representations;
3. identify generators and element orders;
4. publish the integer lattice basis and relation matrix;
5. verify the lattice identity exactly; and
6. publish the Smith invariant factors.

For the standard character vector (1,-4,2,-3,6,0,3), the kernel contains six elements generated by (1,1,1). One exact quotient presentation has Smith invariants (1,1,6), hence faithful kernel Z6.

4. Scope partition

Store separately:

LIE_ALGEBRA_CERTIFICATE
GLOBAL_COVERING_GROUP
MATTER_CHARACTER_TABLE
FAITHFUL_CENTER_KERNEL
ANOMALY_CERTIFICATE

An SG-2 Lie-algebra dossier may defer the final four fields. If an SG-2 gauntlet imports the kernel, its scope record must name the owning downstream artifact and its hash. The import does not make charge derivation an SG-2 result.

5. Required artifacts

GQO_CHARACTER_TABLE.json
GQO_CENTER_ENUMERATION.json
GQO_KERNEL_LATTICE_BASIS.json
GQO_SMITH_CERTIFICATE.json
GQO_SCOPE_IMPORT_RECORD.json
GQO_MUTATION_RESULTS.json

6. Negative controls

• change one character so the actual kernel is Z2 while claiming Z6;
• quote Smith invariants without the matrix;
• change the charge basis without updating the character table;
• infer doublet/triplet availability from Lie algebra alone;
• describe an imported downstream certificate as an independent SG-2 derivation.

7. Evidence decision

The faithful-kernel row passes only when enumeration, lattice relation, and Smith invariants agree. If the row is outside the gate scope, it is absent or typed NOT-APPLICABLE; it may not be contradicted by a challenge specification without an explicit scope-import version.

Upstream authority - SG1 consolidated building block V3 RC

Artifact: FROZEN_INPUTS/SG1_V3/SG1_CONSOLIDATED_BUILDING_BLOCK_V3_RC.md
SHA-256: 613ba09c9808eba826cc6b3319fb88f2defc1aff34f98eb4c43cf559c7b45582

BB-SG1-CLOSURE-3

1. Purpose

This block governs future attempts to close SG-1, the shape-selection gate. It converts every unresolved SG-1 obligation into an exact construction and evidence contract. It also consolidates the reusable rules exposed by the full dossier, the reconstructed gauntlet, the independent execution review, and the later building-block audit.

The block is deliberately fail-closed. It may make the route to closure finite and executable; it may not treat a well-written route as the physical witness produced by that route.

2. Authority and non-overwrite rule

The owner-designated 2026-07-18 22-block source-of-truth remains the parent authority. This V3 block is additive and SG-1-specific.

Before ratification, its authority status is RATIFICATION-CANDIDATE. After explicit owner ratification, it becomes the controlling SG-1 execution block and supersedes only the two earlier SG-1 amendment/execution layers named in the front matter.

It never silently rewrites:

1. the parent shape declaration;
2. the parent block identifiers;
3. a source claim marked OPEN FINITE CONSTRUCTION or CONSTRUCTION-OWED;
4. an evidence status from a public summary page;
5. a reviewer-issued challenge that has not been supplied.

3. Frozen inherited adjudication

The inherited execution state is preserved without promotion:

The internal gauntlet result proves that the reconstructed validator caught its planted defects. It does not prove that the candidate geometry satisfies the 18 open physical obligations.

4. Shared status grammar

Controlling status fields use only:

PASS, FAIL, OPEN, NOT-APPLICABLE, NOT-EVALUATED, CONSTRUCTION-ANCHOR, MEASURED-ANCHOR, CERTIFIED.

Prose qualifiers such as BLOCKED-ON-CONSTRUCTION may explain a status but may not replace it.

4.1 Status/provenance firewall

Every claimed result has two independent fields:

• status: the adjudication;
• provenance_class: one of source-authority, formal-derivation, finite-computation, measured-anchor, construction-anchor, reconstructed-test, or reviewer-issued-test.

No provenance class implies PASS. In particular:

• a source-authority statement can remain OPEN;
• a construction anchor is not a completed construction;
• a reconstructed gauntlet pass is not a reviewer-issued gauntlet pass;
• a measured input is not a prediction.

5. Formulation fork

SG-1 must select exactly one formulation before physical construction begins.

5.1 REDUCED branch

The internal geometry is frozen Stage data and is not varied. The action, admissible variations, observable algebra, and rival shelf must all use that same freeze.

This is the recommended first branch because it may avoid manufacturing a second-class constraint system for variables that the candidate never intended to vary. Recommendation is not selection.

5.2 EMBEDDED branch

The internal geometry is included in the parent variable set and eliminated by a complete constraint system. The candidate must provide the full constraint algebra, its class, the Dirac or reduced bracket, the quotient, determinant factors, boundary behavior, and surviving measure.

5.3 Branch prohibition

The candidate may not derive the action in the embedded theory, freeze variables as if reduced, use the embedded measure, and compare rivals with a different freeze. Any mixed branch is FAIL for the affected rows.

Machine field formulation.selected remains null until an owner/candidate revision selects REDUCED or EMBEDDED and supplies the required revision identifier.

6. Evidence-promotion contract

An open row may become PASS only when all conditions below hold:

1. Every dependency row is PASS or legitimately NOT-APPLICABLE.
2. The exact artifact types listed for the row exist.
3. Each artifact is content-addressed with SHA-256.
4. A deterministic producer command or formal proof path is recorded.
5. An independent validator or reproduction path is recorded.
6. At least one row-specific negative control is executed and caught.
7. The witness scope equals the claim scope.
8. All boundary, domain, quotient, and gauge ownership assumptions are explicit.
9. No measured anchor is counted as a derived prediction.
10. The gate reducer, not dossier prose, computes the terminal status.

If any item is absent, the row remains OPEN. Contradictory evidence makes it FAIL.

7. Reusable amendments consolidated here

7.1 BB-GA-1 - geometric/constraint realization

A constraint slogan is not a realization. The physical objects, allowed moves, full constraints, class of each constraint, quotient, reduced brackets, determinant factors, and boundary/domain data must be constructed in the selected formulation.

7.2 BB-CSDR-1 - CSDR ownership

The surviving gauge group is computed from a specified parent gauge group and a specified isotropy embedding. It is not read from the coset label alone. The required object is the centralizer (C_G(R_G)), with global quotient and faithful kernel tracked separately.

Ordinary electromagnetism is the electroweak descendant and must not be counted as a second independent (U(1)).

7.3 BB-EFG-1 - equal-freeze rival comparison

Every rival must be evaluated with identical formulation, boundary data, domain, regulator, measured anchors, quotient convention, and observer map. A rival eliminated by a special freeze not imposed on the incumbent is not eliminated.

7.4 BB-ESP-1 - evidence/status/provenance separation

Evidence type, authority, and status are independent. Narrative strength, file length, a checksum, or a passing validator for syntax cannot promote a physics row.

7.5 BB-GNT-1 - gauntlet integrity

A valid blind gauntlet requires reviewer-issued manifests, relabeling, a sealed key or escrow, first-hard-failure rules, content-addressed inputs, and an immutable validation report. A reconstructed gauntlet is labeled as reconstructed.

The exact historical mixed-saddle negative control uses a Hessian with positive axis probes but eigenvalues ({-1/3,1}); the validator must test mixed directions or the full spectrum.

7.6 BB-AD-1 amendment - independent AD reproduction

An automatic-differentiation certificate is complete only when the implementation, frozen inputs, precision, variable ordering, Hessian convention, spectrum, and an independent reproduction are content-addressed. Axis-only positivity is insufficient.

7.7 BB-AHG-1 amendment - conditional Actor typing

A top form or Actor term is not automatically part of the physical action. Its inclusion is conditional on the selected formulation, degree/dimension ledger, boundary data, variation rule, gauge ownership, and non-duplication proof. If included, its stress and constraint contributions must appear consistently in W2, W3, W4, and W6.

7.8 Constraint/domain amendment

Formal operators are inseparable from their domains. Self-adjointness, boundary conditions, zero modes, gauge modes, ghosts, kernels, and quotient measures are parts of the witness, not later implementation details.

7.9 QCR/GCR amendment

Quantization and global-consistency claims require an explicit finite record basis, ordered variables, matrices or operators, spectra/invariants, precision, tolerance, and independent reproduction. A qualitative argument cannot stand in for the finite realization.

7.10 Observer/transport amendment

The observer map and regulator transport must be executable on both the incumbent and every rival. It must preserve the declared status and measured-anchor firewall through the complete comparison.

8. Eighteen open-row construction contracts

The canonical machine form is SG1_OPEN_ROW_CONTRACTS_V3.json. The human summary follows.

W8 integrates all rows, regenerates the board, executes the gauntlet controls, and issues the terminal verdict. It owns no shortcut row; it consumes the complete preceding record.

9. Work-package order

W1 formulation + objects + moves + constraints + quotient
 -> W2 action + boundary/domain + variations + Actor decision
 -> W3 modes + kernels + gauge/ghost/zero-mode quotient
 -> W4 CSDR/global descent + complete spectrum
 -> W5 quantum measure + QCR/GCR finite realization
 -> W6 observer + regulator transport
 -> W7 equal-freeze rival shelf
 -> W8 integration + gate reduction + blind/reconstructed challenge bookkeeping

Parallel work is allowed only within a package after all declared dependencies are satisfied. W4 may precompute representation tables, but it cannot promote spectrum rows before W1-W3 freeze the physical object and operator domain.

10. Gate reduction

The terminal reducer is:

1. If any required row is FAIL, SG-1 is FAIL.
2. Else if any required row is OPEN, NOT-EVALUATED, or CONSTRUCTION-ANCHOR, SG-1 is OPEN.
3. Else if every required row is PASS, SG-1 is PASS.
4. MEASURED-ANCHOR may satisfy only a row explicitly typed as measured input; it cannot satisfy a derivation row.
5. CERTIFIED is accepted only where the row contract names a certificate authority and scope.

The current V3 packet has 18 OPEN rows, so its computed result is OPEN.

11. Required final SG-1 closure record

A future PASS package must contain:

• the selected formulation revision;
• all W1-W8 artifacts and their hashes;
• the complete 26-row inherited board with a migration map from the earlier execution package;
• this 18-row contract registry with every row discharged or explicitly replaced by a ratified mapping;
• independent reproduction results;
• negative controls and mutation results;
• regenerated dossier and public summary derived from the board;
• a reviewer-issued blind challenge result if the project retains that as a gate requirement.

12. Current conclusion

This V3 block materially improves SG-1 by consolidating the earlier amendments, selecting a single status/provenance grammar, eliminating formulation mixing, fixing CSDR and electromagnetic ownership, requiring operator domains and quotient data, and making all 18 remaining debts executable.

It does not achieve full physical closure. Its lawful terminal is:

SG-1: OPEN - inherited 8 PASS / 18 OPEN; construction contracts complete, physical witnesses outstanding.

Upstream dependency ledger

Artifact: FROZEN_INPUTS/SG1_TO_SG2_DEPENDENCY_LEDGER.json
SHA-256: f5de6e33e9b36ed4e3513df47db08db6361e23a661e4019711b74b28a9844b47

{
  "schema": "SG1-TO-SG2-DEPENDENCY-LEDGER-1.0",
  "date": "2026-08-01",
  "authority_interpretation": {
    "parent_source_of_truth": "OWNER-DESIGNATED-2026-07-18-22-BLOCK-SOT",
    "upstream_execution_block": "BB-SG1-CLOSURE-3",
    "upstream_version": "3.0.0-rc1",
    "upstream_authority_status": "RATIFICATION-CANDIDATE",
    "use_in_this_package": "OWNER-SUPPLIED-CONTROLLING-UPSTREAM-DEPENDENCY",
    "global_ratification_claimed": false,
    "sg1_package_sha256": "858b08c9186bb8f6ca7010937b7a9d7c61dd51369e27cc844a2536ea3b9cb7bf"
  },
  "upstream_validation": {
    "package_integrity": "PASS",
    "validator": "PASS",
    "inherited_board": {
      "PASS": 8,
      "OPEN": 18
    },
    "gate": "OPEN",
    "formulation_selected": null
  },
  "direct_sg2_physical_readiness_rows": [
    {
      "id": "SG1-O01",
      "status": "OPEN",
      "reason": "SG2 must know whether the metric reduction is REDUCED or EMBEDDED before assigning physical ownership."
    },
    {
      "id": "SG1-O02",
      "status": "OPEN",
      "reason": "Every claimed vector requires a typed parent field, bundle, and ownership record."
    },
    {
      "id": "SG1-O03",
      "status": "OPEN",
      "reason": "Gauge, diffeomorphism, quotient, boundary, and stabilizer moves determine which symmetry directions are physical."
    },
    {
      "id": "SG1-O05",
      "status": "OPEN",
      "reason": "The physical quotient and reduced bracket determine the surviving gauge redundancies."
    },
    {
      "id": "SG1-O06",
      "status": "OPEN",
      "reason": "A complete parent or reduced action is required to distinguish actual gauge carriers from labels."
    },
    {
      "id": "SG1-O07",
      "status": "OPEN",
      "reason": "Operator domains and boundary conditions determine the vector zero-mode kernels."
    },
    {
      "id": "SG1-O09",
      "status": "OPEN",
      "reason": "Conditional top-form or Actor sectors must be included or excluded before the no-extra-vector audit is physical."
    },
    {
      "id": "SG1-O10",
      "status": "OPEN",
      "reason": "The complete mode and kernel inventory is the physical witness for masslessness and multiplicity."
    },
    {
      "id": "SG1-O11",
      "status": "OPEN",
      "reason": "Gauge, ghost, and zero-mode quotient data are required to remove nonphysical modes without deleting physical carriers."
    }
  ],
  "conditional_rows": [
    {
      "ids": ["SG1-O12", "SG1-O13"],
      "condition": "Required if a CSDR or descended-spectrum mechanism is used as the SG2 carrier derivation; not used by the current metric-KK plus principal-U1 conditional certificate."
    },
    {
      "ids": ["SG1-O18"],
      "condition": "Required for architecture-neutral selection or a claim that the gauge pattern uniquely falls out of the selected shape; not required for internal consistency of one frozen candidate manifest."
    }
  ],
  "local_sg2_results": {
    "conditional_candidate_vector_certificate": "PASS",
    "tuple": {
      "dimension": 12,
      "rank": 4,
      "matter_faithful_kernel": "Z6",
      "extra_vectors": 0,
      "missing_vectors": 0
    },
    "internal_reconstructed_gauntlet": "PASS",
    "reviewer_randomized_gauntlet": "NOT-EVALUATED"
  },
  "reducer": {
    "fail_rule": "If any direct upstream row or any local required row is FAIL, SG2 is FAIL.",
    "open_rule": "Otherwise, if any direct upstream row is OPEN, NOT-EVALUATED, or CONSTRUCTION-ANCHOR, SG2 is OPEN.",
    "pass_rule": "SG2 may be PASS only when all direct upstream rows and all local SG2 rows are PASS or legitimately NOT-APPLICABLE.",
    "current_direct_open_count": 9,
    "current_sg2_gate": "OPEN"
  }
}

Upstream SG1 open-row contracts

Artifact: FROZEN_INPUTS/SG1_V3/SG1_OPEN_ROW_CONTRACTS_V3.json
SHA-256: 7651f9a0728d5424ee7b968caeac8f616816a5902657e830323bdb6238024f5d

{
  "schema_version": "3.0.0-rc1",
  "block_id": "BB-SG1-CLOSURE-3",
  "gate": "SG-1",
  "authority_status": "RATIFICATION-CANDIDATE",
  "parent_authority": "OWNER-DESIGNATED-2026-07-18-22-BLOCK-SOT",
  "inherited_board": {"PASS": 8, "OPEN": 18, "gate": "OPEN"},
  "allowed_statuses": ["PASS", "FAIL", "OPEN", "NOT-APPLICABLE", "NOT-EVALUATED", "CONSTRUCTION-ANCHOR", "MEASURED-ANCHOR", "CERTIFIED"],
  "promotion_requirements": [
    "dependencies_discharged",
    "all_required_artifacts_present",
    "sha256_for_each_artifact",
    "deterministic_producer_or_formal_proof",
    "independent_validator_or_reproduction",
    "negative_control_caught",
    "witness_scope_equals_claim_scope",
    "boundary_domain_and_quotient_explicit",
    "measured_anchor_firewall_preserved",
    "terminal_computed_by_reducer"
  ],
  "formulation": {
    "allowed": ["REDUCED", "EMBEDDED"],
    "selected": null,
    "recommendation": "REDUCED",
    "recommendation_is_selection": false
  },
  "work_packages": {
    "W1": "formulation, physical objects, moves, constraints, quotient",
    "W2": "action, boundary/domain, variations, reaction stress, Actor decision",
    "W3": "modes, kernels, gauge/ghost/zero-mode quotient",
    "W4": "CSDR/global descent and complete spectrum",
    "W5": "quantum measure and QCR/GCR finite realization",
    "W6": "observer map and regulator transport",
    "W7": "equal-freeze rival shelf",
    "W8": "integration, reduction, reproducibility, challenge bookkeeping"
  },
  "rows": [
    {
      "id": "SG1-O01", "status": "OPEN", "title": "Formulation selection", "work_packages": ["W1", "W8"], "depends_on": [],
      "claim_scope": "Select exactly one of REDUCED or EMBEDDED in a content-addressed candidate revision.",
      "required_artifacts": ["formulation_decision.md", "candidate_revision.json"],
      "validator": "selected branch is allowed; revision hash exists; recommendation is not treated as selection",
      "negative_control": "reject null, dual, or mixed formulation"
    },
    {
      "id": "SG1-O02", "status": "OPEN", "title": "Physical-object ledger", "work_packages": ["W1", "W8"], "depends_on": ["SG1-O01"],
      "claim_scope": "Enumerate every varied and frozen object with bundle, degree, units, ownership, and boundary behavior.",
      "required_artifacts": ["physical_objects.json", "degree_dimension_ledger.json"],
      "validator": "all symbols used downstream resolve to one typed object",
      "negative_control": "reject an action symbol absent from the object ledger"
    },
    {
      "id": "SG1-O03", "status": "OPEN", "title": "Admissible-move ledger", "work_packages": ["W1", "W8"], "depends_on": ["SG1-O01", "SG1-O02"],
      "claim_scope": "Enumerate gauge, diffeomorphism, boundary, large/global, and stabilizer moves.",
      "required_artifacts": ["moves_and_stabilizers.json", "global_move_classes.json"],
      "validator": "every redundancy and physical deformation is typed and non-overlapping",
      "negative_control": "reject a move typed simultaneously as gauge and physical without quotient rule"
    },
    {
      "id": "SG1-O04", "status": "OPEN", "title": "Full constraint algebra", "work_packages": ["W1", "W8"], "depends_on": ["SG1-O02", "SG1-O03"],
      "claim_scope": "Construct the complete constraints, brackets, rank, closure, and first/second-class split.",
      "required_artifacts": ["constraints.json", "constraint_brackets.mtx", "rank_certificate.json"],
      "validator": "closure and rank reproduced at declared precision and domain",
      "negative_control": "reject omitted constraint or rank-changing perturbation"
    },
    {
      "id": "SG1-O05", "status": "OPEN", "title": "Quotient or reduced phase space", "work_packages": ["W1", "W8"], "depends_on": ["SG1-O03", "SG1-O04"],
      "claim_scope": "Construct quotient map, reduced/Dirac brackets, determinant factors, and surviving measure variables.",
      "required_artifacts": ["quotient_map.json", "reduced_brackets.mtx", "measure_factors.json"],
      "validator": "dimension, rank, stabilizer, and determinant ledgers balance",
      "negative_control": "reject double division by a stabilizer or omitted determinant"
    },
    {
      "id": "SG1-O06", "status": "OPEN", "title": "Complete action", "work_packages": ["W2", "W8"], "depends_on": ["SG1-O01", "SG1-O02", "SG1-O05"],
      "claim_scope": "Provide the complete parent or reduced action with units, signs, coefficients, and ownership.",
      "required_artifacts": ["action.md", "action_terms.json", "unit_sign_check.json"],
      "validator": "every term is typed, dimensionally valid, branch-consistent, and counted once",
      "negative_control": "reject duplicate U(1), missing boundary term, or wrong-sign term"
    },
    {
      "id": "SG1-O07", "status": "OPEN", "title": "Boundary and domain certificate", "work_packages": ["W2", "W3", "W8"], "depends_on": ["SG1-O02", "SG1-O03", "SG1-O06"],
      "claim_scope": "Fix boundary conditions and domains for every operator used by the gate.",
      "required_artifacts": ["boundary_conditions.json", "operator_domains.json", "self_adjointness_certificate.md"],
      "validator": "integration by parts, adjoints, kernels, and boundary fluxes reproduce",
      "negative_control": "reject domain swap that changes kernel or spectrum"
    },
    {
      "id": "SG1-O08", "status": "OPEN", "title": "Variation and reaction-stress certificate", "work_packages": ["W2", "W8"], "depends_on": ["SG1-O06", "SG1-O07"],
      "claim_scope": "Compute first and second variations, equations, reactions, and stress contributions in the selected formulation.",
      "required_artifacts": ["first_variation.md", "second_variation.mtx", "reaction_stress.json"],
      "validator": "analytic and independent symbolic or AD variations agree",
      "negative_control": "reject frozen-variable variation or omitted reaction term"
    },
    {
      "id": "SG1-O09", "status": "OPEN", "title": "Conditional top-form or Actor certificate", "work_packages": ["W2", "W3", "W4", "W6", "W8"], "depends_on": ["SG1-O01", "SG1-O06", "SG1-O07", "SG1-O08"],
      "claim_scope": "Decide inclusion or exclusion and propagate consequences through dynamics, modes, descent, and observers.",
      "required_artifacts": ["actor_decision.json", "degree_boundary_ownership_check.json", "cross_package_propagation.json"],
      "validator": "degree, dimension, gauge ownership, variation, and non-duplication checks all pass",
      "negative_control": "reject an Actor term present in one work package but absent from another"
    },
    {
      "id": "SG1-O10", "status": "OPEN", "title": "Complete mode and kernel inventory", "work_packages": ["W3", "W8"], "depends_on": ["SG1-O07", "SG1-O08", "SG1-O09"],
      "claim_scope": "Enumerate complete kernels, towers, multiplicities, degeneracies, and completeness relations.",
      "required_artifacts": ["mode_inventory.json", "kernel_basis.mtx", "completeness_check.json"],
      "validator": "independent enumeration reproduces dimensions and multiplicities",
      "negative_control": "reject one removed zero mode or one duplicated degeneracy"
    },
    {
      "id": "SG1-O11", "status": "OPEN", "title": "Gauge, ghost, and zero-mode quotient", "work_packages": ["W3", "W5", "W8"], "depends_on": ["SG1-O03", "SG1-O05", "SG1-O07", "SG1-O10"],
      "claim_scope": "Provide gauge fixing, ghost/Jacobian factors, stabilizer volumes, and zero-mode treatment.",
      "required_artifacts": ["gauge_fixing.json", "ghost_jacobian.json", "zero_mode_measure.json"],
      "validator": "quotient and determinant factors agree with W1 and W5",
      "negative_control": "reject double-counted gauge volume or deleted physical zero mode"
    },
    {
      "id": "SG1-O12", "status": "OPEN", "title": "CSDR input and embedding", "work_packages": ["W4", "W8"], "depends_on": ["SG1-O02", "SG1-O03", "SG1-O07", "SG1-O10"],
      "claim_scope": "Specify parent G, isotropy H, embedding R_G into G, centralizer, faithful kernel, and global quotient.",
      "required_artifacts": ["csdr_input.json", "embedding_generators.mtx", "centralizer_and_kernel.json"],
      "validator": "independent Lie-algebra and global-group calculations reproduce",
      "negative_control": "reject inference of surviving group from coset label alone"
    },
    {
      "id": "SG1-O13", "status": "OPEN", "title": "Descended spectrum ledger", "work_packages": ["W4", "W8"], "depends_on": ["SG1-O09", "SG1-O10", "SG1-O11", "SG1-O12"],
      "claim_scope": "Compute charges, chirality, families, anomalies, multiplicities, and no-excess content.",
      "required_artifacts": ["spectrum_ledger.json", "chirality_family_index.json", "anomaly_and_excess_check.json"],
      "validator": "index, representation, anomaly, and full-state counts agree",
      "negative_control": "reject mirror duplication, missing state, or extra vector"
    },
    {
      "id": "SG1-O14", "status": "OPEN", "title": "Quantum measure and finite record basis", "work_packages": ["W5", "W8"], "depends_on": ["SG1-O05", "SG1-O07", "SG1-O10", "SG1-O11", "SG1-O13"],
      "claim_scope": "Freeze basis, variable order, measure, determinants, regulator, precision, and tolerances.",
      "required_artifacts": ["finite_record_basis.json", "quantum_measure.json", "regulator_precision.json"],
      "validator": "basis and measure reconstruct every matrix dimension and determinant",
      "negative_control": "reject basis permutation without corresponding matrix permutation"
    },
    {
      "id": "SG1-O15", "status": "OPEN", "title": "QCR and GCR finite matrices", "work_packages": ["W5", "W8"], "depends_on": ["SG1-O08", "SG1-O11", "SG1-O14"],
      "claim_scope": "Produce explicit ordered matrices/operators, spectra, invariants, and independent AD reproduction.",
      "required_artifacts": ["qcr_matrix.mtx", "gcr_matrix.mtx", "spectra.json", "ad_reproduction.json"],
      "validator": "two independent implementations reproduce spectra and invariants within frozen tolerance",
      "negative_control": "reject the mixed saddle with eigenvalues {-1/3,1} despite positive axis probes"
    },
    {
      "id": "SG1-O16", "status": "OPEN", "title": "Observer map execution", "work_packages": ["W6", "W8"], "depends_on": ["SG1-O07", "SG1-O09", "SG1-O13", "SG1-O15"],
      "claim_scope": "Execute the observer map with units, calibration ownership, uncertainty, and scope.",
      "required_artifacts": ["observer_map.json", "observer_outputs.json", "uncertainty_scope.json"],
      "validator": "independent execution reproduces outputs without promoting measured inputs",
      "negative_control": "reject measured calibration relabeled as prediction"
    },
    {
      "id": "SG1-O17", "status": "OPEN", "title": "Regulator and transport execution", "work_packages": ["W6", "W8"], "depends_on": ["SG1-O14", "SG1-O15", "SG1-O16"],
      "claim_scope": "Show transport/regulator operations commute where claimed and preserve status/provenance.",
      "required_artifacts": ["transport_map.json", "commutation_tests.json", "provenance_transport.json"],
      "validator": "round-trip and order-swap tests pass on incumbent and controls",
      "negative_control": "reject order-dependent status or hidden regulator change"
    },
    {
      "id": "SG1-O18", "status": "OPEN", "title": "Equal-freeze rival shelf", "work_packages": ["W7", "W8"], "depends_on": ["SG1-O01", "SG1-O05", "SG1-O07", "SG1-O13", "SG1-O15", "SG1-O16", "SG1-O17"],
      "claim_scope": "Enumerate rivals and execute a common discriminator with identical freeze, boundary/domain, regulator, anchors, and observer map.",
      "required_artifacts": ["rival_census.json", "equal_freeze_manifest.json", "discriminator_results.json"],
      "validator": "coverage, equal-freeze identity, and first-hard-failure classifications reproduce",
      "negative_control": "reject incumbent-only privilege or omitted admissible rival"
    }
  ]
}

Machine artifact - SG2_STATUS.json

Artifact: EXECUTION/SG2_STATUS.json
SHA-256: 96ece6a963c92046b3d3fdf8f1dde0f4170bcf65d20cd75ec7c74617be90a334

{
  "blocking_dependencies": [
    "SG1-O01",
    "SG1-O02",
    "SG1-O03",
    "SG1-O05",
    "SG1-O06",
    "SG1-O07",
    "SG1-O09",
    "SG1-O10",
    "SG1-O11"
  ],
  "candidate_scope": "declared locked massless-vector manifest plus imported matter-character kernel",
  "conditional_candidate_certificate": "PASS",
  "direct_upstream_open_count": 9,
  "internal_reconstructed_gauntlet": "PASS",
  "nature_selection_claimed": false,
  "physical_gate_status": "OPEN",
  "provenance_class": "construction-anchor",
  "public_wording_ceiling": "SG-2's conditional candidate certificate passes: the frozen manifest reproduces 12 massless vectors, rank 4, faithful kernel Z6, and zero extras. The physical SG-2 gate remains OPEN on nine direct SG1 V3 construction dependencies. The reviewer-randomized package remains NOT-EVALUATED.",
  "reviewer_randomized_gauntlet": "NOT-EVALUATED",
  "schema": "SG2-EXECUTION-STATUS-2.0",
  "upstream_sg1_block": "BB-SG1-CLOSURE-3",
  "upstream_sg1_gate": "OPEN"
}

Machine artifact - SG2_VECTOR_SECTOR_CERTIFICATE.json

Artifact: EXECUTION/SG2_VECTOR_SECTOR_CERTIFICATE.json
SHA-256: 6bb132263e4550233b7e9d5387f816a60cfc44135323659329cfcd1cfa47eb22

{
  "candidate_id": "session-3d9d0d855e9f3acfbd7b",
  "center_enumeration": {
    "domain_size": 36,
    "generator": [
      1,
      1,
      1
    ],
    "kernel_elements": [
      [
        0,
        0,
        0
      ],
      [
        0,
        1,
        3
      ],
      [
        1,
        0,
        4
      ],
      [
        1,
        1,
        1
      ],
      [
        2,
        0,
        2
      ],
      [
        2,
        1,
        5
      ]
    ],
    "kernel_label": "Z6",
    "kernel_size": 6
  },
  "computed": {
    "extra_vectors": 0,
    "geometric_dimension": 11,
    "geometric_rank": 3,
    "missing_vectors": 0,
    "principal_dimension": 1,
    "principal_rank": 1,
    "total_dimension": 12,
    "total_rank": 4
  },
  "duplicate_carriers": {},
  "factor_checks": [
    {
      "algebra": "su3",
      "carrier": "color",
      "dimension": 8,
      "id": "K6",
      "parent": "metric:K6",
      "rank": 2,
      "type": "SU3/T2"
    },
    {
      "algebra": "su2",
      "carrier": "weak",
      "dimension": 3,
      "id": "S2",
      "parent": "metric:S2",
      "rank": 1,
      "type": "S2_round"
    },
    {
      "algebra": "none",
      "carrier": null,
      "dimension": 0,
      "id": "I_chi",
      "parent": "metric:I_chi",
      "rank": 0,
      "type": "I_interval"
    }
  ],
  "no_extra_audit": {
    "boundary_vector_zero_modes": [],
    "independent_nonabelian_connections": [],
    "interval_connected_killing_dimension": 0,
    "p_form_vector_zero_modes": []
  },
  "ownership": {
    "color": [
      "metric:K6"
    ],
    "hypercharge": [
      "independent-principal-U1Y"
    ],
    "weak": [
      "metric:S2"
    ]
  },
  "parity_table": [
    {
      "algebra": "u1",
      "carrier": "hypercharge",
      "claimed_vector_zero_modes": 1,
      "computed_vector_zero_modes": 1,
      "dimension": 1,
      "id": "B_Y",
      "parent": "independent-principal-U1Y",
      "rank": 1,
      "scalar_parity": "odd",
      "vector_parity": "even"
    }
  ],
  "schema": "SG2-VECTOR-SECTOR-CERTIFICATE-1.0",
  "smith_certificate": {
    "basis_convention": "center tuple (a mod 3, b mod 2, k mod 6)",
    "domain_relation_D": [
      [
        3,
        0,
        0
      ],
      [
        0,
        2,
        0
      ],
      [
        0,
        0,
        6
      ]
    ],
    "kernel_group": "Z6",
    "kernel_lift_basis_B": [
      [
        1,
        3,
        0
      ],
      [
        1,
        0,
        2
      ],
      [
        1,
        0,
        0
      ]
    ],
    "phase_rule": "2*triality*a + 3*doublet*b + y6*k = 0 mod 6",
    "quotient_relation_M": [
      [
        0,
        0,
        6
      ],
      [
        1,
        0,
        -2
      ],
      [
        0,
        1,
        -3
      ]
    ],
    "smith_invariants": [
      1,
      1,
      6
    ],
    "verified_B_times_M_equals_D": true
  }
}

Machine artifact - ANSWER_KEY.json

Artifact: EXECUTION/ANSWER_KEY.json
SHA-256: c5f21aea44807f747552d831d3a93faf47197bafd76d6f0a6b9664afeef3902e

{
  "entries": [
    {
      "builder_label": "honest-nonsm-double-sphere",
      "candidate_id": "session-fcebce2758592a4bbbbb",
      "expected_first_failure": "SG2-GNT-07",
      "expected_verdict": "FAIL",
      "role": "INNOCENT-NON-SM"
    },
    {
      "builder_label": "duplicate-su2-decoy",
      "candidate_id": "session-590e8c8330106fe8315b",
      "expected_first_failure": "SG2-GNT-03",
      "expected_verdict": "FAIL",
      "role": "DECOY"
    },
    {
      "builder_label": "factor-swap-decoy",
      "candidate_id": "session-69b5108731469c3ae360",
      "expected_first_failure": "SG2-GNT-01",
      "expected_verdict": "FAIL",
      "role": "DECOY"
    },
    {
      "builder_label": "honest-thirteen-vector",
      "candidate_id": "session-37f6ea093d2cac15c5e9",
      "expected_first_failure": "SG2-GNT-07",
      "expected_verdict": "FAIL",
      "role": "INNOCENT-NON-SM"
    },
    {
      "builder_label": "second-u1-decoy",
      "candidate_id": "session-447b5b256e606e8b30d5",
      "expected_first_failure": "SG2-GNT-02",
      "expected_verdict": "FAIL",
      "role": "DECOY"
    },
    {
      "builder_label": "wrong-global-kernel-decoy",
      "candidate_id": "session-1962e5f81ef9b8287448",
      "expected_first_failure": "SG2-GNT-05",
      "expected_verdict": "FAIL",
      "role": "DECOY"
    },
    {
      "builder_label": "incumbent",
      "candidate_id": "session-3d9d0d855e9f3acfbd7b",
      "expected_first_failure": null,
      "expected_verdict": "PASS",
      "role": "INCUMBENT"
    },
    {
      "builder_label": "odd-hypercharge-decoy",
      "candidate_id": "session-5fa9e02946c18581b740",
      "expected_first_failure": "SG2-GNT-04",
      "expected_verdict": "FAIL",
      "role": "DECOY"
    }
  ],
  "schema": "SG2-GAUNTLET-ANSWER-KEY-1.0",
  "scope": "internally reconstructed from reviewer SG-2 specification"
}

Machine artifact - ESCROWED_VERDICTS.json

Artifact: EXECUTION/ESCROWED_VERDICTS.json
SHA-256: 83616efbc4d581aa16501f9cac2051a42bb34409f6cbc6d18c8d827b0b76b0e6

{
  "entries": [
    {
      "answer_key_commitment": "af4676da2873f26843918dffd27f028d0062590fde797dd094e13639fd0395c5",
      "candidate_id": "session-fcebce2758592a4bbbbb",
      "first_failure": "SG2-GNT-07",
      "manifest_sha256": "0a15b1e385dab572c82c758766af19c6e5aebf89ae18e39020b4895ab0a61a28",
      "verdict": "FAIL",
      "witness": "WITNESSES/session-fcebce2758592a4bbbbb.json",
      "witness_sha256": "a02cfad4ac71ae513af1303f3fa8a53565089776914437ab25990555e777298d"
    },
    {
      "answer_key_commitment": "af4676da2873f26843918dffd27f028d0062590fde797dd094e13639fd0395c5",
      "candidate_id": "session-590e8c8330106fe8315b",
      "first_failure": "SG2-GNT-03",
      "manifest_sha256": "db8837cb351ec806c4e861074c551a6f8edcebdebc304c66d78bf7874ad0048a",
      "verdict": "FAIL",
      "witness": "WITNESSES/session-590e8c8330106fe8315b.json",
      "witness_sha256": "4fdb4b68ef28517c78a3b3816f6104174c0ebddd81a62c2e50344465219e78d5"
    },
    {
      "answer_key_commitment": "af4676da2873f26843918dffd27f028d0062590fde797dd094e13639fd0395c5",
      "candidate_id": "session-69b5108731469c3ae360",
      "first_failure": "SG2-GNT-01",
      "manifest_sha256": "37c1d9f1220a0ebca1105f57190c960dfc6f9b0e92fa0a5723653d1361c489bb",
      "verdict": "FAIL",
      "witness": "WITNESSES/session-69b5108731469c3ae360.json",
      "witness_sha256": "7efbd47453deea84f7aef14cf8ea5dc36db5da18b0388e7e513c6beed2fe885a"
    },
    {
      "answer_key_commitment": "af4676da2873f26843918dffd27f028d0062590fde797dd094e13639fd0395c5",
      "candidate_id": "session-37f6ea093d2cac15c5e9",
      "first_failure": "SG2-GNT-07",
      "manifest_sha256": "4e7c6737c24bcfbb7029d0c1a9ae1deac498b7f64b4d6c789bf40a54e41c3ce7",
      "verdict": "FAIL",
      "witness": "WITNESSES/session-37f6ea093d2cac15c5e9.json",
      "witness_sha256": "be378a1ef5856f38d9ca47aec38cd64d0a13fb0513a56ca420e8d0770ad6a4bd"
    },
    {
      "answer_key_commitment": "af4676da2873f26843918dffd27f028d0062590fde797dd094e13639fd0395c5",
      "candidate_id": "session-447b5b256e606e8b30d5",
      "first_failure": "SG2-GNT-02",
      "manifest_sha256": "1d3a38a0e6d1cdd90651e8246417909a1065291e0d5bb5dac147eb5acda3949c",
      "verdict": "FAIL",
      "witness": "WITNESSES/session-447b5b256e606e8b30d5.json",
      "witness_sha256": "b09e71d595e7c5507a3058316c681108780f5dddbb096e3518781efb72b62a58"
    },
    {
      "answer_key_commitment": "af4676da2873f26843918dffd27f028d0062590fde797dd094e13639fd0395c5",
      "candidate_id": "session-1962e5f81ef9b8287448",
      "first_failure": "SG2-GNT-05",
      "manifest_sha256": "f431dd3bc6ed5b07b9953f690771e991468a398dfa75a529479a45eff52721e1",
      "verdict": "FAIL",
      "witness": "WITNESSES/session-1962e5f81ef9b8287448.json",
      "witness_sha256": "74669c185206f440e17c278eff6847d8bd1339c2b22babf0dca83130c93ce2e1"
    },
    {
      "answer_key_commitment": "af4676da2873f26843918dffd27f028d0062590fde797dd094e13639fd0395c5",
      "candidate_id": "session-3d9d0d855e9f3acfbd7b",
      "first_failure": null,
      "manifest_sha256": "6871b3bc2053d1aa93eb743c14a16e055e436b4c7c628c4955fbf9c6fa741366",
      "verdict": "PASS",
      "witness": "WITNESSES/session-3d9d0d855e9f3acfbd7b.json",
      "witness_sha256": "6bb132263e4550233b7e9d5387f816a60cfc44135323659329cfcd1cfa47eb22"
    },
    {
      "answer_key_commitment": "af4676da2873f26843918dffd27f028d0062590fde797dd094e13639fd0395c5",
      "candidate_id": "session-5fa9e02946c18581b740",
      "first_failure": "SG2-GNT-04",
      "manifest_sha256": "ea9be3765aa1f3defa2744e80c64a6c1acacc3d2476d59df4597489fde454a13",
      "verdict": "FAIL",
      "witness": "WITNESSES/session-5fa9e02946c18581b740.json",
      "witness_sha256": "bb88bfe51da3353ca00abaa0d7639112390091703d4862a76c3264b68b850d4d"
    }
  ],
  "schema": "SG2-GAUNTLET-ESCROW-1.0"
}

Machine artifact - GAUNTLET_COMPARISON.json

Artifact: EXECUTION/GAUNTLET_COMPARISON.json
SHA-256: 69112f3d1d67df65569ec188dcd95e796db94b316346f7ca863c5bfd30ad2c38

{
  "all_matches": true,
  "decoy_count": 5,
  "entries": [
    {
      "actual_first_failure": "SG2-GNT-07",
      "actual_verdict": "FAIL",
      "candidate_id": "session-fcebce2758592a4bbbbb",
      "expected_first_failure": "SG2-GNT-07",
      "expected_verdict": "FAIL",
      "matched": true
    },
    {
      "actual_first_failure": "SG2-GNT-03",
      "actual_verdict": "FAIL",
      "candidate_id": "session-590e8c8330106fe8315b",
      "expected_first_failure": "SG2-GNT-03",
      "expected_verdict": "FAIL",
      "matched": true
    },
    {
      "actual_first_failure": "SG2-GNT-01",
      "actual_verdict": "FAIL",
      "candidate_id": "session-69b5108731469c3ae360",
      "expected_first_failure": "SG2-GNT-01",
      "expected_verdict": "FAIL",
      "matched": true
    },
    {
      "actual_first_failure": "SG2-GNT-07",
      "actual_verdict": "FAIL",
      "candidate_id": "session-37f6ea093d2cac15c5e9",
      "expected_first_failure": "SG2-GNT-07",
      "expected_verdict": "FAIL",
      "matched": true
    },
    {
      "actual_first_failure": "SG2-GNT-02",
      "actual_verdict": "FAIL",
      "candidate_id": "session-447b5b256e606e8b30d5",
      "expected_first_failure": "SG2-GNT-02",
      "expected_verdict": "FAIL",
      "matched": true
    },
    {
      "actual_first_failure": "SG2-GNT-05",
      "actual_verdict": "FAIL",
      "candidate_id": "session-1962e5f81ef9b8287448",
      "expected_first_failure": "SG2-GNT-05",
      "expected_verdict": "FAIL",
      "matched": true
    },
    {
      "actual_first_failure": null,
      "actual_verdict": "PASS",
      "candidate_id": "session-3d9d0d855e9f3acfbd7b",
      "expected_first_failure": null,
      "expected_verdict": "PASS",
      "matched": true
    },
    {
      "actual_first_failure": "SG2-GNT-04",
      "actual_verdict": "FAIL",
      "candidate_id": "session-5fa9e02946c18581b740",
      "expected_first_failure": "SG2-GNT-04",
      "expected_verdict": "FAIL",
      "matched": true
    }
  ],
  "innocent_non_sm_count": 2,
  "schema": "SG2-GAUNTLET-COMPARISON-1.0",
  "session_count": 8
}

Machine artifact - VALIDATION_REPORT.json

Artifact: EXECUTION/VALIDATION_REPORT.json
SHA-256: 3ac381cb88347525d4776973cd06ff44d9ea573493eb8f8f5f317881d1e064d8

{
  "control_count": 16,
  "controls": [
    {
      "control": "Engine compiles",
      "detail": null,
      "result": "PASS"
    },
    {
      "control": "Two byte-identical generated runs",
      "detail": {
        "engine_stdout": "{\"all_matches\": true, \"decoys\": 5, \"dimension\": 12, \"extras\": 0, \"innocents\": 2, \"kernel\": \"Z6\", \"rank\": 4, \"sessions\": 8}",
        "file_count": 24
      },
      "result": "PASS"
    },
    {
      "control": "All JSON parses",
      "detail": {
        "errors": [],
        "json_file_count": 23
      },
      "result": "PASS"
    },
    {
      "control": "Answer-key commitment and escrow",
      "detail": "af4676da2873f26843918dffd27f028d0062590fde797dd094e13639fd0395c5",
      "result": "PASS"
    },
    {
      "control": "All session verdicts match sealed key",
      "detail": {
        "decoys": 5,
        "sessions": 8
      },
      "result": "PASS"
    },
    {
      "control": "Every planted historical defect caught exactly once",
      "detail": [
        "SG2-GNT-01",
        "SG2-GNT-02",
        "SG2-GNT-03",
        "SG2-GNT-04",
        "SG2-GNT-05"
      ],
      "result": "PASS"
    },
    {
      "control": "Honest non-SM candidates fail only SM-match row",
      "detail": [
        "session-37f6ea093d2cac15c5e9",
        "session-fcebce2758592a4bbbbb"
      ],
      "result": "PASS"
    },
    {
      "control": "Vector-sector terminal reproduces 12, rank 4, Z6, zero extras",
      "detail": {
        "dimension": 12,
        "extras": 0,
        "kernel": "Z6",
        "rank": 4
      },
      "result": "PASS"
    },
    {
      "control": "Smith certificate is exact",
      "detail": {
        "basis_convention": "center tuple (a mod 3, b mod 2, k mod 6)",
        "domain_relation_D": [
          [
            3,
            0,
            0
          ],
          [
            0,
            2,
            0
          ],
          [
            0,
            0,
            6
          ]
        ],
        "kernel_group": "Z6",
        "kernel_lift_basis_B": [
          [
            1,
            3,
            0
          ],
          [
            1,
            0,
            2
          ],
          [
            1,
            0,
            0
          ]
        ],
        "phase_rule": "2*triality*a + 3*doublet*b + y6*k = 0 mod 6",
        "quotient_relation_M": [
          [
            0,
            0,
            6
          ],
          [
            1,
            0,
            -2
          ],
          [
            0,
            1,
            -3
          ]
        ],
        "smith_invariants": [
          1,
          1,
          6
        ],
        "verified_B_times_M_equals_D": true
      },
      "result": "PASS"
    },
    {
      "control": "Every incumbent carrier has one declared parent",
      "detail": {
        "color": [
          "metric:K6"
        ],
        "hypercharge": [
          "independent-principal-U1Y"
        ],
        "weak": [
          "metric:S2"
        ]
      },
      "result": "PASS"
    },
    {
      "control": "Frozen session and witness set is exact",
      "detail": {
        "sessions": 8,
        "witnesses": 8
      },
      "result": "PASS"
    },
    {
      "control": "Status and scope firewall",
      "detail": {
        "blocking_dependencies": [
          "SG1-O01",
          "SG1-O02",
          "SG1-O03",
          "SG1-O05",
          "SG1-O06",
          "SG1-O07",
          "SG1-O09",
          "SG1-O10",
          "SG1-O11"
        ],
        "candidate_scope": "declared locked massless-vector manifest plus imported matter-character kernel",
        "conditional_candidate_certificate": "PASS",
        "direct_upstream_open_count": 9,
        "internal_reconstructed_gauntlet": "PASS",
        "nature_selection_claimed": false,
        "physical_gate_status": "OPEN",
        "provenance_class": "construction-anchor",
        "public_wording_ceiling": "SG-2's conditional candidate certificate passes: the frozen manifest reproduces 12 massless vectors, rank 4, faithful kernel Z6, and zero extras. The physical SG-2 gate remains OPEN on nine direct SG1 V3 construction dependencies. The reviewer-randomized package remains NOT-EVALUATED.",
        "reviewer_randomized_gauntlet": "NOT-EVALUATED",
        "schema": "SG2-EXECUTION-STATUS-2.0",
        "upstream_sg1_block": "BB-SG1-CLOSURE-3",
        "upstream_sg1_gate": "OPEN"
      },
      "result": "PASS"
    },
    {
      "control": "Frozen SOT22 input hash",
      "detail": "78e42a81536b1b1c2af79e63ba291eeea1be8767c7421833f36b96da537a031d",
      "result": "PASS"
    },
    {
      "control": "Frozen SG1 V3 package hash",
      "detail": "858b08c9186bb8f6ca7010937b7a9d7c61dd51369e27cc844a2536ea3b9cb7bf",
      "result": "PASS"
    },
    {
      "control": "SG1-to-SG2 dependency reducer",
      "detail": {
        "direct_rows": [
          "SG1-O01",
          "SG1-O02",
          "SG1-O03",
          "SG1-O05",
          "SG1-O06",
          "SG1-O07",
          "SG1-O09",
          "SG1-O10",
          "SG1-O11"
        ],
        "physical_gate": "OPEN"
      },
      "result": "PASS"
    },
    {
      "control": "SG1 V3 validator replay",
      "detail": "VALIDATION PASS\nrows=18 open=18 work_packages=8 mutations=10/10\ngate=OPEN\nregistry_canonical_sha256=37ec18346b5b236d8027902d029940e9abfbadc6d6c75e867378956f818e2c8e",
      "result": "PASS"
    }
  ],
  "failed_controls": [],
  "lawful_states": {
    "conditional_candidate_certificate": "PASS",
    "internal_reconstructed_gauntlet": "PASS",
    "reviewer_randomized_gauntlet": "NOT-EVALUATED",
    "sg2_physical_gate": "OPEN",
    "upstream_sg1_gate": "OPEN"
  },
  "result": "PASS",
  "schema": "SG2-GAUNTLET-VALIDATION-1.0"
}

Machine artifact - FROZEN_INPUT_HASHES.json

Artifact: EXECUTION/FROZEN_INPUT_HASHES.json
SHA-256: 0a8239dd6d653a760a88fd5716f02c88189f8aedf85bc84fbeb452f03d6070f1

{
  "gauntlet_spec": "0cc82d8e33dc80f7d6e811ae2bf426306ac882d0aa01866e8ba021089d08be23",
  "public_dossier": "eefd3eecefa93e25cceda2acba6df6cf1cffc206be5675f6e871d6058e946da0",
  "sg1_to_sg2_dependency_ledger": "f5de6e33e9b36ed4e3513df47db08db6361e23a661e4019711b74b28a9844b47",
  "sg1_v3_package": "858b08c9186bb8f6ca7010937b7a9d7c61dd51369e27cc844a2536ea3b9cb7bf",
  "sot22": "78e42a81536b1b1c2af79e63ba291eeea1be8767c7421833f36b96da537a031d"
}

Executable gauntlet engine

Artifact: EXECUTION/sg2_gauntlet.py
SHA-256: 218eb34cc639541ae83917c934ebfe7f62c578ad8b7db3f5145c6d7a393edbeb

#!/usr/bin/env python3
"""Build and solve the deterministic reconstructed SG-2 gauntlet."""

from __future__ import annotations

import copy
import hashlib
import itertools
import json
import math
import shutil
from pathlib import Path


HERE = Path(__file__).resolve().parent
ROOT = HERE.parent
SOURCE_ZIP = ROOT.parent.parent / "upload/HIKING_PHYSICS_SOURCE_OF_TRUTH_2026-07-18_COMPLETE_REPLACES_PRIOR_VERSIONS(2).zip"
GAUNTLET_SPEC = ROOT.parent.parent / "upload/sg2-sg8-gauntlet-series.md"
PUBLIC_DOSSIER = ROOT / "FROZEN_INPUTS/sg2_public_dossier_2026-08-01.html"
SG1_V3_PACKAGE = ROOT / "FROZEN_INPUTS/SG1_UPDATED_BUILDING_BLOCK_V3_RC_2026-08-01.zip"
SG1_DEPENDENCY_LEDGER = ROOT / "FROZEN_INPUTS/SG1_TO_SG2_DEPENDENCY_LEDGER.json"


FACTOR_CATALOG = {
    "SU3/T2": {"algebra": "su3", "dimension": 8, "rank": 2},
    "S2_round": {"algebra": "su2", "dimension": 3, "rank": 1},
    "I_interval": {"algebra": "none", "dimension": 0, "rank": 0},
    "S1_circle": {"algebra": "u1", "dimension": 1, "rank": 1},
}

STANDARD_CHARACTERS = [
    {"id": "Q", "triality": 1, "doublet": 1, "y6": 1},
    {"id": "u_c", "triality": -1, "doublet": 0, "y6": -4},
    {"id": "d_c", "triality": -1, "doublet": 0, "y6": 2},
    {"id": "L", "triality": 0, "doublet": 1, "y6": -3},
    {"id": "e_c", "triality": 0, "doublet": 0, "y6": 6},
    {"id": "nu_c", "triality": 0, "doublet": 0, "y6": 0},
    {"id": "H", "triality": 0, "doublet": 1, "y6": 3},
]


RULES = [
    "SG2-GNT-01",
    "SG2-GNT-02",
    "SG2-GNT-03",
    "SG2-GNT-04",
    "SG2-GNT-05",
    "SG2-GNT-06",
    "SG2-GNT-07",
]


def sha256(path: Path) -> str:
    digest = hashlib.sha256()
    with path.open("rb") as stream:
        for chunk in iter(lambda: stream.read(1024 * 1024), b""):
            digest.update(chunk)
    return digest.hexdigest()


def canonical_bytes(value: object) -> bytes:
    return json.dumps(value, sort_keys=True, separators=(",", ":")).encode()


def write_json(path: Path, value: object) -> None:
    path.write_text(json.dumps(value, indent=2, sort_keys=True) + "\n")


def candidate_id(seed: str, label: str) -> str:
    return "session-" + hashlib.sha256(f"{seed}:{label}".encode()).hexdigest()[:20]


def base_manifest() -> dict:
    return {
        "schema": "SG2-GAUNTLET-CANDIDATE-1.0",
        "candidate_id": "UNASSIGNED",
        "presentation_role": "UNDISCLOSED",
        "scope": "massless vector sector plus reviewer-requested faithful matter kernel",
        "geometry": {
            "factors": [
                {"id": "K6", "type": "SU3/T2", "carrier": "color"},
                {"id": "S2", "type": "S2_round", "carrier": "weak"},
                {"id": "I_chi", "type": "I_interval", "carrier": None},
            ]
        },
        "principal_connections": [
            {
                "id": "B_Y",
                "algebra": "u1",
                "dimension": 1,
                "rank": 1,
                "carrier": "hypercharge",
                "parent": "independent-principal-U1Y",
                "vector_parity": "even",
                "scalar_parity": "odd",
                "claimed_vector_zero_modes": 1,
            }
        ],
        "parent_nonabelian_connections": [],
        "boundary_vector_zero_modes": [],
        "p_form_vector_zero_modes": [],
        "matter_characters": copy.deepcopy(STANDARD_CHARACTERS),
        "claims": {
            "connected_components": [
                {"carrier": "color", "algebra": "su3", "dimension": 8, "rank": 2},
                {"carrier": "weak", "algebra": "su2", "dimension": 3, "rank": 1},
                {"carrier": "hypercharge", "algebra": "u1", "dimension": 1, "rank": 1},
            ],
            "principal_connection_count": 1,
            "total_dimension": 12,
            "total_rank": 4,
            "extra_vectors": 0,
            "missing_vectors": 0,
            "matter_kernel": "Z6",
            "sm_match": True,
        },
    }


def phase_numerator(character: dict, a: int, b: int, k: int) -> int:
    return (
        2 * character["triality"] * a
        + 3 * character["doublet"] * b
        + character["y6"] * k
    ) % 6


def center_kernel(characters: list[dict]) -> list[tuple[int, int, int]]:
    return [
        (a, b, k)
        for a, b, k in itertools.product(range(3), range(2), range(6))
        if all(phase_numerator(item, a, b, k) == 0 for item in characters)
    ]


def element_order(item: tuple[int, int, int]) -> int:
    a, b, k = item
    orders = [1 if a == 0 else 3, 1 if b == 0 else 2, 1 if k == 0 else 6 // math.gcd(k, 6)]
    return math.lcm(*orders)


def kernel_label(kernel: list[tuple[int, int, int]]) -> str:
    if len(kernel) == 1:
        return "TRIVIAL"
    max_order = max(element_order(item) for item in kernel)
    if max_order == len(kernel):
        return f"Z{len(kernel)}"
    return f"ORDER-{len(kernel)}-ABELIAN"


def det3(matrix: list[list[int]]) -> int:
    a, b, c = matrix[0]
    d, e, f = matrix[1]
    g, h, i = matrix[2]
    return a * (e * i - f * h) - b * (d * i - f * g) + c * (d * h - e * g)


def smith_invariants_3x3(matrix: list[list[int]]) -> list[int]:
    entries = [abs(value) for row in matrix for value in row]
    d1 = math.gcd(*entries)
    minors = []
    for rows in itertools.combinations(range(3), 2):
        for cols in itertools.combinations(range(3), 2):
            a, b = matrix[rows[0]][cols[0]], matrix[rows[0]][cols[1]]
            c, d = matrix[rows[1]][cols[0]], matrix[rows[1]][cols[1]]
            minors.append(abs(a * d - b * c))
    delta2 = math.gcd(*minors)
    delta3 = abs(det3(matrix))
    return [d1, delta2 // d1, delta3 // delta2]


def smith_certificate() -> dict:
    # L = D Z^3 + Z(1,1,1) is the lift of the faithful center kernel.
    # B is a basis of L; D = B M.  SNF(M)=diag(1,1,6), hence L/DZ^3=Z6.
    basis_B = [[1, 3, 0], [1, 0, 2], [1, 0, 0]]
    relation_M = [[0, 0, 6], [1, 0, -2], [0, 1, -3]]
    domain_D = [[3, 0, 0], [0, 2, 0], [0, 0, 6]]
    product = [
        [sum(basis_B[r][j] * relation_M[j][c] for j in range(3)) for c in range(3)]
        for r in range(3)
    ]
    invariants = smith_invariants_3x3(relation_M)
    return {
        "basis_convention": "center tuple (a mod 3, b mod 2, k mod 6)",
        "phase_rule": "2*triality*a + 3*doublet*b + y6*k = 0 mod 6",
        "kernel_lift_basis_B": basis_B,
        "domain_relation_D": domain_D,
        "quotient_relation_M": relation_M,
        "verified_B_times_M_equals_D": product == domain_D,
        "smith_invariants": invariants,
        "kernel_group": "Z6" if invariants == [1, 1, 6] else "UNEXPECTED",
    }


def factor_contributions(manifest: dict) -> list[dict]:
    output = []
    for factor in manifest["geometry"]["factors"]:
        computed = FACTOR_CATALOG[factor["type"]]
        output.append({**factor, **computed, "parent": f"metric:{factor['id']}"})
    return output


def parity_zero_modes(connection: dict) -> int:
    return 1 if connection["vector_parity"] == "even" else 0


def compute_certificate(manifest: dict) -> dict:
    factors = factor_contributions(manifest)
    principal = []
    for item in manifest["principal_connections"]:
        principal.append({**item, "computed_vector_zero_modes": parity_zero_modes(item)})
    nonabelian = manifest["parent_nonabelian_connections"]
    boundary = manifest["boundary_vector_zero_modes"]
    pforms = manifest["p_form_vector_zero_modes"]

    geometric_dim = sum(item["dimension"] for item in factors)
    geometric_rank = sum(item["rank"] for item in factors)
    principal_dim = sum(item["dimension"] * parity_zero_modes(item) for item in principal)
    principal_rank = sum(item["rank"] * parity_zero_modes(item) for item in principal)
    nonabelian_dim = sum(item["dimension"] for item in nonabelian)
    nonabelian_rank = sum(item["rank"] for item in nonabelian)
    boundary_dim = sum(item["dimension"] for item in boundary)
    boundary_rank = sum(item["rank"] for item in boundary)
    pform_dim = sum(item["dimension"] for item in pforms)
    pform_rank = sum(item["rank"] for item in pforms)
    total_dim = geometric_dim + principal_dim + nonabelian_dim + boundary_dim + pform_dim
    total_rank = geometric_rank + principal_rank + nonabelian_rank + boundary_rank + pform_rank

    ownership: dict[str, list[str]] = {}
    for item in factors + principal + nonabelian + boundary + pforms:
        carrier = item.get("carrier")
        if carrier:
            ownership.setdefault(carrier, []).append(item.get("parent", item["id"]))

    kernel = center_kernel(manifest["matter_characters"])
    return {
        "schema": "SG2-VECTOR-SECTOR-CERTIFICATE-1.0",
        "candidate_id": manifest["candidate_id"],
        "factor_checks": factors,
        "parity_table": principal,
        "ownership": ownership,
        "duplicate_carriers": {key: value for key, value in ownership.items() if len(value) != 1},
        "no_extra_audit": {
            "independent_nonabelian_connections": nonabelian,
            "boundary_vector_zero_modes": boundary,
            "p_form_vector_zero_modes": pforms,
            "interval_connected_killing_dimension": sum(
                item["dimension"] for item in factors if item["type"] == "I_interval"
            ),
        },
        "computed": {
            "geometric_dimension": geometric_dim,
            "geometric_rank": geometric_rank,
            "principal_dimension": principal_dim,
            "principal_rank": principal_rank,
            "total_dimension": total_dim,
            "total_rank": total_rank,
            "extra_vectors": max(0, total_dim - 12),
            "missing_vectors": max(0, 12 - total_dim),
        },
        "center_enumeration": {
            "domain_size": 36,
            "kernel_size": len(kernel),
            "kernel_elements": [list(item) for item in kernel],
            "kernel_label": kernel_label(kernel),
            "generator": [1, 1, 1] if (1, 1, 1) in kernel else None,
        },
        "smith_certificate": smith_certificate() if kernel_label(kernel) == "Z6" else None,
    }


def component_claims(manifest: dict) -> dict[str, tuple[str, int, int]]:
    return {
        item["carrier"]: (item["algebra"], item["dimension"], item["rank"])
        for item in manifest["claims"]["connected_components"]
    }


def solve(manifest: dict) -> tuple[str | None, dict]:
    certificate = compute_certificate(manifest)
    claims = manifest["claims"]
    claimed_components = component_claims(manifest)

    # SG2-GNT-01: every active geometric Killing contribution must be declared.
    for factor in certificate["factor_checks"]:
        carrier = factor.get("carrier")
        if factor["dimension"] == 0:
            continue
        expected = (factor["algebra"], factor["dimension"], factor["rank"])
        if carrier not in claimed_components or claimed_components[carrier] != expected:
            return "SG2-GNT-01", certificate

    # SG2-GNT-02: the principal-connection inventory must match its claim.
    if len(manifest["principal_connections"]) != claims["principal_connection_count"]:
        return "SG2-GNT-02", certificate

    # SG2-GNT-03: each massless carrier has exactly one parent.
    if certificate["duplicate_carriers"]:
        return "SG2-GNT-03", certificate

    # SG2-GNT-04: parity claim and actual vector zero-mode count agree.
    for item in certificate["parity_table"]:
        if item["claimed_vector_zero_modes"] != item["computed_vector_zero_modes"]:
            return "SG2-GNT-04", certificate

    # SG2-GNT-05: the faithful matter kernel is exactly recomputed when claimed.
    if claims["matter_kernel"] != "NOT-CLAIMED":
        if claims["matter_kernel"] != certificate["center_enumeration"]["kernel_label"]:
            return "SG2-GNT-05", certificate

    # SG2-GNT-06: total dimension, rank, missing and extra counts match.
    for key in ["total_dimension", "total_rank", "extra_vectors", "missing_vectors"]:
        if claims[key] != certificate["computed"][key]:
            return "SG2-GNT-06", certificate

    # SG2-GNT-07: separate consistency from Standard-Model match.
    sm_components = {
        "color": ("su3", 8, 2),
        "weak": ("su2", 3, 1),
        "hypercharge": ("u1", 1, 1),
    }
    actual_sm = (
        claimed_components == sm_components
        and certificate["computed"]["total_dimension"] == 12
        and certificate["computed"]["total_rank"] == 4
        and certificate["computed"]["extra_vectors"] == 0
        and certificate["center_enumeration"]["kernel_label"] == "Z6"
    )
    if claims["sm_match"] != actual_sm or not actual_sm:
        return "SG2-GNT-07", certificate
    return None, certificate


def build_candidates(seed: str) -> list[tuple[str, dict, str | None, str]]:
    candidates = []

    def add(label: str, manifest: dict, failure: str | None, role: str) -> None:
        manifest["candidate_id"] = candidate_id(seed, label)
        candidates.append((label, manifest, failure, role))

    manifest = base_manifest()
    add("incumbent", manifest, None, "INCUMBENT")

    manifest = base_manifest()
    manifest["geometry"]["factors"][-1] = {"id": "S1_Y", "type": "S1_circle", "carrier": "circle-u1"}
    add("factor-swap-decoy", manifest, "SG2-GNT-01", "DECOY")

    manifest = base_manifest()
    manifest["principal_connections"].append(
        {
            "id": "B_X",
            "algebra": "u1",
            "dimension": 1,
            "rank": 1,
            "carrier": "extra-u1",
            "parent": "undeclared-second-principal-U1",
            "vector_parity": "even",
            "scalar_parity": "odd",
            "claimed_vector_zero_modes": 1,
        }
    )
    add("second-u1-decoy", manifest, "SG2-GNT-02", "DECOY")

    manifest = base_manifest()
    manifest["parent_nonabelian_connections"].append(
        {
            "id": "A_weak_independent",
            "algebra": "su2",
            "dimension": 3,
            "rank": 1,
            "carrier": "weak",
            "parent": "independent-principal-SU2",
        }
    )
    add("duplicate-su2-decoy", manifest, "SG2-GNT-03", "DECOY")

    manifest = base_manifest()
    manifest["principal_connections"][0]["vector_parity"] = "odd"
    add("odd-hypercharge-decoy", manifest, "SG2-GNT-04", "DECOY")

    manifest = base_manifest()
    manifest["matter_characters"][0]["y6"] = -3
    add("wrong-global-kernel-decoy", manifest, "SG2-GNT-05", "DECOY")

    manifest = base_manifest()
    manifest["geometry"]["factors"] = [
        {"id": "S2_A", "type": "S2_round", "carrier": "weak-a"},
        {"id": "S2_B", "type": "S2_round", "carrier": "weak-b"},
        {"id": "I_chi", "type": "I_interval", "carrier": None},
    ]
    manifest["matter_characters"] = []
    manifest["claims"].update(
        {
            "connected_components": [
                {"carrier": "weak-a", "algebra": "su2", "dimension": 3, "rank": 1},
                {"carrier": "weak-b", "algebra": "su2", "dimension": 3, "rank": 1},
                {"carrier": "hypercharge", "algebra": "u1", "dimension": 1, "rank": 1},
            ],
            "total_dimension": 7,
            "total_rank": 3,
            "extra_vectors": 0,
            "missing_vectors": 5,
            "matter_kernel": "NOT-CLAIMED",
            "sm_match": False,
        }
    )
    add("honest-nonsm-double-sphere", manifest, "SG2-GNT-07", "INNOCENT-NON-SM")

    manifest = base_manifest()
    manifest["geometry"]["factors"][-1] = {"id": "S1_X", "type": "S1_circle", "carrier": "extra-u1"}
    manifest["claims"].update(
        {
            "connected_components": manifest["claims"]["connected_components"]
            + [{"carrier": "extra-u1", "algebra": "u1", "dimension": 1, "rank": 1}],
            "total_dimension": 13,
            "total_rank": 5,
            "extra_vectors": 1,
            "missing_vectors": 0,
            "sm_match": False,
        }
    )
    add("honest-thirteen-vector", manifest, "SG2-GNT-07", "INNOCENT-NON-SM")
    return candidates


def main() -> None:
    for generated_directory in [HERE / "SESSIONS", HERE / "WITNESSES"]:
        if generated_directory.exists():
            shutil.rmtree(generated_directory)
        generated_directory.mkdir(parents=True)

    hashes = {
        "sot22": sha256(SOURCE_ZIP),
        "gauntlet_spec": sha256(GAUNTLET_SPEC),
        "public_dossier": sha256(PUBLIC_DOSSIER),
        "sg1_v3_package": sha256(SG1_V3_PACKAGE),
        "sg1_to_sg2_dependency_ledger": sha256(SG1_DEPENDENCY_LEDGER),
    }
    seed = hashlib.sha256("".join(hashes.values()).encode()).hexdigest()
    candidates = build_candidates(seed)
    order = sorted(range(len(candidates)), key=lambda index: hashlib.sha256(f"{seed}:order:{index}".encode()).hexdigest())

    answer_entries = []
    session_order = []
    for index in order:
        label, manifest, failure, role = candidates[index]
        session_id = manifest["candidate_id"]
        directory = HERE / "SESSIONS" / session_id
        directory.mkdir(parents=True, exist_ok=True)
        write_json(directory / "manifest.json", manifest)
        session_order.append(session_id)
        answer_entries.append(
            {
                "candidate_id": session_id,
                "builder_label": label,
                "role": role,
                "expected_verdict": "PASS" if failure is None else "FAIL",
                "expected_first_failure": failure,
            }
        )

    answer_key = {
        "schema": "SG2-GAUNTLET-ANSWER-KEY-1.0",
        "scope": "internally reconstructed from reviewer SG-2 specification",
        "entries": answer_entries,
    }
    commitment = hashlib.sha256(canonical_bytes(answer_key)).hexdigest()
    (HERE / "ANSWER_KEY_COMMITMENT.txt").write_text(commitment + "\n")
    write_json(HERE / "SESSION_ORDER.json", {"schema": "SG2-SESSION-ORDER-1.0", "seed": seed, "order": session_order})

    escrow_entries = []
    witness_directory = HERE / "WITNESSES"
    witness_directory.mkdir(exist_ok=True)
    for session_id in session_order:
        manifest_path = HERE / "SESSIONS" / session_id / "manifest.json"
        manifest = json.loads(manifest_path.read_text())
        failure, certificate = solve(manifest)
        certificate_path = witness_directory / f"{session_id}.json"
        write_json(certificate_path, certificate)
        escrow_entries.append(
            {
                "candidate_id": session_id,
                "manifest_sha256": sha256(manifest_path),
                "verdict": "PASS" if failure is None else "FAIL",
                "first_failure": failure,
                "witness": certificate_path.relative_to(HERE).as_posix(),
                "witness_sha256": sha256(certificate_path),
                "answer_key_commitment": commitment,
            }
        )
    escrow = {"schema": "SG2-GAUNTLET-ESCROW-1.0", "entries": escrow_entries}
    write_json(HERE / "ESCROWED_VERDICTS.json", escrow)
    write_json(HERE / "ANSWER_KEY.json", answer_key)

    by_id = {item["candidate_id"]: item for item in answer_entries}
    comparison_entries = []
    for actual in escrow_entries:
        expected = by_id[actual["candidate_id"]]
        matched = (
            actual["verdict"] == expected["expected_verdict"]
            and actual["first_failure"] == expected["expected_first_failure"]
        )
        comparison_entries.append(
            {
                "candidate_id": actual["candidate_id"],
                "expected_verdict": expected["expected_verdict"],
                "actual_verdict": actual["verdict"],
                "expected_first_failure": expected["expected_first_failure"],
                "actual_first_failure": actual["first_failure"],
                "matched": matched,
            }
        )
    comparison = {
        "schema": "SG2-GAUNTLET-COMPARISON-1.0",
        "session_count": len(comparison_entries),
        "decoy_count": sum(item[3] == "DECOY" for item in candidates),
        "innocent_non_sm_count": sum(item[3] == "INNOCENT-NON-SM" for item in candidates),
        "all_matches": all(item["matched"] for item in comparison_entries),
        "entries": comparison_entries,
    }
    write_json(HERE / "GAUNTLET_COMPARISON.json", comparison)

    incumbent = next(item for item in candidates if item[0] == "incumbent")[1]
    incumbent_certificate = compute_certificate(incumbent)
    write_json(HERE / "SG2_VECTOR_SECTOR_CERTIFICATE.json", incumbent_certificate)
    dependency = json.loads(SG1_DEPENDENCY_LEDGER.read_text())
    blocking_dependencies = [
        item["id"]
        for item in dependency["direct_sg2_physical_readiness_rows"]
        if item["status"] in {"OPEN", "NOT-EVALUATED", "CONSTRUCTION-ANCHOR"}
    ]
    status = {
        "schema": "SG2-EXECUTION-STATUS-2.0",
        "conditional_candidate_certificate": "PASS",
        "candidate_scope": "declared locked massless-vector manifest plus imported matter-character kernel",
        "internal_reconstructed_gauntlet": "PASS" if comparison["all_matches"] else "FAIL",
        "reviewer_randomized_gauntlet": "NOT-EVALUATED",
        "upstream_sg1_gate": dependency["upstream_validation"]["gate"],
        "upstream_sg1_block": dependency["authority_interpretation"]["upstream_execution_block"],
        "direct_upstream_open_count": len(blocking_dependencies),
        "blocking_dependencies": blocking_dependencies,
        "physical_gate_status": "OPEN" if comparison["all_matches"] else "FAIL",
        "provenance_class": "construction-anchor",
        "nature_selection_claimed": False,
        "public_wording_ceiling": (
            "SG-2's conditional candidate certificate passes: the frozen manifest "
            "reproduces 12 massless vectors, rank 4, faithful kernel Z6, and zero extras. "
            "The physical SG-2 gate remains OPEN on nine direct SG1 V3 construction "
            "dependencies. The reviewer-randomized package remains NOT-EVALUATED."
        ),
    }
    write_json(HERE / "SG2_STATUS.json", status)
    write_json(HERE / "FROZEN_INPUT_HASHES.json", hashes)
    print(
        json.dumps(
            {
                "sessions": len(session_order),
                "decoys": comparison["decoy_count"],
                "innocents": comparison["innocent_non_sm_count"],
                "all_matches": comparison["all_matches"],
                "dimension": incumbent_certificate["computed"]["total_dimension"],
                "rank": incumbent_certificate["computed"]["total_rank"],
                "kernel": incumbent_certificate["center_enumeration"]["kernel_label"],
                "extras": incumbent_certificate["computed"]["extra_vectors"],
            },
            sort_keys=True,
        )
    )


if __name__ == "__main__":
    main()

Independent execution validator

Artifact: EXECUTION/verify_gauntlet.py
SHA-256: 66af29addaef47bc4695a564e6d1c35f6970ff11386f82c589645de9e1671f2e

#!/usr/bin/env python3
"""Verify SG-2 execution determinism, arithmetic, and status firewalls."""

from __future__ import annotations

import hashlib
import json
import py_compile
import subprocess
import sys
from pathlib import Path


HERE = Path(__file__).resolve().parent
REPORT = HERE / "VALIDATION_REPORT.json"


def sha256(path: Path) -> str:
    digest = hashlib.sha256()
    with path.open("rb") as stream:
        for chunk in iter(lambda: stream.read(1024 * 1024), b""):
            digest.update(chunk)
    return digest.hexdigest()


def generated_files() -> list[Path]:
    names = [
        "ANSWER_KEY.json",
        "ANSWER_KEY_COMMITMENT.txt",
        "ESCROWED_VERDICTS.json",
        "FROZEN_INPUT_HASHES.json",
        "GAUNTLET_COMPARISON.json",
        "SESSION_ORDER.json",
        "SG2_STATUS.json",
        "SG2_VECTOR_SECTOR_CERTIFICATE.json",
    ]
    files = [HERE / name for name in names]
    files.extend(sorted((HERE / "SESSIONS").glob("*/manifest.json")))
    files.extend(sorted((HERE / "WITNESSES").glob("*.json")))
    return files


def snapshot() -> dict[str, str]:
    return {path.relative_to(HERE).as_posix(): sha256(path) for path in generated_files()}


def main() -> None:
    controls: list[dict[str, object]] = []

    def add(name: str, passed: bool, detail: object = None) -> None:
        controls.append({"control": name, "result": "PASS" if passed else "FAIL", "detail": detail})

    try:
        py_compile.compile(str(HERE / "sg2_gauntlet.py"), doraise=True)
        add("Engine compiles", True)
    except Exception as exc:
        add("Engine compiles", False, str(exc))

    first = subprocess.run(
        [sys.executable, str(HERE / "sg2_gauntlet.py")],
        cwd=HERE,
        text=True,
        stdout=subprocess.PIPE,
        stderr=subprocess.STDOUT,
        check=False,
    )
    snap1 = snapshot() if first.returncode == 0 else {}
    second = subprocess.run(
        [sys.executable, str(HERE / "sg2_gauntlet.py")],
        cwd=HERE,
        text=True,
        stdout=subprocess.PIPE,
        stderr=subprocess.STDOUT,
        check=False,
    )
    snap2 = snapshot() if second.returncode == 0 else {}
    add(
        "Two byte-identical generated runs",
        first.returncode == 0 and second.returncode == 0 and snap1 == snap2,
        {"file_count": len(snap2), "engine_stdout": second.stdout.strip()},
    )

    json_errors = []
    json_files = [path for path in HERE.rglob("*.json") if path != REPORT]
    for path in json_files:
        try:
            json.loads(path.read_text())
        except Exception as exc:
            json_errors.append(f"{path.relative_to(HERE)}: {exc}")
    add("All JSON parses", not json_errors, {"json_file_count": len(json_files), "errors": json_errors})

    answer_key = json.loads((HERE / "ANSWER_KEY.json").read_text())
    commitment = hashlib.sha256(
        json.dumps(answer_key, sort_keys=True, separators=(",", ":")).encode()
    ).hexdigest()
    escrow = json.loads((HERE / "ESCROWED_VERDICTS.json").read_text())
    add(
        "Answer-key commitment and escrow",
        (HERE / "ANSWER_KEY_COMMITMENT.txt").read_text().strip() == commitment
        and all(item["answer_key_commitment"] == commitment for item in escrow["entries"]),
        commitment,
    )

    comparison = json.loads((HERE / "GAUNTLET_COMPARISON.json").read_text())
    add(
        "All session verdicts match sealed key",
        comparison["all_matches"]
        and comparison["session_count"] == 8
        and all(item["matched"] for item in comparison["entries"]),
        {"sessions": comparison["session_count"], "decoys": comparison["decoy_count"]},
    )
    decoy_failures = sorted(
        item["expected_first_failure"]
        for item in answer_key["entries"]
        if item["role"] == "DECOY"
    )
    add(
        "Every planted historical defect caught exactly once",
        decoy_failures == [f"SG2-GNT-0{index}" for index in range(1, 6)],
        decoy_failures,
    )
    innocent_ids = {
        item["candidate_id"] for item in answer_key["entries"] if item["role"] == "INNOCENT-NON-SM"
    }
    actual_by_id = {item["candidate_id"]: item for item in comparison["entries"]}
    add(
        "Honest non-SM candidates fail only SM-match row",
        len(innocent_ids) == 2
        and all(actual_by_id[item]["actual_first_failure"] == "SG2-GNT-07" for item in innocent_ids),
        sorted(innocent_ids),
    )

    cert = json.loads((HERE / "SG2_VECTOR_SECTOR_CERTIFICATE.json").read_text())
    add(
        "Vector-sector terminal reproduces 12, rank 4, Z6, zero extras",
        cert["computed"]["total_dimension"] == 12
        and cert["computed"]["total_rank"] == 4
        and cert["computed"]["extra_vectors"] == 0
        and cert["center_enumeration"]["kernel_label"] == "Z6",
        {
            "dimension": cert["computed"]["total_dimension"],
            "rank": cert["computed"]["total_rank"],
            "kernel": cert["center_enumeration"]["kernel_label"],
            "extras": cert["computed"]["extra_vectors"],
        },
    )
    smith = cert["smith_certificate"]
    add(
        "Smith certificate is exact",
        smith["verified_B_times_M_equals_D"]
        and smith["smith_invariants"] == [1, 1, 6]
        and smith["kernel_group"] == "Z6",
        smith,
    )
    add(
        "Every incumbent carrier has one declared parent",
        not cert["duplicate_carriers"]
        and all(len(parents) == 1 for parents in cert["ownership"].values()),
        cert["ownership"],
    )

    order = json.loads((HERE / "SESSION_ORDER.json").read_text())["order"]
    actual_session_dirs = sorted(path.name for path in (HERE / "SESSIONS").iterdir() if path.is_dir())
    actual_witnesses = sorted(path.stem for path in (HERE / "WITNESSES").glob("*.json"))
    add(
        "Frozen session and witness set is exact",
        sorted(order) == actual_session_dirs == actual_witnesses,
        {"sessions": len(actual_session_dirs), "witnesses": len(actual_witnesses)},
    )

    status = json.loads((HERE / "SG2_STATUS.json").read_text())
    add(
        "Status and scope firewall",
        status["conditional_candidate_certificate"] == "PASS"
        and status["internal_reconstructed_gauntlet"] == "PASS"
        and status["reviewer_randomized_gauntlet"] == "NOT-EVALUATED"
        and status["physical_gate_status"] == "OPEN"
        and status["provenance_class"] == "construction-anchor"
        and status["upstream_sg1_gate"] == "OPEN"
        and status["direct_upstream_open_count"] == 9
        and not status["nature_selection_claimed"],
        status,
    )

    input_hashes = json.loads((HERE / "FROZEN_INPUT_HASHES.json").read_text())
    add(
        "Frozen SOT22 input hash",
        input_hashes["sot22"] == "78e42a81536b1b1c2af79e63ba291eeea1be8767c7421833f36b96da537a031d",
        input_hashes["sot22"],
    )

    expected_sg1_hash = "858b08c9186bb8f6ca7010937b7a9d7c61dd51369e27cc844a2536ea3b9cb7bf"
    add(
        "Frozen SG1 V3 package hash",
        input_hashes.get("sg1_v3_package") == expected_sg1_hash,
        input_hashes.get("sg1_v3_package"),
    )

    dependency = json.loads((HERE.parent / "FROZEN_INPUTS/SG1_TO_SG2_DEPENDENCY_LEDGER.json").read_text())
    direct = dependency["direct_sg2_physical_readiness_rows"]
    add(
        "SG1-to-SG2 dependency reducer",
        dependency["upstream_validation"]["gate"] == "OPEN"
        and len(direct) == 9
        and all(item["status"] == "OPEN" for item in direct)
        and dependency["reducer"]["current_sg2_gate"] == "OPEN"
        and status["blocking_dependencies"] == [item["id"] for item in direct],
        {"direct_rows": [item["id"] for item in direct], "physical_gate": status["physical_gate_status"]},
    )

    sg1_validation = subprocess.run(
        [sys.executable, "validate_sg1_v3.py"],
        cwd=HERE.parent / "FROZEN_INPUTS/SG1_V3",
        text=True,
        stdout=subprocess.PIPE,
        stderr=subprocess.STDOUT,
        check=False,
    )
    add(
        "SG1 V3 validator replay",
        sg1_validation.returncode == 0
        and "VALIDATION PASS" in sg1_validation.stdout
        and "gate=OPEN" in sg1_validation.stdout,
        sg1_validation.stdout.strip(),
    )

    failed = [item["control"] for item in controls if item["result"] != "PASS"]
    report = {
        "schema": "SG2-GAUNTLET-VALIDATION-1.0",
        "result": "PASS" if not failed else "FAIL",
        "control_count": len(controls),
        "failed_controls": failed,
        "controls": controls,
        "lawful_states": {
            "conditional_candidate_certificate": "PASS",
            "internal_reconstructed_gauntlet": "PASS",
            "reviewer_randomized_gauntlet": "NOT-EVALUATED",
            "upstream_sg1_gate": "OPEN",
            "sg2_physical_gate": "OPEN",
        },
    }
    REPORT.write_text(json.dumps(report, indent=2, sort_keys=True) + "\n")
    print(f"{len(controls)} controls {report['result']}")
    raise SystemExit(0 if not failed else 1)


if __name__ == "__main__":
    main()

Reviewer SG-2 challenge specification

SG-2 - Gauge structure

Scoped claim under test: the locked architecture yields exactly twelve massless vectors - su(3)⊕su(2) from the Killing fields of K₆×S², one u(1)_Y from the independent bundle with even parity - rank 4, matter kernel ℤ₆, zero extras, each carrier with exactly one declared parent.

Historical weak point: ownership. The original gate claimed the interval had a U(1) isometry it doesn’t have, and a later execution showed symmetry support is not ownership.

The decoys (each a complete manifest; the machinery must certify whether its claims match its own geometry): a factor swap that restores a circle isometry, so a thirteenth massless vector exists that the manifest doesn’t declare · a second, undeclared U(1) bundle · a duplicate SU(2) actor alongside the geometric connection - the provenance ledger must catch one carrier with two parents · an odd-parity hypercharge connection, so the claimed twelfth vector has no zero mode and the true count is eleven · a manifest whose claimed global form disagrees with its own center arithmetic. Innocents: manifests of non-SM gauge structures whose claims exactly match their geometry - they must PASS the consistency rows and fail only the SM-match row, for the computed reason. Plus the incumbent, relabeled.

The generated artifact: the full vector-sector certificate as a script - Killing-algebra dimensions per factor, the parity table, the no-extra-vector audit as a generated checklist, and the ℤ₆ kernel via Smith normal form with the matrix and basis conventions published - reproducing the SG-2 certificate (12, rank 4, ℤ₆, 0 extras) from the manifest alone.

Integrity ledger

Final controlling terminal

SG-2 CONDITIONAL CANDIDATE CERTIFICATE: PASS
SG1 V3 UPSTREAM GATE: OPEN (8 PASS / 18 OPEN)
SG-2 PHYSICAL GATE: OPEN (9 DIRECT UPSTREAM DEPENDENCIES OPEN)
INTERNAL RECONSTRUCTED GAUNTLET: PASS (8/8)
REVIEWER-RANDOMIZED GAUNTLET: NOT-EVALUATED
NATURE-SELECTION: NOT-CLAIMED

The dossier completely records the conditional calculation, cumulative dependency reduction, and remaining physical debts. It does not convert the passing conditional tuple into a physical closure claim, does not claim a reviewer test that was never delivered, and does not claim that nature selected the candidate.