UQF-3 — Reflection Positivity

Complete scoped-closure dossier

Decision: The available sources support a complete scoped construction of reflection positivity for the accepted finite-floor Actor–Co-Actor branch. The construction is not an unconditional derivation from Shape and is not a universal continuum chiral-gauge theorem. Canonical promotion remains blocked by missing authoritative execution objects: the current UQF-4 terminal, the complete Actor term/domain ledger, the complete Co-Actor map registry, the controlling dossier checklist, an independent randomized gauntlet, the merge, and owner ratification.

The strongest honest terminal is therefore

CLOSED-SCOPED / POSITIVE REALIZATION GIVEN
Xi_RP dashv Xi_RP^vee / CONSTRUCTION-ANCHOR /
OWNER-REVIEW CANDIDATE

This document is deliberately more conservative than the public v4.0 headline. It closes every obligation that can be closed from the evidence provided, and it keeps every unavailable authority object visible rather than manufacturing a pass.

Part I — Gate definition and decision frame

1. The exact question

UQF-3 asks whether the accepted interacting physical theory admits a positive Hilbert-space reconstruction. At finite operational resolution \(\Delta>0\), let \(\mathcal A_{\Delta,+}\) denote the physical observable algebra supported at positive Euclidean time, let \(\Theta_E\) be external Euclidean-time reflection, and let \(\omega_\Delta\) be the physical state. The gate predicate is

\[ \boxed{ \omega_\Delta(F^{\Theta_E}F)\ge0 \quad\text{for every }F\in\mathcal A_{\Delta,+}. } \]

This predicate must hold on the complete accepted physical algebra after gauge, chirality, discrete-symmetry, heavy-mode, and observer reductions. It is not enough to prove positivity sector by sector and then assume that the reductions compose.

2. Questions that are not UQF-3

The gate does not require:

• a strictly positive Yang–Mills mass gap;
• confinement;
• a unique vacuum;
• pointwise positivity of a chiral determinant or Pfaffian;
• positivity of gauge-fixed ghosts before the physical quotient;
• reflection positivity of every regulator that could be imagined;
• an ontic zero-spacing continuum limit;
• nonlinear quantum gravity above the accepted cutoff;
• a proof that Nature uniquely selects the constructed representation.

Each item can be important, but none is identical to the physical OS quadratic form. Loading them into UQF-3 would make the gate larger than its actual obligation and would hide the smaller, testable failure conditions.

3. Four wrong-object substitutions

3.1 Determinant for state

Integrating out fermions can produce a complex determinant. That determinant is a compressed computational weight. It is not the complete parent state and is not a list of physical probabilities. A complex reduced weight may create a Monte Carlo sign problem without creating a negative-norm physical state.

3.2 Internal parity for Euclidean-time reflection

The orbifold map \(y\mapsto-y\) selects the chiral domain. Osterwalder–Schrader reflection reverses Euclidean time \(\tau\mapsto-\tau\) and includes adjunction. They act on different coordinates, carry different spin/tensor data, and have different owners. Internal parity helps only if its projector is orthogonal and commutes with the external reflection on the complete domain.

3.3 Mass gap for positivity

Reflection positivity requires a nonnegative shifted Hamiltonian \(K=H-E_0\ge0\), not a strictly positive gap above the ground sector. A massless or vacuum-degenerate theory can satisfy the norm-square identity.

3.4 Sector certificates for global composition

Positive gauge, fermion, scalar, and boundary sectors do not automatically yield a positive observer theory. Every quotient, compression, trace, restriction, and record map must be typed. The Co-Actor exists to close this composition gap.

4. Closure axes

This dossier uses four independent axes.

A pass on one axis cannot silently promote another. In particular, a correct construction theorem is not a blind derivation, and a polished dossier is not package authority.

Part II — Authority reconstruction

5. Available authority

The available source-of-truth package supplies the following load-bearing blocks.

The public UQF-3 v4.0 page contributes the proposed \(\Xi_{\rm RP}\dashv\Xi_{\rm RP}^{\vee}\) mechanism. Its front section claims full closure, while its archive retains older open residuals. Under the supplied execution protocol, a dossier is below package authority, registry rows, and execution evidence. The mechanism can therefore be reconstructed and tested, but its headline cannot self-ratify.

6. Missing controlling artifacts

No available file supplies:

1. the current controlling gates-test-checklist.md;
2. UQF-3 gate-scoped evidence states;
3. the current controlling UQF-3 registry row;
4. a content-addressed current UQF-4 anomaly terminal;
5. the complete current Hamiltonian term/domain ledger;
6. the complete current reduction-map registry;
7. an independent randomized UQF-3 gauntlet execution;
8. an owner ratification record.

These are not all physical theorem gaps. Some are authority and execution gaps. They nevertheless block the word canonical because the protocol explicitly requires them.

7. Reconciliation of the old and new dossiers

The archived dossier carried a family of residuals. The Actor–Co-Actor repair changes several of their types rather than merely relabeling them.

The repair is legitimate only with construction provenance. It would be illegitimate to present the new Actor as though Shape had forced it before the old residuals were known.

8. Authority decision

The current deliverable is a new development branch. It preserves the previous open evidence and encodes the repair in nine new building blocks. The branch is eligible for shadow replay and owner review. It is not canonically ratified by its own creation.

Part III — The positive Actor

9. Actor definition

At operational resolution \(\Delta\), define

\[ \Xi_{\rm RP}(\Delta)= (\mathcal H_\Delta,\mathcal D_\Delta,H_\Delta, \mathcal A_{\Delta,+},\Theta_E,\Omega_\Delta, \mathfrak T_\Delta,\mathfrak C_\Delta). \]

The tuple owns the positive carrier, common domain, complete finite-floor Hamiltonian, positive-time physical algebra, external time reflection, interacting state, transfer semigroup, and constraint/boundary interfaces. Leaving any item implicit would permit a false pass in which the proof applies to a smaller or different theory.

10. Sector carrier audit

10.1 Compact gauge links

For compact gauge factors, a link carrier is \(L^2(G,d\mu_{\rm Haar})\). The electric term is built from positive quadratic Casimirs with positive kinetic coefficients. The magnetic Wilson term is real and bounded below at finite resolution. These familiar ingredients admit a positive transfer representation when the action and reflection plane satisfy the usual conditions. Lüscher constructed a self-adjoint positive transfer matrix for Euclidean lattice gauge theory, and Menotti–Pelissetto proved OS positivity for the Wilson action at general separations. These results support the gauge sector but do not by themselves prove the project-specific chiral parent and observer map.

10.2 Chiral parent fermions

The parent construction uses a positive CAR Fock carrier and a self-adjoint higher-dimensional Dirac/domain-wall operator on the frozen orbifold domain. Chirality is read from the internal projector and index sector rather than from an isolated four-dimensional Weyl determinant treated as a positive measure.

This distinction is important. Rigorous reflection-positivity results for free overlap fermions and certain nongauge interactions do not automatically extend to the full interacting non-Abelian chiral gauge branch. The dossier therefore uses them only as cross-checks, never as the load-bearing theorem.

10.3 Scalars and Higgs

The scalar carrier has a positive kinetic metric. The finite-floor potential must be real and bounded below on the accepted domain. A negative Hessian at a particular Shape stationary point is a stability verdict about that point; it does not by itself create a negative Hilbert norm. Conversely, an actually unbounded accepted potential would destroy the lower-bound contract and reopen UQF-3.

10.4 Yukawa and flavor terms

Complex phases do not violate Hermiticity when interactions appear with their adjoints:

\[ H_Y=\Psi_L^\dagger Y\Phi\Psi_R+ \Psi_R^\dagger\Phi^\dagger Y^\dagger\Psi_L. \]

The finite flavor Actor changes matrix entries but does not change the positive carrier. The required witness is the adjoint pair on the common gauge/chiral domain, together with an appropriate relative-bound estimate.

10.5 Discrete gauge and higher-form sectors

A finite discrete gauge group acts unitarily and has an orthogonal group-average projector. A compact higher-form rotor uses a positive kinetic coefficient and a real bounded-below periodic potential. Neither sector creates a negative norm when its interfaces satisfy the same domain and reflection rules.

10.6 Counterterms and fixed-set terms

Symmetry permission is not positivity. A fixed-set or matching term enters \(H_\Delta\) only after it is shown Hermitian, relatively lower-bounded, domain preserving, gauge/chiral compatible, and externally reflection compatible. An uncomputed heat-kernel coefficient is therefore not automatically fatal, but neither can it be silently assumed benign.

11. Hamiltonian normal form

The construction organizes the complete current Hamiltonian as

\[ H_\Delta=H_G+H_F+H_H+H_Y+H_{B_3}+H_{\Theta9} +H_{\partial}+H_{\rm match}. \]

For every term \(H_i\), the owed row is

actor owner
carrier and common domain
adjoint witness
relative form bound
lower-bound contribution
gauge commutator
chiral-domain commutator
external-reflection interface
source hash
destructive control
reopen trigger

The available sources support this admissibility normal form but do not supply the complete exact matrix realization required by BB-QCR-1. That is the main reason the result remains a construction anchor.

12. Self-adjointness and lower bound

The physical requirement is a lower bound

\[ \langle\psi,H_\Delta\psi\rangle \ge E_0(\Delta)\lVert\psi\rVert^2, \qquad \psi\in\mathcal D_\Delta. \]

Defining \(K_\Delta=H_\Delta-E_0(\Delta)\mathbf1\) gives \(K_\Delta\ge0\) and the transfer semigroup

\[ T_\Delta(t)=e^{-tK_\Delta},\qquad t\ge0. \]

This is a self-adjoint contraction. It remains well defined when the ground space is degenerate and when no strictly positive spectral gap exists.

13. Anti-circularity finding

The positive-state sign is an explicit foundational input. UQF-3 does not derive it from geometry. The nontrivial task is to show that the accepted interacting Dynamics, constraints, projections, reductions, and observer maps do not destroy it.

The construction would become circular if the Actor were defined by choosing only those operators whose final OS form happened to be positive after seeing the answer. The repair avoids that failure only if its exact interfaces are versioned, its target-known history is disclosed, its removal downgrades the gate, and its terminal says CONSTRUCTION-ANCHOR.

Part IV — Reflection and constraints

14. The external reflection

External Euclidean-time reflection acts schematically as

\[ \Theta_E:(\tau,\mathbf x,y,k)\mapsto(-\tau,\mathbf x,y,k) \]

together with adjunction and representation-specific spin/tensor conjugation. It maps the positive-time algebra to the negative-time algebra.

15. The internal orbifold map

The internal reflection acts as

\[ \mathscr R_\chi:(\tau,\mathbf x,y,k) \mapsto(\tau,\mathbf x,-y,k) \]

with species parity. It defines an internal projector

\[ P_\chi=\tfrac12(1+\mathscr R_\chi), \qquad P_\chi^2=P_\chi=P_\chi^\dagger. \]

The UQF-3 seam requires

\[ [P_\chi,H_\Delta]=0, \qquad [P_\chi,\Theta_E]=0 \]

on the complete common domain. Merely observing that \(\tau\) and \(y\) are different coordinates is not enough; the spinor matrices, fixed-set boundary conditions, and species parities must also commute.

16. Gauge projection

For a compact gauge group with consistent unitary action \(U(g)\), define

\[ P_G=\int_Gd\mu(g)\,U(g). \]

Haar invariance yields

\[ P_G^2=P_G=P_G^\dagger. \]

If \([P_G,H_\Delta]=0\), the physical gauge space is the closed positive subspace \(P_G\mathcal H_\Delta\).

17. Why anomaly closure is an imported theorem

An anomaly can obstruct a consistent unitary representation of the gauge constraint. In that case the formal group average does not define the intended physical quotient. UQF-3 therefore imports the anomaly terminal from UQF-4; it does not recompute it and may not replace it with a public summary.

The available files do not contain a content-addressed current UQF-4 terminal. The gauge-projector theorem is consequently valid given that terminal, but the global package cannot be promoted to unconditional canonical closure.

Local cancellation is not enough. Fixed-set/inflow consistency and global determinant-line or bordism obstructions must match the exact global gauge form and chiral domain used by this branch.

18. Combined physical projector

When the exact seam witnesses pass, the combined projector is

\[ P_{\rm phys}=P_GP_\chi P_{B_3}\cdots, \]

with commuting orthogonal factors. Then

\[ P_{\rm phys}^2=P_{\rm phys}=P_{\rm phys}^\dagger, \qquad \mathcal H_{\rm phys}=P_{\rm phys}\mathcal H_\Delta, \]

so the physical space inherits a positive inner product.

19. BRST firewall

The perturbative BRST description

\[ \mathcal H_{\rm BRST}^{\rm phys}=\ker Q/\operatorname{im}Q \]

remains a useful cross-check. It is not the load-bearing definition of the nonperturbative physical carrier in this dossier. This avoids treating an indefinite gauge-fixed space as the final probability space and avoids making Gribov questions the primary UQF-3 predicate. It does not erase the need for a consistent anomaly-free gauge quotient.

Part V — The interacting norm-square theorem

20. Positive-time words

Let

\[ F=\sum_{j=1}^{n}c_j A_j(t_j),\qquad t_j>0, \]

where every \(A_j\) belongs to the physical observable algebra and preserves the required domains. In the Hamiltonian representation,

\[ A_j(t_j)=e^{t_jK_\Delta}A_je^{-t_jK_\Delta}. \]

External reflection reverses the time order and applies the physical adjoint. For an invariant normalized vacuum vector \(\Omega_\Delta\), the reflected quadratic form becomes

\[ \begin{aligned} \omega_\Delta(F^{\Theta_E}F) &= \sum_{i,j}\overline{c_i}c_j \langle\Omega_\Delta, A_i^\dagger e^{-(t_i+t_j)K_\Delta}A_j \Omega_\Delta\rangle\\ &= \left\lVert \sum_jc_j e^{-t_jK_\Delta}A_j\Omega_\Delta \right\rVert^2\\ &\ge0. \end{aligned} \]

This identity is the central mathematical closure. It is exact for the Hamiltonian placed inside the Actor; it does not expand in a coupling or loop order.

21. What “interacting” means here

The word interacting is justified only when \(H_\Delta\) is the complete accepted finite-floor Hamiltonian, including gauge, fermion, scalar, Yukawa, discrete, higher-form, boundary, and matching terms. If the ledger omits a current interaction, the theorem proves positivity for a smaller Actor and the gate remains incomplete.

The norm-square algebra itself is not the weak point. Completeness and domain control are. This dossier therefore separates the theorem row from the Actor realization row.

22. Degenerate ground sectors

A unique vacuum is unnecessary. If the ground sector carries a positive trace-class state \(\rho_0\), define

\[ \omega_0(A)=\operatorname{Tr}(\rho_0A). \]

The reflected form can be represented as a Hilbert–Schmidt norm square after purifying \(\rho_0\) or using its spectral decomposition. Vacuum degeneracy changes superselection and state-selection questions, not the sign of the inner product.

23. Massless negative control

Choose

\[ K=\operatorname{diag}(0,0,2). \]

The ground space is two-dimensional and the spectral gap from one ground state to another is zero. For any finite family of vectors \(X_j=e^{-t_jK}A_j\Omega\), the Gram matrix

\[ G_{ij}=\langle X_i,X_j\rangle \]

is positive semidefinite because it is a Gram matrix. This fixture directly kills both false dependencies: “a strictly positive gap is required” and “the vacuum must be unique.”

24. Local-net extension

The finite operational floor does not require a finite total universe. Let \(\mathcal A_\Lambda\) be the algebra of a bounded operational region \(\Lambda\), with compatible embeddings for nested regions. Require

\[ \omega_{\Lambda'}\circ\iota_{\Lambda\Lambda'}=\omega_\Lambda \]

and reflection-compatible embeddings. Every finite record then has a positive realization, and positivity extends to the closure of the inductive record algebra. A mismatch on overlapping regions is a finite, testable reopen trigger.

25. Relation to classical lattice theorems

The construction is consistent with three standard landmarks:

1. Lüscher’s positive transfer-matrix construction for Euclidean lattice gauge theory;
2. Osterwalder–Seiler and Menotti–Pelissetto reflection positivity for Wilson gauge actions;
3. Kikukawa–Usui reflection positivity for free overlap fermions and selected nongauge interactions.

The project-specific result is not attributed to any of these papers. The papers establish important sector and representation results. The present construction adds the project’s higher-dimensional chiral domain, finite-floor Actor, anomaly seam, complete reduction typing, and observer map. Recent work continues to treat general chiral lattice-gauge anomaly realization as a nontrivial subject; that reinforces the construction-provenance ceiling.

Part VI — The positivity-preserving Co-Actor

26. Why a Co-Actor is necessary

The parent carrier may be positive while an invalid quotient or reduction produces a nonpositive observer assignment. The Co-Actor owns the complete dual inventory of state- and algebra-changing maps:

\[ \Xi_{\rm RP}^{\vee}= (\mathbb E_G,\mathbb E_\chi,\mathbb E_{B_3}, \mathbb E_{\rm obs},\operatorname{Tr}_{\rm heavy}, \mathcal R_{13\to4},\mathfrak M_{\rm adm},\mathfrak M_{\rm kill}). \]

The theorem that CP maps compose is elementary. The difficult project-specific question is whether every actual accepted operation has been enumerated and correctly typed.

27. Gauge conditional expectation

In the observable picture,

\[ \mathbb E_G(A)=\int_Gd\mu(g)\,U(g)AU(g)^\dagger. \]

For a consistent compact unitary action, \(\mathbb E_G\) is unital, completely positive, idempotent, and maps onto the gauge-invariant algebra. Its validity inherits the UQF-4 anomaly seam.

28. Discrete averaging and chiral compression

Finite-group averaging is UCP. The chiral map

\[ A\mapsto P_\chi AP_\chi \]

is completely positive and unital on the reduced algebra whose identity is \(P_\chi\). Calling it unital on the unreduced parent algebra would be a type error.

29. Restriction and partial trace

A positive state remains positive when restricted to a unital \(*\)-subalgebra. If a justified subsystem factorization is available, partial trace is CPTP:

\[ \rho_{\rm light} =\operatorname{Tr}_{\rm heavy}(\rho_{\rm parent}). \]

The state may be entangled; no product-state assumption is required.

30. KK modes and the factorization firewall

A KK label does not automatically define an independent tensor factor after interactions and constraints. The reduction must use one of two lawful routes:

• restrict the parent positive state to the light observable subalgebra; or
• prove a subsystem inclusion/factorization and take the partial trace.

Simple deletion of all rows and columns carrying a heavy label is not a channel unless it is shown to be an adjoint-closed compression with the correct normalization.

31. Thirteen-to-four-dimensional record map

The candidate record map is

\[ \mathcal R_{13\to4} =\mathbb E_{\rm obs}\circ\Phi_{\rm heavy} \circ\mathbb E_\chi\circ\mathbb E_G. \]

If every factor is CP and trace preserving in the state picture, their composition is CPTP. If an observable-picture conditional expectation is used, the dual state map must be normalized consistently.

The map also owes a reflection intertwiner. A CP channel that mixes positive and negative Euclidean-time supports may preserve density-matrix positivity while failing the typed OS interface. Complete positivity and reflection support are separate rows.

32. Canonical non-CP control

For the Bell state

\[ |\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt2, \]

partial trace produces \(I/2\), a positive density matrix. Partial transpose produces eigenvalues

\[ \left\{-\tfrac12,\tfrac12,\tfrac12,\tfrac12\right\}. \]

A validator that checks a map only on unentangled states can miss this failure. The partial-transpose fixture is therefore the nearest destructive control for the Co-Actor.

33. Map completeness debt

The public v4.0 dossier names the principal generators, but the available package does not provide a complete content-addressed map registry. The new building block defines the required schema:

map id and physical purpose
owner
domain and codomain algebras
state or observable picture
CP/UCP/CPTP witness
normalization rule
external-reflection intertwiner
dependencies and source hashes
destructive control
reopen trigger

Until every accepted operation has a row, the CP composition theorem passes but the exhaustion claim remains conditional.

Part VII — Granularity and claim scope

34. Finite-floor obligation

BB-GCN-1 defines the physical theory at an exact finite operational floor. UQF-3 must therefore cover every admitted finite record and every compatible bounded region. It need not construct an ontic infinite-resolution continuum that no admitted record can distinguish.

35. Granularity is not a universal solvent

Granularity cannot erase:

• a negative finite OS Gram eigenvalue;
• a negative kinetic coefficient;
• a non-Hermitian accepted interaction;
• an unbounded-below accepted Hamiltonian;
• an anomalous physical gauge projector;
• a non-CP observer channel;
• incompatible states on overlapping operational regions.

Each is a finite falsifier. The package includes a destructive control that deliberately introduces a negative finite Gram matrix and requires a failure even though the continuum ontology is non-gating.

36. Continuum claim ceiling

This dossier does not prove a universal OS reconstruction for every continuum chiral gauge theory. It does not solve the Clay Yang–Mills problem. It does not show that all regulators are reflection positive. It shows that the accepted finite-floor branch has a positive realization when its complete Actor and Co-Actor interfaces pass.

The continuum extension may remain an open mathematical research program without reopening the physical finite-floor gate. If the project later defines itself only as a limit of regulator theories, or claims a literal continuum ontology, the scope changes and UQF-3 must be replayed.

Part VIII — Evidence-row adjudication

37. Atomic registry

38. Why conditional rows do not invalidate the construction theorem

Rows R04, R05, and R09 identify what the construction must consume. The mathematical implications are sound: a lower-bounded self-adjoint Hamiltonian gives the norm-square form; a consistent compact unitary gauge action gives an orthogonal projector; CP maps compose. The missing objects prevent verifying that the current project instance satisfies every premise. They do not refute the conditional theorem.

For this reason the result is neither FAILED nor FULLY CANONICAL. It is a complete scoped construction candidate with precisely named promotion obligations.

39. Why the public “open blockers = 0” field is not adopted

The public page is a dossier/public-summary layer. The supplied protocol places package authority, registry rows, execution evidence, and generated board state above it. Because the controlling checklist and row-state artifacts are absent, this document cannot independently reproduce the public zero-blocker rollup. It preserves the proposed mechanism but emits its own fail-closed row states.

Part IX — Adversarial validation

40. Mandatory mutation suite

41. Static execution result

The packaged deterministic validator executes representative finite-dimensional fixtures for the norm-square Gram matrix, a zero-gap degenerate ground sector, partial trace, partial transpose, CP-map composition, commuting and noncommuting reflections, a negative kinetic mutation, and the provenance ceiling. It also checks the required files, nine unique block identifiers, fourteen atomic evidence rows, and the presence of fail-closed dependency states.

This static run is meaningful but is not an independent randomized gauntlet. The latter remains a promotion condition because the execution agent and the construction author are not decorrelated in this deliverable.

42. Reopen triggers

UQF-3 reopens on any of the following:

1. addition or modification of an Actor;
2. addition or modification of a Hamiltonian, boundary, or matching term;
3. changed gauge group, global form, or anomaly ledger;
4. changed internal parity or chiral domain;
5. a new state or observable reduction;
6. failure of CP, trace preservation, adjunction, or reflection intertwining;
7. a negative exact physical OS Gram matrix;
8. a changed operational floor or regulator;
9. incompatibility between regional positive states;
10. an upstream source-hash change;
11. a public claim stronger than the machine state.

Part X — Dependency and ownership graph

43. Dependency graph

flowchart TD
    B["Born sign"] --> A["Xi_RP Actor"]
    D["Finite-floor Dynamics"] --> A
    S["Shape / chiral domain"] --> A
    U["UQF-4 anomaly terminal"] --> G["Gauge projector"]
    A --> N["OS norm-square theorem"]
    G --> N
    C["Xi_RP co-Actor"] --> O["13D to 4D record"]
    N --> O

The graph has no lawful arrow from the UQF-3 output back into the Actor that generated the same output. The Actor and Co-Actor are explicit retrospective construction anchors on a new branch.

44. Ownership table

Part XI — Final decision

45. Technical decision

The Actor–Co-Actor mechanism solves the correctly scoped mathematical challenge:

1. a positive finite-floor carrier and lower-bounded self-adjoint Hamiltonian produce a contraction semigroup;
2. the OS form of any positive-time physical word is a norm square;
3. a strictly positive mass gap and unique vacuum are unnecessary;
4. gauge and chiral constraints preserve positivity when represented by commuting orthogonal projectors;
5. every lawful reduction is required to be CP/UCP/CPTP or a positive subalgebra restriction;
6. compositions of the registered maps remain positive;
7. the four-dimensional observer record therefore inherits positivity;
8. Granularity removes only the literal continuum-ontology demand, never a finite negative witness.

46. Evidence decision

The current instance is not completely witnessed. The exact current Hamiltonian/domain ledger, exact current map inventory, and UQF-4 terminal are missing. BB-QCR-1 independently warns that the exact quantization matrices are not present in the available source authority. Therefore the theorem may be applied only as a construction contract, not reported as a completed blind derivation.

47. Authority decision

The controlling checklist is unavailable and owner ratification has not occurred. The deliverable is suitable for development authority and shadow replay, not self-promotion to canonical authority.

48. Machine-readable terminal

GATE=UQF-3
PREDICATE=omega_phys(F^Theta_E F)>=0
ACTOR_COACTOR_PAIR=Xi_RP_dashturn_Xi_RP_dual
EXTERNAL_INTERNAL_REFLECTIONS_DISTINCT=PASS
OS_NORM_SQUARE_THEOREM=PASS_GIVEN_ACTOR
MASS_GAP_REQUIRED=NO
UNIQUE_VACUUM_REQUIRED=NO
POINTWISE_DETERMINANT_POSITIVITY_REQUIRED=NO
CP_GENERATOR_CLASSES=PASS
CP_COMPOSITION=PASS
FINITE_FLOOR_SCOPE=PASS
COMPLETE_ACTOR_REALIZATION=CONDITIONAL
UQF4_ANOMALY_TERMINAL=ABSENT_FROM_AVAILABLE_INPUTS
COMPLETE_COACTOR_MAP_REGISTRY=CONDITIONAL
CONTROLLING_DOSSIER_CHECKLIST=ABSENT
INDEPENDENT_RANDOMIZED_GAUNTLET=PENDING
OWNER_RATIFICATION=PENDING
TECHNICAL_TERMINAL=CLOSED_SCOPED_CONSTRUCTION_ANCHOR_CANDIDATE
CANONICAL_TERMINAL=PENDING
UNCONDITIONAL_CLOSURE_AUTHORIZED=NO

49. Promotion checklist

To promote this branch, the owner or merge steward must:

1. freeze the exact parent branch and source hashes;
2. populate the Hamiltonian term/domain ledger;
3. populate the reduction-map registry;
4. attach the current UQF-4 terminal;
5. execute the randomized mutation suite with committed answer key;
6. generate gate-scoped row states;
7. run the controlling gates-test-checklist.md;
8. merge the new blocks and amendments;
9. rebuild the board deterministically;
10. ratify the construction and its public-language ceiling.

No additional conceptual invention is required to begin these steps. The remaining work is exact instantiation, execution, authority, and owner action.

Part XII — Technical reference register

50. Primary references

1. M. Lüscher, “Construction of a Selfadjoint, Strictly Positive Transfer Matrix for Euclidean Lattice Gauge Theories,” Communications in Mathematical Physics 54 (1977), 283–292.
   https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-54/issue-3/Construction-of-a-selfadjoint-strictly-positive-transfer-matrix-for-euclidean/cmp/1103900872.full

2. K. Osterwalder and E. Seiler, “Gauge Field Theories on a Lattice,” Annals of Physics 110 (1978), 440–471.
   https://www.sciencedirect.com/science/article/pii/0003491678900398

3. P. Menotti and A. Pelissetto, “General Proof of Osterwalder–Schrader Positivity for the Wilson Action,” Communications in Mathematical Physics 113 (1987), 369–373.
   https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-113/issue-3/General-proof-of-Osterwalder-Schrader-positivity-for-the-Wilson-action/cmp/1104160284.pdf

4. D. B. Kaplan, “A Method for Simulating Chiral Fermions on the Lattice,” Physics Letters B 288 (1992), 342–347.
   https://arxiv.org/abs/hep-lat/9206013

5. M. Lüscher, “Abelian Chiral Gauge Theories on the Lattice with Exact Gauge Invariance,” Nuclear Physics B 549 (1999), 295–334.
   https://arxiv.org/abs/hep-lat/9811032

6. M. Lüscher, “Weyl Fermions on the Lattice and the Non-Abelian Gauge Anomaly,” Nuclear Physics B 568 (2000), 162–179.
   https://arxiv.org/abs/hep-lat/9904009

7. Y. Kikukawa and K. Usui, “Reflection Positivity of Free Overlap Fermions,” Physical Review D 82 (2010), 114503.
   https://arxiv.org/abs/1005.3751

8. Y. Tu, “Anomalies of Global Symmetries on the Lattice” (2025). This is a contemporary anomaly-classification reference, not a proof of the project’s UQF-4 terminal.
   https://arxiv.org/abs/2507.21209

51. Project sources

The attached 2026-07-18 source-of-truth pack, updated Shape and SG1 packages, execution protocol, gate-execution handoff, and live UQF-3 v4.0 mechanism are the project-specific sources. Exact local hashes are carried in the companion building-block package.

52. Final statement

UQF-3 is solved at the level that the available evidence genuinely supports: a complete, falsifiable finite-floor positive-realization construction with an explicit Actor, explicit Co-Actor, exact norm-square theorem, CP composition law, finite/continuum firewall, and reopen registry. The remaining ceiling is not hidden. Canonical closure awaits exact project-instance witnesses and owner ratification.