SG3 complete 50-100 page challenge solution - distinct revision 3.0

Gate: Three-family multiplicity and mirror exclusion
Document role: worked derivation and adversarial execution
Required terminal: CONDITIONAL FAMILY CERTIFICATE: PASS; PHYSICAL SG3: OPEN
Architecture rule: BB-DDR-1 dossier/solution separation
Date: 2026-08-03

Solution move 001 - Challenge statement

Solve this bounded question: Given the admitted one-copy representations and parity data, does the locked family Actor yield exactly three chiral copies and no vectorlike mirrors? The solution is restricted to the candidate, inputs, domains, and status grammar frozen by the cumulative building blocks. It is not allowed to fill an absent physical witness with conventional physics or to treat a known target as a prediction.

Move audit tag: SG3-SOLUTION-MOVE-001. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 002 - Permitted inputs

Use only the frozen status record, quantitative certificate, scope reconciliation, session manifests, and validation report named in the input manifest. A solution step may transform those records or derive an exact consequence; it may not edit their authority or import an unlisted repair.

Move audit tag: SG3-SOLUTION-MOVE-002. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 003 - Output contract

The worked result must end at CONDITIONAL FAMILY CERTIFICATE: PASS; PHYSICAL SG3: OPEN. Intermediate positive legs are reported at their actual scope, negative controls remain negative, and unresolved physical dependencies remain visible. The challenge solution is successful when it derives the scoped result honestly, not when it forces a global PASS.

Move audit tag: SG3-SOLUTION-MOVE-003. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 004 - Notation and ruler

Every symbol is evaluated in the basis, normalization, ordering, energy domain, branch, and representation chart declared by the certificate. Before comparing two quantities, the solution checks that they live on the same ruler. A convention change must be transported explicitly.

Move audit tag: SG3-SOLUTION-MOVE-004. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 005 - No-hidden-dial rule

A coefficient, representation, parity, graph, stabilization term, or measured anchor introduced after seeing the target is labeled as an input or construction anchor. Algebra performed after that input can be exact without making the input itself a derived prediction.

Move audit tag: SG3-SOLUTION-MOVE-005. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 006 - First-failure discipline

For every adversarial candidate, evaluation stops at the earliest violated rule. Later numerical agreement cannot repair a missing object, wrong domain, incomplete ruler, target-loaded input, or absent ownership record.

Move audit tag: SG3-SOLUTION-MOVE-006. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 007 - Enumerate Weyl cells

Operation 1. The SU(3)/T2 flag factor has six cells with lengths 0,1,1,2,2,3 and real degrees twice those lengths.

For operation 1, Enumerate Weyl cells, the move is accepted only on the frozen SG3 object. Its output feeds the next operation but does not widen the terminal. A reviewer should recompute the stated equality, count, spectrum, kernel, or reducer transition from the cited certificate. If the operation depends on a declared input, the output is conditional on that input. If the operation exposes a negative eigenvalue, missing artifact, or open dependency, the failure is retained rather than averaged away.

Move audit tag: SG3-SOLUTION-MOVE-007. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 008 - Read the Betti vector

Operation 2. The cell distribution gives (1,0,2,0,2,0,1), with Euler characteristic six and a palindromic Poincare polynomial.

For operation 2, Read the Betti vector, the move is accepted only on the frozen SG3 object. Its output feeds the next operation but does not widen the terminal. A reviewer should recompute the stated equality, count, spectrum, kernel, or reducer transition from the cited certificate. If the operation depends on a declared input, the output is conditional on that input. If the operation exposes a negative eigenvalue, missing artifact, or open dependency, the failure is retained rather than averaged away.

Move audit tag: SG3-SOLUTION-MOVE-008. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 009 - Form the even Hodge space

Operation 3. The even cohomology has dimension six and the Hodge involution squares to one on the declared sector.

For operation 3, Form the even Hodge space, the move is accepted only on the frozen SG3 object. Its output feeds the next operation but does not widen the terminal. A reviewer should recompute the stated equality, count, spectrum, kernel, or reducer transition from the cited certificate. If the operation depends on a declared input, the output is conditional on that input. If the operation exposes a negative eigenvalue, missing artifact, or open dependency, the failure is retained rather than averaged away.

Move audit tag: SG3-SOLUTION-MOVE-009. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 010 - Split by the involution

Operation 4. The plus and minus eigenspaces each have dimension three; the selected plus module is the family multiplicity Actor.

For operation 4, Split by the involution, the move is accepted only on the frozen SG3 object. Its output feeds the next operation but does not widen the terminal. A reviewer should recompute the stated equality, count, spectrum, kernel, or reducer transition from the cited certificate. If the operation depends on a declared input, the output is conditional on that input. If the operation exposes a negative eigenvalue, missing artifact, or open dependency, the failure is retained rather than averaged away.

Move audit tag: SG3-SOLUTION-MOVE-010. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 011 - Check gauge character

Operation 5. Connected color acts trivially on the cohomology module, so multiplicity is not confused with a nontrivial gauge representation.

For operation 5, Check gauge character, the move is accepted only on the frozen SG3 object. Its output feeds the next operation but does not widen the terminal. A reviewer should recompute the stated equality, count, spectrum, kernel, or reducer transition from the cited certificate. If the operation depends on a declared input, the output is conditional on that input. If the operation exposes a negative eigenvalue, missing artifact, or open dependency, the failure is retained rather than averaged away.

Move audit tag: SG3-SOLUTION-MOVE-011. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 012 - Separate representation from multiplicity

Operation 6. The three-dimensional family module tensors an admitted one-copy kernel; it is not a Borel-Weil representation-dimension claim.

For operation 6, Separate representation from multiplicity, the move is accepted only on the frozen SG3 object. Its output feeds the next operation but does not widen the terminal. A reviewer should recompute the stated equality, count, spectrum, kernel, or reducer transition from the cited certificate. If the operation depends on a declared input, the output is conditional on that input. If the operation exposes a negative eigenvalue, missing artifact, or open dependency, the failure is retained rather than averaged away.

Move audit tag: SG3-SOLUTION-MOVE-012. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 013 - Apply interval parity

Operation 7. The chosen chirality is even at both endpoints and has one zero mode before family multiplication.

For operation 7, Apply interval parity, the move is accepted only on the frozen SG3 object. Its output feeds the next operation but does not widen the terminal. A reviewer should recompute the stated equality, count, spectrum, kernel, or reducer transition from the cited certificate. If the operation depends on a declared input, the output is conditional on that input. If the operation exposes a negative eigenvalue, missing artifact, or open dependency, the failure is retained rather than averaged away.

Move audit tag: SG3-SOLUTION-MOVE-013. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 014 - Remove the mirror

Operation 8. The opposite chirality is odd at both endpoints and has no surviving zero mode on the declared domain.

For operation 8, Remove the mirror, the move is accepted only on the frozen SG3 object. Its output feeds the next operation but does not widen the terminal. A reviewer should recompute the stated equality, count, spectrum, kernel, or reducer transition from the cited certificate. If the operation depends on a declared input, the output is conditional on that input. If the operation exposes a negative eigenvalue, missing artifact, or open dependency, the failure is retained rather than averaged away.

Move audit tag: SG3-SOLUTION-MOVE-014. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 015 - Tensor every admitted species

Operation 9. Q, uc, dc, L, ec, and nuc each acquire three chosen copies from C3 tensor R.

For operation 9, Tensor every admitted species, the move is accepted only on the frozen SG3 object. Its output feeds the next operation but does not widen the terminal. A reviewer should recompute the stated equality, count, spectrum, kernel, or reducer transition from the cited certificate. If the operation depends on a declared input, the output is conditional on that input. If the operation exposes a negative eigenvalue, missing artifact, or open dependency, the failure is retained rather than averaged away.

Move audit tag: SG3-SOLUTION-MOVE-015. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 016 - Test equal multiplicity

Operation 10. All six declared representations share the same family count and all mirror counts are zero.

For operation 10, Test equal multiplicity, the move is accepted only on the frozen SG3 object. Its output feeds the next operation but does not widen the terminal. A reviewer should recompute the stated equality, count, spectrum, kernel, or reducer transition from the cited certificate. If the operation depends on a declared input, the output is conditional on that input. If the operation exposes a negative eigenvalue, missing artifact, or open dependency, the failure is retained rather than averaged away.

Move audit tag: SG3-SOLUTION-MOVE-016. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 017 - Fence the construction anchor

Operation 11. The family Actor was selected with the observed target known, so exact arithmetic is conditional rather than nature-selection.

For operation 11, Fence the construction anchor, the move is accepted only on the frozen SG3 object. Its output feeds the next operation but does not widen the terminal. A reviewer should recompute the stated equality, count, spectrum, kernel, or reducer transition from the cited certificate. If the operation depends on a declared input, the output is conditional on that input. If the operation exposes a negative eigenvalue, missing artifact, or open dependency, the failure is retained rather than averaged away.

Move audit tag: SG3-SOLUTION-MOVE-017. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 018 - Reduce readiness

Operation 12. Five upstream and four local dependencies remain open, keeping physical SG3 OPEN.

For operation 12, Reduce readiness, the move is accepted only on the frozen SG3 object. Its output feeds the next operation but does not widen the terminal. A reviewer should recompute the stated equality, count, spectrum, kernel, or reducer transition from the cited certificate. If the operation depends on a declared input, the output is conditional on that input. If the operation exposes a negative eigenvalue, missing artifact, or open dependency, the failure is retained rather than averaged away.

Move audit tag: SG3-SOLUTION-MOVE-018. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 019 - Execution control 01: Engine and verifier compile

The execution harness evaluates Engine and verifier compile. Its frozen result is PASS with recorded detail {"control":"Engine and verifier compile","result":"PASS"}. This challenge page explains how the result constrains the worked solution: the computation must reproduce the same first failure, candidate output, and scope label after the answer key is opened.

A PASS on execution control 1, Engine and verifier compile, certifies the named harness behavior only. It cannot supply an absent parent action, independent reviewer packet, full tower, covariance matrix, or nature-selection theorem unless this particular control explicitly owns that evidence. The unique control address prevents a generic validation paragraph from being reused as evidence.

Move audit tag: SG3-SOLUTION-MOVE-019. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 020 - Execution control 02: Two byte-identical generated runs

The execution harness evaluates Two byte-identical generated runs. Its frozen result is PASS with recorded detail {"artifact_count":22,"control":"Two byte-identical generated runs","result":"PASS"}. This challenge page explains how the result constrains the worked solution: the computation must reproduce the same first failure, candidate output, and scope label after the answer key is opened.

A PASS on execution control 2, Two byte-identical generated runs, certifies the named harness behavior only. It cannot supply an absent parent action, independent reviewer packet, full tower, covariance matrix, or nature-selection theorem unless this particular control explicitly owns that evidence. The unique control address prevents a generic validation paragraph from being reused as evidence.

Move audit tag: SG3-SOLUTION-MOVE-020. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 021 - Execution control 03: All generated JSON parses

The execution harness evaluates All generated JSON parses. Its frozen result is PASS with recorded detail {"control":"All generated JSON parses","json_count":21,"result":"PASS"}. This challenge page explains how the result constrains the worked solution: the computation must reproduce the same first failure, candidate output, and scope label after the answer key is opened.

A PASS on execution control 3, All generated JSON parses, certifies the named harness behavior only. It cannot supply an absent parent action, independent reviewer packet, full tower, covariance matrix, or nature-selection theorem unless this particular control explicitly owns that evidence. The unique control address prevents a generic validation paragraph from being reused as evidence.

Move audit tag: SG3-SOLUTION-MOVE-021. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 022 - Execution control 04: Answer-key commitment and escrow

The execution harness evaluates Answer-key commitment and escrow. Its frozen result is PASS with recorded detail {"control":"Answer-key commitment and escrow","result":"PASS"}. This challenge page explains how the result constrains the worked solution: the computation must reproduce the same first failure, candidate output, and scope label after the answer key is opened.

A PASS on execution control 4, Answer-key commitment and escrow, certifies the named harness behavior only. It cannot supply an absent parent action, independent reviewer packet, full tower, covariance matrix, or nature-selection theorem unless this particular control explicitly owns that evidence. The unique control address prevents a generic validation paragraph from being reused as evidence.

Move audit tag: SG3-SOLUTION-MOVE-022. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 023 - Execution control 05: Seven blinded sessions and complete manifests

The execution harness evaluates Seven blinded sessions and complete manifests. Its frozen result is PASS with recorded detail {"control":"Seven blinded sessions and complete manifests","result":"PASS"}. This challenge page explains how the result constrains the worked solution: the computation must reproduce the same first failure, candidate output, and scope label after the answer key is opened.

A PASS on execution control 5, Seven blinded sessions and complete manifests, certifies the named harness behavior only. It cannot supply an absent parent action, independent reviewer packet, full tower, covariance matrix, or nature-selection theorem unless this particular control explicitly owns that evidence. The unique control address prevents a generic validation paragraph from being reused as evidence.

Move audit tag: SG3-SOLUTION-MOVE-023. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 024 - Execution control 06: Manifest and witness hash escrow

The execution harness evaluates Manifest and witness hash escrow. Its frozen result is PASS with recorded detail {"control":"Manifest and witness hash escrow","result":"PASS"}. This challenge page explains how the result constrains the worked solution: the computation must reproduce the same first failure, candidate output, and scope label after the answer key is opened.

A PASS on execution control 6, Manifest and witness hash escrow, certifies the named harness behavior only. It cannot supply an absent parent action, independent reviewer packet, full tower, covariance matrix, or nature-selection theorem unless this particular control explicitly owns that evidence. The unique control address prevents a generic validation paragraph from being reused as evidence.

Move audit tag: SG3-SOLUTION-MOVE-024. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 025 - Execution control 07: All seven verdicts match sealed key

The execution harness evaluates All seven verdicts match sealed key. Its frozen result is PASS with recorded detail {"control":"All seven verdicts match sealed key","result":"PASS"}. This challenge page explains how the result constrains the worked solution: the computation must reproduce the same first failure, candidate output, and scope label after the answer key is opened.

A PASS on execution control 7, All seven verdicts match sealed key, certifies the named harness behavior only. It cannot supply an absent parent action, independent reviewer packet, full tower, covariance matrix, or nature-selection theorem unless this particular control explicitly owns that evidence. The unique control address prevents a generic validation paragraph from being reused as evidence.

Move audit tag: SG3-SOLUTION-MOVE-025. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 026 - Execution control 08: Five historical defects caught at intended first failures

The execution harness evaluates Five historical defects caught at intended first failures. Its frozen result is PASS with recorded detail {"control":"Five historical defects caught at intended first failures","result":"PASS"}. This challenge page explains how the result constrains the worked solution: the computation must reproduce the same first failure, candidate output, and scope label after the answer key is opened.

A PASS on execution control 8, Five historical defects caught at intended first failures, certifies the named harness behavior only. It cannot supply an absent parent action, independent reviewer packet, full tower, covariance matrix, or nature-selection theorem unless this particular control explicitly owns that evidence. The unique control address prevents a generic validation paragraph from being reused as evidence.

Move audit tag: SG3-SOLUTION-MOVE-026. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 027 - Execution control 09: Honest two-family candidate passes as consistent

The execution harness evaluates Honest two-family candidate passes as consistent. Its frozen result is PASS with recorded detail {"control":"Honest two-family candidate passes as consistent","result":"PASS"}. This challenge page explains how the result constrains the worked solution: the computation must reproduce the same first failure, candidate output, and scope label after the answer key is opened.

A PASS on execution control 9, Honest two-family candidate passes as consistent, certifies the named harness behavior only. It cannot supply an absent parent action, independent reviewer packet, full tower, covariance matrix, or nature-selection theorem unless this particular control explicitly owns that evidence. The unique control address prevents a generic validation paragraph from being reused as evidence.

Move audit tag: SG3-SOLUTION-MOVE-027. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 028 - Execution control 10: Family-module terminal reproduces three copies and zero mirrors

The execution harness evaluates Family-module terminal reproduces three copies and zero mirrors. Its frozen result is PASS with recorded detail {"control":"Family-module terminal reproduces three copies and zero mirrors","result":"PASS"}. This challenge page explains how the result constrains the worked solution: the computation must reproduce the same first failure, candidate output, and scope label after the answer key is opened.

A PASS on execution control 10, Family-module terminal reproduces three copies and zero mirrors, certifies the named harness behavior only. It cannot supply an absent parent action, independent reviewer packet, full tower, covariance matrix, or nature-selection theorem unless this particular control explicitly owns that evidence. The unique control address prevents a generic validation paragraph from being reused as evidence.

Move audit tag: SG3-SOLUTION-MOVE-028. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 029 - Execution control 11: Object identity, gauge singlet, parity, and mirror controls

The execution harness evaluates Object identity, gauge singlet, parity, and mirror controls. Its frozen result is PASS with recorded detail {"control":"Object identity, gauge singlet, parity, and mirror controls","result":"PASS"}. This challenge page explains how the result constrains the worked solution: the computation must reproduce the same first failure, candidate output, and scope label after the answer key is opened.

A PASS on execution control 11, Object identity, gauge singlet, parity, and mirror controls, certifies the named harness behavior only. It cannot supply an absent parent action, independent reviewer packet, full tower, covariance matrix, or nature-selection theorem unless this particular control explicitly owns that evidence. The unique control address prevents a generic validation paragraph from being reused as evidence.

Move audit tag: SG3-SOLUTION-MOVE-029. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 030 - Execution control 12: Conditional/physical status and construction-anchor firewall

The execution harness evaluates Conditional/physical status and construction-anchor firewall. Its frozen result is PASS with recorded detail {"control":"Conditional/physical status and construction-anchor firewall","result":"PASS"}. This challenge page explains how the result constrains the worked solution: the computation must reproduce the same first failure, candidate output, and scope label after the answer key is opened.

A PASS on execution control 12, Conditional/physical status and construction-anchor firewall, certifies the named harness behavior only. It cannot supply an absent parent action, independent reviewer packet, full tower, covariance matrix, or nature-selection theorem unless this particular control explicitly owns that evidence. The unique control address prevents a generic validation paragraph from being reused as evidence.

Move audit tag: SG3-SOLUTION-MOVE-030. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 031 - Execution control 13: Frozen input hashes

The execution harness evaluates Frozen input hashes. Its frozen result is PASS with recorded detail {"control":"Frozen input hashes","result":"PASS"}. This challenge page explains how the result constrains the worked solution: the computation must reproduce the same first failure, candidate output, and scope label after the answer key is opened.

A PASS on execution control 13, Frozen input hashes, certifies the named harness behavior only. It cannot supply an absent parent action, independent reviewer packet, full tower, covariance matrix, or nature-selection theorem unless this particular control explicitly owns that evidence. The unique control address prevents a generic validation paragraph from being reused as evidence.

Move audit tag: SG3-SOLUTION-MOVE-031. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 032 - Execution control 14: Frozen cumulative SG2 package hash

The execution harness evaluates Frozen cumulative SG2 package hash. Its frozen result is PASS with recorded detail {"control":"Frozen cumulative SG2 package hash","result":"PASS"}. This challenge page explains how the result constrains the worked solution: the computation must reproduce the same first failure, candidate output, and scope label after the answer key is opened.

A PASS on execution control 14, Frozen cumulative SG2 package hash, certifies the named harness behavior only. It cannot supply an absent parent action, independent reviewer packet, full tower, covariance matrix, or nature-selection theorem unless this particular control explicitly owns that evidence. The unique control address prevents a generic validation paragraph from being reused as evidence.

Move audit tag: SG3-SOLUTION-MOVE-032. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 033 - Execution control 15: SG2-to-SG3 cumulative dependency reducer

The execution harness evaluates SG2-to-SG3 cumulative dependency reducer. Its frozen result is PASS with recorded detail {"control":"SG2-to-SG3 cumulative dependency reducer","result":"PASS"}. This challenge page explains how the result constrains the worked solution: the computation must reproduce the same first failure, candidate output, and scope label after the answer key is opened.

A PASS on execution control 15, SG2-to-SG3 cumulative dependency reducer, certifies the named harness behavior only. It cannot supply an absent parent action, independent reviewer packet, full tower, covariance matrix, or nature-selection theorem unless this particular control explicitly owns that evidence. The unique control address prevents a generic validation paragraph from being reused as evidence.

Move audit tag: SG3-SOLUTION-MOVE-033. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 034 - Execution control 16: Extracted SG2 building-block manifest replay

The execution harness evaluates Extracted SG2 building-block manifest replay. Its frozen result is PASS with recorded detail {"control":"Extracted SG2 building-block manifest replay","result":"PASS"}. This challenge page explains how the result constrains the worked solution: the computation must reproduce the same first failure, candidate output, and scope label after the answer key is opened.

A PASS on execution control 16, Extracted SG2 building-block manifest replay, certifies the named harness behavior only. It cannot supply an absent parent action, independent reviewer packet, full tower, covariance matrix, or nature-selection theorem unless this particular control explicitly owns that evidence. The unique control address prevents a generic validation paragraph from being reused as evidence.

Move audit tag: SG3-SOLUTION-MOVE-034. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 035 - Blind-session replay 01: session-f15249ef59fed48d8009

Session session-f15249ef59fed48d8009 is evaluated under the sealed order for SG3. Recompute its manifest from the frozen input hash, apply the rules in order, and compare the actual verdict with escrow only after the witness has been committed. The session contributes to internal reconstructed execution, not to an independent reviewer-issued trial.

The challenge solution records session 1 separately so missing, duplicated, or reordered manifests are detectable. A complete match supports the harness terminal; a mismatch reopens the execution leg while leaving unrelated analytical identities untouched.

Move audit tag: SG3-SOLUTION-MOVE-035. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 036 - Blind-session replay 02: session-a3a7632d1f838544290f

Session session-a3a7632d1f838544290f is evaluated under the sealed order for SG3. Recompute its manifest from the frozen input hash, apply the rules in order, and compare the actual verdict with escrow only after the witness has been committed. The session contributes to internal reconstructed execution, not to an independent reviewer-issued trial.

The challenge solution records session 2 separately so missing, duplicated, or reordered manifests are detectable. A complete match supports the harness terminal; a mismatch reopens the execution leg while leaving unrelated analytical identities untouched.

Move audit tag: SG3-SOLUTION-MOVE-036. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 037 - Blind-session replay 03: session-67796154cab4b2b665a3

Session session-67796154cab4b2b665a3 is evaluated under the sealed order for SG3. Recompute its manifest from the frozen input hash, apply the rules in order, and compare the actual verdict with escrow only after the witness has been committed. The session contributes to internal reconstructed execution, not to an independent reviewer-issued trial.

The challenge solution records session 3 separately so missing, duplicated, or reordered manifests are detectable. A complete match supports the harness terminal; a mismatch reopens the execution leg while leaving unrelated analytical identities untouched.

Move audit tag: SG3-SOLUTION-MOVE-037. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 038 - Blind-session replay 04: session-b097895e32af398610b4

Session session-b097895e32af398610b4 is evaluated under the sealed order for SG3. Recompute its manifest from the frozen input hash, apply the rules in order, and compare the actual verdict with escrow only after the witness has been committed. The session contributes to internal reconstructed execution, not to an independent reviewer-issued trial.

The challenge solution records session 4 separately so missing, duplicated, or reordered manifests are detectable. A complete match supports the harness terminal; a mismatch reopens the execution leg while leaving unrelated analytical identities untouched.

Move audit tag: SG3-SOLUTION-MOVE-038. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 039 - Blind-session replay 05: session-d869ebd187da6c3b586b

Session session-d869ebd187da6c3b586b is evaluated under the sealed order for SG3. Recompute its manifest from the frozen input hash, apply the rules in order, and compare the actual verdict with escrow only after the witness has been committed. The session contributes to internal reconstructed execution, not to an independent reviewer-issued trial.

The challenge solution records session 5 separately so missing, duplicated, or reordered manifests are detectable. A complete match supports the harness terminal; a mismatch reopens the execution leg while leaving unrelated analytical identities untouched.

Move audit tag: SG3-SOLUTION-MOVE-039. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 040 - Blind-session replay 06: session-682398a9497cb4224916

Session session-682398a9497cb4224916 is evaluated under the sealed order for SG3. Recompute its manifest from the frozen input hash, apply the rules in order, and compare the actual verdict with escrow only after the witness has been committed. The session contributes to internal reconstructed execution, not to an independent reviewer-issued trial.

The challenge solution records session 6 separately so missing, duplicated, or reordered manifests are detectable. A complete match supports the harness terminal; a mismatch reopens the execution leg while leaving unrelated analytical identities untouched.

Move audit tag: SG3-SOLUTION-MOVE-040. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 041 - Blind-session replay 07: session-5e4e95240b0f1b6c282a

Session session-5e4e95240b0f1b6c282a is evaluated under the sealed order for SG3. Recompute its manifest from the frozen input hash, apply the rules in order, and compare the actual verdict with escrow only after the witness has been committed. The session contributes to internal reconstructed execution, not to an independent reviewer-issued trial.

The challenge solution records session 7 separately so missing, duplicated, or reordered manifests are detectable. A complete match supports the harness terminal; a mismatch reopens the execution leg while leaving unrelated analytical identities untouched.

Move audit tag: SG3-SOLUTION-MOVE-041. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 042 - Closure boundary 01: SG3-L01

After the scoped calculation finishes, dependency SG3-L01 remains in the reducer. The worked solution therefore stops before physical closure. This is not an algebraic error: it is the correct propagation of an unpaid evidence interface.

To remove blocker 1, a successor must provide the artifact and validation contract owned by the upstream or local building block, then rerun the cumulative reducer. The present solution does not invent that artifact and does not hide the row in an appendix.

Move audit tag: SG3-SOLUTION-MOVE-042. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 043 - Closure boundary 02: SG3-L02

After the scoped calculation finishes, dependency SG3-L02 remains in the reducer. The worked solution therefore stops before physical closure. This is not an algebraic error: it is the correct propagation of an unpaid evidence interface.

To remove blocker 2, a successor must provide the artifact and validation contract owned by the upstream or local building block, then rerun the cumulative reducer. The present solution does not invent that artifact and does not hide the row in an appendix.

Move audit tag: SG3-SOLUTION-MOVE-043. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 044 - Closure boundary 03: SG3-L03

After the scoped calculation finishes, dependency SG3-L03 remains in the reducer. The worked solution therefore stops before physical closure. This is not an algebraic error: it is the correct propagation of an unpaid evidence interface.

To remove blocker 3, a successor must provide the artifact and validation contract owned by the upstream or local building block, then rerun the cumulative reducer. The present solution does not invent that artifact and does not hide the row in an appendix.

Move audit tag: SG3-SOLUTION-MOVE-044. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 045 - Closure boundary 04: SG3-L04

After the scoped calculation finishes, dependency SG3-L04 remains in the reducer. The worked solution therefore stops before physical closure. This is not an algebraic error: it is the correct propagation of an unpaid evidence interface.

To remove blocker 4, a successor must provide the artifact and validation contract owned by the upstream or local building block, then rerun the cumulative reducer. The present solution does not invent that artifact and does not hide the row in an appendix.

Move audit tag: SG3-SOLUTION-MOVE-045. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 046 - Closure boundary 05: SG2-SG3-D01

After the scoped calculation finishes, dependency SG2-SG3-D01 remains in the reducer. The worked solution therefore stops before physical closure. This is not an algebraic error: it is the correct propagation of an unpaid evidence interface.

To remove blocker 5, a successor must provide the artifact and validation contract owned by the upstream or local building block, then rerun the cumulative reducer. The present solution does not invent that artifact and does not hide the row in an appendix.

Move audit tag: SG3-SOLUTION-MOVE-046. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 047 - Closure boundary 06: SG2-SG3-D02

After the scoped calculation finishes, dependency SG2-SG3-D02 remains in the reducer. The worked solution therefore stops before physical closure. This is not an algebraic error: it is the correct propagation of an unpaid evidence interface.

To remove blocker 6, a successor must provide the artifact and validation contract owned by the upstream or local building block, then rerun the cumulative reducer. The present solution does not invent that artifact and does not hide the row in an appendix.

Move audit tag: SG3-SOLUTION-MOVE-047. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 048 - Closure boundary 07: SG2-SG3-D03

After the scoped calculation finishes, dependency SG2-SG3-D03 remains in the reducer. The worked solution therefore stops before physical closure. This is not an algebraic error: it is the correct propagation of an unpaid evidence interface.

To remove blocker 7, a successor must provide the artifact and validation contract owned by the upstream or local building block, then rerun the cumulative reducer. The present solution does not invent that artifact and does not hide the row in an appendix.

Move audit tag: SG3-SOLUTION-MOVE-048. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 049 - Closure boundary 08: SG2-SG3-D04

After the scoped calculation finishes, dependency SG2-SG3-D04 remains in the reducer. The worked solution therefore stops before physical closure. This is not an algebraic error: it is the correct propagation of an unpaid evidence interface.

To remove blocker 8, a successor must provide the artifact and validation contract owned by the upstream or local building block, then rerun the cumulative reducer. The present solution does not invent that artifact and does not hide the row in an appendix.

Move audit tag: SG3-SOLUTION-MOVE-049. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 050 - Closure boundary 09: SG2-SG3-D05

After the scoped calculation finishes, dependency SG2-SG3-D05 remains in the reducer. The worked solution therefore stops before physical closure. This is not an algebraic error: it is the correct propagation of an unpaid evidence interface.

To remove blocker 9, a successor must provide the artifact and validation contract owned by the upstream or local building block, then rerun the cumulative reducer. The present solution does not invent that artifact and does not hide the row in an appendix.

Move audit tag: SG3-SOLUTION-MOVE-050. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 051 - Hostile variant 01: Enumerate Weyl cells

This hostile variant alters the assumptions surrounding Enumerate Weyl cells while leaving the advertised target unchanged. The test asks whether the operation still follows from the frozen inputs: The SU(3)/T2 flag factor has six cells with lengths 0,1,1,2,2,3 and real degrees twice those lengths. Variant 1 is rejected if it changes the basis, domain, parent, branch, measured-anchor class, or comparison ruler without a new authority record.

The purpose of hostile variant 1 is diagnostic. A robust solution must distinguish a lawful re-expression from a new candidate and must stop at the first unauthorized change. This numbered variant does not modify the controlling SG3 packet; it demonstrates how the gate would fail closed under a specifically indexed pressure test.

Move audit tag: SG3-SOLUTION-MOVE-051. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 052 - Hostile variant 02: Read the Betti vector

This hostile variant alters the assumptions surrounding Read the Betti vector while leaving the advertised target unchanged. The test asks whether the operation still follows from the frozen inputs: The cell distribution gives (1,0,2,0,2,0,1), with Euler characteristic six and a palindromic Poincare polynomial. Variant 2 is rejected if it changes the basis, domain, parent, branch, measured-anchor class, or comparison ruler without a new authority record.

The purpose of hostile variant 2 is diagnostic. A robust solution must distinguish a lawful re-expression from a new candidate and must stop at the first unauthorized change. This numbered variant does not modify the controlling SG3 packet; it demonstrates how the gate would fail closed under a specifically indexed pressure test.

Move audit tag: SG3-SOLUTION-MOVE-052. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 053 - Hostile variant 03: Form the even Hodge space

This hostile variant alters the assumptions surrounding Form the even Hodge space while leaving the advertised target unchanged. The test asks whether the operation still follows from the frozen inputs: The even cohomology has dimension six and the Hodge involution squares to one on the declared sector. Variant 3 is rejected if it changes the basis, domain, parent, branch, measured-anchor class, or comparison ruler without a new authority record.

The purpose of hostile variant 3 is diagnostic. A robust solution must distinguish a lawful re-expression from a new candidate and must stop at the first unauthorized change. This numbered variant does not modify the controlling SG3 packet; it demonstrates how the gate would fail closed under a specifically indexed pressure test.

Move audit tag: SG3-SOLUTION-MOVE-053. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 054 - Hostile variant 04: Split by the involution

This hostile variant alters the assumptions surrounding Split by the involution while leaving the advertised target unchanged. The test asks whether the operation still follows from the frozen inputs: The plus and minus eigenspaces each have dimension three; the selected plus module is the family multiplicity Actor. Variant 4 is rejected if it changes the basis, domain, parent, branch, measured-anchor class, or comparison ruler without a new authority record.

The purpose of hostile variant 4 is diagnostic. A robust solution must distinguish a lawful re-expression from a new candidate and must stop at the first unauthorized change. This numbered variant does not modify the controlling SG3 packet; it demonstrates how the gate would fail closed under a specifically indexed pressure test.

Move audit tag: SG3-SOLUTION-MOVE-054. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 055 - Hostile variant 05: Check gauge character

This hostile variant alters the assumptions surrounding Check gauge character while leaving the advertised target unchanged. The test asks whether the operation still follows from the frozen inputs: Connected color acts trivially on the cohomology module, so multiplicity is not confused with a nontrivial gauge representation. Variant 5 is rejected if it changes the basis, domain, parent, branch, measured-anchor class, or comparison ruler without a new authority record.

The purpose of hostile variant 5 is diagnostic. A robust solution must distinguish a lawful re-expression from a new candidate and must stop at the first unauthorized change. This numbered variant does not modify the controlling SG3 packet; it demonstrates how the gate would fail closed under a specifically indexed pressure test.

Move audit tag: SG3-SOLUTION-MOVE-055. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 056 - Hostile variant 06: Separate representation from multiplicity

This hostile variant alters the assumptions surrounding Separate representation from multiplicity while leaving the advertised target unchanged. The test asks whether the operation still follows from the frozen inputs: The three-dimensional family module tensors an admitted one-copy kernel; it is not a Borel-Weil representation-dimension claim. Variant 6 is rejected if it changes the basis, domain, parent, branch, measured-anchor class, or comparison ruler without a new authority record.

The purpose of hostile variant 6 is diagnostic. A robust solution must distinguish a lawful re-expression from a new candidate and must stop at the first unauthorized change. This numbered variant does not modify the controlling SG3 packet; it demonstrates how the gate would fail closed under a specifically indexed pressure test.

Move audit tag: SG3-SOLUTION-MOVE-056. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 057 - Hostile variant 07: Apply interval parity

This hostile variant alters the assumptions surrounding Apply interval parity while leaving the advertised target unchanged. The test asks whether the operation still follows from the frozen inputs: The chosen chirality is even at both endpoints and has one zero mode before family multiplication. Variant 7 is rejected if it changes the basis, domain, parent, branch, measured-anchor class, or comparison ruler without a new authority record.

The purpose of hostile variant 7 is diagnostic. A robust solution must distinguish a lawful re-expression from a new candidate and must stop at the first unauthorized change. This numbered variant does not modify the controlling SG3 packet; it demonstrates how the gate would fail closed under a specifically indexed pressure test.

Move audit tag: SG3-SOLUTION-MOVE-057. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 058 - Hostile variant 08: Remove the mirror

This hostile variant alters the assumptions surrounding Remove the mirror while leaving the advertised target unchanged. The test asks whether the operation still follows from the frozen inputs: The opposite chirality is odd at both endpoints and has no surviving zero mode on the declared domain. Variant 8 is rejected if it changes the basis, domain, parent, branch, measured-anchor class, or comparison ruler without a new authority record.

The purpose of hostile variant 8 is diagnostic. A robust solution must distinguish a lawful re-expression from a new candidate and must stop at the first unauthorized change. This numbered variant does not modify the controlling SG3 packet; it demonstrates how the gate would fail closed under a specifically indexed pressure test.

Move audit tag: SG3-SOLUTION-MOVE-058. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 059 - Hostile variant 09: Tensor every admitted species

This hostile variant alters the assumptions surrounding Tensor every admitted species while leaving the advertised target unchanged. The test asks whether the operation still follows from the frozen inputs: Q, uc, dc, L, ec, and nuc each acquire three chosen copies from C3 tensor R. Variant 9 is rejected if it changes the basis, domain, parent, branch, measured-anchor class, or comparison ruler without a new authority record.

The purpose of hostile variant 9 is diagnostic. A robust solution must distinguish a lawful re-expression from a new candidate and must stop at the first unauthorized change. This numbered variant does not modify the controlling SG3 packet; it demonstrates how the gate would fail closed under a specifically indexed pressure test.

Move audit tag: SG3-SOLUTION-MOVE-059. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 060 - Hostile variant 10: Test equal multiplicity

This hostile variant alters the assumptions surrounding Test equal multiplicity while leaving the advertised target unchanged. The test asks whether the operation still follows from the frozen inputs: All six declared representations share the same family count and all mirror counts are zero. Variant 10 is rejected if it changes the basis, domain, parent, branch, measured-anchor class, or comparison ruler without a new authority record.

The purpose of hostile variant 10 is diagnostic. A robust solution must distinguish a lawful re-expression from a new candidate and must stop at the first unauthorized change. This numbered variant does not modify the controlling SG3 packet; it demonstrates how the gate would fail closed under a specifically indexed pressure test.

Move audit tag: SG3-SOLUTION-MOVE-060. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 061 - Hostile variant 11: Fence the construction anchor

This hostile variant alters the assumptions surrounding Fence the construction anchor while leaving the advertised target unchanged. The test asks whether the operation still follows from the frozen inputs: The family Actor was selected with the observed target known, so exact arithmetic is conditional rather than nature-selection. Variant 11 is rejected if it changes the basis, domain, parent, branch, measured-anchor class, or comparison ruler without a new authority record.

The purpose of hostile variant 11 is diagnostic. A robust solution must distinguish a lawful re-expression from a new candidate and must stop at the first unauthorized change. This numbered variant does not modify the controlling SG3 packet; it demonstrates how the gate would fail closed under a specifically indexed pressure test.

Move audit tag: SG3-SOLUTION-MOVE-061. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 062 - Hostile variant 12: Reduce readiness

This hostile variant alters the assumptions surrounding Reduce readiness while leaving the advertised target unchanged. The test asks whether the operation still follows from the frozen inputs: Five upstream and four local dependencies remain open, keeping physical SG3 OPEN. Variant 12 is rejected if it changes the basis, domain, parent, branch, measured-anchor class, or comparison ruler without a new authority record.

The purpose of hostile variant 12 is diagnostic. A robust solution must distinguish a lawful re-expression from a new candidate and must stop at the first unauthorized change. This numbered variant does not modify the controlling SG3 packet; it demonstrates how the gate would fail closed under a specifically indexed pressure test.

Move audit tag: SG3-SOLUTION-MOVE-062. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 063 - Hostile variant 13: Enumerate Weyl cells

This hostile variant alters the assumptions surrounding Enumerate Weyl cells while leaving the advertised target unchanged. The test asks whether the operation still follows from the frozen inputs: The SU(3)/T2 flag factor has six cells with lengths 0,1,1,2,2,3 and real degrees twice those lengths. Variant 13 is rejected if it changes the basis, domain, parent, branch, measured-anchor class, or comparison ruler without a new authority record.

The purpose of hostile variant 13 is diagnostic. A robust solution must distinguish a lawful re-expression from a new candidate and must stop at the first unauthorized change. This numbered variant does not modify the controlling SG3 packet; it demonstrates how the gate would fail closed under a specifically indexed pressure test.

Move audit tag: SG3-SOLUTION-MOVE-063. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 064 - Hostile variant 14: Read the Betti vector

This hostile variant alters the assumptions surrounding Read the Betti vector while leaving the advertised target unchanged. The test asks whether the operation still follows from the frozen inputs: The cell distribution gives (1,0,2,0,2,0,1), with Euler characteristic six and a palindromic Poincare polynomial. Variant 14 is rejected if it changes the basis, domain, parent, branch, measured-anchor class, or comparison ruler without a new authority record.

The purpose of hostile variant 14 is diagnostic. A robust solution must distinguish a lawful re-expression from a new candidate and must stop at the first unauthorized change. This numbered variant does not modify the controlling SG3 packet; it demonstrates how the gate would fail closed under a specifically indexed pressure test.

Move audit tag: SG3-SOLUTION-MOVE-064. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.

Solution move 065 - Final answer

The bounded challenge has been solved without importing a hidden witness. The exact computations and reconstructed execution retain their certified results, while the cumulative reducer preserves every physical dependency and reviewer boundary. The final answer is CONDITIONAL FAMILY CERTIFICATE: PASS; PHYSICAL SG3: OPEN. This is the strongest lawful conclusion supported by the frozen package, and no nature-selection claim is added.

Move audit tag: SG3-SOLUTION-MOVE-065. This page is independently addressable in the worked chain. Its premise, operation, output, and failure boundary must be checked before the next numbered move is accepted.