Gate dossier — UQF-14 — Above-Cutoff Causality

Final controlling closure dossier — v15 ratification candidate

Gate question: Does the descended theory preserve cause and effect through the complete physically admitted energy domain, including the finite microscopic floor?

Date: 2026-07-18
Controlling Shape: v2.10, including the Relational Local-Unitary Actor, the Constraint–Refoliation–Composition Co-Actor, the global-anomaly pair, the chiral-domain pair, and the current Scale synchronization.
Method: TECRAC v1.2, Gate Closure Constitution, Canonical Gate Dossier Reconstruction and Verification Protocol v3.0.
Evidence strength: B — empirically anchored reconstruction / explicit finite construction.

Proposed controlling endpoint

CLOSED / REALIZED-GIVEN-RELATIONAL-LOCAL-UNITARY-DYNAMICS–CONSTRAINT–REFOLIATION–COMPOSITION-CO-ACTOR / EXACT-CAUSAL-DIAMOND NO-SIGNALLING / ALL-PHYSICALLY-ADMITTED-KK-AND-ACTOR-BLOCKS INCLUDED / NO-PHYSICALLY-DISTINCT ABOVE-CUTOFF SECTOR / POSITIVE CONSTRUCTION / RESOLVED +0.

The earlier below-cutoff-only result is retained as a correct infrared shadow. The previously open “above-cutoff” branch is closed by the current finite microscopic Dynamics already adopted under UQF-5C: every physical state belongs to a finite relational causal-diamond Hilbert space; every legal update is local and unitary; incomparable events commute; all legal foliations of one causal diamond compose to the same boundary map; and the Scale projector excludes any record-distinct state above the accepted operational cutoff. Conventional regulator-independent continuum quantum gravity is not claimed.

Reader-first result

The old UQF-14 dossier asked a question that the old branch could not answer: what happens after the four-dimensional effective field theory reaches the compactification cutoff and perturbation theory is no longer reliable? The July 12 correction correctly refused to extrapolate dispersion relations, eikonal time delay, perturbative BRST cancellation, or the observed low-energy graviton speed into that regime.

The current Shape has since changed in a decisive way. Shape v2.9 added the Relational Local-Unitary Quantum-Geometry Actor \(\Xi_{\rm RLU}\) and the Constraint–Refoliation–Composition Co-Actor \(\Xi_{\rm CRC}^{\vee}\). They do not merely regulate an otherwise continuous theory. They define the microscopic physical theory on each bounded operational causal diamond:

\[ \mathcal H_D=\ell^2(\mathcal Q_*(D)), \qquad |\mathcal Q_*(D)|<\infty, \]

with legal local moves generated by bounded Hermitian operators,

\[ \nu_m=e^{-i\delta\tau_*h_m/\hbar}, \qquad \nu_m^\dagger\nu_m=I. \]

A legal history is not an arbitrary global ordering. It is a linear extension of a finite causal partial order. If two events are incomparable, their gates commute. Any two legal linear extensions differ by swaps of adjacent incomparable events, so every legal foliation gives exactly the same physical boundary map. This is the microscopic refoliation theorem.

The same structure yields exact causality. A local observable can be changed only by gates in its causal cone. Gates at incomparable events commute through the observable and cancel from the conjugation. Therefore an operation in region \(A\) cannot alter any observable in region \(B\) before a directed causal path from \(A\) reaches \(B\). The result is exact, not an asymptotic Lieb–Robinson estimate.

Granularity and Scale then finish the scope question. The physical carrier is already quotiented by record-invisible refinements, and the accepted source domain is projected to quasienergy at or below \(M_*\). There is no second physical Hilbert sector “above the cutoff” waiting for a causal law. Attempted finer or higher-energy descriptions are either the same operational record, an excluded parent label used only for matching, or a versioned change of Shape. Thus the universal demand “prove causality at arbitrarily high energy in an infinitely refinable continuum” is not the gate’s physical obligation. The actual obligation is causality on the complete admitted microscopic domain, and that obligation is now satisfied by construction.

This is a construction-grade closure. It does not derive the finite local-unitary quantization map uniquely from bare Einstein dynamics. It does not prove that every continuum quantum-gravity proposal is causal. It proves that the project’s current finite microscopic branch is exactly unitary, exactly causal in its declared poset sense, refoliation coherent, anomaly compatible, and matched to the causal four-dimensional infrared theory.

Table of contents

1. Part I — Authority, charter, and supersession
   
2. Part II — The old open branch and the wrong-object diagnosis
   
3. Part III — Thought experiments and cross-building-block simplification
   
4. Part IV — Complete microscopic object
   
5. Part V — Exact causal theorems
   
6. Part VI — Finite calculations and deterministic certificate
   
7. Part VII — Full admitted KK and dependency closure
   
8. Part VIII — Infrared matching and observational cross-checks
   
9. Part IX — Anchor, same-ruler, and scope audits
   
10. Part X — Hostile review and destructive controls
    
11. Part XI — Final terminal and propagation contracts
    
12. Technical appendices
    
13. Archive firewall — superseded technical record

Part I — Authority, charter, and supersession

1. Exact gate obligation

The physical obligation is narrower and stronger than the phrase “causality above the cutoff” initially suggests. The gate must determine whether every physically admitted state, operation, and observable in the controlling construction obeys a consistent causal order. It must also show that the four-dimensional descended theory is the infrared shadow of that same causal object, rather than an unrelated effective model pasted onto an undefined high-energy region.

The owed certificate therefore has five legs:

1. a positive physical state space and unitary microscopic updates;
2. a finite, explicit causal domain for every update;
3. exact no-signalling outside the directed causal future;
4. refoliation-independent composition of one causal diamond;
5. a lawful infrared matching map preserving the external light cone and the admitted particle/KK domain.

The gate does not owe a proof about every imaginable ultraviolet theory. It owes a proof about the current branch. Conversely, it may not declare the branch causal merely because no violation has yet been measured. The Dynamics itself must carry the causal restriction.

2. Controlling authority order

This dossier applies the following precedence:

1. owner-ratified constitutional and gate-governance rules;
2. current Shape v2.10 and its embedded Dynamics pair;
3. current Scale and Granularity synchronization;
4. accepted UQF-3, UQF-4, UQF-7, UQF-9, and UQF-10 construction branches;
5. this UQF-14 governing section;
6. the archived UQF-14 technical record.

The July 12 correction controls over the older blanket CERTIFIED-IRREDUCIBLE wording. The correction was right for the branch that existed at that time: below-cutoff causality was established and the microscopic branch was absent. Shape v2.9 subsequently supplied the missing microscopic object. The present close is therefore not a reinterpretation of the old evidence. It is a new construction on a changed constitutional branch.

3. Supersession rule

The following statements are retained:

• twice-subtracted forward-dispersion positivity is a valid scoped infrared check;
• positive eikonal time delay is a valid scoped check of particular high-energy scattering regimes below the construction’s loss of perturbative control;
• the graviton zero mode is observationally luminal to approximately one part in \(10^{15}\);
• ordinary perturbative microcausality and BRST cancellation remain valid in their stated domains;
• none of those results alone proves microscopic causality.

The following statements are superseded:

• “the above-cutoff branch is necessarily open because no microscopic Dynamics exists”; the current Shape contains one;
• “the full KK branch is open because UQF-10 is open”; the accepted UQF-10 construction supplies a coercive admitted internal-metric operator;
• “the mirror and global-anomaly legs are open”; the current UQF-7 and UQF-4 pairs close those source domains;
• “a conventional continuum UV fixed point is required to close causality”; that is not required by the project’s finite physical ontology.

4. Evidence vocabulary

DERIVED is reserved for consequences of the frozen construction. REALIZED-GIVEN-ACTOR means a mechanism exists because an explicit Actor or Dynamics law was added. MEASURED-ANCHOR denotes an empirical input, not a derivation. CLOSED-NEGATIVE records a failed branch. DISSOLVED-GIVEN-root is used only when the demand is not a physical observable in the controlling ontology.

The present closure is mixed but cleanly typed:

• finite local-unitary microscopic Dynamics: construction anchor;
• unitarity of every legal gate and history: derived given that construction;
• exact causal-cone/no-signalling theorem: derived;
• refoliation independence: derived;
• absence of a distinct above-cutoff physical sector: derived given Scale + Granularity + source projection;
• low-energy graviton luminality: measured cross-check;
• uniqueness of the microscopic quantization map: not claimed.

5. Boot certification

The dossier booted the Theory Constitution, Shape, Scale, Granularity, Dynamics, current dependency graph, master assumptions ledger, TECRAC v1.2, Interdependence v4, and BB-TS-1. The controlling versions are mutually compatible on the causal object:

• Shape provides the Lorentzian Stage and relational causal diamonds;
• Scale supplies \(M_*\), \(\delta\tau_*\), and the quasienergy branch;
• Granularity supplies the finite operational record quotient and rejects record-invisible refinements as new states;
• Dynamics supplies local invertible moves and their bounded Hermitian generators;
• Interdependence forbids a primitive tensor-product decomposition that could conceal a nonlocal coupling;
• time synchronization forbids replacing causal order with one preferred global clock;
• TECRAC requires bounded capacity, phase retention, quotient-first constraints, causal-diamond coherence, and primary Lorentzian Dynamics.

No boot item failed. Review is authorized.

Part II — The old open branch and the wrong-object diagnosis

6. What the old branch actually proved

The older UQF-14 branch established a strong but limited result. At finite Kaluza–Klein truncation and below the compactification cutoff, the descended four-dimensional theory inherited ordinary perturbative QFT causality. Free commutators vanished outside the external light cone; Epstein–Glaser causal factorization preserved microcausality order by order; the physical BRST quotient removed negative-norm gauge states under the stated nilpotency assumptions; and the effective action remained local as a derivative expansion.

Those statements remain correct. The problem was scope, not mathematics. The older dossier had no object representing the microscopic theory after perturbative EFT stopped being an adequate description. Its honest endpoint was therefore CLOSED-CONDITIONAL below cutoff / OPEN above cutoff.

7. Why dispersion positivity could not close the old branch

A forward-dispersion relation is a consistency condition on an amplitude under assumptions that already include analyticity, unitarity, crossing behavior, polynomial boundedness, and an appropriate subtraction structure. A positive dimension-eight coefficient can rule out some low-energy EFTs. It does not construct the high-energy state space or prove that all microscopic operations are causally local.

The thought experiment is simple: two different microscopic theories can produce the same first several Wilson coefficients. One may be a local unitary completion and the other may fail at a higher threshold. Matching the low-order coefficients cannot decide between them. Therefore positivity was retained as an infrared diagnostic and removed from the role of microscopic existence proof.

8. Why eikonal delay could not close the old branch

A nonnegative eikonal delay is an important scattering check. It can expose higher-derivative interactions that create resolvable time advances. But it is a statement about a controlled semiclassical scattering regime, not an exhaustive theorem for every finite-floor update. The old branch was tempted to treat “no time advance in the calculated channel” as “no acausal process exists.” TECRAC identifies that as a wrong-object promotion.

The correct microscopic object must say which operations are legal before a scattering amplitude is computed. In the present branch, that object is the causal-diamond move grammar. Eikonal delay becomes a downstream shadow and negative control.

9. Why a measured luminal zero mode could not close the old branch

The joint gravitational-wave and gamma-ray observation constrains the propagation of the observed graviton zero mode over one astrophysical baseline. It does not inspect Planck-scale moves, internal KK transitions, or alternative microscopic histories. The measurement is therefore a powerful record-interface anchor but cannot be promoted to a theorem about unobserved states.

The present construction reverses the logic. Microscopic causality follows from the legal-move algebra. Low-energy luminality checks that the external graviton readout belongs to the correct infrared cone.

10. The wrong-object diagnosis

The phrase “above-cutoff causality” bundled together three different questions:

1. Does an infinitely refinable continuum remain causal at arbitrarily high energy?
2. Does the project possess a physical theory at its smallest admitted resolution?
3. Does the four-dimensional EFT match that microscopic theory without a causal mismatch?

Question 1 is not the controlling ontology and is not owed. Question 2 was genuinely open before Shape v2.9 and is now answered by the Relational Local-Unitary pair. Question 3 is a finite matching and observer-map obligation. Separating these questions is the decisive simplification.

Part III — Thought experiments and cross-building-block simplification

11. Child-level thought experiment: dominoes on arrows

Imagine a floor covered with tiles connected by arrows. A domino on tile \(A\) may knock down a domino on tile \(B\) only when an arrow points from \(A\) to \(B\). Some tiles have no arrow between them. Their dominoes can fall in either order, but neither can knock over the other.

A legal quantum history works the same way. The arrows form the causal partial order. Local unitary gates replace domino falls. If two events have no causal arrow or directed path between them, their gates commute and cannot be used to send a signal. The order in which an observer writes those independent events on a page is only a foliation convention.

12. The two chore lists

Two children clean two separate rooms. One list says “clean room A, then room B.” Another says “clean room B, then room A.” If the rooms are independent, both lists leave the same final house. If changing the list changes the result, the rooms were not independent or the instructions secretly shared an object.

This is the refoliation test. Different legal linear extensions of one causal diamond must give the same boundary map. The Co-Actor does not assume this globally. It enforces the local condition that incomparable-event gates commute, and the finite-poset swap theorem then proves equality of all legal foliations.

13. The locked top shelf

A library has books on shelves numbered up to \(M_*\). A request for shelf \(M_*+1\) does not reveal an unknown room. It is not a valid library address. A librarian may use archived catalog information to improve the summaries of books on the admitted shelves, but no reader can check out a book from a nonexistent shelf.

Scale and Granularity play the librarian’s role. Parent labels above the operational threshold may influence matching coefficients when the Rulebook permits it, but they are not physical external states or signal carriers. The causal theorem is required on every admitted shelf and nowhere else.

14. Shape simplification

Shape supplies a Lorentzian external causal order and a finite internal geometry. The microscopic causal order is not invented by a lattice laid over spacetime. It is a relation among complete relational Shape records in bounded causal diamonds. A legal move changes only a bounded causal neighborhood and preserves the physical constraint domain.

Because the Stage already distinguishes timelike/null relation from spacelike incomparability, the gate never needs to derive causality from a global clock. The causal order is structural; the Dynamics must respect it.

15. Scale simplification

Scale does two jobs and no more. First, it freezes the cell update scale \(\delta\tau_*\) and the accepted quasienergy branch. Second, it defines the source projector \(Q_{\le M_*}\). It does not prove causality by making high-energy effects small. Causality comes from support and composition. Scale only states which states and updates are physical.

The current synchronized cutoff is

\[ M_*=7.952716123567312\times10^{16}\ {\rm GeV}. \]

The accepted UQF-10 first internal-metric excitation is

\[ m_{\rm int,1}=1.3573231578625674\times10^{17}\ {\rm GeV} =1.7067416173\,M_*. \]

It is outside the admitted source domain. This is a source-domain statement, not a claim that an omitted mode never appears in a matching coefficient.

16. Granularity simplification

Granularity removes the endless-refinement regress only after the bounded-capacity test passes. Shape v2.9 does not infer finiteness from a finite alphabet alone. It defines \(\mathcal Q_*(D)\) after quotienting gauge copies, empty subdivisions, and record-invisible refinements, and it requires the resulting set to be finite for every bounded operational causal diamond.

This turns the existence problem into finite algebra. It does not replace the causal proof. A finite nonlocal unitary could still signal instantly. The local-move and incomparable-commutation conditions remain essential.

17. Dynamics simplification

The primary Dynamics is Lorentzian and phase retaining:

\[ \nu_m=e^{-i\delta\tau_*h_m/\hbar}. \]

The earlier positive-transfer candidate \(A^\dagger A\) was retained only as a negative control because it erased physical phases. TECRAC’s phase-retention test requires the complete parent-action phase to enter the microscopic gate. Reflection positivity and Euclidean transfer are derived later from the self-adjoint generator; they are not substituted for the real-time causal law.

18. Interdependence simplification

Interdependence forbids starting with a primitive product \(\mathcal H_A\otimes\mathcal H_B\) and declaring the factors independent. The parent object is one finite causal-diamond Hilbert space. Local algebras and approximate subsystems are derived from support and the constraint quotient.

This avoids a common fake causality proof: assuming the remote system is an independent tensor factor, then concluding a local operation cannot affect it. Here the no-signalling theorem is proved from the action of the complete unitary on the complete algebra. Gauge and gravitational dressing are part of the localization definition.

19. Time synchronization simplification

BB-TS-1 separates synchronization from signal propagation. A foliation is a legal ordering of events, not a physical channel. No observer-independent global clock is required. The microscopic proof uses only the partial order and the commuting relation of incomparable events.

This removes a hidden preferred-frame risk. Two observers may choose different linear extensions while agreeing on the same causal relation and boundary map. The construction is refoliation coherent even when no single global simultaneity partition is physically privileged.

20. Anomaly-descent and chiral-domain simplification

UQF-4 and UQF-7 remove two hidden sources of causal failure. The anomaly pair guarantees that the physical symmetry and constraint action is globally well defined. The chiral-domain pair requires

\[ [h_m,\Pi_{\rm adm}]=0 \]

for every legal local move. Thus microscopic evolution cannot leak into an excluded mirror domain or change the parity assignment while pretending to remain on the same Shape branch.

21. UQF-9 simplification

UQF-9 now supplies the exact finite order-six matching data in the declared graviton/ghost sector. UQF-14 does not use the sign of one heat coefficient as a causal theorem. Its relevance is narrower: it makes the finite EFT matching ledger explicit and prevents an uncomputed boundary block from being mistaken for a hidden causal sector.

The bulk and fixed-set coefficients remain typed by support dimension and are not added into a fake scalar total. This same-ruler discipline carries into UQF-14: local microscopic support and four-dimensional observer support are related by a declared map, not conflated.

22. UQF-10 simplification

The accepted compactification construction removes the volume singlet by a multiplier equation and makes the admitted internal-metric KK operator coercive. UQF-14 therefore does not need to enumerate a possibly unstable infinite tower. It needs only check that the current causal update algebra preserves the constrained physical domain and that no admitted tachyonic block remains.

The current Scale correction strengthens the source-domain result: the first internal-metric mode remains above \(M_*\) by a factor of approximately \(1.70674\).

23. Final elegance result

After all building blocks are applied, the gate no longer requires a new UV dispersion calculation, a full continuum fixed point, an infinite KK sum, or a novel causal Actor. It reduces to three finite theorems on an already accepted object:

1. products of local unitary gates are unitary;
2. commuting incomparable gates make the boundary map independent of foliation;
3. conjugation by a local circuit expands operator support only along directed causal chains.

Everything else is source completeness, matching, and destructive control.

Part IV — Complete microscopic object

24. Frozen Stage

The metric Stage remains

\[ \mathcal M_4\times K_6\times S^2\times I_\chi, \qquad K_6=SU(3)/T^2, \qquad I_\chi=S^1_\chi/\mathbb Z_2. \]

Only the Stage carries metric dimension. The Rulebook and Actors are zero-metric-dimensional but physically load bearing. The causal order used in this gate is the relation among complete relational records induced by the Lorentzian external Stage together with the legal internal and boundary domains.

25. Physical causal-diamond carrier

For every bounded operational causal diamond \(D\), the controlling carrier is

\[ \mathcal H_D=\ell^2(\mathcal Q_*(D)). \]

A basis label \(q\in\mathcal Q_*(D)\) is a complete relational record, not a field value at a background coordinate point. The quotient has already removed gauge and diffeomorphism copies, empty subdivisions, and refinements that produce no new admitted record. Bounded capacity is a constitutional condition:

\[ |\mathcal Q_*(D)|<\infty. \]

The finite-dimensionality is therefore earned by the full quotient and capacity rule, not assumed from a lattice picture.

26. Legal local move tuple

A legal move is a tuple

\[ m=(S_m,\,h_m,\,\nu_m,\,m^{-1}), \]

where \(S_m\) is the bounded causal support, \(h_m=h_m^\dagger\) is the bounded physical generator on the constrained domain, \(\nu_m=\exp(-i\delta\tau_*h_m/\hbar)\), and \(m^{-1}\) is the inverse move. The move must preserve every current projector and domain condition, including gauge, anomaly, chiral parity, fixed-set, and Scale admissibility.

A move that fails any one of those tests is not a small correction. It is not a legal move of this theory.

27. Causal event poset

A physical diamond is a finite poset \((E_D,\preceq)\). The event relation is reflexive, antisymmetric, and transitive. Comparable events may influence one another through a directed chain. Incomparable events are spacelike for the microscopic causal grammar.

A history slicing \(\sigma=(m_1,\ldots,m_N)\) is legal only when it is a linear extension of \(\preceq\). The associated boundary map is

\[ U_D(\sigma)=\overrightarrow{\prod_{m\in\sigma}}\nu_m. \]

28. Constraint–Refoliation–Composition Co-Actor

The Co-Actor owns five obligations:

• closure of the physical constraint domain under every move;
• commutation of gates assigned to incomparable events;
• equality of all legal foliation compositions;
• consistent gluing of adjacent diamonds and their edge/corner data;
• preservation of observer and heavy-mode reduction as completely positive lawful maps.

The key microscopic causal axiom is

\[ m\parallel n\quad\Longrightarrow\quad [\nu_m,\nu_n]=0, \]

where \(m\parallel n\) means the events are incomparable in the causal poset and their complete dressed supports are causally disjoint.

29. Source projector and physical cutoff

The full accepted microscopic source domain is

\[ \mathcal H_{\rm phys} =Q_{\le M_*}\Pi_{\rm adm}\mathcal H_{\rm parent}. \]

The projector is not allowed to be an arbitrary nonlocal spectral knife. The chiral-domain contract requires it to commute with every legal local generator on the accepted branch. Parent labels outside the source domain may contribute only through symmetry-preserving local matching data. They are not external asymptotic states and cannot be used as hidden signal carriers.

30. Observer algebra and localization

For a causally complete region \(R\subseteq D\), let \(\mathfrak A(R)\) be the algebra of gauge-invariant, constraint-preserving observables whose complete dressed support lies in \(R\). Isotony follows from inclusion of supports. Locality is not defined by bare coordinate support; all required dressing and boundary data are included.

The observer map

\[ \mathcal O_F:\mathcal R_{13}\rightarrow\mathcal R_{{\rm obs},F} \]

is frozen after the causal and Scale domains are fixed. It may coarse-grain or trace out inaccessible records through a UCP/CPTP map, but it may not create a causal influence absent from the parent algebra.

Part V — Exact causal theorems

31. Theorem UQF14-U — unitary history

Statement. Let \(D\) be a bounded physical causal diamond and let \(\sigma\) be any legal linear extension. If every local move generator is bounded and Hermitian, then \(U_D(\sigma)\) is unitary.

Proof. Each gate satisfies

\[ \nu_m^\dagger\nu_m =e^{+i\delta\tau_*h_m/\hbar}e^{-i\delta\tau_*h_m/\hbar}=I. \]

A finite ordered product of unitary operators is unitary:

\[ U_D(\sigma)^\dagger U_D(\sigma) =\nu_{m_1}^\dagger\cdots\nu_{m_N}^\dagger \nu_{m_N}\cdots\nu_{m_1}=I. \]

Finiteness of \(E_D\) ensures no domain ambiguity from an unperformed infinite product. \(\square\)

32. The adjacent-swap lemma

Any two linear extensions of a finite poset are connected by a finite sequence of swaps of adjacent incomparable elements. This is the combinatorial engine of refoliation independence. It means the Co-Actor need not compare every pair of global foliations. It needs only certify the finite local relation associated with one adjacent incomparable swap.

This is the same elegance pattern used throughout the project: prove a result on a generating set and extend by composition.

33. Theorem UQF14-R — refoliation independence

Statement. If gates assigned to incomparable events commute, then any two legal linear extensions \(\sigma,\sigma'\) of one finite causal diamond produce the same boundary map.

Proof. By the adjacent-swap lemma, write the transformation from \(\sigma\) to \(\sigma'\) as finitely many swaps of adjacent incomparable events. For one swap,

\[ \cdots\nu_m\nu_n\cdots =\cdots\nu_n\nu_m\cdots \]

because \([\nu_m,\nu_n]=0\). Each swap leaves the product unchanged. Induction over the finite swap sequence gives

\[ U_D(\sigma)=U_D(\sigma'). \]

Thus the physical boundary map is a property of the causal diamond and boundary records, not of a preferred foliation. \(\square\)

34. Support update rule

For an observable \(A\) with support \(S(A)\), conjugation by a gate \(\nu_m\) has only two possibilities:

• if \(S_m\) is causally disjoint from \(S(A)\), then \([\nu_m,A]=0\) and the support is unchanged;
• otherwise the new support is contained in the causal closure of \(S(A)\cup S_m\).

This statement is exact for the declared local algebra. It replaces a brute-force commutator calculation for every pair of fields.

35. Theorem UQF14-C — exact microscopic causal cone

Statement. Let \(A\in\mathfrak A(R_A)\). Under a legal history, the evolved observable has support only in the directed causal closure of \(R_A\). If \(R_B\) lies outside that closure, then every \(B\in\mathfrak A(R_B)\) commutes with the evolved \(A\).

Proof. Order the gates by any legal linear extension and apply the support update rule inductively. A gate not causally connected to the current support commutes through the observable and does not enlarge support. A gate can enlarge support only when its event is connected to the current support by the causal grammar. After finitely many steps, support is contained in the union of events reachable by a directed path from \(R_A\). If \(R_B\) is outside that union, the complete dressed supports remain disjoint and

\[ [A(\sigma),B]=0. \]

The result is foliation independent by Theorem UQF14-R. \(\square\)

36. Operational no-signalling theorem

Let \(\mathcal E_{A,x}\) and \(\mathcal E_{A,x'}\) be two admissible local protocol choices in \(A\). Let \(B\) be a remote observable outside the causal future of \(A\) during the tested interval. Because the dual channel acts trivially on \(B\),

\[ \mathcal E_{A,x}^*(B)=B=\mathcal E_{A,x'}^*(B). \]

Consequently every remote outcome distribution is independent of the local setting before causal arrival:

\[ p(b\mid x,y,\lambda)=p(b\mid x',y,\lambda). \]

Entanglement does not invalidate the theorem. It may change correlations conditional on a later comparison of records, but it does not create a controllable remote marginal change outside the causal cone.

37. Primitive-causality composition

Let \(D_1\) and \(D_2\) be composable diamonds with a complete shared boundary record. Edge/corner gluing requires

\[ U_{D_2\circ D_1}=U_{D_2}U_{D_1} \]

on the common physical domain. Associativity follows from operator composition. Together with refoliation independence, this supplies a finite analogue of primitive causality: data on a complete admissible boundary determine one interior-to-boundary map, independent of the bookkeeping slice.

38. Theorem UQF14-KK — KK-basis invariance of the external causal cone

A Kaluza–Klein expansion is a unitary change of basis in the internal Hilbert factor or its constrained finite analogue. It reorganizes internal labels but does not change external support. Let \(W_{\rm KK}\) be the internal basis transform. For an external local algebra,

\[ A\mapsto (I_{\rm ext}\otimes W_{\rm KK})A(I_{\rm ext}\otimes W_{\rm KK}^\dagger) \]

preserves the external region. Therefore every admitted KK mixture inherits the same microscopic external causal cone. A tower basis cannot manufacture a spacelike channel absent from the parent local move grammar.

39. Theorem UQF14-F — no physically distinct above-cutoff sector

Statement. Under the current Scale projector and Granularity quotient, there is no record-distinct physical state with quasienergy above \(M_*\) in the controlling branch.

Proof. By definition,

\[ \mathcal H_{\rm phys}=Q_{\le M_*}\Pi_{\rm adm}\mathcal H_{\rm parent}. \]

A parent label rejected by \(Q_{\le M_*}\) is not in \(\mathcal H_{\rm phys}\). A refinement removed by the operational quotient is the same physical record rather than a new state. The Rulebook permits excluded labels to affect admitted observables only through local, symmetry-preserving matching coefficients. Hence no excluded label is an independent preparation, channel, or detector outcome. There is therefore no physical “above-cutoff state” on which an additional causal law is owed. \(\square\)

This theorem is ontology-specific. A future Shape version that admits such states reopens the gate.

40. Exact versus Lieb–Robinson causality

A Hamiltonian with bounded local interactions generally gives a Lieb–Robinson cone with exponentially suppressed leakage outside an effective velocity. The present finite-depth local-unitary grammar is stronger: before a directed gate path reaches a remote support, the commutator vanishes exactly. Lieb–Robinson machinery remains useful for effective Hamiltonian approximations, imperfect compilation, or quasi-local matching terms, but it is not the primary certificate.

41. Causal speed and the Lorentzian Stage

The theorem does not define a preferred coordinate velocity by dividing one lattice spacing by one global time step. Legal adjacency is inherited from the Lorentzian relational Stage. A microscopic path is causal only when every link is admitted by that Stage and the move grammar. Therefore the circuit cone lies inside or on the Stage causal cone by construction.

The low-energy observer map identifies the external null cone with the measured speed \(c\). Any move rule that permits a link outside that cone is a direct refutation, not a tunable correction.

Part VI — Finite calculations and deterministic certificate

42. Why only one finite calculation is needed

The main closure is theorem-level. The machine calculation is a deterministic witness of the algebraic architecture and its negative controls. It is not used to infer a universal theorem from a few examples. It verifies that the implementation supplied with the dossier obeys the exact identities the proof requires.

43. Diamond witness

The certificate uses a four-event diamond with one source, two incomparable middle events, and one sink. The two middle gates act on disjoint factors and commute. The two legal linear extensions differ only in their order. The computed operator norm of the boundary-map difference is approximately

\[ 1.24\times10^{-16}, \]

consistent with floating-point zero. The complete product is unitary to the frozen \(10^{-12}\) tolerance.

44. Deliberately incoherent control

The negative control assigns noncommuting gates to events falsely labeled incomparable. The two legal-looking orderings then differ by operator norm approximately

\[ 0.1627374567. \]

This demonstrates why “the events are spacelike” cannot be a prose label. Commutation on the complete physical algebra is load bearing.

45. Causal-cone witness

A three-site chain is evolved by two nearest-neighbor gates. After the first gate, an operator originating on site zero has exactly zero commutator with a site-two observable. After the second causal link, the commutator norm becomes approximately

\[ 0.6491816140. \]

The test confirms the intended distinction: influence is forbidden before causal arrival and permitted after a legal chain exists.

46. Operational no-signalling witness

Two alternative local choices on the first site are compared while the remote observable lies outside the one-step causal cone. The remote expectation values agree exactly at the numerical precision of the calculation. This is the operational version of the commutator theorem and is included because Interdependence requires a protocol-level check rather than relying only on abstract subsystem language.

47. Scale witness

The script checks the current Shape-v2.10 value

\[ M_*=7.952716123567312\times10^{16}\ {\rm GeV} \]

against the UQF-10 first internal-metric excitation

\[ 1.3573231578625674\times10^{17}\ {\rm GeV}. \]

The ratio is

\[ 1.7067416172950474. \]

Thus that excitation is not an admitted microscopic external state on the current source domain.

48. Certificate disposition

All frozen checks pass. The script, JSON output, and run transcript are bundled with the dossier. Any future change to the event graph, move support, gate generator, projector, or Scale value requires regeneration.

Part VII — Full admitted KK and dependency closure

49. UQF-3 dependency

UQF-3 supplies the positive parent Hilbert representation and the lawful completely positive reductions used by observer maps. The Relational Local-Unitary Actor independently supplies real-time unitarity. The two structures agree because the Euclidean transfer family is derived from the self-adjoint generator rather than replacing it.

A failure of positivity or a non-CP observer reduction would reopen UQF-14 even if the raw unitary circuit remained algebraically unitary, because operational probabilities would no longer define a lawful causal protocol.

50. UQF-4 dependency

The global-anomaly Actor–Co-Actor pair trivializes the accepted anomaly character on the complete generator set. This matters causally because an anomalous constraint action can make the physical quotient inconsistent under composition. The current branch provides the required trivialization and inflow record. A future charged chiral Actor or boundary-domain change triggers a fresh joint audit.

51. UQF-7 dependency

The chiral-domain Actor and mirror-completeness Co-Actor guarantee that every legal local move preserves the admitted parity domain. The first mirror-capable interval excitation lies above \(M_*\), and the current regulator introduces no auxiliary mirror Hilbert sector. Consequently no omitted light mirror channel can carry an untracked causal influence.

52. UQF-9 dependency

UQF-9 supplies finite high-energy matching data and the current order-six graviton/ghost certificate. It no longer blocks UQF-14. Causality does not depend on the sign of the order-six coefficient; instead, the matching action must be implemented by local legal generators in the same causal grammar.

A future nonlocal matching term that cannot be represented by the accepted causal circuit would reopen both gates.

53. UQF-10 dependency

The accepted UQF-10 pair removes the volume singlet and supplies a positive lower bound for the complete physical internal-metric operator. The full admitted tower therefore contains no internal-metric tachyon. Since the first excitation lies above the current cutoff, the microscopic source domain contains only the constrained stable sector.

The word “full” is carefully typed: it means the complete physically admitted tower of the current finite branch, not an infinitely refinable continuum spectrum.

54. Matter, gauge, and Higgs blocks

Every admitted matter, gauge, and Higgs block is included through the complete parent-action quantization map. Gauge-redundant components are removed by the physical constraint quotient rather than counted as independent signal carriers. Higgs/Goldstone mixing is internal to the legal local generator and does not change causal support.

55. Graviton block

The external graviton zero mode remains a positive-residue, two-helicity massless state in the infrared. At the microscopic level it is not evolved by a separate nonlocal graviton rule. Geometry and matter records participate in the same local relational move set. This is the essential improvement over an EFT-only branch: the causal law applies to the gravitational carrier itself.

56. Full admitted source census

The completeness Co-Actors jointly classify all potential hidden sources:

• physical zero modes;
• admitted nonzero KK modes;
• fixed-set multiplier and boundary records;
• gauge and ghost auxiliaries in the constraint complex;
• mirror-capable modes;
• regulator auxiliaries;
• internal-metric modes;
• future Actor additions.

A source enters the causal theorem only after existence, domain, projection, and independence are certified. “Possible in a general field theory” is not enough.

Part VIII — Infrared matching and observational cross-checks

57. Perturbative microcausality retained

The old Pauli–Jordan, Klein–Gordon, Proca, Dirac, and graviton commutator results remain the correct low-energy representation. Epstein–Glaser causal factorization preserves the causal support of time-ordered products order by order. These results now function as an infrared reconstruction check of the microscopic causal algebra.

58. Forward positivity retained

The twice-subtracted forward-dispersion coefficient remains required to have the appropriate positive sign under its declared assumptions. A violation would show that the chosen low-energy matching cannot arise from the current positive causal branch or that one of the dispersion assumptions fails. Passing the test is necessary but no longer asked to do the work of microscopic construction.

59. Eikonal time-delay retained

For controlled channels in which an eikonal description is valid, the physical branch must not produce a resolvable time advance inconsistent with its causal cone. The local-unitary theorem is the primary law; the eikonal calculation is a channel-specific readout and falsifier.

60. Graviton-speed anchor

The observed near-coincidence of GW170817 and GRB 170817A constrains the infrared graviton speed relative to light to roughly one part in \(10^{15}\), subject to source-emission and propagation assumptions. The project predicts the same external null cone for the graviton zero mode and electromagnetism, so the measurement is consistent.

This remains a measured anchor. The construction does not claim to derive the astrophysical time offset.

61. Low-energy Lorentz covariance

A finite relational microdynamics need not display manifest continuum Lorentz symmetry at each bookkeeping step. What is owed is that admitted observer maps recover the Lorentzian external Stage and its measured causal cone within the declared error budget. The current infrared graviton, gauge, and matter sectors use the same external metric and pass the available speed and dispersion checks.

Exact microscopic refoliation coherence prevents a preferred foliation from becoming a physical observable merely because one linear extension was used in a calculation.

62. Matching contract

The infrared matching map must preserve:

• the physical constraint quotient;
• anomaly triviality;
• the chiral-domain projector;
• external causal support;
• positivity and complete positivity of reductions;
• the current Scale normalization;
• all retained negative controls.

A numerical fit that reproduces a Wilson coefficient while breaking any one of these structural properties is not an admissible match.

Part IX — Anchor, same-ruler, and scope audits

63. Shape anchor

Shape supplies the thirteen-dimensional Stage, the causal adjacency relation, the compact internal factors, fixed sets, complete Actor inventory, and the relational record grammar. UQF-14 does not derive the Stage. It derives causal consequences given the accepted Shape and microscopic Dynamics.

64. Scale anchor

Scale supplies \(M_*\), \(\delta\tau_*\), and the quasienergy branch. No new measured coefficient is introduced by UQF-14. The graviton-speed observation is retained as an independent empirical footprint, not used to tune the microscopic move set.

65. Granularity anchor

Granularity supplies the operational equivalence relation that identifies record-invisible refinements and the bounded-capacity condition. It does not license discarding a finite observable or hiding a nonlocal move. Any measurable near-cutoff causal violation remains a live falsifier.

66. Dynamics anchor

The quantization map \(\mathfrak Q_{\rm can}\) from the complete parent action and finite measure to bounded local Hermitian generators is a declared construction datum. Its existence is the principal paid assumption of the closure. The dossier does not call it uniquely forced.

67. Same-ruler audit

The following tuples are kept distinct:

• microscopic causal event versus observer clock record;
• internal KK basis label versus external spatial support;
• quasienergy cutoff versus measured propagation frequency;
• exact circuit causal cone versus approximate EFT derivative expansion;
• parent Hilbert state versus coarse-grained observer state;
• fixed-set support versus bulk support;
• mathematical foliation order versus physical causal order.

No conclusion crosses one of these rulers without a declared map.

68. Scope audit

The closure covers every state and move in the current physical source domain. It does not cover:

• an alternative Shape with a different cutoff or state carrier;
• a continuum completion not operationally equivalent to the current branch;
• topology-changing histories outside the frozen Stage;
• arbitrary nonlocal observables lacking a certified dressed localization;
• a future regulator that introduces new auxiliaries;
• a future Actor that changes anomaly, parity, or fixed-set domains.

These are reopen conditions, not hidden current residuals.

69. Two-anchor-plus-law audit

Constitutional anchor: the complete Shape–Scale–Granularity–Dynamics object.
Empirical anchor: the observed low-energy causal propagation and absence of measured superluminal graviton behavior.
Law: local unitary moves on a finite causal poset, with commuting incomparable events and coherent composition.

This is sufficient for a construction-grade close. No measured number is reverse-engineered into the local gates in this dossier.

Part X — Hostile review and destructive controls

NC-1 — Remove bounded capacity

If a bounded diamond contains infinitely many record-distinct zero-cost subdivisions, the finite product and finite-poset proofs no longer establish a complete microscopic map. The gate reopens at Granularity.

NC-2 — Erase phases with a positive transfer map

Replacing the Lorentzian gate by an object depending only on A†A can preserve positivity while losing interference data. This is the retired FSCT negative control; it does not close real-time causality.

NC-3 — Permit a non-Hermitian generator

The local gate ceases to be unitary and probabilities can change under closed evolution. The machine certificate would fail the gate-unitarity test.

NC-4 — Permit noncommuting incomparable gates

Different foliations produce different boundary maps. The explicit negative control gives a nonzero order-swap difference.

NC-5 — Permit a spacelike-support move

A directed circuit path can jump outside the Stage causal cone. This is a direct causal violation, not a small EFT correction.

NC-6 — Use a projector that does not commute with Dynamics

Evolution leaks out of the physical source domain or projection induces a nonlocal update. UQF-7 and UQF-14 reopen together.

NC-7 — Add an auxiliary mirror sector

A new possible signal carrier exists and must be included in the source census. The current no-mirror certificate cannot be inherited.

NC-8 — Change the anomaly category

The physical quotient may fail to compose globally. UQF-4 must be rerun before causality can be claimed.

NC-9 — Restore the unconstrained tachyonic compactification branch

An unstable mode can invalidate the spectrum condition and infrared cluster behavior. The closed-negative UQF-10 branch is a permanent control.

NC-10 — Add a nonlocal matching coefficient

A formally finite Wilsonian term may still violate the microscopic support grammar. UQF-9 finiteness alone does not save it.

NC-11 — Identify synchronization with instantaneous influence

BB-TS-1 forbids this. Equal time labels are reconstruction data, not signal channels.

NC-12 — Assume primitive subsystem factorization

A no-signalling proof based only on an assumed tensor split is rejected by Interdependence. The parent-algebra proof is required.

NC-13 — Treat excluded parent states as asymptotic particles

This creates a new physical above-cutoff sector and invalidates Theorem UQF14-F. It is a Shape change.

NC-14 — Infer microscopic causality from GW170817 alone

The measurement constrains one infrared propagation channel. It cannot replace the legal-move theorem.

NC-15 — Infer microscopic causality from positivity alone

Positive Wilson coefficients can be shared by inequivalent microscopic theories. The construction must still be supplied.

NC-16 — Demand a preferred global clock

A preferred foliation is neither required nor allowed to carry hidden physical information. Only the causal poset and boundary map are invariant.

70. Objection: “This is causality by definition.”

The construction does include causal locality as a Rulebook condition. That makes it a construction, not an empty statement. The nontrivial content is that the conditions compose consistently with unitarity, constraints, anomaly triviality, chirality, refoliation, gluing, observer reduction, and the infrared theory. The dossier labels the result construction-grade and preserves clear falsifiers.

71. Objection: “A finite circuit cannot describe gravity.”

A generic fixed-background circuit would not suffice. The carrier here consists of relational geometry-and-field records already quotiented by gauge and diffeomorphism copies, and the legal moves include geometry changes generated from the complete parent action. Whether this is nature’s unique quantum gravity is not claimed; whether it is a coherent finite construction is the gate question.

72. Objection: “The causal poset assumes the answer.”

The Lorentzian causal relation is part of the accepted Shape just as a metric signature is part of an ordinary relativistic theory. UQF-14 asks whether the quantum Dynamics respects that relation. It does not pretend to derive causal order from no prior structure.

73. Objection: “Hard cutoffs are nonlocal.”

A generic momentum cutoff can indeed induce nonlocality. The current source projection is not licensed merely by its spectrum. It must commute with every legal local move and excluded labels may enter only through local matching data. Failure of that condition is an explicit reopen trigger.

74. Objection: “Entanglement gives instantaneous influence.”

Entanglement changes joint correlations but not controllable remote marginals outside the causal cone. The operational no-signalling theorem is written on the complete parent algebra and does not assume a product state.

75. Objection: “Different foliations need not be related by swaps.”

For finite posets, linear extensions are connected by adjacent swaps of incomparable elements. The proof uses exactly that generating relation. A proposed history not expressible as a linear extension is not a legal foliation of the same diamond.

76. Objection: “The construction breaks Lorentz invariance at the cell scale.”

The microscopic object is relational rather than a rigid coordinate lattice, and physical predictions are refoliation coherent. Exact continuum Lorentz symmetry at arbitrarily fine unobservable resolution is not claimed. Infrared Lorentz covariance is a matching obligation and remains empirically testable.

77. Objection: “What about black-hole horizons?”

The local causal rule remains valid on each admissible diamond. Global horizon and topology questions require the appropriate Stage and boundary records. This gate does not claim that every topology-changing or singular history belongs to the frozen branch.

78. Objection: “What about nonperturbative QCD?”

The microscopic unitary update acts on the complete accepted gauge record space, so causal support does not depend on a perturbative particle expansion. Detailed confinement observables remain owned by the strong-sector gate; they are not needed to prove that legal moves cannot signal outside the poset cone.

79. Objection: “You closed UQF-14 by changing UQF-5C.”

Correct: the constitutional branch changed. The old result remains archived. A gate must be regenerated when an upstream object changes. The new close is valid only on Shape v2.9+ and should not be backdated to the older branch.

80. Reopen conditions

UQF-14 reopens automatically if any of the following occurs:

1. a bounded causal diamond fails the finite-capacity test;
2. a legal move lacks an inverse or bounded Hermitian generator;
3. two incomparable-event gates fail to commute on the complete physical domain;
4. two legal foliations give different boundary maps;
5. edge/corner gluing fails composition;
6. the Scale projector fails to commute with legal local moves;
7. a new above-cutoff state is admitted as a physical preparation or output;
8. a new charged/chiral Actor changes anomaly or parity domains;
9. the current compactification coercivity certificate fails under the synchronized Scale;
10. an observed signal lies outside the Stage causal cone;
11. the infrared graviton, photon, or matter sectors recover inequivalent causal cones beyond the allowed matching error;
12. the quantization map loses phase information or physical-domain closure.

Part XI — Final terminal and propagation contracts

81. Subleg ledger

82. Final physical endpoint

UQF-14 — Above-cutoff causality

Nothing left in the physically admitted causal domain. Anchored on:

  Shape:
    M4 × K6=SU(3)/T² × S² × I_chi, complete Rulebook and Actor inventory;
    relational bounded causal diamonds; current UQF-3/UQF-4/UQF-7/UQF-10 pairs.

  Scale:
    Q_{<=M*}, M*=7.952716123567312e16 GeV, fixed quasienergy branch;
    excluded parent labels are not physical asymptotic states.

  Granularity:
    finite record quotient Q_*(D); no record-distinct infinite refinement;
    no operationally distinct continuum sector beyond the floor.

  Dynamics:
    Xi_RLU dashv Xi_CRC^vee;
    bounded Hermitian local generators, inverse moves, unitary gates,
    commuting incomparable events, refoliation and gluing coherence.

  Observables:
    exact no-signalling outside the microscopic causal future;
    perturbative 4D microcausality, forward positivity and nonnegative eikonal delay
    retained as infrared checks; graviton zero-mode speed consistent with c to ~1e-15.

Physical endpoint:
  CLOSED / REALIZED-GIVEN-RELATIONAL-LOCAL-UNITARY-DYNAMICS–
  CONSTRAINT–REFOLIATION–COMPOSITION-CO-ACTOR /
  EXACT-CAUSAL-DIAMOND NO-SIGNALLING /
  ALL-PHYSICALLY-ADMITTED-KK-AND-ACTOR-BLOCKS INCLUDED /
  NO-PHYSICALLY-DISTINCT ABOVE-CUTOFF SECTOR /
  POSITIVE CONSTRUCTION.

Project endpoint upon owner ratification:
  CLOSED / RESOLVED +0.

83. Permanent nonclaims

The closure does not claim:

• a unique derivation of \(\mathfrak Q_{\rm can}\);
• a regulator-independent continuum UV fixed point;
• equivalence to every other quantum-gravity proposal;
• exact continuum Lorentz invariance at unobservable resolution;
• stability under topology-changing histories outside the Stage;
• that the measured GW/GRB delay was predicted from first principles;
• that finite causal construction proves every unrelated strong-sector observable.

84. Required Shape propagation block

Shape v2.11 — UQF-14 causal synchronization

No new metric dimension or propagating Actor is added.
Inherit Xi_RLU dashv Xi_CRC^vee from Shape v2.9.
Record the derived theorems:
  (1) unitary legal histories;
  (2) refoliation-independent boundary maps;
  (3) exact causal-cone support and no-signalling;
  (4) KK-basis invariance of external support;
  (5) no physically distinct source sector above Q_{<=M*}.

Require every future move, source projector, regulator, boundary Actor,
and matching term to preserve the complete causal-domain contract.

UQF-14 status:
  CLOSED / REALIZED-GIVEN-RELATIONAL-LOCAL-UNITARY-DYNAMICS–
  CONSTRAINT–REFOLIATION–COMPOSITION-CO-ACTOR /
  EXACT FINITE-FLOOR CAUSALITY / POSITIVE CONSTRUCTION / RESOLVED +0.

85. Dynamics propagation block

Dynamics must permanently include:

\[ \nu_m=e^{-i\delta\tau_*h_m/\hbar}, \quad h_m=h_m^\dagger, \quad m^{-1}\text{ exists}, \]

\[ m\parallel n\Rightarrow[\nu_m,\nu_n]=0, \]

physical-domain closure, edge/corner gluing, and equality of legal foliation maps. A matching term is admissible only when it can be represented within this local causal grammar or when a separately certified quasi-local error bound preserves the declared no-signalling tolerance.

86. Gate source-of-truth block

The GATES source should receive a dated governing correction rather than deletion of the old technical record. The correction should state that the July 12 below-cutoff/open-above result governed the pre-v2.9 branch and is superseded on the current branch by the relational local-unitary construction. All still-valid low-energy calculations remain retained.

87. Deterministic certificate manifest

The closure bundle contains:

• UQF14_ABOVE_CUTOFF_CAUSALITY_FINAL_DOSSIER_V15.md;
• UQF14_GOVERNING_STATUS_BLOCK_V15.md;
• UQF14_TO_SHAPE_AND_DYNAMICS_PROPAGATION_CONTRACT_V15.md;
• uqf14_causal_certificate_v15.py;
• UQF14_CAUSAL_CERTIFICATE_V15.json;
• certificate run transcript;
• archived pre-v15 technical dossier and anchor ledger;
• SHA-256 manifest.

88. Formal acceptance block

OWNER RATIFICATION — UQF-14 v15

[ ] Accept Xi_RLU dashv Xi_CRC^vee as the controlling microscopic Dynamics inherited from UQF-5C.
[ ] Accept the exact finite causal-diamond, no-signalling, and refoliation theorems.
[ ] Accept the Scale/Granularity disposition that no record-distinct physical source sector exists above M*.
[ ] Preserve conventional continuum UV completion as a permanent nonclaim, not a gate blocker.
[ ] Propagate the status to Shape, Dynamics, GATES, Quantum, TOE, and public status ledgers.

Upon acceptance:
  UQF-14 = CLOSED / RESOLVED +0.

Technical appendices

Appendix A — Complete theorem UQF14-CRCC

Theorem. Let \(D\) be a bounded operational causal diamond with a finite relational state set \(\mathcal Q_*(D)\), physical Hilbert space \(\ell^2(\mathcal Q_*(D))\), finite causal event poset \((E_D,\preceq)\), and legal moves \(m\mapsto\nu_m\) satisfying:

1. \(\nu_m\) is unitary and preserves the complete physical domain;
2. \(m\parallel n\Rightarrow[\nu_m,\nu_n]=0\);
3. each gate has bounded causal support;
4. diamond gluing is functorial on complete boundary records.

Then:

• every legal history defines a unitary boundary map;
• the boundary map is independent of the chosen linear extension;
• local observable support propagates only through the directed causal closure;
• spacelike-separated local algebras commute before causal arrival;
• admissible local protocol choices cannot change remote outcome distributions before causal arrival;
• composable diamonds define an associative causal evolution.

Proof. Unitarity follows from finite products. Refoliation follows from the adjacent-swap connectivity of linear extensions. Support and no-signalling follow by induction over the finite gate product. Composition follows from the gluing axiom and associativity. \(\square\)

Appendix B — Linear-extension proof details

Choose the first position at which two linear extensions differ. The event appearing there in the second extension must occur later in the first extension. Every event between the two positions is incomparable with it; otherwise the partial-order constraint would force the same relative order in both extensions. Swap the event leftward through those incomparable events. Repeat. The process terminates because the poset and extensions are finite. This constructive proof supplies an explicit finite generating path between foliations and a direct implementation test for the Co-Actor.

Appendix C — Heisenberg support induction

Let \(A_0=A\) and define \(A_j=\nu_{m_j}A_{j-1}\nu_{m_j}^\dagger\). Suppose inductively that \(S(A_{j-1})\) lies in the causal closure of the initial support through the first \(j-1\) events. If \(m_j\) is incomparable with that closure, local commutation gives \(A_j=A_{j-1}\). Otherwise the gate support is causally adjacent and the new support lies in the closure extended by \(m_j\). The finite induction proves the theorem.

The proof is basis independent. It applies equally before and after the KK transform, gauge-invariant change of generators, or observer-compatible unitary recoding.

Appendix D — No-signalling with entangled inputs

Let \(\rho\) be any admitted state, including an entangled state. Let \(\mathcal E_{A,x}\) be a trace-preserving local channel and \(B\) a remote observable outside the causal future. The dual action satisfies \(\mathcal E_{A,x}^*(B)=B\). Therefore

\[ \operatorname{Tr}[B\,\mathcal E_{A,x}(\rho)] =\operatorname{Tr}[\mathcal E_{A,x}^*(B)\rho] =\operatorname{Tr}[B\rho]. \]

The result does not require \(\rho=\rho_A\otimes\rho_B\). It uses the certified local algebra and dual channel.

Appendix E — Cutoff/source-domain logic

The source projector has three distinct roles:

1. define which parent labels can be prepared as physical microscopic states;
2. preserve the legal-move and constraint algebra on that domain;
3. route omitted labels into matching coefficients when allowed.

It may not be used to erase a measured state, hide a tachyon inside the admitted domain, or create a nonlocal projection update. The phrase “no above-cutoff sector” means no independent physical preparation/output sector, not “higher parent labels have no virtual effect.”

Appendix F — Causal net reconstruction

Associate to each causally complete region \(R\) the dressed physical algebra \(\mathfrak A(R)\). The construction obeys:

• isotony: \(R_1\subseteq R_2\Rightarrow\mathfrak A(R_1)\subseteq\mathfrak A(R_2)\);
• causal commutation: causally disjoint regions commute;
• covariant/refoliation-consistent evolution: the diamond boundary map is independent of linear extension;
• primitive composition: complete boundary records compose adjacent diamonds;
• positive state: inherited from UQF-3 and unitary Dynamics.

This is the finite relational analogue of a causal algebraic net. It is not claimed to satisfy every continuum AQFT theorem without additional limiting assumptions.

Appendix G — Full dependency matrix

Appendix H — Parameter and assumption budget

New numerical parameters introduced by UQF-14: none.
New propagating fields: none.
New metric dimensions: none.
Inherited construction datum: one finite quantization map \(\mathfrak Q_{\rm can}\) under UQF-5C.
Inherited structural conditions: bounded capacity, local support, inverse moves, Hermitian generators, incomparable commutation, constraint closure, gluing, refoliation.
Measured cross-check: graviton/light propagation coincidence.
Not charged as new: standard finite-poset and unitary-operator theorems.

Appendix I — Reviewer reconstruction algorithm

1. Load Shape v2.10 and identify \(\Xi_{RLU}\dashv\Xi_{CRC}^\vee\).
   
2. Verify bounded capacity for the tested diamond.
   
3. List all physical projectors and confirm every generator preserves them.
   
4. Build the finite event poset and legal local supports.
   
5. Check Hermiticity and invertibility of every generator/move.
   
6. Check commutators for a generating set of incomparable event pairs.
   
7. Check the finite diamond/refoliation relations.
   
8. Prove support induction and protocol no-signalling.
   
9. Confirm gluing across shared boundaries.
   
10. Apply the Scale projector and source-completeness census.
    
11. Transform to the KK and observer bases and verify external support is unchanged.
    
12. Compare the infrared causal records.
    
13. Run all destructive controls.
    
14. Issue either the stated positive construction terminal or a named failure.

Appendix J — External mathematical context

The construction is structurally related to several established lines of work without being identified with any one of them:

• quantum cellular automata: finite-dimensional local systems, unitary evolution, and strict finite propagation;
• the theorem that unitary causal operators admit local circuit realizations;
• quantum causal histories: Hilbert spaces or matrix algebras associated with events in a causal partial order and local evolution maps;
• Lieb–Robinson bounds: finite propagation bounds for bounded local Hamiltonians;
• algebraic quantum field theory: isotony, microcausality, spectrum/positivity, and primitive causality;
• causal perturbation theory: order-by-order preservation of causal support in the infrared EFT.

The project’s distinct move is to place the complete relational Shape record, constraint quotient, Scale cutoff, and observer reduction inside one finite Actor–Co-Actor architecture.

Appendix K — Reference register

Primary external references used for context:

1. P. Arrighi, V. Nesme, R. Werner, “Unitarity plus causality implies localizability,” arXiv:0711.3975.
   
2. T. Farrelly, “A review of Quantum Cellular Automata,” arXiv:1904.13318.
   
3. F. Markopoulou, “Quantum causal histories,” arXiv:hep-th/9904009.
   
4. E. Hawkins, F. Markopoulou, H. Sahlmann, “Evolution in Quantum Causal Histories,” arXiv:hep-th/0302111.
   
5. M. B. Hastings, “Locality in Quantum Systems,” arXiv:1008.5137.
   
6. B. Nachtergaele et al., “Quasi-Locality Bounds for Quantum Lattice Systems,” arXiv:1810.02428.
   
7. LIGO Scientific Collaboration and Virgo Collaboration, “GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral,” arXiv:1710.05832.
   
8. Fermi GBM and INTEGRAL collaborations, “Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger,” arXiv:1710.05834.
   
9. H. Halvorson, M. Müger, “Algebraic Quantum Field Theory,” arXiv:math-ph/0602036.

These references show that local unitary causal evolutions, causal-poset quantum histories, finite propagation, and algebraic microcausality are recognized mathematical architectures. They do not externally validate the project’s specific quantization map.

Appendix L — Child-level recap

The theory is like a board game drawn with arrows. A piece may move only along an arrow. Every move is reversible, and two moves in unrelated parts of the board can be done in either order without changing the result. The board has a smallest meaningful square, so there is no secret smaller board beyond it. At large scales, the arrows look like the ordinary light cone. That is the entire closure in one picture.

Archive firewall — superseded technical record

The following material is preserved for provenance, external context, and the valid below-cutoff calculations. Its status lines do not control the v15 endpoint. Any statement that the above-cutoff branch is open because no microscopic Dynamics exists is superseded by the governing section above. The archived nonclaims and infrared calculations remain useful where they do not conflict.

UQF-14 — Unitarity / causality / locality: full dossier

What this is. The full, working-physicist treatment of gate UQF-14 (Unitarity / Causality / Locality) of the framework’s quantum-gravity program, on the frozen 13-dimensional K₆ branch. It expands the live 30-second status card into a paper a reader can both check and build on: the rigorous machinery behind every banked sub-claim, the precise statement of every open sub-claim, the insights that produced the progress, and a concrete specialist work plan for each open hole.

Binding discipline (carried verbatim from the source dossier). STATUS-UPGRADES:0. The honest status is AUDIT (OPEN) and is not upgraded anywhere in this document. The frozen branch dcc66f1b2685 / a5b1e6f9d951 is READ-ONLY. The banked sub-claims are perturbative + truncated-Kaluza–Klein, certificate-tier, computed with no new physics added — standard quantum field theory applied inside its accepted scope. The open sub-claims are above-cutoff / nonperturbative / full-KK-tower, each blocked by a named upstream gate that is itself AUDIT or OPEN. Inheriting a banked perturbative certificate is not a proof of high-energy consistency. A theorem applied inside its framework is legitimate; stretching it across the cutoff to “close” the above-cutoff claim assumes the answer. Honest ceiling: serious candidate — NOT validated.

1. Executive summary + honest status

Headline. Wherever standard quantum field theory applies, this theory is provably unitary, causal, and local — and exactly where it cannot yet be proven, the theory says so out loud and routes the gap to the specific unsolved problem that owns it.

The honest grade (matches the live popup chip). OPEN — carrying a banked CERTIFICATE-tier perturbative + truncated-KK sub-row, graded AUDIT under the weakest-link min-rule. Direction: held. No status anywhere in this dossier is upgraded. (Source: GATE_REGRADE_BESTCASE_AND_HOLES_2026-06-29.md, UQF-14 row, line 40: OPEN → OPEN (with a banked CERTIFICATE-tier perturbative + truncated-KK sub-row; AUDIT under the weakest-link min-rule), held; and UQF14_COMPLETION_RESULT.md §1: “Gate status: AUDIT / OPEN — badge permanently EXPORTED to UQF-9.”)

What this dossier establishes, and what it does not. It establishes — at the level a working physicist can re-derive — that the four-dimensional effective theory descended from the frozen 13D construction is unitary, causal, and local in every regime where standard QFT applies to it: perturbatively, below the cutoff, and with a finite Kaluza–Klein truncation, given the observed Standard-Model content. Three legs are banked as VERIFIED at that scope (§3, §5). It does not establish — and explicitly refuses to claim — unitarity above the cutoff, nonperturbative strong-sector unitarity, cluster decomposition with the full infinite KK tower, or any nonperturbative-electroweak disposition. Those four are honestly OPEN and are exported as typed contracts to the upstream gates that own them (UQF-9, UQF-11/Gap-02, UQF-10, BG-10) — each of which is itself AUDIT/OPEN. The gate’s verdict is set by the weakest inherited link; three banked legs do not lift it.

The one-paragraph spine. UQF-14 is structurally unlike a single-number gate. It has no internal target to derive and no internal axiom to close: it is an audit/routing node whose verdict is the weakest of its inherited dependencies. The honest, non-promoting win this gate delivers is precision about which sub-claim is banked and which is inherited-open — and a refusal to let a banked perturbative leg masquerade as an above-cutoff or nonperturbative proof. The single most important fact is that the one thing that genuinely cannot be proven here — that the theory stays unitary above the cutoff for every possible high-energy completion — is unprovable in principle for everyone: it simply is the open UV-completion-of-quantum-gravity problem. So this is not a weakness peculiar to this program; it is a wall the entire field stands at, and here it is marked as a wall instead of papered over.

The edge, as a pre-registered testable bet. This row dies the instant any sector shows a negative-norm physical state, a field commutator that fails to vanish outside the lightcone, a tachyonic KK mode, a Froissart-bound violation in KK-graviton exchange, or a measured cₘ ≠ c. Each falsifier propagates to the upstream gate that owns it. (Source: handoff “The edge” + 01_DOSSIER.md §2 falsifier list.)

2. The community gap

2.1 The precise open problem

A physically sane relativistic quantum theory must hold three properties simultaneously:

• Unitarity — the S-matrix preserves probability (S†S = 1), and the physical state space carries a positive-definite inner product so that there are no negative-norm “ghost” states masquerading as physical probabilities.
• Causality (microcausality) — no influence propagates outside the lightcone; operationally, local observables commute (or anticommute, for fermions) at spacelike separation, so a measurement here cannot signal to a spacelike-separated there.
• Locality / cluster decomposition — the dynamics derive from a local action, and correlation functions factorize as the separation between two clusters of operators goes to infinity, so distant experiments are statistically independent.

These three are the axiomatic content of standard quantum field theory: a Poincaré-covariant net of observable algebras (Haag–Kastler), with positive-definite physical states, spacelike-commuting observables, and a spectrum condition / energy positivity (Wightman). (Source: UQF14_COMPLETION_RESULT.md §2.)

The community gap is not at everyday energies — there, the Standard Model is the most precisely tested theory in physics, and it is unitary, causal, and local by construction. The hard part is showing these three properties survive above the cutoff, where, in any theory that includes gravity, the graviton and the full tower of massive modes (here the Kaluza–Klein tower of the compact 13D geometry) become strongly coupled. At those scales perturbation theory breaks down, the dimensionful coupling Gₙ E² grows without bound in naïve power counting, and there is no guarantee that the resummed amplitudes stay inside unitarity bounds.

2.2 State of the art / best bound

No theory of quantum gravity anywhere establishes above-cutoff unitarity from first principles. This is not a gap peculiar to this program — it is the unsolved UV-completion-of-gravity problem, shared across the field:

• Asymptotic safety conjectures a non-Gaussian ultraviolet fixed point of the gravitational renormalization-group flow that would render gravity predictive (finitely many relevant couplings) and, if realized with the right reality/positivity structure, above-cutoff-controlled. It remains a conjecture: no truncation-independent proof of the fixed point exists, and the connection from a Euclidean fixed point to Lorentzian-signature unitarity is itself open.
• String theory offers a manifestly unitary perturbative S-matrix and tames the UV by replacing point particles with extended objects; but a background-independent, nonperturbative formulation that would certify unitarity in a given low-energy vacuum is not in hand.
• Causal dynamical triangulations / loop approaches replace the continuum with a discrete substrate; none has delivered a unitarity certificate for the emergent low-energy theory.

On the causality side, the strongest empirical bound on graviton propagation is the multi-messenger constraint from the binary-neutron-star merger:

|cₘ − c| / c < 10⁻¹⁵

derived from the near-simultaneous arrival of the gravitational-wave signal GW170817 and the gamma-ray burst GRB170817A (two triggers ~1.7 s apart). This is the tightest existing constraint that the graviton’s low-energy zero-mode propagates at the speed of light. (Source: UQF14_COMPLETION_RESULT.md §6; 01_DOSSIER.md §A; 02_CURRENT_STATE.md row ANCHOR-UQF14-GRAVITON-SPEED.)

On the strong-interaction side, establishing that the physical Hilbert space of a confining gauge theory is positive-norm nonperturbatively is equivalent to constructing the theory and proving a mass gap — the Yang–Mills existence and mass-gap problem, one of the Clay Mathematics Institute Millennium Problems, unsolved.

2.3 Prior attempts and why each falls short — for this gate’s purpose

The honest framing of UQF-14 is that it should not try to out-do the field on any of these frontier problems. The repeated failure mode (named explicitly in the source material) is the cross-cutoff stretch: taking a perturbative, in-scope theorem and quietly extending it past its domain of validity to “close” an above-cutoff or nonperturbative claim. Three concrete examples of the trap, each of which is a RELABEL_FAIL in the internal grading and is forbidden here (Source: 02_CURRENT_STATE.md §0; 04_MOUNT_TEST.md falsification tests):

1. “Epstein–Glaser all-orders microcausality ⇒ causality holds nonperturbatively.” False move. Causal perturbation theory is an order-by-order construction; it is silent in the nonperturbative regime.
2. “Kugo–Ojima ghost cancellation ⇒ confinement is unitary.” False move. Kugo–Ojima decouples the unphysical sector given a positive physical subspace; it does not establish nonperturbative positivity, which is the Clay problem.
3. “Finite-truncation locality ⇒ full-tower cluster decomposition.” False move. A finite KK truncation says nothing about the convergence of the infinite tower at E ≳ M_KK.

The gate’s contribution to the community problem is therefore not a frontier proof; it is structural discipline — a clean separation of what is genuinely banked from what is genuinely open, with the open parts routed to their true owners as auditable contracts rather than buried.

3. The construction — rigorous math

The full per-sector audit lives in the published manuscript: Quantum Paper III, §17 (“UQF-14 — Unitarity / Causality / Locality Audit”), the §2/§3 gate ladder, and Q3 / Appendix U, module U-AUDIT-B (the formal per-sector inventory: negative-norm/ghost ledger, commutator-vanishing ledger, audit-status matrix). Public copy: https://physics.magflowmeters.com/articles/Quantum.html. This section is the attack-grade recap; the manuscript controls the common material. (Source: 01_DOSSIER.md §0.2.)

3.1 The descent chain (how the geometry supplies the inputs)

The 13D action is local in 13D — a property of the frozen construction, not a derived result. At each four-dimensional spacetime point, integrating out the compact factors K₆ × S² × S¹_Y/ℤ₂ produces a four-dimensional effective action. The chain, named so a reader can attack each link (Source: 01_DOSSIER.md §1.1; GATE_BRIEF_UQF14.md):

 13D local action
    --integrate out K₆ × S² × S¹_Y/ℤ₂ per spacetime point-->
 4D effective action
    (local at any FINITE KK truncation; analytic O(E²/M_KK²) corrections only)
    --descend each carrier-->
 graviton zero-mode, gluons, W/Z, photon, Higgs, chiral fermions
    --inherit per carrier-->
 commutator structure (microcausality) + BRST/positivity bookkeeping (unitarity)
    --apply standard QFT theorems IN SCOPE-->
 banked perturbative / truncated-KK certificate
    --above the cutoff / nonperturbatively-->
 INHERITS upstream blockers (UQF-9 / UQF-10 / UQF-11 / BG-10)

The four load-bearing facts, with their built-in attack handles:

1. 13D locality is given. The action is local in 13D. Attack handle: locality in 13D plus finite-truncation locality in 4D does not establish locality with the full infinite KK tower — that is residual R3, owned by UQF-10.
2. Finite-truncation 4D locality. At any finite KK cutoff the 4D EFT is local with only analytic, EFT-suppressed O(E²/M_KK²) corrections. Attack handle: this is an EFT statement valid at E ≪ M_KK; it says nothing about E ≳ M_KK where the tower is strongly coupled.
3. Per-carrier positivity & microcausality bookkeeping. Each descended field inherits a standard ghost/positivity structure and a standard (anti)commutator. Attack handle: the bookkeeping is perturbative; the nonperturbative physical Hilbert space (confinement) is residual R2, owned by UQF-11.
4. The graviton zero-mode propagation speed equals c to the measured precision. Attack handle: this is the zero-mode, low-energy; above-cutoff graviton unitarity is residual R1, owned by UQF-9.

A one-line discipline statement governs the whole section: “The geometry is given” is NOT “the high-energy theory is proven consistent.” Supplying a consistent set of fields below the cutoff does not derive the theory’s behavior above it.

3.2 The decisive reduction: three pillars collapse to one class-membership statement

The first and most economical move in the campaign is a collapse, not a stretch (Source: UQF14_COMPLETION_RESULT.md §2; specialist_reply.txt §1). Unitarity, causality, and locality are not three independent properties each needing its own separate proof. They are three faces of one statement: that the descended theory is a member of the standard axiomatic-QFT class — a Poincaré-covariant net of observable algebras with

• a positive-definite physical state space (⇒ unitarity, no negative-norm ghosts),
• spacelike-commuting observables (⇒ microcausality), and
• an energy-positivity / spectrum condition (⇒ stability and, with Lorentz covariance and a local action, cluster decomposition).

Dropping the hidden “three independent guarantees” premise dissolves the apparent three-pillar problem into a single causal-net class-membership statement. That class-membership statement is then itself derived, given the observed Standard-Model content E, by finite-KK truncation below the cutoff from 13D locality. The net reduction is:

3 properties + 5 banked cancellation theorems → 1 class-membership statement, derived (given the observed Standard-Model content) by finite Kaluza–Klein truncation below the cutoff from 13-dimensional locality, resting on 2 named structural posits + 1 substrate posit + 1 measured anchor.

(Source: UQF14_COMPLETION_RESULT.md §2, verbatim reduction line.)

A discipline note carried from the cert reconciliation: this collapse is recorded publicly as a DERIVED-GIVEN-E conjecture / theorem-debt, not as a proven theorem, precisely so it is not over-read. (Source: 08_CERT_RECONCILIATION.md, deploy-prep TODO #1.)

3.3 The banked legs — rigorous content

The residual register splits the gate into seven named residuals R1–R7. Three of them — R5, R6, R7 — are banked as VERIFIED, each a standard QFT result applied strictly inside its accepted scope (perturbative / finite-tower / below-cutoff), given E. (Source: 01_DOSSIER.md §2; 01_BANKED_R5_R6_R7.md.)

R5 — Finite-truncation 4D EFT locality / cluster decomposition

• Object. The descended 4D effective field theory of the finite KK truncation of the frozen-branch 13D-local action.
• Scope. E ≪ M_KK (below cutoff, finite tower).
• Theorem. A 13D-local action, with the compact factors integrated out at finite KK truncation, yields a 4D-local effective action obeying cluster decomposition, with only analytic O(E²/M_KK²) corrections. The non-localities induced by integrating out massive KK modes are analytic in external momenta and power-suppressed by the heavy scale — i.e. they are ordinary local higher-dimension operators in the EFT expansion, not genuine non-localities.
• Why cluster decomposition follows. With a local action, Lorentz covariance, and the spectral condition (energy positivity), cluster decomposition is the standard consequence: correlation functions factorize at large spacelike separation. (Source: 01_DOSSIER.md §0.1, R5; 02_CURRENT_STATE.md (iii).)
• Modal status. DERIVED-GIVEN-E (perturbative / EFT).
• What it does NOT prove. It does not close R3. Finite truncation ≠ full infinite tower; strong coupling / convergence of the tower at E ≳ M_KK is exported to UQF-10. The falsification test “R5 ⇒ close R3” is a RELABEL_FAIL. (Source: 01_BANKED_R5_R6_R7.md, R5 falsification test.)

R6 — Per-sector perturbative ghost / unphysical-polarization decoupling

• Object. Per-sector decoupling of unphysical states from the physical subspace, on the physical Hilbert space H_phys = ker Q_BRST / im Q_BRST.
• Scope. Perturbative, per-sector.
• Theorems (the five banked cancellation results).
• Gluons: Kugo–Ojima quartet mechanism — the BRST quartet (the two unphysical gluon polarizations plus the ghost/antighost) cancels in pairs on H_phys.
• W/Z (weak bosons): the Higgs mechanism plus the Equivalence Theorem — the longitudinal polarizations of the massive vectors are controlled by the would-be Goldstone bosons, and unphysical modes decouple from physical amplitudes at high energy.
• Photon: Gupta–Bleuler / BRST — the timelike and longitudinal photon polarizations cancel on the physical subspace.
• Fermions: spin-statistics with no surviving mirror fermions — the chirality projection of the frozen geometry leaves no light mirror partners, consistent with the LEP neutrino count and the no-light-mirror structure banked upstream at UQF-7. (Source: 01_DOSSIER.md §0.1, R6; GATE_BRIEF_UQF14.md.)
• Modal status. DERIVED-GIVEN-E and GIVEN AXIOM-PHYSICAL-POSITIVITY (perturbative kinematic positivity).
• The load-bearing caveat (do not lose it). These theorems decouple the unphysical sector GIVEN a positive physical subspace; they do not by themselves establish that positivity. Nonperturbative 4D Yang–Mills positivity sits upstream of all five theorems and is exported to R2 → UQF-11/Gap-02. The dossier prose “cancellation on the physical Hilbert space” must inherit this caveat. (Source: 01_BANKED_R5_R6_R7.md, R6; specialist_reply.txt R6 note; 08_CERT_RECONCILIATION.md deploy-prep TODO #2: scope AXIOM-PHYSICAL-POSITIVITY to perturbative kinematics only.)
• What it does NOT prove. It does not close R2. Perturbative ghost-decoupling ≠ nonperturbative confinement / physical-Hilbert-space unitarity. Falsification test “R6 ⇒ close R2” is a RELABEL_FAIL; so is relabeling nonperturbative-YM-positivity as a terminal corpus axiom.

R7 — Perturbative all-orders microcausality + the measured graviton-speed anchor

• Object. (a) all-orders perturbative microcausality; (b) measured zero-mode graviton luminality.
• Scope. (a) perturbative, all orders; (b) graviton zero-mode / low-energy / single astrophysical baseline only.
• Theorem / anchor.
• \(a\) Each free-field carrier (graviton zero-mode and each KK mode, gluon, W, Z, photon, Higgs, each fermion and KK partner) has a Pauli–Jordan / Proca / Klein–Gordon / Dirac commutator vanishing outside the lightcone. This is extended to all loop orders by Epstein–Glaser causal perturbation theory — the rigorous, regulator-free inductive construction of the time-ordered products that builds microcausality in order by order. Kept perturbative (silent nonperturbatively / above-cutoff). (Source: 01_DOSSIER.md §0.1, R7; handoff “Banked” item 2.)
• \(b\) The measured graviton-speed anchor from GW170817 + GRB170817A — see §3.4 for the number hygiene that must travel with it.
• Modal status. (a) DERIVED-GIVEN-E (perturbative); (b) measured-but-irreducible zero-mode anchor.
• What it does NOT prove. It does not close R1. Below-cutoff ≠ above-cutoff; Epstein–Glaser is kept perturbative; the measured cₘ covers the graviton zero-mode / low-energy / single-baseline only and is not a quantum-gravity causality certificate. Falsification tests “R7 ⇒ close R1” and “measured cₘ ⇒ full quantum-gravity causality” are RELABEL_FAILs. (Source: 01_BANKED_R5_R6_R7.md, R7.)

3.4 Number hygiene: the GW170817 / GRB170817A anchor (the one measured invariant)

The headline graviton-speed bound is

|cₘ − c| / c < 10⁻¹⁵

derived from the near-simultaneous arrival of GW170817 and GRB170817A (two triggers ~1.7 s apart). This bound is not itself the measured atom. The bare measured invariant is narrower: two detector triggers ~1.7 s apart of laboratory clock time at a shared apparatus. Converting that coincidence into a fractional speed bound imports three named, non-irreducible co-premises, which must travel with the number wherever it is shown (Source: UQF14_COMPLETION_RESULT.md §6; specialist_reply.txt R7; 01_BANKED_R5_R6_R7.md R7(b)):

1. a common-emission-time model (that the two messengers left one source with a bounded intrinsic emission-time offset);
2. the ~40 Mpc baseline, itself a distance-ladder / redshift-inferred quantity;
3. shared causal order — needed even to call two propagation rates “equal.”

The bare clock-coincidence is the measured atom; the speed bound is a derived statement given those three co-premises. The bound is a single low-energy, single-baseline datum and cannot be promoted to cover the strongly-coupled, above-cutoff graviton sector — that is R1 / UQF-9.

3.5 The weakest-link split (the one honest table)

The badge is the weakest link. Three banked legs do not lift the gate; four inherited-open legs hold it at AUDIT (OPEN). (Source: 01_DOSSIER.md §2.1; 02_CURRENT_STATE.md (ii).)

4. The insights we used

These are the moves that made the progress believable and reproducible — shareable in full.

4.1 The three-pillars collapse (dissolving a false multiplicity)

The first insight is recognizing that “unitarity AND causality AND locality” is not a conjunction of three independent proof obligations but a single class-membership statement in axiomatic QFT (§3.2). The hidden premise — “three independent guarantees, three separate proofs” — is what makes the problem look three times as hard as it is. Dropping it does not weaken anything; it clarifies that there is one object to establish (membership in the Wightman / Haag–Kastler class) with three observable consequences. This is a genuine economy, and it is recorded honestly as a DERIVED-GIVEN-E conjecture, not as a finished theorem.

4.2 The weakest-link / min-rule discipline

The second insight is the min-rule: an audit gate’s verdict is the least-closed of its residuals. This is what keeps the gate honest. It would be easy — and wrong — to advertise “three legs banked!” as if the gate were two-thirds closed. Under the min-rule, the three banked legs are real but do not move the badge; the four inherited-open legs do. The min-rule is the formal mechanism that prevents grade inflation on a gate that genuinely has strong sub-results. (Source: 06_GATE_GRADING_RUBRIC.md via 02_CURRENT_STATE.md; 01_DOSSIER.md §2.1.)

4.3 Export-as-contract (routing, not burying)

The third insight is to treat each open leg not as a vague prose pointer (“this is hard, see quantum gravity”) but as a typed bridge contract with an explicit shape: {required object · required theorem/certificate · must-NOT-rely-on · falsifiers/rejection-triggers · mount criteria · owner · disposition-until-mounted · gate effect}. An export is then auditable: a reader can check that the right object is being asked of the right owner, and that the open leg is not silently resting on an already-failed upstream. (Source: 07_CONVERGED_CLOSURE_PATH.md; 02_BRIDGE_CONTRACTS_R1_R4.md.)

4.4 The mount test (when an upstream certificate may actually discharge a leg)

The fourth insight is the five-axiom mount test: an upstream certificate may lift one of R1–R4 only if it passes all five of ANCHOR, OBJECT IDENTITY, FAITHFUL BRIDGE, COMPOSITION, MODAL STATUS. This is the guard against a plausible-looking but mis-targeted upstream result being waved through. In particular MODAL STATUS forces the certificate to prove the required modal class (above-cutoff / nonperturbative / full-tower), not mere admissibility; and COMPOSITION forces “sector → whole” to be proven, not assumed (e.g. finite-truncation locality must be shown to compose to the full tower). (Source: 04_MOUNT_TEST.md.)

4.5 The meta-guardrail: convergence ≠ proof

The fifth and subtlest insight is recorded as a META-GUARDRAIL: convergence of our reasoning is not proof of the physics — it is proof the bookkeeping has stabilized. The campaign reached a fixed point (no sixth axiom needed; the reduction bottoms out cleanly). That fixed point is bookkeeping stability, not closure. The gate stays AUDIT (OPEN) until the upstream bridge objects are actually built and mounted. This is the guard against the deepest trap — mistaking a stable, self-consistent internal account for an external physical result. (Source: 07_CONVERGED_CLOSURE_PATH.md; 04_MOUNT_TEST.md META-GUARDRAIL.)

4.6 Why the dominant blind spot is “anchoring on the representation”

The recurring near-miss the discipline is built to catch is anchoring on the representation: treating the finite / perturbative / below-cutoff / descended-4D-EFT certificate as if it were the full / nonperturbative / above-cutoff / infinite-KK physical theory. Every banked leg has a precisely-stated scope, and every falsification test is a specific instance of refusing this slide (R5↛R3, R6↛R2, R7↛R1, cₘ↛full-QG-causality). Naming the blind spot explicitly is what lets a reader audit that no leg was banked too broadly. (Source: 07_CONVERGED_CLOSURE_PATH.md “Dominant blind spot”; 01_BANKED_R5_R6_R7.md banking-scope summary.)

5. Evidence & reproducibility

5.1 The certificate of record

The campaign produced a certificate packet returning verdict AUDIT_COMPLETE_OPEN (cert id wm1z47vzm, 2026-06-26), a clean confirmation of the converged endpoint with no over- or under-claim. (Source: 08_CERT_RECONCILIATION.md.) The packet lives at …/UQF14_COMPLETION_HANDOFF/certificates/UQF14_AUDIT_COMPLETE/ and contains:

• 00_README_VERDICT.md — the verdict header.
• 01_BANKED_R5_R6_R7.md — the three banked legs, each with explicit scope and falsification test.
• 02_BRIDGE_CONTRACTS_R1_R4.md — the four typed export contracts.
• 03_WALL_RECORDS.md — the four wall records (one per open residual).
• 04_MOUNT_TEST.md — the five-axiom mount test and current certificate states.
• 05_FINAL_ROLLUP.md — the gate roll-up.

The verdict’s meaning is precise and is reproduced verbatim so it is not mis-read: “audit-complete but physically OPEN.” “Completion” means the reduction reached its honest fixed point — no further-reducible step remains inside the gate’s scope. It does not mean the gate is solved or passes. (Source: UQF14_COMPLETION_RESULT.md status note; 07_CONVERGED_CLOSURE_PATH.md.)

5.2 Frozen geometry hashes

All work rides the frozen branch dcc66f1b2685 (branch) / a5b1e6f9d951 (manifest meta), READ-ONLY and unmodified. A silent frozen-branch change is itself a falsification test trigger (→ REFUTED / restart). (Source: 01_DOSSIER.md §0; 04_MOUNT_TEST.md falsification tests.)

5.3 Current upstream-gate standings (must travel with every export)

So the gate cannot be read as resting on already-closed upstreams, the four exported legs carry their current upstream standing verbatim (Source: 02_BRIDGE_CONTRACTS_R1_R4.md; specialist_reply.txt §5 fix #4; 08_CERT_RECONCILIATION.md TODO #3):

All four upstream certificates are currently MISSING; the gate does not rise.

5.4 The named-posit floor (the irreducibility judgment)

The reduction bottoms out on the following set. A posit is irreducible only if it is a directly measured invariant or is proven irreducible; plausibly-deep is not sufficient, and the default under uncertainty is not irreducible. (Source: UQF14_COMPLETION_RESULT.md §5; specialist_reply.txt §2A.)

Fixed-point judgment: exactly one posit is claimed irreducible (the bare coincidence anchor), satisfying the floor of at least one measured invariant; every other member is in a terminal disposition. Continuing to “reduce” further would mean either reducing the corpus GRANULARITY/SCALE/SHAPE posits inside this gate (out of scope) or deriving a measurement from structure (circular). The reduction has reached its honest fixed point.

5.5 How a reader re-runs / re-derives

There is no numerical simulation to re-run for this gate — it is an audit gate, and its evidence is a chain of in-scope theorem applications plus one measured anchor. A reader checks it by:

1. Re-deriving each banked leg in scope. Confirm R5 (finite-truncation EFT locality, analytic O(E²/M_KK²) corrections), R6 (the five cancellation theorems on H_phys, given physical positivity), and R7 (Pauli–Jordan/Dirac commutator support + Epstein–Glaser all-orders) against any standard QFT reference, and confirm each is applied inside its stated scope.
2. Re-running the four falsification tests. For each of R5→R3, R6→R2, R7→R1, and cₘ→full-QG-causality, confirm the cross is forbidden — i.e. that no banked leg is used to close an open leg.
3. Checking the number hygiene. Confirm |cₘ − c|/c < 10⁻¹⁵ is never shown without its three co-premises, and that the bare measured atom is the ~1.7 s two-trigger coincidence.
4. Echoing the upstream standings. Confirm the four exports name UQF-9 (AUDIT), UQF-11/Gap-02 (Precisely-OPEN), UQF-10 (AUDIT), BG-10 (0-of-4).
5. The independent open task — re-deriving the U-AUDIT-B inventory tables line-by-line — is itself one of the open holes (see §6, Hole H5); it is not yet done, and is marked OPEN.

6. Open gaps + closure path (the specialist work plan)

The most load-bearing section. Each open hole is a work-package a specialist can pick up and start. STATUS-UPGRADES:0 — a refuting result is a valid close. The decisive context-setting fact: there is no self-contained closure for UQF-14. The gate rises only as its upstream dependencies rise; the correct next move is the upstream bridge program, not re-attacking UQF-14. (Source: 07_CONVERGED_CLOSURE_PATH.md strongest endpoint.) The holes below are stated in the order of the specialist hole-queue. (Source: SPECIALIST_HOLE_QUEUE_2026-06-29.md lines 206–217; GATE_REGRADE_BESTCASE_AND_HOLES_2026-06-29.md lines 789–828.)

Hole H1 (R1) — Above-cutoff graviton-sector unitarity → UQF-9

(a) Precise statement. Establish that the graviton sector remains unitary — no negative-norm physical states; high-energy amplitude growth respecting unitarity / Froissart bounds — above the perturbative cutoff, where the graviton and the full KK tower are strongly coupled, for the frozen branch.

(b) Why it’s hard / prior-attempt lessons / traps. This is the badge-setting residual and it is a global open problem of quantum gravity, not a defect of this program. No in-framework theorem reaches above the cutoff. The required object — a nonperturbative / UV-complete above-cutoff graviton-unitarity certificate — does not exist on the frozen branch. Traps the verifier flagged, do not repeat: - The cardinal-sin move “substrate granularity UV-regulates the above-cutoff graviton sector, therefore the badge lifts” assumes the answer to UQF-9 and is a RELABEL_FAIL. Those UV consequences are owned by UQF-9. (Source: specialist_reply.txt §3 cardinal-sin guard; 04_MOUNT_TEST.md.) - Do not use Epstein–Glaser, finite-EFT locality, or the measured cₘ as an above-cutoff proof. (Source: 02_BRIDGE_CONTRACTS_R1_R4.md R1 must-NOT-rely-on.) - This hole is correctly flagged DO-NOT-CHASE: unicorn in the cross-gate UQF-5C queue — above-cutoff unitarity cannot be settled until the UV completion is, and it cannot be reframed as malformed without first proving the relevant prerequisite. (Source: SPECIALIST_HOLE_QUEUE_2026-06-29.md line 295.)

(c) Exactly what closes it. A UV-completion certificate for UQF-9 / B3 — e.g. a truncation-independent non-Gaussian fixed point, or a defensible non-continuum replacement — that validates above-cutoff graviton unitarity / Froissart-bound / Landau-pole control, and that passes the five-axiom mount test (in particular MODAL STATUS: it must prove the above-cutoff status, not mere admissibility). Success: mounted, R1 (and the badge) lift. A refuting result that closes it negatively: uncontrolled high-E amplitudes, a negative-norm physical state above the cutoff, or a regulator-dependent conclusion — any of these falsifies the above-cutoff unitarity claim and is a valid (negative) close. (Source: 02_BRIDGE_CONTRACTS_R1_R4.md R1; SPECIALIST_HOLE_QUEUE_2026-06-29.md line 209.)

(d) Machinery & inputs. Start from the UQF-9 dossier and the shared-blocker B3 ledger. The concretely-buildable sub-object on the frozen branch is the 6th heat-kernel coefficient a₆ (a necessary-not-sufficient input to the fixed-point question); its current state is AUDIT_UNVERIFIED — two computation routes disagreed, and only the dimensionless 124/315 ratio is clean (DERIVED-PENDING-INDEPENDENT-TARGET-BLIND-REPRODUCTION); the dimensionful magnitude is scheme/scale-anchored, and an order-6 mixed Neumann+Dirichlet boundary coefficient for S¹_Y/ℤ₂ does not exist in the literature (the boundary tower stops at a₅). (Source: MASTER_AXIOM_FLOOR_AND_WALL_LEDGER.md B1/B3 rows; GATE_REGRADE… lines 1073–1077.) Do not fabricate a₆ magnitudes, a positivity functional pass, or the orbifold-defect total.

(e) Leverage. R1 and R3 both bottom on B3 (the universal UV wall). Closing UQF-9 lifts R1 (the badge) here, lifts the downstream UQF-10 leg (R3), and feeds Gap-01, Gap-13, the graviton certificate (UQF-5A/5B), and the UV-divergence-class dissolution recorded at UQF-9. This is the single highest-leverage node for the whole quantum-gravity column. (Source: MASTER_AXIOM_FLOOR_AND_WALL_LEDGER.md B3 row; 02_BRIDGE_CONTRACTS_R1_R4.md dependency graph.)

Hole H2 (R2) — Nonperturbative strong-sector unitarity → UQF-11 / Gap-02

(a) Precise statement. Establish that the strong sector’s physical Hilbert space is positive-norm nonperturbatively — i.e. color confinement and a mass gap fixing what the asymptotic states actually are.

(b) Why it’s hard / prior-attempt lessons / traps. The nonperturbative certificate is the Yang–Mills mass gap — a Clay Millennium problem. The UV framework supplies UV data only and structurally cannot supply the IR certificate. Two traps the verifier explicitly flagged: - Do not rely on perturbative BRST / Kugo–Ojima alone: those presuppose positivity, they do not establish it. (Source: 02_BRIDGE_CONTRACTS_R1_R4.md R2 must-NOT-rely-on; 01_BANKED_R5_R6_R7.md R6.) - The Clay-level framing is itself a retired continuum unicorn — do not chase it as stated. The verifier note (Source: GATE_REGRADE… line 828(c)) records that pinning the close-condition to “the level the Clay Millennium problem requires” = the continuum a→0 uniform inequality, which the corpus’s own COMMON_BLINDSPOTS.md (BS-1, lines 21–24) and MASTER_AXIOM_FLOOR_AND_WALL_LEDGER.md (W-MG row) name as a retired continuum unicorn: REDUCED-TO-AXIOM (granularity), axiom-conditional. The honest bounded target is the finite-resolution z* < 1 face of Marginal-KP — the finite IR gap that survives granularity as computation-debt (≈ the measured glueball mass). The dissolution is axiom-conditional on the granularity posit P1, never “solved,” and you must do the finite computation anyway with a negative control (the κ³/π lesson — do not let granularity make the result true-by-construction). (Source: COMMON_BLINDSPOTS.md BS-1 honesty guards; MASTER_AXIOM_FLOOR_AND_WALL_LEDGER.md W-MG.) - Hidden co-gate: the verifier flags that 4D SU(3) continuum-measure existence is a separate, independent OPEN co-gate (Gap-02 R3) that the loosely-worded “spectral-gap lower bound” can absorb and hide. Treat measure-existence as its own object, not folded into the gap inequality. (Source: GATE_REGRADE… line 828(b).)

(c) Exactly what closes it. Bounded form (correct): a finite-resolution (z* < 1) spectral-gap result for the relevant SU(3) gauge theory — the finite IR gap, axiom-conditional on granularity, with a negative control — plus an explicit disposition of 4D SU(3) measure-existence. Frontier form (the unicorn, do not target): the continuum a→0 uniform inequality. Success: a finite gap certificate that passes the mount test discharges R2. Refuting close: nonperturbative positivity fails / a negative-norm physical state survives confinement → valid negative close. (Source: SPECIALIST_HOLE_QUEUE_2026-06-29.md line 211; 02_BRIDGE_CONTRACTS_R1_R4.md R2; COMMON_BLINDSPOTS.md BS-1.)

(d) Machinery & inputs. Gap-02 dossier; the “uniform clustering ⇒ gap” implication is already rigorous — the open object is one marginal-coercivity inequality in the d=4 band (Source: 01_DOSSIER.md R2). The W-MG ledger row gives the exact split: continuum half retired by P1; finite IR gap survives as computation-debt. Lattice-style finite computation is the natural tool; do not fabricate a gap and do not axiomatize a pass.

(e) Leverage. Closing the strong-sector IR certificate lifts R2 here and discharges the UQF-11/Gap-02 dependence shared with Born, Λ, and (conditionally) UQF-14. (Source: MASTER_AXIOM_FLOOR_AND_WALL_LEDGER.md W-MG affected-row list.)

Hole H3 (R3) — Full infinite-KK-tower cluster decomposition → UQF-10

(a) Precise statement. Establish that locality / cluster decomposition survives when the full infinite KK tower is kept (not just a finite truncation) — i.e. that finite-truncation locality composes to the full tower with no tachyonic/runaway mode.

(b) Why it’s hard / prior-attempt lessons / traps. The full-tower statement requires the compact geometry to be stable as a quantum theory — UQF-10, itself downstream of UQF-9 (it needs loop control of the moduli potential, which needs the UV completion). Traps: - Do not rely on finite KK truncation alone (falsification test R5 → R3 is RELABEL_FAIL). The COMPOSITION axiom of the mount test must be proven, not assumed. (Source: 02_BRIDGE_CONTRACTS_R1_R4.md R3; 04_MOUNT_TEST.md.) - This hole is additionally gated by the F1 shape-doublet saddle, currently LEANING REFUTED — a live negative signal a specialist must reconcile, not ignore. (Source: 03_WALL_RECORDS.md WALL R3.)

(c) Exactly what closes it. A compactification-stability certificate for the full KK tower — moduli mass-matrix positivity, no tachyonic/runaway modes, from a performed RG calculation — establishing the spectral condition holds with the infinite tower, plus a proof that finite-truncation locality composes to the full-tower cluster decomposition. Success: mounted, R3 lifts. Refuting close: a tachyonic KK mode (the named falsifier) → valid negative close. (Source: SPECIALIST_HOLE_QUEUE_2026-06-29.md line 213; 02_BRIDGE_CONTRACTS_R1_R4.md R3.)

(d) Machinery & inputs. UQF-10 dossier; the moduli potential and its loop control; the F1 shape-doublet-saddle finding (LEANING REFUTED) must be confronted first. The B3/UQF-9 UV completion sits below it.

(e) Leverage. R3 bottoms on B3 (shared with R1), so the UQF-9 program partially unblocks it; conversely a clean UQF-10 stability result closes the full-tower cluster-decomposition claim here.

Hole H4 (R4) — Nonperturbative electroweak (instantons / sphalerons) → BG-10

(a) Precise statement. Provide the unitarity/causality bookkeeping for nonperturbative EW processes — instantons, sphalerons, high-T baryon-number violation — currently excluded by the claim boundary.

(b) Why it’s hard / prior-attempt lessons / traps. These effects sit outside this paper’s scope and inherit BG-10, which is 0-of-4 at certificate grade. Trap: do not rely on perturbative EW alone, and do not let an anomaly-blind class-membership claim stand in for an anomaly-aware nonperturbative treatment. (Source: 02_BRIDGE_CONTRACTS_R1_R4.md R4 falsifiers.)

(c) Exactly what closes it. Bring the nonperturbative EW sector in-scope and discharge the BG-10 sphaleron/instanton ledger at certificate grade — a finished, basis-invariant treatment (anomaly-aware SMG-class certificate) — so the excluded sub-claim becomes a banked one. Refuting close: a genuine nonperturbative-EW unitarity/causality violation → valid negative close, propagated to BG-10. (Source: SPECIALIST_HOLE_QUEUE_2026-06-29.md line 215; 02_BRIDGE_CONTRACTS_R1_R4.md R4.)

(d) Machinery & inputs. BG-10 dossier (baryogenesis / nonperturbative EW); the SMG-class anomaly machinery (cf. the B2 Dai–Freed boundary-anomaly construction and the shared-blocker B4 dynamical-mirror-decoupling work). (Source: MASTER_AXIOM_FLOOR_AND_WALL_LEDGER.md B2/B4 rows.)

(e) Leverage. Routed, not closed; closing BG-10 discharges this scoped-out export and feeds the broader baryogenesis column.

Hole H5 — Independent re-derivation of the U-AUDIT-B per-sector inventory

(a) Precise statement. The banked perturbative claim rests on per-sector inventory tables — the negative-norm/ghost ledger, the commutator-vanishing ledger, and the per-sector audit-status matrix — cited to Q3 Appendix U, module U-AUDIT-B. A specialist has not independently re-derived these for completeness and coverage of every carrier, including all KK partners. (Source: SPECIALIST_HOLE_QUEUE_2026-06-29.md line 216; handoff open-hole #5.)

(b) Why it’s hard / prior-attempt lessons / traps. It is laborious rather than deep: the risk is an asserted entry — a carrier listed as “commutator vanishes” or “ghost cancels” without the explicit calculation behind it — or a missing carrier (especially a KK partner) not in the table at all. The trap is signing off the table because it “looks complete.” This is the one open hole that is genuinely internal to UQF-14 and closeable without an upstream wall falling.

(c) Exactly what closes it. An independent line-by-line audit reproducing the U-AUDIT-B tables: verify every carrier’s commutator-vanishing entry and ghost-cancellation entry, confirm no sector is missing and no entry is asserted without the explicit calculation. Success: every entry independently reproduced, coverage confirmed for all carriers including KK partners → R6/R7’s banking is independently corroborated (still in-scope; does not lift the gate). Refuting close: an asserted-but-false entry, or a missing carrier → a real defect in the banked legs, fed back into R6/R7. (Source: SPECIALIST_HOLE_QUEUE_2026-06-29.md line 217.)

(d) Machinery & inputs. Quantum Paper III §17 + Q3 Appendix U module U-AUDIT-B; standard QFT references for each cancellation theorem (Kugo–Ojima, Higgs+ET, Gupta–Bleuler/BRST, spin-statistics) and each commutator (Pauli–Jordan/Proca/Klein–Gordon/Dirac). The frozen-branch carrier list (graviton zero-mode + each KK mode, gluon, W, Z, photon, Higgs, each fermion + KK partner) is the coverage checklist. (Source: 01_DOSSIER.md §0.2, R7 carrier list.)

(e) Leverage. Self-contained: it corroborates (or refutes) R6/R7 directly. It is the cleanest near-term specialist win on this gate because it requires no upstream certificate.

Hole H6 — Order-by-order BRST nilpotency of the descended 4D theory → UQF-4

(a) Precise statement. BRST nilpotency for the descended theory (Q²_BRST = 0 at all loop orders) is asserted as a falsifier condition, but order-by-order nilpotency of the descended/compactified BRST is verified upstream only partially — it couples to UQF-4, which is OPEN. A nilpotency failure at some loop order would falsify the unitarity sub-claim here. (Source: handoff open-hole #6; SPECIALIST_HOLE_QUEUE_2026-06-29.md UQF-14 list.)

(b) Why it’s hard / prior-attempt lessons / traps. Trap the verifier flagged: this hole is routed to UQF-4 with an optimistic bounded:true, but the open-walls record marks the UQF-4 coset-twist / descended-anomaly construction as NO-KNOWN-ROUTE (“no short path exists today”). It is still genuinely bounded / specialist-closeable, but bounded:true should carry the no-known-route caveat — do not present it as a near-term certainty. (Source: GATE_REGRADE… line 828, minor note.)

(c) Exactly what closes it. An order-by-order BRST-nilpotency verification for the descended 4D theory (closing the relevant UQF-4 leg). Success: Q²_BRST = 0 confirmed to the required loop order → the falsifier condition is discharged. Refuting close: a nilpotency failure at some loop order → falsifies the unitarity sub-claim and propagates to UQF-4. (Source: handoff open-hole #6.)

(d) Machinery & inputs. UQF-4 dossier; the descended/compactified BRST construction; the coset-twist / descended-anomaly machinery (currently NO-KNOWN-ROUTE). Connect to the B2 Dai–Freed boundary-anomaly invariant where the descended anomaly bookkeeping lives. (Source: MASTER_AXIOM_FLOOR_AND_WALL_LEDGER.md B2 row.)

(e) Leverage. Couples UQF-14’s unitarity bookkeeping to UQF-4; closing the UQF-4 leg discharges this falsifier and strengthens R6’s foundation.

6.7 Closure-path summary (dependency graph)

• R1 + R3 both bottom on B3 / UQF-9 (the universal UV wall) — the single highest-leverage node.
• R2 → UQF-11 / Gap-02 — bounded target = finite-resolution gap (axiom-conditional on granularity); the continuum a→0 inequality is a retired unicorn; watch the hidden measure-existence co-gate.
• R3 → UQF-10 (downstream of UQF-9); falsifier = tachyonic KK mode; F1 shape-doublet saddle LEANING REFUTED must be reconciled.
• R4 → BG-10 (scoped-out, 0-of-4).
• H5 (U-AUDIT-B re-derivation) and H6 (BRST nilpotency, → UQF-4 with no-known-route caveat) are the two holes most internal to / closest to UQF-14.

The correct next move is the upstream bridge program (B3 / Gap-02 / UQF-10 / BG-10 / UQF-4), not re-attacking UQF-14. (Source: 07_CONVERGED_CLOSURE_PATH.md strongest endpoint.)

7. Honest ceiling & scope

The honest ceiling. UQF-14 is a serious candidate — NOT validated. Unitarity, causality, and locality are demonstrated where standard QFT supports them (perturbative ghost cancellation, all-orders microcausality, finite-truncation EFT locality, the measured graviton-speed zero-mode bound — all given the observed SM content, all in-scope) and named-and-routed where nonperturbative subtlety enters. This is structural discipline, not a deep nonperturbative proof. (Source: 02_CURRENT_STATE.md (iv); 01_DOSSIER.md §5.)

What is explicitly NOT claimed. - No above-cutoff unitarity proof. The graviton sector’s unitarity above the cutoff is conditional on UQF-9 and is not proven here. This is the one thing that cannot be proven here — and it is unprovable in principle for everyone, because it simply is the open UV-completion-of-gravity problem. The bounded statement — unitary where standard QFT applies, conditional above the cutoff on a UV completion — is the ceiling, not a hedge. - No nonperturbative strong-sector unitarity. Banked Kugo–Ojima is perturbative and presupposes physical positivity; it does not establish it. The nonperturbative certificate is the Clay mass gap. - No full-tower cluster decomposition. Banked locality is the finite-truncation EFT statement; the infinite-tower claim is exported to UQF-10. - No nonperturbative-EW disposition. Instantons/sphalerons are scoped out and exported to BG-10. - No proof that no future nonperturbative effect could violate microcausality or cluster decomposition. That is an open-ended universal negative; the correct bounded form is the pre-registered falsifier set (negative-norm state, spacelike-commutator non-vanishing, tachyonic KK mode, Froissart violation, cₘ ≠ c). - No Λ / vacuum-energy / cosmological-constant cancellation. Explicitly disclaimed; Λ is recorded separately and remains open (Weinberg-open). (Source: 01_DOSSIER.md §5 binding rules; UQF14_COMPLETION_RESULT.md §7.)

The distinctions that keep this honest. Dissolved ≠ solved (the three-pillar collapse is an economy of statement, not a closure of physics). Selection ≠ derivation. Given-E ≠ derivation of E (the carrier content is the observed SM spectrum, not derived here). Anchored ≠ derived. A banked perturbative leg ≠ a nonperturbative proof. Audit-complete ≠ physically closed (the reduction reaching a fixed point is bookkeeping stability, not proof). Convergence ≠ proof. (Source: 04_METHOD_AXIOM_CLOSURE_AND_DISCIPLINE.md via 01_DOSSIER.md §5; 07_CONVERGED_CLOSURE_PATH.md meta-guardrail.)

The anchors paid. Exactly one measured invariant carries the irreducible floor: the bare GW170817 + GRB170817A two-trigger ~1.7 s clock-coincidence (the speed bound |cₘ − c|/c < 10⁻¹⁵ is derived from it given three named co-premises — common-emission-time model, ~40 Mpc baseline, shared causal order). Two value-free structural roots (shared causal order; physical positivity) and one substrate posit reduce to named corpus posits at the corpus level, never as in-gate derivations. The class-membership umbrella is DERIVED-GIVEN-E. (Source: UQF14_COMPLETION_RESULT.md §5, §8; specialist_reply.txt §2A.)

The final roll-up. The gate stays AUDIT (OPEN); the badge is permanently EXPORTED to UQF-9 (R1 carries it). Three banked legs are genuine in-scope verifications; four inherited-open legs set the badge. The only path to a higher grade is the upstream walls rising — no move inside UQF-14 lifts it. STATUS-UPGRADES:0. Frozen branch dcc66f1b2685 / a5b1e6f9d951 READ-ONLY and unmodified.

Sources synthesized (all read in full): …/UQF14_COMPLETION_HANDOFF/01_DOSSIER.md, 02_CURRENT_STATE.md, 07_CONVERGED_CLOSURE_PATH.md, 08_CERT_RECONCILIATION.md, specialist_reply.txt, COMMON_BLINDSPOTS.md (BS-1), MASTER_AXIOM_FLOOR_AND_WALL_LEDGER.md (B1/B2/B3/B4/W-MG rows), and certificates/UQF14_AUDIT_COMPLETE/{01_BANKED_R5_R6_R7, 02_BRIDGE_CONTRACTS_R1_R4, 03_WALL_RECORDS, 04_MOUNT_TEST}.md; plus …/rendered/TOE/GATE_REGRADE_BESTCASE_AND_HOLES_2026-06-29.md (UQF-14 row + verifier note, lines 40, 789–828), SPECIALIST_HOLE_QUEUE_2026-06-29.md (lines 206–217, 295), and the published artifacts articles/GATE_BRIEF_UQF14.md + articles/UQF14_COMPLETION_RESULT.md. Live manuscript of record: Quantum Paper III §17 + Q3 Appendix U (U-AUDIT-B), https://physics.magflowmeters.com/articles/Quantum.html. No number in this dossier is fabricated; each traces to one of these files.

Archived anchor ledger

Second archive firewall — canonical GATES pre-v15 source excerpt

The following source-of-truth excerpt governed the branch before the Relational Local-Unitary causal closure was propagated into UQF-14. It is preserved because it contains the complete below-cutoff derivation, dependency routing, and negative controls. Its terminal label and statements that the microscopic branch is absent are superseded by Parts I–XI above. Its perturbative and observational calculations remain retained.

=== GATE: UQF-14 (above-cutoff causality) ===

Gate dossier — UQF-14 — above-cutoff causality

Question: Is the descended theory a proper, cause-respecting quantum theory?
Status (fixed): CERTIFIED-IRREDUCIBLE · RESOLVED +0
Full 13-dimensional treatment: the frozen arena M4 x K6=SU(3)/T2 x S2 x S1_Y/Z2, all three layers, full precision. Self-contained — every value derived or stated inline.

Executive summary & honest status

Headline. On the frozen thirteen-dimensional arena 𝔅_active = [M₄ × K₆ × S² × S¹_Y/ℤ₂]× ⊕ [F⁺_finite ⊕ C_admiss]⊕ ⊗ [E_matter ⊕ E_gauge ⊕ E_Higgs ⊕ E_proton]_⊗, with K₆ = SU(3)/T² and D = 4 + 6 + 2 + 1 = 13, the 4D theory that descends by finite Kaluza–Klein (KK) truncation is a bona fide member of the axiomatic-QFT class wherever standard quantum field theory is entitled to speak: its physical Hilbert space H_phys = ker Q_BRST / im Q_BRST carries no negative-norm state, every commutator of local observables vanishes at spacelike separation to all perturbative orders, and the effective 4D action is local up to analytic O(E²/M_KK²) corrections at any finite truncation. This is not a new theorem manufactured for the occasion; it is the disciplined, sector-by-sector application of standard, decades-old QFT machinery — Kugo–Ojima BRST quartet cancellation, the Higgs mechanism plus the Equivalence Theorem, Gupta–Bleuler/BRST quotienting for the photon, spin-statistics with no surviving mirror fermions, and Epstein–Glaser causal perturbation theory extending Pauli–Jordan/Dirac microcausality to all loop orders — run forward from the given carrier content E, never backward from a wanted answer. Layered onto this perturbative certificate is exactly one empirical fact that does not derive from any of the geometry: the graviton zero-mode propagates at the speed of light to the precision |c_g − c|/c < 10⁻¹⁵, fixed by the ~1.7 s clock-coincidence between the LIGO/Virgo gravitational-wave trigger GW170817 and the Fermi/INTEGRAL gamma-ray trigger GRB170817A. The three classical sanity pillars of a physical theory — unitarity, causality, locality — are the three defining conditions of membership in the axiomatic-QFT class, and “C-net-membership(E_frozen)” is the name for their conjunction (each leg proven separately and separately scoped), banked below the cutoff at finite KK truncation and resting on a floor of exactly one measured atom. Membership is the conclusion assembled from the three legs, never a premise the legs are read off from — the derivation runs one way only, U∧C∧L ⇒ (the label) C-net-membership, not the converse.

The precise claim. Wherever standard QFT applies, the descended theory is provably unitary, causal, and local, and wherever a property cannot yet be proven, the theory names the exact unsolved problem that owns the gap and routes to it rather than papering over it. Four sub-claims carry this, each pinned to its exact ⊗-Actors object and rulebook:

• Unitarity below cutoff, per sector (DERIVED-GIVEN-E ∧ CONDITIONAL-ON-H6). Ghost and unphysical-polarization cancellation is banked carrier by carrier — with the honest tag that the cohomological definition \(\mathcal H_{\rm phys}=\ker Q_{\rm BRST}/\mathrm{im}\,Q_{\rm BRST}\) presumes order-\(\hbar\) BRST nilpotency \(Q_{\rm BRST}^2=0\) (established at tree level, uncomputed at loop level = the exported residual H6), so this leg is conditional on that internal premise, not unconditionally derived: the Kugo–Ojima BRST quartet mechanism removes the longitudinal/timelike gluon pair channel by channel (the same quartet mechanism, invoked once, also covers the photon’s would-be third polarization — a Nonseparability audit screen that PASSES only because this shared object is counted once, not double-booked as two independent wins); the Higgs mechanism together with the Equivalence Theorem unitarizes the longitudinal W and Z at high energy; the Gupta–Bleuler/BRST indefinite-metric quotient removes the photon’s negative-norm timelike state; and spin-statistics forbids a negative-norm outcome from wrong-statistics assignment, consistent with the corpus’s own no-light-mirror structure — the Atiyah–Singer–Patodi index on the orbifold interval [0, π] gives n_L = +3, n_R = 0, three left-handed families and no surviving mirror zero mode, matching the observed absence of extra light neutrino species at LEP.
• Causality — microcausality to all loop orders (DERIVED-GIVEN-E, perturbative all-orders). Every free-field carrier that descends from the 13D arena — the graviton zero-mode and each KK partner, gluons, W, Z, photon, Higgs, and each chiral fermion with its KK tower — carries a Pauli–Jordan (vector), Proca (massive vector), Klein–Gordon (scalar), or Dirac (fermion) commutator/anticommutator that vanishes identically outside the lightcone at tree level, and Epstein–Glaser causal perturbation theory extends this vanishing to all orders in perturbation theory without introducing any non-causal counterterm.
• Locality / cluster decomposition at finite KK truncation, E ≪ M_KK (DERIVED-GIVEN-E, perturbative/EFT). The 13D action is local by construction; integrating out K₆ × S² × S¹_Y/ℤ₂ pointwise over M₄ produces a 4D effective action that is local at any finite KK truncation, with corrections to locality appearing only as an analytic power series in E²/M_KK² — no branch cuts, no non-analytic long-range tails. Cluster decomposition (factorization of well-separated measurements) then follows from the spectral condition (energy positivity — no tachyon in the retained tower) plus Lorentz covariance plus the local action, exactly as in any standard EFT.
• No superluminal graviton (MEASURED-ANCHOR, irreducible, floor ≥ 1). The graviton zero-mode’s propagation speed matches c to the bound |c_g − c|/c < 10⁻¹⁵ set by the joint GW170817/GRB170817A observation. This single measured number carries three explicitly named, non-irreducible co-premises that travel with it whenever it is quoted: (a) a common-emission-time model for the binary neutron star merger, (b) the ~40 Mpc distance-ladder baseline over which the two signals raced, and (c) the prior assumption of a shared causal order needed even to say that “two propagation speeds” can be compared. None of these three is claimed to reduce further inside this gate — the ~1.7 s two-detector coincidence at a shared apparatus is the atomic measured fact, and treating it as anything other than a floor (e.g. trying to “derive” it from geometry) would be the cardinal sin of anchoring on the target.

Each of these four legs is pinned across all three layers of the frozen arena as required: the × Stage identifies which manifold/bundle factor the carrier lives on (M₄ for the flat lightcone structure; K₆, S², S¹_Y/ℤ₂ for the internal wavefunction and KK mass operator); the ⊕ Rulebook fixes the scheme (BRST/Gribov gauge-fixing and cohomology, the orbifold ℤ₂ parity convention, the finite-KK-truncation admissibility rule); and the ⊗ Actors layer supplies the operator content actually doing the work — the graviton transverse-traceless (TT) projector, the Fierz–Pauli ghost-avoidance structure, the KK mass operator built from the volume modulus, the Kugo–Ojima BRST quartet, the Higgs/Goldstone doublet, the Gupta–Bleuler photon indefinite-metric quotient, and the chirality projector P_χ = ½(1 + γ₅Γ₈) on the internal 8-dimensional spinor bundle S(K₆) ⊗ S(S²) ⊗ S(S¹_Y). The KK spectrum that the microcausality and locality legs ride on is itself pinned to full precision: quadratic Casimir C₂(p,q) = (p² + q² + pq + 3p + 3q)/3 and dimension dim(p,q) = (p+1)(q+1)(p+q+2)/2 for SU(3)/T² representations, vector KK masses m²_(p,q),vec = (C₂(p,q) + Δ_vec)/R₆², Dirac KK masses m²_(p,q),Dirac = (C₂(p,q) + ‖ρ‖² + Δ_spin^c)/R₆² with the exact Killing-norm value ‖ρ‖² = 2 (half-sum of positive roots ρ = (1,0,−1) of A₂ = 𝔰𝔲(3)), the lowest nonzero scalar harmonic sitting at the adjoint (1,1) with dimension 8 and C₂ = 3 exactly, and the compactification radius at the chamber center R₆ = R₀ = (2πM_U)⁻¹ = 1.591549430918954 × 10⁻¹⁷ GeV⁻¹ with M_U = 1.0 × 10¹⁶ GeV and M_* = 7.467050992135091 × 10¹⁶ GeV. At finite truncation this tower is manifestly non-tachyonic — the honest open item is only the full-infinite-tower non-tachyon statement, which is a named exported residual (R3 below), not silently assumed.

The explicit non-claims — stated plainly, not buried. This gate does not claim: (1) unitarity, causality, or locality above the cutoff, for any or every possible ultraviolet completion of quantum gravity — that is the open, field-wide UV-completion-of-quantum-gravity problem, addressed below as the gate’s one unicorn; (2) a nonperturbative, positive-norm physical-Hilbert-space certificate for confining QCD — that is precisely the Clay Millennium Yang–Mills mass-gap problem, owned by a different gate entirely; (3) a proof of cluster decomposition for the full infinite KK tower — only the finite-truncation statement is established here; (4) a derivation of the carrier content E itself (the Standard Model spectrum and hypercharges Y(Q_L) = +1/6, Y(u_R) = +2/3, Y(d_R) = −1/3, Y(L_L) = −1/2, Y(e_R) = −1, Y(H) = +1/2) — every leg of this gate is explicitly GIVEN-E, and given-E is not the same claim as derived-E; (5) any vacuum-energy or cosmological-constant cancellation — Λ remains recorded elsewhere as Weinberg-open, and this gate’s ledger explicitly forbids smuggling a Λ-cancellation claim in under cover of a unitarity/causality argument; and (6) that the measured graviton-speed bound extends to the strongly-coupled, above-cutoff graviton sector — it is a single-baseline, zero-mode, low-energy statement only, and it carries the three named co-premises above whenever quoted. A companion specificity diagnostic makes the scoping explicit and falsifiable: four forbidden logical crosses are named and refused — finite-truncation locality does not give full-infinite-tower cluster decomposition; perturbative ghost cancellation does not give nonperturbative QCD confinement/positivity; all-orders-perturbative, below-cutoff microcausality does not give above-cutoff causality; and the single-baseline graviton-speed bound does not cover the strongly-coupled graviton. Asserting any of these four crosses would be a fabrication, and none is asserted anywhere in this gate.

The honest current grade, stated plainly. UQF-14 is graded CERTIFIED-IRREDUCIBLE / RESOLVED +0, read as TERMINAL with residuals shown — a reached endpoint, not a hedge, and not subject to being talked back down to “open” by a reviewer’s framing. This grade is fixed by the workflow and is not revisited, upgraded, or downgraded in this dossier. It is earned, not asserted, because UQF-14 is structurally unlike a compute gate: it has no internal number of its own to predict and no internal axiom of its own left to close. It functions as an audit and routing node — it tests whether the descended 4D theory belongs to the axiomatic-QFT class (a Poincaré-covariant net of observable algebras on a positive-definite physical state space, with spacelike-commuting observables and energy positivity), it books each of the three sanity properties honestly in its exact scope, and it routes every piece that standard QFT genuinely cannot settle to the specific frontier gate that owns that piece. Three legs are proven in-scope and are DERIVED-GIVEN-E (unitarity below cutoff per sector; causality to all perturbative orders; locality/cluster decomposition at finite truncation); one leg is a measured anchor, irreducible by construction, carrying floor ≥ 1 (the graviton-speed bound); and the entire residual openness — above-cutoff graviton/KK-tower unitarity, nonperturbative strong-sector positivity, the full infinite-tower cluster-decomposition statement, nonperturbative electroweak sphaleron/instanton effects, and order-by-order BRST nilpotency of the descended theory — is named, none of it is dissolved prematurely, and each piece is routed to a genuinely tracked owning gate (UQF-9/B3, UQF-11/Gap-02, UQF-10, BG-10, and UQF-4 respectively). The axiom floor underneath all of this converges to a stable fixed point: exactly one measured atom (the ~1.7 s two-trigger coincidence), two logically independent structural posits (a shared causal order; positive-definite physical-state-space kinematics), one substrate/granularity disposition posit, one derivation (DERIVED-GIVEN-E class membership via finite-KK truncation, of which the three banked legs are in-scope instances), and one discipline meta-rule (no cross-cutoff lift; export rather than paper over). No sixth reduction is available without either reducing corpus-level Shape/Scale/Granularity from inside this gate — out of scope here — or deriving a measurement from structure, which is forbidden on principle. Every Layer-2 audit screen passes on the banked legs: gauge/BRST invariance, a genuine finite-observable record interface (S†S = 1 order by order; [O(x), O(y)] = 0 at spacelike separation), forward (not target-loaded) causal order of the reasoning, and correct non-double-counting of the shared BRST/Kugo–Ojima quartet across the gluon and photon sectors.

What this dossier establishes, and what it does not. This dossier establishes that the frozen 13D construction, once its Standard Model carrier content is taken as given, produces a 4D descendant that is a certified member of the axiomatic-QFT class in exactly the domain where axiomatic QFT is a meaningful target — below the cutoff, at finite KK truncation, order by order in perturbation theory — using only textbook, field-tested QFT theorems applied without stretching any one of them past its proper domain, plus one clean, high-significance astrophysical measurement for the graviton sector. It does not establish, and does not claim to establish, that this good behavior survives intact once the theory is pushed to energies at or above the compactification/unification scale where the graviton and the full KK tower become strongly coupled; that question is identical to the unsolved problem of finding any consistent UV completion of quantum gravity, a wall the entire field stands at today, not a defect specific to this program. Nor does it establish nonperturbative confinement-sector positivity (Clay-level), full-infinite-tower behavior, or nonperturbative electroweak dynamics — each of these is a named, live, falsifiable research question pointed at its own gate, carrying a concrete pre-registered falsifier (a negative-norm physical state; a spacelike commutator that fails to vanish; a tachyonic KK mode; a Froissart-bound violation in KK-graviton exchange; or a measured c_g ≠ c), any one of which would immediately propagate back and revoke the corresponding claim.

Single-sentence endpoint preview. Unitarity, causality, and locality are proven as standard, textbook QFT theorems below the cutoff given the observed particle content — at finite KK truncation, order by order in perturbation theory — plus one measured graviton-speed floor from GW170817/GRB170817A, while the fully-quantum-gravitational, nonperturbative, and full-tower pieces are named as honestly open and routed to the specific frontier gates that own them, making this a reached terminal with its residuals shown rather than a gap papered over.

The community gap & state of the art

1. What the wider field is actually asking

Strip away formalism and the question UQF-14 answers is the oldest sanity check a quantum theory has to pass: does it conserve probability, does it refuse to send signals outside the lightcone, and do measurements made far apart fail to influence one another instantaneously? In axiomatic language this is the question of whether a theory belongs to the Wightman class (a Poincaré-covariant field theory on a Hilbert space with positive-definite inner product, a unique vacuum, spectral condition, and microcausal — spacelike-commuting — local fields) or, in the algebraic reformulation, the Haag–Kastler class (a net of local observable algebras assigned to spacetime regions, isotone, obeying spacelike commutativity/locality, covariant under the Poincaré group, and admitting a positive-energy vacuum representation). These are not decorative axioms; they are the minimum a theory must satisfy before it is entitled to be called “a quantum field theory” in the sense the community has used the term since Wightman’s reconstruction theorem and the Haag–Kastler axioms were laid down in the 1950s–60s.

The reason this remains a live, unfinished question — for every candidate theory of quantum gravity, string-derived or otherwise, not just for the 13D construction audited here — is that the three pillars are comparatively easy to secure below some cutoff scale and dramatically hard to secure above it. Below cutoff, one has weakly coupled fields, a controlled loop expansion, and a mature toolkit (canonical quantization with BRST cohomology, causal perturbation theory, dispersion relations) that has been battle-tested since the 1950s. Above cutoff — where the graviton self-coupling, the full Kaluza–Klein (KK) tower, and whatever new degrees of freedom complete the UV behavior of gravity all turn on simultaneously — none of that toolkit is known to apply, and no one has a nonperturbative, background-independent construction of an interacting quantum gravity theory in four or more dimensions to check it against. This is the UV-completion-of-quantum-gravity problem, and it is universally acknowledged (Wilsonian effective field theory practitioners, string theorists, loop quantum gravity practitioners, and asymptotic-safety practitioners alike) to be open. UQF-14’s job is to state, with full precision, exactly how much of the three-pillar package the frozen 13D construction can prove using only mature below-cutoff QFT machinery, and to name — rather than paper over — the exact point at which the argument runs out of runway and hands off to that field-wide open problem.

2. History of the three pillars, one at a time

Unitarity. The requirement that the S-matrix be unitary (S†S = 1, so that transition probabilities sum to one and no negative-norm state propagates as an asymptotic particle) has a well-charted history of near-misses and their repairs. Massive non-Abelian gauge theories without a Higgs mechanism violate perturbative unitarity at high energy in longitudinal vector-boson scattering — the amplitude grows like E² unless a scalar of the right coupling is present to cancel the bad high-energy behavior (Cornwall–Levin–Tiktopoulos and Lee–Quigg–Thacker in the mid-1970s codified this “unitarity bound” argument, which is exactly the argument later used to bound the Higgs mass from above). The Standard Electroweak sector avoids this failure only because the Higgs doublet is present with precisely the coupling dictated by the gauge symmetry; the Equivalence Theorem (Cornwall–Levin–Tiktopoulos; Chanowitz–Gaillard) formalizes the fact that at high energy the longitudinal W/Z amplitude equals the corresponding Goldstone-boson amplitude, which is what makes the cancellation transparent order by order. On the gauge-fixing side, quantizing a non-Abelian gauge theory covariantly introduces unphysical longitudinal and timelike gluon/photon polarizations and, for the photon, an indefinite-metric Hilbert space; Gupta (1950) and Bleuler (1950) solved this for QED by quotienting to a subsidiary-condition subspace, and Kugo and Ojima (1979) generalized the mechanism to non-Abelian gauge theories via the BRST quartet mechanism, showing that BRST-exact quartets (the unphysical gauge and ghost/antighost degrees of freedom) decouple from the physical spectrum order by order in perturbation theory, leaving a positive-definite physical subquotient H_phys = ker Q_BRST / im Q_BRST. This machinery is precisely what the gate leans on for the gluon and photon sectors.

None of these unitarity-restoration mechanisms, however, says anything about what happens when the theory is strongly coupled or when new towers of massive states (a KK tower, string oscillator modes, or whatever completes quantum gravity) turn on near or above a UV cutoff. The textbook worry is graviton-graviton scattering: the tree-level amplitude for longitudinal-graviton (helicity-0, i.e., the “extra” polarization that a massive or KK graviton carries beyond the massless case) scattering grows with energy in a way exactly parallel to the pre-Higgs W_LW_L problem, and whether any UV completion tames that growth without violating the Froissart bound (the model-independent statement, proved by Froissart in 1961 for elastic hadron scattering from unitarity plus analyticity/polynomial-boundedness, that total cross sections cannot grow faster than log²s at fixed sub-asymptotic energy) is precisely the unsolved graviton-unitarization problem. This is a field-wide open question — it is the same obstruction that motivates the entire asymptotic-safety program (Weinberg’s 1979 non-Gaussian UV fixed-point conjecture, actively pursued via functional renormalization group methods since the 1990s–2000s) and the entire string-theoretic UV-completion program (where the Regge tower of massive string states is precisely the mechanism believed to soften high-energy graviton scattering, e.g., in the classic Gross–Mende / Gross–Manes analyses of fixed-angle string amplitudes at high energy). No result in the literature demonstrates a truncation-independent non-Gaussian fixed point for a realistic (matter-coupled, four-generation) theory, nor a background-independent nonperturbative quantization of a Kaluza–Klein graviton tower of the kind the 13D construction produces. That absence is not a defect of this program specifically; it is the state of the entire field.

Causality / microcausality. The demand that field operators at spacelike separation commute (bosons) or anticommute (fermions) — [O(x), O(y)] = 0 for (x−y)² < 0 — is the field-theoretic proxy for “no signal outside the lightcone,” and it was established at tree level for free fields the moment canonical quantization was formulated: the Pauli–Jordan function for a free scalar, the analogous Proca commutator for a massive vector, and the Dirac anticommutator for spin-1/2 fields all vanish identically outside the lightcone by direct construction from the mode expansion, a fact traceable to Jordan and Pauli’s 1928 commutation-function paper. What is much harder, and was not settled until decades later, is showing that this tree-level statement survives to all orders in perturbation theory once interactions and loop corrections are included, where naive time-ordered products can generate apparent acausal pieces that must be shown to cancel. Epstein and Glaser (1973) solved this rigorously with causal perturbation theory: by constructing the perturbative S-matrix order by order as an operator-valued distribution satisfying a causal factorization condition from the start (rather than regularizing a divergent Feynman-diagram sum and hoping causality survives), they proved microcausality holds to all orders for a wide class of theories, with UV divergences handled via a mathematically rigorous causal splitting of distributions rather than momentum-space regularization. This Epstein–Glaser framework is exactly the “all-orders” tool the gate invokes for its microcausality leg, and it is a genuine, complete, non-approximate theorem — within its domain, which is the perturbative expansion of the theory below whatever scale defines convergence of that expansion (in practice, below the cutoff where new strongly-coupled or nonperturbative physics is expected to intervene).

What causal perturbation theory does not do, and was never claimed to do, is establish microcausality nonperturbatively or above the cutoff. Perturbation series in QFT are, generically, asymptotic rather than convergent (a fact traceable to Dyson’s 1952 argument about the analytic structure of QED in the coupling constant), so “all orders” is a statement about the term-by-term structure of a formal series, not a nonperturbative existence-and-causality proof for the resummed or exact theory. This gap — all-orders-perturbative causality versus true nonperturbative causality — is exactly as open for the Standard Model itself as it is for any extension of it; it is bound up with the general unsolved problem of nonperturbative QFT construction (the same problem responsible for the Yang–Mills mass-gap question, discussed below). No result in the literature closes this gap for any interacting four-dimensional gauge theory, let alone for a compactified higher-dimensional gravitational extension of one.

Locality / cluster decomposition. The physical content of locality beyond microcausality is cluster decomposition: correlators of operators localized in mutually far-separated regions factorize, ⟨O_1(x)O_2(y)⟩ → ⟨O_1(x)⟩⟨O_2(y)⟩ as the separation grows, which is what makes physics “extensive” and rules out the kind of long-range acausal correlation that would let a measurement here instantaneously affect statistics there. In an effective field theory obtained by integrating out heavy or compact degrees of freedom (exactly the situation here, integrating out K₆ × S² × S¹_Y/ℤ₂), the standard and well-understood result is that the resulting 4D action is local up to a derivative (equivalently, momentum-power) expansion: heavy-mode exchange produces only analytic, higher-derivative corrections suppressed by powers of E²/M² where M is the heavy/compactification scale, and these are the well-known local EFT operators one always gets from integrating out a mass gap (Wilsonian effective-action reasoning, matched-asymptotic / heat-kernel expansion techniques standard since the 1970s–80s). At any finite order in this expansion (equivalently, at any finite KK truncation) the resulting 4D theory is a perfectly local EFT in the ordinary sense, and cluster decomposition follows from the standard combination of the spectral condition (positivity of energy), Lorentz covariance, and locality of the action (this chain of implication is itself part of the axiomatic-QFT canon, going back to the Wightman reconstruction theorem and its corollaries).

What is emphatically not established by this standard EFT reasoning is what happens when the full infinite KK tower is resummed rather than truncated. Summing an infinite tower of higher-derivative corrections can, in principle, either resolve into a nonlocal but still causal object (as is believed to happen in string theory, where the infinite tower of stringy corrections is widely understood to produce a specific, softly-nonlocal but consistent UV completion) or fail to resum at all in a controlled way. Demonstrating that a specific infinite KK tower resums to a consistent, causal, and local (or controllably nonlocal) theory — as opposed to merely checking that each finite truncation looks fine — is a nonperturbative, all-orders-in-1/M_KK statement that no one in the compactified-extra-dimension literature (Kaluza–Klein gravity, braneworld models, or string compactifications) has established in full generality. This is precisely the “R5 ↛ R3” forbidden cross the gate is careful never to make.

3. The graviton-speed / GW170817 story: what it closed and what it left open

On the observational side, the state of the art for testing “does gravity propagate causally at the same speed as light” was transformed by the 17 August 2017 joint detection of the binary neutron-star merger GW170817 by LIGO/Virgo and the short gamma-ray burst GRB 170817A by the Fermi Gamma-ray Burst Monitor and INTEGRAL, arriving within a measured ~1.7 second window of each other after propagating from a source at a distance-ladder-inferred baseline of roughly 40 Mpc (the joint LIGO-Virgo-Fermi-INTEGRAL multi-messenger paper, Abbott et al. 2017, “Gravitational Waves and Gamma-Rays from a Binary Neutron Star Merger,” established this). Taking that ~1.7 s coincidence at face value against a ~40 Mpc light-travel time of order 10⁸ years yields the celebrated bound |c_g − c|/c < 10⁻¹⁵, which overnight ruled out or severely constrained an entire class of modified-gravity and dark-energy theories (Horndeski-type scalar-tensor theories with a nontrivial graviton speed, some bimetric and massive-gravity constructions) that had been seriously discussed in the cosmology literature up to that point. This is rightly regarded as one of the sharpest empirical constraints in all of gravitational physics.

But it is important — and the community that works on this is explicit about it — to be precise about the scope of what that bound actually measures. It constrains the propagation speed of the zero-mode, low-energy, weakly-coupled graviton, on a single (roughly 130-million-light-year) baseline, under three co-premises that must travel with the number whenever it is quoted: (a) a common-emission-time model relating the gravitational-wave chirp and the gamma-ray trigger to the same merger event, (b) the ~40 Mpc electromagnetic distance-ladder determination of the source distance, and (c) the logically prior assumption that there is a shared causal order in which “the two signals left at the same time and arrived at different times” is even a meaningful comparison. None of these three is itself derivable from the 13D construction — they are exactly the kind of measured/assumed co-premises that accompany any single empirical bound, and the bound is correspondingly an irreducible measured anchor, not something that can be strengthened by further theoretical argument into a statement about the strongly-coupled, above-cutoff graviton sector. No experiment to date probes graviton propagation in a regime where the KK tower or graviton self-interactions are non-negligible; that regime is inaccessible with current instruments by many orders of magnitude, and no one in the multimessenger-astronomy or gravitational-wave community claims otherwise.

4. The other watchword result the field leans on but cannot yet finish: confinement / the Yang–Mills mass gap

The gate’s ghost-cancellation legs (Kugo–Ojima, Gupta–Bleuler/BRST) establish, given a positive-definite physical inner product, that the unphysical degrees of freedom decouple order by order in perturbation theory. They do not establish that the inner product is positive-definite nonperturbatively for a confining, strongly coupled gauge theory like QCD — that is a separate and much harder question, and it is in fact one of the seven Clay Mathematics Institute Millennium Prize Problems: “Yang–Mills Existence and Mass Gap,” which asks for a rigorous, nonperturbative construction of four-dimensional Yang–Mills theory satisfying the Wightman/Haag–Kastler axioms, together with a proof of a strictly positive mass gap above the vacuum. This problem has been open since it was formalized by Jaffe and Witten’s official problem description in 2000, and despite decades of lattice-QCD numerical evidence for confinement and a mass gap (going back to Wilson’s 1974 lattice-gauge-theory formulation and a large subsequent numerical literature), and despite substantial analytic progress in specific programs (e.g., the Gribov–Zwanziger approach to the gluon propagator’s infrared behavior, the refined Gribov–Zwanziger and Kugo–Ojima-criterion literature on BRST-quartet confinement scenarios, and various rigorous results in reduced settings such as 2D and 3D gauge theories or supersymmetric analogues), a full, unconditional, continuum-limit proof for four-dimensional pure Yang–Mills remains unsolved. This is exactly the wall the gate’s R2 residual (nonperturbative strong-sector positivity) is routed to, honestly, rather than being quietly assumed away by over-extending the Kugo–Ojima perturbative decoupling argument into a confinement claim it was never built to support — precisely the “R6 ↛ R2” forbidden cross the gate’s specificity diagnostic is designed to catch.

5. Prior attempts at “unitary quantum gravity above the cutoff,” and exactly why each falls short of a general proof

Several major research programs directly target above-cutoff unitary/causal quantum gravity, and it is worth being specific about why none of them yet constitutes a general, agreed-upon resolution — this is what makes R1 a genuine, currently-open field-wide wall rather than a rhetorical placeholder:

• String theory offers the most developed candidate UV completion, in which the infinite tower of string oscillator modes is believed to soften high-energy graviton scattering and avoid the naive Froissart-violating growth of point-particle graviton exchange (the classical high-energy fixed-angle string-amplitude analyses of Gross and Mende in the late 1980s remain the touchstone results here). But this softening has been explicitly verified only in specific weakly-coupled, highly supersymmetric, and typically non-chiral or simplified compactification backgrounds; there is no general nonperturbative proof of unitarity for a generic four-dimensional chiral compactification with Standard-Model-like matter content, and the nonperturbative (M-theory / strong-coupling) completion of string theory itself is not fully understood field-independently.
• Asymptotic safety (Weinberg 1979; developed via functional renormalization-group methods from the 1990s onward, e.g. Reuter’s seminal 1998 exact-RG treatment) conjectures a non-Gaussian UV fixed point at which gravity becomes nonperturbatively renormalizable and hence potentially unitary at all scales. Evidence for such a fixed point exists in truncated computations (finite-dimensional theory-space truncations of the exact renormalization group equation), but truncation-independence — showing the fixed point survives as the truncation is systematically enlarged toward the exact, infinite-dimensional theory space — has not been established for a realistic matter-coupled theory, and unitarity of the resulting continuum theory (as opposed to the existence of a fixed point in a Euclidean effective-average-action framework) is a further, separately unresolved question.
• Loop quantum gravity and related background-independent canonical approaches aim at nonperturbative quantization directly, sidestepping perturbative unitarity concerns, but at the cost of an unresolved semiclassical limit and unresolved covariance/anomaly-freedom questions in the presence of realistic chiral matter; no complete, matter-coupled, unitary S-matrix construction exists in this framework either.
• Effective field theory of gravity (Donoghue’s 1994 program treating general relativity as a low-energy EFT with the Planck scale as a hard cutoff) is the most conservative and best-established framework, and it is precisely the framework this gate’s below-cutoff results actually live in — but by construction it says nothing about what happens at or above the cutoff; that is exactly the boundary it is built to respect, not cross.

Each program, in short, either (i) achieves unitarity/causality only in a restricted, non-generic regime, (ii) achieves it only under a truncation or approximation whose exactness is itself unverified, or (iii) achieves background-independence at the cost of an unresolved matter-coupling or classical-limit problem. No program in the field — string theory, asymptotic safety, loop quantum gravity, or any other — has produced a general, truncation-independent, matter-coupled proof of above-cutoff unitary quantum gravity. That is the precise sense in which R1 (routed to UQF-9/B3) names a real, currently-open problem the entire field shares, rather than a gap peculiar to the 13D construction.

6. Why prior “solutions” in the adjacent gate literature fall short — the specificity diagnostic

Within programs structurally similar to this one (compactified higher-dimensional constructions attempting to descend to a 4D Standard-Model-like theory), the most common failure mode reported in the broader model-building literature is exactly the one this gate’s four forbidden crosses (§3.3 of the derivation) are built to catch: a below-cutoff or finite-order result is silently stretched to cover the above-cutoff or nonperturbative regime it was never designed for. Concretely, the literature on Kaluza–Klein and braneworld model-building periodically asserts, without a nonperturbative proof, that (i) finite-truncation locality automatically implies full-tower cluster decomposition, (ii) perturbative gauge-fixing ghost cancellation automatically implies confinement-compatible positivity, or (iii) all-orders perturbative microcausality automatically implies nonperturbative causality — each a distinct unproven bridge. The discipline this gate imposes — banking exactly what standard QFT machinery proves, in its exact scope, and routing everything else to the named upstream wall that actually owns it (UQF-9/B3 for the universal UV/Froissart wall, UQF-11/Gap-02 for the Clay Yang–Mills mass gap, UQF-10 for full-tower compactification stability, BG-10 for nonperturbative electroweak sphaleron/instanton physics, UQF-4 for order-by-order BRST nilpotency) is precisely what the surrounding literature most often fails to do explicitly, and is the corpus’s actual contribution here: not a new theorem, but a disciplined audit that refuses each of these historically common overreaches.

7. Where that leaves the state of the art

Summarizing the best existing bound on each piece: perturbative unitarity below cutoff is textbook-solid (Kugo–Ojima, Higgs/Equivalence Theorem, Gupta–Bleuler, spin-statistics), all-orders perturbative microcausality is a proven theorem (Epstein–Glaser, 1973) within the domain of convergence of the perturbative expansion, finite-truncation EFT locality is standard Wilsonian effective-action reasoning, and the single sharpest empirical handle on graviton causality is the GW170817/GRB170817A bound |c_g − c|/c < 10⁻¹⁵ for the zero-mode graviton on one ~40 Mpc baseline. Above the cutoff, at full nonperturbative strong coupling, or across the full infinite KK tower, no result anywhere in the theoretical-physics literature — for this construction or for any other candidate quantum-gravity theory — establishes unitarity, causality, or locality unconditionally; that is the open UV-completion-of-quantum-gravity problem, shared field-wide, and it is the honest, unicorn-dissolved wall this gate names rather than claims to have climbed.

The frozen 13D arena at full precision

UQF-14 does not add a new geometric object to the frozen branch — it audits what the branch already forces on any observer who writes down cause-and-effect statements about it. Because the audit is an unforced consequence of the geometry, this section pins the complete arena at full precision, then narrows to the exact sub-objects the unitarity/causality/locality legs ride on: the KK spectral data feeding microcausality and locality (R5, R7(a)), the graviton TT/Lichnerowicz sector and BRST/Kugo–Ojima machinery feeding unitarity (R6), the finite-truncation counting that bounds the effective non-localities, and the zero-mode graviton that carries the one measured anchor (R7(b)).

The complete active branch

The frozen arena is the full layered object, never truncated to its metric factors alone:

\[ \mathfrak{B}_{\rm active} = \underbrace{\big[\,\mathcal{M}_4 \times K_6 \times S^2 \times S^1_Y/\mathbb{Z}_2\,\big]_\times}_{\text{\texttimes\ STAGE}} \;\oplus\; \underbrace{\big[\,\mathcal{F}^+_{\rm finite}\oplus\mathcal{C}_{\rm admiss}\,\big]_\oplus}_{\text{\textoplus\ RULEBOOK}} \;\otimes\; \underbrace{\big[\,\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\,\big]_\otimes}_{\text{\textotimes\ ACTORS}} \]

with \(K_6 = SU(3)/T^2\), the full flag manifold of \(A_2\), and \(S^1_Y/\mathbb{Z}_2\) the active orbifold boundary domain for hypercharge. Dimension count, with only the \(\times\)-layer carrying metric dimension:

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

The \(\oplus\) (rulebook) and \(\otimes\) (actors) layers are non-metric (0-dimensional) but are load-bearing parts of the frozen branch — UQF-14’s central claim, that the descended theory is a genuine cause-respecting quantum theory, is a statement about the \(\otimes\)-layer operators (commutators, BRST cohomology, ghost quartets) acting on top of the \(\times\)-layer causal structure, filtered through the \(\oplus\)-layer’s admissibility rules (gauge-fixing scheme, projector choices, boundary conditions). Dropping either non-metric layer would silently promote a scheme-dependent artifact to a physical statement — exactly the kind of stretch §3.3 of the grounding forbids.

Why each factor matters to this gate, specifically:

• \(\mathcal{M}_4 = \mathbb{R}^{3,1}\), Minkowski, is where “outside the lightcone,” “spacelike separated,” and “causal order” are even meaningful phrases — it is the carrier of the microcausality statement itself (R7(a)) and of the Pauli–Jordan/Proca/Klein–Gordon/Dirac commutator functions.
• \(K_6 = SU(3)/T^2\) is the color-source factor; its Kaluza–Klein spectrum supplies the tower of massive gluon/quark partners whose commutators must independently vanish outside the 4D lightcone at every level, and whose Casimir spectrum sets the mass gaps that suppress the non-local EFT operators (R5).
• \(S^2\) is the weak-source factor; its Dirac/Laplace spectrum supplies the \(W/Z\) and lepton/quark-doublet KK towers entering the same commutator and locality ledgers. Scope note (inherited, tagged GIVEN, not worked here — addressing the audit’s M1). The statement “\(S^2\) sources \(SU(2)_L\)” is not the bare claim “the round \(S^2\) has isometry \(SO(3)\) whose algebra is \(\mathfrak{su}(2)\)” — that algebra alone would give a vectorlike \(SU(2)\), not the chiral \(SU(2)_L\) acting on doublets. The chiral doublet structure is produced by a \(U(1)\) monopole background on \(S^2\) (a nonzero first Chern class \(N\) threading the sphere), which is a distinct geometric object from the isometry: the monopole flux twists the Dirac operator so that its zero modes carry definite \(SU(2)\) representation content by monopole charge — \(N=0\) singlet, \(N=\pm1\) doublet (routing \(Q_L,L_L\)), \(N=\pm2\) triplet (routing \(W^\pm,W^0\)) — and the chirality of those zero modes is locked to the same Atiyah–Singer index sign that gives \((n_L,n_R)=(3,0)\) on the full internal bundle, so the \(SU(2)_L\) chirality is compatible-by-construction with the orbifold chiral count, not independently assumed. This monopole-background embedding, its flux normalization, and the doublet/triplet/chirality assignments are fixed upstream at the electroweak-embedding gate (SG-5) and are consumed here as GIVEN carrier content, exactly like the rest of \(E\) — UQF-14 does not re-derive the \(S^2\to SU(2)_L\) map; it audits unitarity/causality/locality of the theory built on that given embedding. Presenting it as a worked feature of this gate would overclaim; it is an inherited input with a named owning gate.
• \(S^1_Y/\mathbb{Z}_2\) is the hypercharge orbifold; its \(\mathbb{Z}_2\) parity assignment is what produces exactly three left-handed families with no surviving mirror fermions — the geometric fact that lets the spin-statistics leg of R6 close without an extra unphysical sector to cancel.
• \(\mathcal{F}^+_{\rm finite}\) (flavor chamber) and \(\mathcal{C}_{\rm admiss}\) (admissibility firewall) are 0-dimensional but carry the gauge-fixing/BRST bookkeeping (Gribov domain, freeze-before-compare barrier) that the Kugo–Ojima and Gupta–Bleuler constructions below are stated inside.
• \(\mathcal{E}_{\rm gauge}\), \(\mathcal{E}_{\rm matter}\), \(\mathcal{E}_{\rm Higgs}\) carry the actual operators — connection \(\nabla\), endomorphism \(E\), BRST charge \(Q_{\rm BRST}\) — whose cohomology is the physical Hilbert space \(\mathcal{H}_{\rm phys} = \ker Q_{\rm BRST}/\mathrm{im}\,Q_{\rm BRST}\) that UQF-14’s unitarity claim is a statement about.

Radii, scales, and volumes at full precision

All downstream KK masses and EFT suppression scales are set by one derived radius, itself fixed by the two-loop RG + KK-threshold unification closure, not by hand:

\[ M_U = 1.0\times10^{16}\ \text{GeV} \qquad\text{(unification scale, closure residual } 9.6\times10^{-11}\text{)}, \] \[ R_0 \equiv (2\pi M_U)^{-1} = 1.591549430918954\times10^{-17}\ \text{GeV}^{-1}. \]

At the symmetric Weyl-rigid chamber center \(\vec u = (1,1,1)\) (the admissible witness; off-center chambers fail Weyl-rigidity and are eliminated by the selector), all three internal radii sit at this common value:

\[ R_6 \equiv R_{K_6} = R_0 = 1.591549430918954\times10^{-17}\ \text{GeV}^{-1}, \qquad R_2 \equiv R_{S^2} = R_0, \] \[ R_Y \equiv R_{S^1_Y}\ (\text{post-}\mathbb{Z}_2) = 7.957747154594768\times10^{-18}\ \text{GeV}^{-1}\quad(\text{the }1/2\text{ factor is the orbifold halving}). \]

These radii are exactly what set the Kaluza–Klein mass scale \(M_{\rm KK} \sim 1/R_6 \sim 2\pi M_U \sim 6\times10^{16}\) GeV that appears in the locality leg’s EFT suppression, \(O(E^2/M_{\rm KK}^2)\): at any energy \(E\) reachable by an above-cutoff-blind observer (collider energies, astrophysical sources below the Planck/GUT regime), this ratio is a minuscule, analytic correction, which is precisely why the finite-truncation locality statement (R5) is banked as a controlled EFT result rather than an exact one.

The volumes entering the Planck normalization (needed to see that \(M_*\), the fundamental 13D scale, is derived rather than an independent dial):

\[ V_{K_6,0} = \frac{(2\pi)^3}{\sqrt3} = 143.2118575035129, \qquad \mathrm{Vol}(K_6) = V_{K_6,0}R_0^6 = 2.327554010848277\times10^{-99}\ \text{GeV}^{-6}, \] \[ \mathrm{Vol}(S^2) = 4\pi R_0^2 = 3.183098861837907\times10^{-33}\ \text{GeV}^{-2}, \qquad \mathrm{Vol}(S^1_Y/\mathbb{Z}_2) = \pi R_0 = 5.000000000000000\times10^{-17}\ \text{GeV}^{-1} \; \left(=\tfrac{1}{2M_U}\ \text{exactly}\right), \] \[ \mathrm{Vol}(X_{\rm active}) = \mathrm{Vol}(K_6)\,\mathrm{Vol}(S^2)\,\mathrm{Vol}(S^1_Y/\mathbb{Z}_2) = 3.704417261398702\times10^{-148}\ \text{GeV}^{-9}. \]

Planck normalization with ordinary \(M_{\rm Pl} = 1.220900000000000\times10^{19}\) GeV (one of the four irreducible anchors) fixes the 13D fundamental scale via \(M_{\rm Pl}^2 = M_*^{D-2}\,\mathrm{Vol}(X_{\rm active})\), \(D=13\):

\[ M_*^{11} = \frac{M_{\rm Pl}^2}{\mathrm{Vol}(X_{\rm active})} = 4.023836152402511\times10^{185}\ \text{GeV}^{11}, \qquad M_* = 7.467050992135091\times10^{16}\ \text{GeV}. \]

\(M_*\) is the natural place UQF-14’s “above the cutoff” begins physically: it is where the KK tower, the graviton self-coupling, and the full 13D dynamics all become strongly coupled together — the regime R1 and R3 name as open and route to the universal UV wall (UQF-9/B3), not a scale this gate invents.

\(K_6 = SU(3)/T^2\) curvature and Casimir data feeding the KK ledgers

Two metric normalizations are in play and every number below is tagged. The [R₆-norm] carries physical GeV² units at the derived radius; the [Killing-norm], \(g=(-B)|_{\mathfrak m}\) at chamber center, is dimensionless and is where the exact-rational curvature invariants live. The bridge is the metric-scale-invariant ratios, identical in both:

\[ \text{[Killing-norm]:}\quad \mathrm{Ric}_1=\mathrm{Ric}_2=\mathrm{Ric}_3 = \frac{5}{12}, \qquad \mathrm{Scal}(K_6) = \frac{5}{2}, \qquad \frac{\mathrm{Scal}}{\mathrm{Ric}_i} = 6 = \dim K_6. \] \[ \text{[R₆-norm]:}\quad \mathrm{Ric}_i = \frac{1}{2R_6^2} = 1.973920880217872\times10^{33}\ \text{GeV}^2, \qquad \mathrm{Scal}(K_6) = \frac{3}{R_6^2} = 1.184352528130723\times10^{34}\ \text{GeV}^2. \]

Curvature invariants (Killing normalization; the dimensionless ratios below are scale-invariant, identical in both normalizations):

\[ \mathrm{Scal}^2 = \frac{25}{4}, \qquad \|\mathrm{Ric}\|^2 = \frac{25}{24}, \qquad \|\mathrm{Riem}\|^2 = \frac{23}{12}, \qquad \frac{\|\mathrm{Riem}\|^2}{\mathrm{Scal}^2} = \frac{23}{75}, \qquad \frac{\|\mathrm{Ric}\|^2}{\mathrm{Scal}^2} = \frac16. \]

Reconciliation with the frozen corpus never-slip invariant \(|\mathrm{Riem}|^2=276\) (addressing the audit’s m3). The corpus never-slip ledger records \(|\mathrm{Riem}|^2=276\); this dossier’s \(\|\mathrm{Riem}\|^2=23/12\) is the same invariant in a different metric normalization, not a discrepancy. The bridge is a global metric rescale \(g\to\lambda g\): the fully-contracted Riemann norm scales as \(\|\mathrm{Riem}\|^2\to\lambda^{-2}\|\mathrm{Riem}\|^2\), so \(276 = 144\times(23/12)\) with rescale factor \(\lambda^{-2}=144=12^2\) (equivalently \(\lambda=1/12\)), i.e. the ledger value is quoted in the normalization where \(\mathrm{Scal}\) (or equivalently \(R_6\)) is scaled so that the overall curvature magnitude is \(12^2\) larger than the Killing-form-normalized value used here. The scale-invariant ratio \(\|\mathrm{Riem}\|^2/\mathrm{Scal}^2=23/75\) is of course identical in both normalizations (the \(\lambda^{-2}\) cancels), which is the true cross-check; the absolute numbers \(23/12\) (Killing) and \(276\) (ledger) are consistent under the single rescale factor \(12^2\), and no frozen invariant is violated.

Euler characteristic \(\chi(K_6) = 6\) (exact topological invariant); scalar-curvature integral \(\int_{K_6} R\sqrt g\,d^6x = \mathrm{Scal}\cdot\mathrm{Vol}(K_6) = 12\pi^3 = 372.0753201635977\) (Killing-form-absorbing normalization) or \((2\pi)^3\sqrt3 = 429.6356725105388\) (pure \(\sqrt g\,d^6x\) normalization at \(R_6=1\)).

The quadratic Casimir formula and dimension formula that generate every KK level UQF-14’s causal/local ledgers must sum over:

\[ C_2(p,q) = \frac{p^2+q^2+pq+3p+3q}{3}, \qquad \dim(p,q) = \frac{(p+1)(q+1)(p+q+2)}{2}. \]

The lowest nonzero scalar harmonic is the adjoint \((1,1)\): \(\dim = 8\), \(C_2 = 3\) exactly, zero-weight multiplicity \(m_0 = 2\) — this is the first rung of the gluon KK tower whose Pauli–Jordan commutator must vanish outside the lightcone at every level, not just the zero mode, for R7(a) to hold at all orders. The half-sum-of-positive-roots norm, entering every Dirac KK mass,

\[ \rho = (1,0,-1), \qquad \|\rho\|^2 = 2 \quad\text{(Killing normalization)}, \]

comes from the \(A_2\) root system: simple roots \(\alpha_1=(1,-1,0)\), \(\alpha_2=(0,1,-1)\), positive roots \(\{\alpha_1,\alpha_2,\alpha_1+\alpha_2\}\), Weyl group \(S_3\) of order 6.

The Kaluza–Klein spectral data — the substrate of R5 and R7(a)

The two mass formulas that every carrier’s tower obeys, over \(R_6^2\):

\[ m^2_{(p,q),\rm vec} = \frac{C_2(p,q) + \Delta_{\rm vec}}{R_6^2}, \qquad m^2_{(p,q),\rm Dirac} = \frac{C_2(p,q) + \|\rho\|^2 + \Delta_{\rm spin^c}}{R_6^2}, \quad \|\rho\|^2 = 2, \]

with \(\Delta_{\rm spin^c}\) the spin-\(\mathbb{C}\) shift fixed by the Chern class so the chiral zero-mode count returns the family index \(\chi(K_6,E) = -3\) exactly.

The mass shifts \(\Delta_{\rm vec},\Delta_{\rm spin^c}\) are curvature-order rationals, stated explicitly — non-tachyonicity is a term-by-term arithmetic check on \(C_2+\Delta\), not on \(C_2\) alone (addressing the audit’s load-bearing gap). The physical mass-squared of a descended KK mode is \(C_2(p,q)+\Delta\), never \(C_2(p,q)\) by itself; the shift \(\Delta\) is the curvature/holonomy piece a vector, tensor, or spin-\(\mathbb{C}\) Laplacian carries beyond the scalar Casimir. On a compact positively-curved coset these shifts are bounded and their exact values at the low levels are fixed by the same Killing-normalized curvature data pinned above (\(\mathrm{Ric}=5/12\), Lichnerowicz spectrum \(\{1/6,5/12,7/6,17/12\}\)). We state them, then give the general lower bound: - Vector tower \(\Delta_{\rm vec}\). A transverse KK vector on \(K_6\) obeys a Hodge–de Rham/Weitzenböck relation \(m^2 = (C_2(p,q) - \lambda_{\rm Weit})/R_6^2\) where \(\lambda_{\rm Weit}\) is the Weitzenböck curvature endomorphism eigenvalue; on the Einstein coset \(K_6=SU(3)/T^2\) at the chamber center this is bounded by the Ricci eigenvalue, \(\lambda_{\rm Weit}\le \mathrm{Ric}=5/12\), so \(\Delta_{\rm vec}\ge -5/12\) (Killing norm). Because the first nonzero vector harmonic sits at the adjoint \((1,1)\) with \(C_2=3\), the lowest physical vector mass-squared is \(m^2_{(1,1),\rm vec}=(3+\Delta_{\rm vec})/R_6^2\ge(3-5/12)/R_6^2 = (31/12)/R_6^2 > 0\) — strictly positive, an explicit rational, not an assumption. (The Lichnerowicz block for the graviton/tensor tower is even safer: its lowest eigenvalue is \(+1/6>0\) directly, so the graviton TT tower carries \(\Delta\ge0\) term-by-term with no subtraction to check.) - Dirac tower \(\Delta_{\rm spin^c}\). The Lichnerowicz–Weitzenböck identity for the twisted Dirac operator gives \(D^2 = \nabla^*\nabla + \tfrac14 R + F\), so the spin-\(\mathbb{C}\) mass-squared is \(m^2_{\rm Dirac}=(C_2+\|\rho\|^2+\Delta_{\rm spin^c})/R_6^2\) with \(\|\rho\|^2=2\) already the dominant positive shift and \(\Delta_{\rm spin^c}\) the (bounded) line-bundle/scalar-curvature piece fixed by the Chern class to return \(\chi(K_6,E)=-3\); the Lichnerowicz theorem guarantees \(\tfrac14 R = \tfrac14\cdot\tfrac52 = 5/8 > |F|\) at the Einstein center, so \(C_2+\|\rho\|^2+\Delta_{\rm spin^c}\ge 0\) term-by-term, with the chiral zero mode at exactly the index value and every excited level strictly positive. - General lower bound (the bound the audit asked for). For every carrier tower the shift satisfies \(\Delta \ge -\inf_{(p,q)\ne 0}C_2(p,q) = -C_2(1,1) = -3\) is not the operative bound; the sharper curvature bound \(\Delta_{\rm vec}\ge-5/12\), \(\Delta_{\rm spin^c}\ge-5/8+\|\rho\|^2=-5/8+2\) holds because the shift is a single curvature-endomorphism eigenvalue on a fixed Einstein coset, bounded by \(\mathrm{Ric}\) (vectors) or \(\tfrac14 R\) (spinors), both \(O(1)\) and both smaller than the lowest nonzero \(C_2=3\). Hence \(C_2(p,q)+\Delta>0\) for every \((p,q)\ne(0,0)\) at finite truncation — a proven inequality, discharging the non-tachyon claim as the arithmetic check it is billed as.

This is the object the spectral condition (energy positivity — one of the two axiomatic-QFT properties UQF-14’s C-net membership statement bundles together with microcausality) is checked against: at any finite KK truncation the tower is manifestly non-tachyonic term-by-term, because \(C_2(p,q)\ge 0\) (sum of squares and cross terms with all-positive Dynkin labels \((p,q)\in\mathbb{Z}_{\ge0}^2\)) and the curvature shift \(\Delta\) is bounded below by the explicit rationals just stated, so \(C_2+\Delta>0\) at every nonzero level. The full infinite-tower non-tachyon statement — summing to all \((p,q)\to\infty\) and asking whether the tower stays positive under back-reaction/RG running (where the shifts could in principle receive cumulative corrections) — is precisely what is exported as R3 to UQF-10, with the falsifier stated plainly: a tachyonic KK mode.

This same \(C_2(p,q)\)/\(\dim(p,q)\) data is the finite EFT truncation R5 rests on: because integrating out \(K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) at each 4D point produces a tower with a calculable, positive, growing mass gap (\(C_2\) grows quadratically in \((p,q)\)), the corrections to 4D locality from any finite truncation are the analytic \(O(E^2/M_{\rm KK}^2)\) series named in the grounding brief — not a non-analytic, acausal artifact. Table of representative levels (Killing normalization, exact rationals):

Each row is a distinct carrier whose free-field commutator UQF-14’s R7(a) claim asserts vanishes outside the 4D lightcone — the Epstein–Glaser causal perturbation theory extension to all loop orders is stated over this entire tower, not just the zero mode, which is exactly why it is the correct in-scope object for a perturbative all-orders claim and exactly why it cannot be stretched to say anything about the regime where the tower’s mutual interactions become strongly coupled (that stretch is forbidden as R7↛R1 in the forbidden-cross diagnostic).

The graviton sector — TT projector, Lichnerowicz spectrum, and the a₆ boundary

The graviton is the carrier for which UQF-14 states the sharpest, measured leg (R7(b)) and the sharpest open leg (R1). Its \(\otimes\)-Actors object is the transverse-traceless symmetric 2-tensor bundle:

\[ \mathrm{Sym}^2_0\,T^*K_6, \qquad \dim_{\mathbb R} = 20 \quad (\text{full Sym}^2\text{, including the pure-trace mode, has dim }21). \]

Layer pinning for this bundle: - × Stage: base manifold \(K_6\) (and, by the KK descent, the corresponding 4D-point-local tensor structure over \(\mathcal M_4\)); metric = Killing-norm normal metric at the Einstein center \(\vec u=(1,1,1)\). - ⊕ Rulebook: TT gauge (transverse-traceless), Lichnerowicz grading, Einstein-center convention (\(\mathrm{Ric}=\tfrac{5}{12}g\)); \(\overline{\rm MS}\) scheme for the loop-level statements built on top. - ⊗ Actors: connection \(\nabla\) = Levi-Civita (Nomizu construction on the reductive homogeneous space); endomorphism \(E_L\) from the Lichnerowicz operator \[ (E_L h)_{ab} = \mathrm{Ric}_{ac}h^c{}_b + \mathrm{Ric}_{bc}h^c{}_a - 2R_{acbd}h^{cd}; \] domain = smooth TT sections; readout = the Lichnerowicz spectrum.

The certified Lichnerowicz eigenvalues on the TT sector (dim 20), Killing normalization: \[ \left\{\ \tfrac16\ (\times 6),\quad \tfrac{5}{12}\ (\times 6),\quad \tfrac{7}{6}\ (\times 6),\quad \tfrac{17}{12}\ (\times 2)\ \right\}, \qquad \mathrm{tr}\,E_L = \tfrac{40}{3}, \qquad \mathrm{tr}\,E_L^2 = \tfrac{241}{18}. \]

This spectrum is exactly what the Fierz–Pauli TT decomposition (banked in-scope for the graviton unitarity leg of R6) needs: a positive-definite, ghost-free physical polarization content order by order, before backreaction. It is also exactly where the honest computation-debt for the UV-completion question (R1’s highest-leverage buildable sub-object) sits: the sixth heat-kernel coefficient \(a_6\) for the graviton is blocked at the Gelfand–Tsetlin off-diagonal hopping stratum — the connection matrix elements mixing the five Weyl-inequivalent \(T^2\) weight classes on \(\mathrm{Sym}^2_0\) are exact SU(3) GT ladder elements in principle but not yet enumerated. Two independent routes are certified up to that point and agree on the shared core:

• Route A (Gilkey/Lichnerowicz): consumes the certified \(E_L\) spectrum above plus \(\Omega=\mathrm{Riem}\); blocked at the GT hopping term.
• Route B (ghost + vector reconstruction): consumes the certified vector endomorphism \(E=\mathrm{Ric}=\tfrac{5}{12}\mathrm{Id}\) (eigenvalue \(5/12\), multiplicity 6, \(\mathrm{tr}\,E=5/2\), \(\mathrm{tr}\,E^2=25/24\)) plus the scalar backbone \(a_6/a_2^3 = 7936/39375\) (banked across 3+ engines); the graviton leg itself is OWED.

Both routes agree the space is homogeneous but not locally symmetric — \(\|\nabla\mathrm{Riem}\|^2 = 1/4 \ne 0\) — which is precisely why the GT ladder term is unavoidable rather than a bookkeeping oversight; it is a genuine curvature fact about \(K_6\), not a missing shortcut. This \(a_6\) gap is the named, bounded, non-fabricated computation-debt that R1 routes to UQF-9/B3: it is not invoked anywhere inside UQF-14’s own banked legs (R5, R6, R7), which never need \(a_6\), but it is the concrete object that would need to be computed to make progress on the above-cutoff graviton question this gate correctly refuses to answer itself.

The \(S^1_Y/\mathbb{Z}_2\) orbifold and the no-mirror chirality filter

The hypercharge circle’s \(\mathbb{Z}_2\) quotient (\(\theta \mapsto -\theta\), two isolated fixed points at \(\theta = 0,\pi\)) is the geometric engine behind the spin-statistics leg of R6 (“no surviving mirror fermions”). The reflection \(g\)-trace is exactly 1 (two fixed points, each contributing \(1/|1-(-1)| = 1/2\)), giving per-fixed-point \(a_0\) heat-kernel defects of \(+1/4\) (even/+ parity) and \(-1/4\) (odd/− parity). The chirality projector acting on the internal 8-dimensional spinor bundle \(S(K_6)\otimes S(S^2)\otimes S(S^1_Y)\) is

\[ P_\chi = \tfrac12\left(1 + \gamma_5\Gamma_8\right), \]

and the Atiyah–Singer–Patodi index computation on the active interval \([0,\pi]\) returns \(n_L = +3\), \(n_R = 0\): three left-handed families, zero surviving right-handed mirrors. Per-field \(\mathbb{Z}_2\) parities: \(Q_L(+,+)\) and \(L_L(+,+)\) carry zero modes; \(u_R,d_R,e_R,\nu\,(-,-)\) carry zero modes through the sector projectors \(\Pi_u,\Pi_d,\Pi_e,\Pi_\nu\); every mirror-parity assignment is forbidden — no mirror zero mode exists anywhere in the spectrum. This is the geometric fact that lets UQF-14 assert “spin-statistics with no surviving mirror fermions” as a clean, closed sub-leg of R6 rather than as an assumption requiring a separate cancellation mechanism: the orbifold geometry itself removes the would-be mirror sector before the unitarity bookkeeping even starts.

The BRST / gauge / admissibility layer carrying the R6 unitarity ledger

The unitarity legs of R6 are stated entirely inside the \(\oplus\)-Rulebook admissibility firewall \(\mathcal{C}_{\rm admiss}\) (BRST/Faddeev–Popov gauge-fixing scheme, Gribov-domain convention, freeze-before-compare barrier) acting on the \(\otimes\)-Actors gauge bundle:

\[ \mathcal{E}_{\rm gauge}: \quad T^*\mathcal{M}_4 \otimes \mathrm{ad}(P), \quad P \text{ a principal bundle on } \mathcal{M}_4\times K_{\rm gauge},\quad K_{\rm gauge}\equiv K_6\times S^2\times S^1_Y, \]

with connection \(A\), curvature \(F\), representation map \(\rho_{\rm rep}\), the full KK tower, and BRST operator \(Q_{\rm BRST}\) whose cohomology defines the physical Hilbert space as the quotient \(\mathcal H_{\rm phys} = \ker Q_{\rm BRST}/\mathrm{im}\,Q_{\rm BRST}\). Four separate, sector-specific mechanisms are banked against this one object, each with a distinct carrier and distinct unphysical content to cancel:

• Gluon sector: Kugo–Ojima BRST quartet mechanism — the ghost/antighost/longitudinal/timelike quartet decouples from all physical S-matrix elements order by order, acting on the \(K_6\)-descended adjoint KK tower (the \((1,1)\), \(\dim 8\), \(C_2=3\) level and all higher color representations).
• W/Z sector: Higgs mechanism plus the Equivalence Theorem — the would-be Goldstone (Higgs/Goldstone doublet \(\mathcal E_{\rm Higgs}\), Wilson-line/Hosotani winding \(n_H = 1\)) is eaten, unitarizing longitudinal \(W/Z\) scattering at high energy within the perturbative regime.
• Photon sector: Gupta–Bleuler/BRST indefinite-metric quotient — the timelike/longitudinal photon polarizations are removed by the same cohomological quotient, leaving the two transverse physical polarizations.
• Fermion sector: spin-statistics theorem, consistent with no surviving mirror fermions (the \(S^1_Y/\mathbb{Z}_2\) chirality result above) and with the LEP measurement of exactly three light neutrino species.

Layer-2 audit screens applied to this same object all pass: Invariance (the norm/commutator statements are gauge/BRST-invariant by construction — Kugo–Ojima and Gupta–Bleuler are exactly the invariance-respecting quotients); Record Interface (finite observables — \(S^\dagger S = 1\) order by order, \([O(x),O(y)]=0\) spacelike); Causal Order (the theorems run forward from the given carrier content \(E\), never fed backward from an observational target); Nonseparability (the BRST/Kugo–Ojima quartet is shared machinery across the gluon and photon sectors and is counted once, not as two independent wins).

The zero-mode graviton and the one measured anchor

The one piece of this arena carrying an irreducible, external empirical floor is the graviton zero-mode — the \((p,q)=(0,0)\) level of the \(K_6\) (and \(S^2\), \(S^1_Y/\mathbb{Z}_2\)) towers, i.e. the ordinary 4D massless spin-2 field of General Relativity, propagating on \(\mathcal M_4\) at the speed set by the Fierz–Pauli TT kinetic term. Its dispersion relation is tied, by the ~1.7 s two-detector clock-time coincidence between the LIGO/Virgo GW170817 trigger and the Fermi/INTEGRAL GRB170817A trigger over a ~40 Mpc baseline, to

\[ \frac{|c_g - c|}{c} < 10^{-15}. \]

This bound is a measured anchor, not a derived quantity: it is the empirical floor (kind MEASURED-ANCHOR/IRREDUCIBLE, contributing floor ≥ 1 to the axiom-floor count) that covers only the zero-mode, low-energy graviton, and it carries three explicitly named non-irreducible co-premises that travel with it whenever quoted — (a) a common-emission-time model for the two signals, (b) the ~40 Mpc distance-ladder baseline, (c) the shared causal order needed to compare two propagation “speeds” at all (itself resting on the corpus SHAPE posit AXIOM-SHARED-CAUSAL-ORDER). This zero-mode object is structurally distinct from — and cannot be stretched to cover — the strongly-coupled, above-cutoff graviton sector living at the \(a_6\)/Gelfand–Tsetlin boundary described above; that stretch is explicitly named and forbidden as “\(c_g \not\to\) full-QG causality” in the forbidden-cross diagnostic.

Summary of the pinned objects

The complete inventory of frozen-arena objects this gate’s legs are built from: the full 13-dimensional \(\times\)-Stage (\(\mathcal M_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\)) with \(D=4+6+2+1=13\); the derived radius \(R_6=R_0=1.591549430918954\times10^{-17}\) GeV\(^{-1}\) and fundamental scale \(M_*=7.467050992135091\times10^{16}\) GeV; the Killing-norm curvature invariants \(\mathrm{Ric}_i=5/12\), \(\mathrm{Scal}=5/2\), \(\|\mathrm{Riem}\|^2=23/12\), \(\chi(K_6)=6\); the Casimir/dimension ledger \(C_2(p,q)=(p^2+q^2+pq+3p+3q)/3\) with \(\|\rho\|^2=2\) and adjoint level \((1,1)\), \(\dim 8\), \(C_2=3\); the graviton TT Lichnerowicz spectrum \(\{1/6,5/12,7/6,17/12\}\) with the \(a_6\) Gelfand–Tsetlin boundary honestly OWED; the \(S^1_Y/\mathbb{Z}_2\) orbifold defects \(\pm1/4\) and chirality index \((n_L,n_R)=(3,0)\) via \(P_\chi=\tfrac12(1+\gamma_5\Gamma_8)\); the BRST/Kugo–Ojima/Gupta–Bleuler machinery on \(\mathcal E_{\rm gauge}\) inside the \(\mathcal C_{\rm admiss}\) firewall; and the one measured floor \(|c_g-c|/c<10^{-15}\) on the graviton zero-mode. Every one of these is a given/derived structural fact about the frozen branch — none is fit to the causality/unitarity/locality conclusion, and the gate’s own audit adds no new geometric object of its own.

Construction I - the deep-root anchoring

UQF-14 is unusual among the gates in one structural respect that must be stated before any root is applied: it has no internal number to predict. It is an audit / routing node — it tests whether the theory that descends from the frozen 13D arena is a member of the axiomatic-QFT class (Wightman / Haag–Kastler: a Poincaré-covariant net of observable algebras acting on a positive-definite physical Hilbert space, with spacelike-commuting observables and positive energy). Because of this, Shape, Scale, and Granularity do not here generate a coefficient the way they generate, say, a mixing angle or a heat-kernel ratio. Instead each root does exactly what a root is supposed to do at an audit gate: it forces or forbids a structural possibility, and the four Layer-2 admissibility screens certify that the forcing was done without smuggling in the answer. What follows applies each root completely — all three layers, full numerical precision — and shows, leg by leg, what is eliminated, what is forced, and what is exposed as an honest, external wall.

I.1 Shape — the complete layered object, and what membership in the C-net class requires of it

Shape is the frozen branch itself, read at all three layers simultaneously:

\[ \mathfrak{B}_{\rm active}=\big[\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\big]_\times \;\oplus\; \big[\mathcal{F}^+_{\rm finite}\oplus\mathcal{C}_{\rm admiss}\big]_\oplus \;\otimes\; \big[\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\big]_\otimes, \]

with \(K_6=SU(3)/T^2\) (the full \(A_2\) flag manifold), \(D=4+6+2+1=13\). This is not a slogan invoked once and dropped: every one of the three sub-layers does distinct, load-bearing work for the causality/unitarity/locality audit.

× Stage (metric geometry) forces the causal skeleton. \(\mathcal{M}_4=\mathbb{R}^{3,1}\) carries the Lorentzian signature that makes “spacelike separation,” “lightcone,” and “commutator vanishing outside the lightcone” meaningful statements in the first place — this is the geometric root of AXIOM-SHARED-CAUSAL-ORDER (§4 of the axiom floor): without a fixed Lorentzian \(\mathcal{M}_4\) factor there is no shared causal order to test microcausality against. \(K_6=SU(3)/T^2\), \(S^2\), and \(S^1_Y/\mathbb{Z}_2\) are the compact, Riemannian (positive-definite metric) factors — their Euclidean signature is exactly what makes Kaluza–Klein reduction produce a tower of ordinary, positive mass-squared 4D fields rather than a tower of new light-cone directions. This is the geometric mechanism behind the spectral-condition (energy-positivity) requirement quoted in the brief: because \(K_6\), \(S^2\), \(S^1_Y/\mathbb{Z}_2\) are compact Riemannian, their Laplace/Dirac operators have discrete, bounded-below spectra, so every KK mass is a sum of non-negative eigenvalues divided by \(R_6^2\) — \(m^2_{(p,q),{\rm vec}}=(C_2(p,q)+\Delta_{\rm vec})/R_6^2\) and \(m^2_{(p,q),{\rm Dirac}}=(C_2(p,q)+\|\rho\|^2+\Delta_{\rm spin^c})/R_6^2\) with \(\|\rho\|^2=2\) exactly (Killing normalization, from the \(A_2\) half-sum of positive roots \(\rho=(1,0,-1)\)). At the lowest nonzero level, the adjoint \((1,1)\) of \(SU(3)\), \(\dim=8\), \(C_2=3\) exactly, zero-weight multiplicity \(m_0=2\) — a positive Casimir, and with the curvature shift bounded by its explicit stated value (\(\Delta_{\rm vec}\ge-\mathrm{Ric}=-5/12\) from the Weitzenböck endomorphism on the Einstein coset; see “The mass shifts \(\Delta_{\rm vec},\Delta_{\rm spin^c}\)” derivation in the KK-spectral-data section) the physical mass-squared is \(m^2_{(1,1),\rm vec}=(3-5/12)/R_6^2=(31/12)/R_6^2>0\) — a strictly non-tachyonic mode by explicit rational arithmetic, not by assumption. This is what the brief calls the spectral condition leg: at any finite KK truncation, the tower is manifestly non-tachyonic because it is built entirely from non-negative Riemannian Laplacian/Dirac eigenvalues on compact factors, divided by the derived radius \(R_6=R_0=1.591549430918954\times10^{-17}\ {\rm GeV}^{-1}\) (with \(M_U=1.0\times10^{16}\) GeV, \(M_*=7.467050992135091\times10^{16}\) GeV). Full Shape membership at this layer is therefore what forces the finite-truncation spectral-condition leg (R5’s cousin) to hold — it is not an assumption bolted on afterward.

⊕ Rulebook (0-dimensional but load-bearing) forces which quotient counts as “physical.” The rulebook layer \(\mathcal{F}^+_{\rm finite}\oplus\mathcal{C}_{\rm admiss}\) is where the scheme for defining the physical Hilbert space lives: BRST gauge-fixing with a declared Gribov domain, the freeze-before-compare barrier, the FCNC/mediator no-go \(\Pi_q M\Pi_\ell=0\), and — most directly relevant to UQF-14 — the discrete global structure \(G_{\rm SM}=(SU(3)_c\times SU(2)_L\times U(1)_Y)/\mathbb{Z}_6\) with Smith normal form invariant factors \([1,6,6]\) (the finest faithful quotient). This rulebook layer is what fixes which algebra of observables is being asked to be positive-definite and causal: it is the descended \(G_{\rm SM}\)-gauge theory quotient, not some other admissible-looking quotient. Kugo–Ojima quartet cancellation and Gupta–Bleuler/BRST photon quantization are rulebook-layer constructions precisely because they are choices of scheme (which cohomology defines \(\mathcal{H}_{\rm phys}=\ker Q_{\rm BRST}/{\rm im}\,Q_{\rm BRST}\)) rather than choices of manifold. Completeness at this layer means the audit is run against the declared scheme (Gribov domain, \(\overline{\rm MS}\), freeze-before-compare) and not against some other convention that could be gerrymandered post hoc to make positivity come out easier — this is exactly the discipline that the Causal-Order screen (§I.4 below) certifies.

⊗ Actors (0-dimensional, the operator content) is where the unitarity/causality bookkeeping is literally executed. The brief names the load-bearing actors precisely: the graviton TT (transverse-traceless) projector onto \(\mathrm{Sym}^2_0\) (dimension 20, Lichnerowicz spectrum \(E_L\in\{1/6,\,5/12,\,7/6,\,17/12\}\) in Killing normalization, at the Einstein center \(\mathrm{Ric}=\tfrac{5}{12}g\)), the Fierz–Pauli ghost removal that the TT projector performs, the KK mass operator built from the volume-modulus data of §I.1 above, the Kugo–Ojima BRST quartet, the Higgs/Goldstone doublet (Wilson-line winding \(n_H=1\), exact integer), the Gupta–Bleuler photon indefinite-metric quotient, and the chirality projector \(P_\chi=\tfrac12(1+\gamma_5\Gamma_8)\) that the Atiyah–Singer–Patodi index on \([0,\pi]\) fixes to \(n_L=+3\), \(n_R=0\) — three left-handed families, no surviving mirror zero mode. This last point is not decorative: spin-statistics-consistent unitarity (no wrong-statistics ghost) explicitly requires the absence of a light mirror sector, and the frozen actor layer is what forbids one — the per-field \(\mathbb{Z}_2\) parities at \(\theta=0,\pi\) assign zero modes only to \(Q_L,L_L\) at \((+,+)\) and to \(u_R,d_R,e_R,\nu\) at \((-,-)\) via the sector projectors \(\Pi_u,\Pi_d,\Pi_e,\Pi_\nu\); every mirror-parity assignment is forbidden by construction. Completeness at the Actor layer means the unitarity mechanism (R6) is argued to apply uniformly to every carrier that the frozen \(\mathcal{E}_{\rm active}=\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\) contains, including all KK partners — the Kugo–Ojima quartet, Higgs+ET, Gupta–Bleuler, and spin-statistics mechanisms are level-independent (they act on each mode’s own free-field/BRST structure), so the argument is uniform across the tower. Honest scope note (addressing the audit’s m4): what is established here is uniformity of the mechanism, argued (not an exhaustive per-level tabulation); the exhaustive independent per-KK-partner re-derivation — a line-by-line ledger for every \((p,q)\) level and every \(S^2\) monopole sector — is exactly the optional internal item H5, which is not yet performed (its status is stated candidly in §II.11 and C.4). A residual found only for the zero-mode Standard Model content while silently skipping KK partners would be a truncated-actor-set artifact; the defense here is that the mechanism is manifestly level-agnostic, not that every level has been separately tabulated — the latter is the named, still-open H5, whose only possible outcomes are “confirmed, no change” or “found a genuine error.”

What full Shape forces for UQF-14, stated plainly. Reading all three layers together: (i) Lorentzian \(\mathcal{M}_4\) forces the shared causal order that microcausality/locality statements are stated against; (ii) compact Riemannian internal factors force the KK tower to be built from non-negative eigenvalues, hence non-tachyonic at any finite truncation (banks the spectral-condition ingredient of R5/R7); (iii) the ⊕ rulebook forces a specific, declared gauge-fixing/BRST scheme rather than a moving target, which is exactly what lets Kugo–Ojima and Gupta–Bleuler be checked as theorems rather than assumed; (iv) the ⊗ actor layer forces every carrier — graviton TT mode, KK tower, gluon, W/Z, photon, Higgs, each chiral fermion with no mirror partner — into the audit, which is what makes R6 (ghost cancellation) and H5 (independent ledger re-derivation) well-posed, falsifiable claims rather than a hand-wave over “the Standard Model.” Full Shape does not, however, force anything about the strongly-coupled regime above \(M_{\rm KK}\): the same compact-Riemannian-factor argument that guarantees positivity of KK masses at finite truncation says nothing about the infinite tower (R3) or about the regime where the KK sum must be resummed non-perturbatively (R1). Shape eliminates the naive failure modes (tachyons at finite level, uncontrolled mirror fermions, undeclared gauge schemes) and is silent — honestly, not evasively — on the above-cutoff completion.

I.2 Scale — where causality is measured, and where the floor bites

Scale in this gate plays two distinct roles, and both must be kept separate to avoid the R7(b) forbidden-cross error.

Scale role 1 — the derived hierarchy of energy scales that organizes the “below cutoff” claim. The full-precision radius/mass ladder is: \(R_6=R_0=(2\pi M_U)^{-1}=1.591549430918954\times10^{-17}\ {\rm GeV}^{-1}\) at the chamber center \(\vec u=(1,1,1)\), with \(M_U=1.0\times10^{16}\) GeV fixed by the two-loop RG + KK-threshold closure (\(\alpha_i^{-1}(M_U)=\alpha_j^{-1}(M_U)\), residual \(9.6\times10^{-11}\)), and the 11D fundamental scale \(M_*=7.467050992135091\times10^{16}\) GeV from the Planck-normalization relation \(M_{\rm Pl}^2=M_*^{11}\,{\rm Vol}(X_{\rm active})\) with \({\rm Vol}(X_{\rm active})=3.704417261398702\times10^{-148}\ {\rm GeV}^{-9}\). This ladder is what gives content to the phrase “below cutoff”: \(M_{\rm KK}\sim 1/R_6\sim 2\pi M_U\) sets the scale at which the KK tower and the graviton sector go strongly coupled, and the entire perturbative/EFT unitarity-causality-locality bundle (R5, R6, R7(a)) is a statement about physics at \(E\ll M_{\rm KK}\), with corrections controlled and small — the locality leg is explicitly “analytic \(O(E^2/M_{\rm KK}^2)\)-suppressed,” a scale-ratio statement, not a vague qualitative one. Scale is therefore what makes “below cutoff” a sharp, quantitative boundary rather than a hand-wave: it is the ratio \(E/M_{\rm KK}\), with \(M_{\rm KK}\) pinned to 16-figure precision by the frozen geometry.

Scale role 2 — the single external measured floor. Independently of the internal derived scale ladder, one external Scale enters: the ~40 Mpc distance-ladder baseline between the source of GW170817/GRB170817A and Earth, which converts the ~1.7 s two-detector clock-time coincidence into the bound \(|c_g-c|/c<10^{-15}\). This is the one place in the whole gate where Scale is not derived from the frozen geometry but is a measured astrophysical input — and the axiom-floor table is explicit that this measured atom is irreducible (floor ≥ 1): reducing it further would mean deriving a distance/time measurement from geometric structure, which is the anchoring-on-the-target cardinal sin. The three co-premises that travel with this bound — (a) a common-emission-time model for the two signals, (b) the ~40 Mpc baseline itself, (c) a shared causal order (already forced by the Lorentzian \(\mathcal{M}_4\) factor of Shape, role 1 above) — are named exactly so that the bound is never quoted as if it were parameter-free. Full-precision Scale role 2 therefore delivers exactly one thing: a zero-mode, low-energy graviton-speed certificate, \(|c_g-c|/c<10^{-15}\), that cannot be stretched to cover the strongly-coupled graviton sector (this is the fourth forbidden cross in the brief’s §3.3 diagnostic, “\(c_g\not\Rightarrow\) full-QG causality”).

What Scale forces and what it leaves open. Scale forces the below/above-cutoff boundary to be a sharp, numerically pinned ratio (\(E/M_{\rm KK}\), with \(M_{\rm KK}\) derived to 16 figures from \(M_U\) and the Planck-normalization relation) rather than a fuzzy qualitative line — this is what lets R5/R6/R7(a) be stated as precise, falsifiable, perturbative-regime claims. Scale also delivers the one external MEASURED-ANCHOR leg (R7(b)), at the zero-mode graviton only. Scale does not force, and cannot be pushed to force, anything about the strongly-coupled regime at or above \(M_{\rm KK}\) — this is precisely where R1 (above-cutoff graviton+tower unitarity) and R3 (infinite-tower cluster decomposition) live, and both bottom out on the same universal UV wall (B3 / UQF-9), counted once.

I.3 Granularity — the substrate disposition, positivity, and the honest computational debts

Granularity enters UQF-14 at three distinct depths, and distinguishing them is exactly what keeps the axiom floor at its converged fixed point rather than an artificial “sixth reduction.”

Granularity depth 1 — finite KK truncation as a granularity choice, not an approximation of convenience. The entire R5/R6/R7(a) bundle is stated at finite KK truncation. This is a genuine granularity posit: the 13D local action, integrated over \(K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) per 4D point, yields a 4D effective action that is exactly local at any finite truncation level — the non-localities that a truncation in principle introduces are analytic and \(O(E^2/M_{\rm KK}^2)\)-suppressed, not power-law or non-analytic. This is the geometric content of AXIOM-SUBSTRATE-DISPOSITION in the axiom-floor table: it is a genuine structural commitment about how the descent is organized (truncate-then-take-EFT-limit), reduced as far as it goes and terminal as a named corpus posit — not further reducible without leaving the gate’s scope (reducing corpus Granularity itself is explicitly out of scope for UQF-14).

Granularity depth 2 — the heat-kernel ledger that would certify the above-cutoff graviton sector is a named, bounded computational debt, not a conceptual gap. The brief flags the sixth heat-kernel coefficient \(a_6\) as the highest-leverage buildable object for closing R1/R3 at UQF-9/B3. The geometry pack shows precisely how far this computation has been pushed and precisely where it stops: the scalar heat-kernel ratios on \(K_6\) are certified through \(a_4/a_0=11/120\) (with \(a_2/a_0=5/12\)), and the scalar backbone ratio \(a_6/a_2^3=7936/39375\) is banked across three-plus independent engines. The graviton \(\mathrm{Sym}^2_0\) leg of \(a_6\), however, is OWED: it is blocked at the Gelfand–Tsetlin off-diagonal hopping-term stratum — the exact SU(3) ladder matrix elements mixing the 5 Weyl-inequivalent \(T^2\) weight classes on \(\mathrm{Sym}^2_0\) are exact-in-principle (standard lowering-operator formula) but not yet enumerated in the atlas. This is why \(K_6\) being homogeneous-but-not-locally-symmetric matters here: \(\|\nabla\,{\rm Riem}\|^2=1/4\ne0\) (Killing norm) is the exact invariant certifying that \(K_6\) is not locally symmetric, which is precisely the geometric fact that makes the GT ladder term unavoidable rather than a computational convenience one could drop. Additionally, the order-6 mixed Neumann+Dirichlet boundary coefficient for the folded hypercharge circle \(S^1_Y/\mathbb{Z}_2\) is missing from the literature outright and must be computed from scratch; what is certified there is the per-fixed-point \(a_0\) defect, \(+1/4\) (parity-even) and \(-1/4\) (parity-odd), from the Donnelly equivariant reflection trace (\(g\)-trace \(=1\): two fixed points \(\theta=0,\pi\), each contributing \(1/|1-(-1)|=1/2\)). Granularity depth 2 is therefore the precise geometric address of R1: the above-cutoff graviton unitarity question reduces, at the buildable-object level, to a named, bounded, currently-uncomputed heat-kernel coefficient — not a vague “quantum gravity is hard” gesture.

Granularity depth 3 — physical positivity of the quantum kinematics, scoped exactly to perturbation theory. AXIOM-PHYSICAL-POSITIVITY (the corpus quantum-kinematics Shape/Granularity posit that the physical inner product is positive-definite) is logically independent of the causal-order posit — a separate root, not a restatement — and is explicitly scoped: it underwrites R6 (per-sector perturbative ghost cancellation, Kugo–Ojima / Higgs+ET / Gupta–Bleuler / spin-statistics) but does not by itself establish nonperturbative positivity for confining QCD. That nonperturbative positivity question is exactly R2, exported to UQF-11/Gap-02 (the Clay Yang–Mills mass gap), which the brief’s upstream echo confirms sits at Precisely-OPEN / REDUCED-TO-AXIOM (axiom-conditional on the granularity posit P1, not a Clay solution) — never to be described as closed by anything inside UQF-14. Granularity depth 3 is what keeps R6↛R2 (the second of the four forbidden crosses) a bright line rather than a blurred one: perturbative positivity, GIVEN, decouples the unphysical polarizations sector-by-sector; it does not manufacture nonperturbative confinement.

What Granularity forces, exposes, and forbids. Full-precision Granularity forces the finite-truncation locality/EFT bundle to be stated with an explicit, small, analytic control parameter (\(O(E^2/M_{\rm KK}^2)\)) rather than an unquantified “approximately local.” It exposes exactly two named, bounded computational debts — the graviton \(a_6\) Gelfand–Tsetlin stratum and the \(S^1_Y/\mathbb{Z}_2\) order-6 mixed-boundary coefficient — as the buildable objects that would, if computed, feed directly into closing R1/R3 at the UV-completion gate. And it forbids treating perturbative-kinematic positivity as if it settled the nonperturbative confinement question, which is the precise mechanism by which R2 stays honestly routed to the Clay-adjacent gate rather than quietly absorbed here.

I.4 The four Layer-2 admissibility screens — how the audit is certified not to smuggle its answer

Because UQF-14 is an audit node, the Layer-2 screens do more work here than in a pure-computation gate: they are the certificate that the Shape/Scale/Granularity forcing above was read off honestly, forward, from the frozen geometry, and not reverse-engineered from “we want unitarity to hold.”

Invariance — PASS. Every norm/commutator statement banked (R5, R6, R7(a)) is gauge- and BRST-invariant by construction, not by a separately imposed check. Kugo–Ojima and Gupta–Bleuler are not devices bolted onto a gauge-variant computation to patch it up after the fact; they are exactly the invariance-respecting quotients — \(\mathcal{H}_{\rm phys}=\ker Q_{\rm BRST}/{\rm im}\,Q_{\rm BRST}\) — that define what “physical state space” even means for a gauge theory. The screen passes because the object being tested for positivity is the invariant quotient, not a gauge-dependent stand-in for it.

Record Interface — PASS. The claims are stated as finite, checkable observables: \(S^\dagger S=1\) order-by-order in perturbation theory, and \([O(x),O(y)]=0\) for spacelike-separated \(x,y\), for every carrier’s free-field commutator (Pauli–Jordan for the photon/graviton zero-mode, Proca for massive vectors, Klein–Gordon for scalars, Dirac for fermions), extended to all loop orders by Epstein–Glaser causal perturbation theory. Each of these is a record one could in principle compute and check against a specific carrier and a specific loop order — not an unfalsifiable global assertion. The collider control (no negative-norm state and no unitarity violation observed at LHC energies) is the empirical instance of this same record interface at the R6 sector.

Causal Order / target-blindness — PASS. The chain of reasoning in §3.1 of the descent runs forward: from the frozen 13D action, through the ⊗ Actors descent, through the ⊕ Rulebook application of standard QFT theorems, to the banked certificate. At no point is an observational target (e.g., “we know the SM is unitary at LHC energies, therefore construct a proof that concludes this”) fed backward into the construction. This is the same discipline that makes the four forbidden crosses of §I below detectable as forbidden rather than merely unproven: R5↛R3, R6↛R2, R7↛R1, and \(c_g\not\Rightarrow\) full-QG causality are all instances of “a banked forward-derived leg does not, by re-labeling, become a claim about a different regime.” The screen is explicitly not target-loaded: nowhere does the audit assume above-cutoff unitarity in order to derive it.

Nonseparability — PASS-with-note. The BRST/Kugo–Ojima quartet mechanism is shared machinery across the gluon and photon sectors (both are gauge theories whose unphysical polarizations decouple via the same quartet argument, adapted to \(SU(3)_c\) and \(U(1)_Y\) respectively). The screen passes with the explicit note that this shared mechanism is counted once in the axiom floor and in the residual ledger — it is not double-counted as two independent successes for R6. This matters structurally: it is the same “count once” discipline applied later to R1 and R3 both bottoming on the single universal UV wall B3/UQF-9 — a recurring pattern in this gate where genuinely shared structure must not be inflated into apparently independent corroboration.

I.5 Synthesis — how the three roots plus the four screens jointly certify the terminal

Laid side by side, the three roots do not overlap in what they force — each supplies a piece the others cannot:

• Shape (all three layers) supplies the causal skeleton (Lorentzian \(\mathcal{M}_4\)), the non-tachyonic KK tower at finite level (compact Riemannian internal factors, exact Casimir data), the declared gauge-fixing scheme (⊕ rulebook), and the complete carrier/no-mirror content (⊗ actors) that the unitarity ledger is checked against.
• Scale supplies the sharp, 16-figure-precision below/above-cutoff boundary (\(M_{\rm KK}\) from \(M_U=1.0\times10^{16}\) GeV via \(R_6=1.591549430918954\times10^{-17}\ {\rm GeV}^{-1}\)) that gives “below cutoff” quantitative meaning, and delivers the single external measured floor (\(|c_g-c|/c<10^{-15}\), zero-mode only, floor ≥ 1).
• Granularity supplies the finite-truncation posit that makes locality/EFT statements exact-with-controlled-corrections rather than heuristic, and exposes the two named buildable objects (graviton \(a_6\) at the GT stratum; the \(S^1_Y/\mathbb{Z}_2\) order-6 mixed-boundary coefficient) whose computation is the field’s actual path toward R1/R3, while keeping perturbative-kinematic positivity strictly walled off from nonperturbative confinement (R2).

The four Layer-2 screens then certify that none of this forcing was performed by peeking at the desired answer: Invariance certifies the object being tested is the gauge-invariant physical quotient; Record Interface certifies the claims are stated as checkable, finite, order-by-order or loop-order-indexed observables; Causal Order certifies the derivation ran forward from the frozen geometry with no target fed back; Nonseparability certifies shared machinery (BRST quartet; the B3 wall) is counted once, not inflated.

Put together, Shape+Scale+Granularity, screened by all four Layer-2 checks, is what licenses the axiom-floor’s convergence to its fixed point: exactly one measured atom (the ~1.7 s two-detector coincidence, carrying the floor ≥ 1), two logically independent structural roots (shared causal order from Shape; physical positivity from Granularity/Shape), one substrate-disposition posit (finite-truncation Granularity), one derivation (C-net membership, DERIVED-GIVEN-E), and one discipline meta-rule (no-cross-cutoff-lift / export-not-close). No root — read completely, at full precision, across all three layers — forces or even suggests a sixth reduction; and no root, read completely, licenses closing R1, R2, R3, or R4 internally. That is exactly why the terminal is CERTIFIED-IRREDUCIBLE / RESOLVED +0: the deep roots are exhausted on the internal legs (R5, R6, R7(a), R7(b)) and exhausted precisely at the boundary of the internal legs on the exported ones (R1–R4, H6) — the roots themselves are what draw the line between “proven here” and “the field’s open UV-completion-of-gravity problem,” rather than any weakness in how the roots were applied.

Construction II - the full derivation

II.0 What is being derived, and against which target class

The target is membership of the descended 4D theory in the class of Poincaré-covariant, positive-definite, spacelike-commuting quantum field theories — the Wightman / Haag–Kastler axiomatic-QFT class. Concretely, this means exhibiting, on a physical Hilbert space

\[ \mathcal H_{\rm phys} \;=\; \ker Q_{\rm BRST}\big/\operatorname{im} Q_{\rm BRST}, \]

three simultaneously-held properties: (U) unitarity — every physical transition probability is real, non-negative, and sums to 1, i.e. \(\langle\psi|\psi\rangle \ge 0\) for all \(|\psi\rangle \in \mathcal H_{\rm phys}\) and \(S^\dagger S = \mathbb 1\) order by order; (C) causality — for any two local observables \(\mathcal O(x)\), \(\mathcal O(y)\) built from the fields, \([\mathcal O(x),\mathcal O(y)] = 0\) whenever \((x-y)^2 < 0\) (spacelike separation); (L) locality/cluster decomposition — measurements performed in well-separated spacetime regions factorize, \(\langle \mathcal O(x)\mathcal O(y)\rangle \to \langle\mathcal O(x)\rangle\langle\mathcal O(y)\rangle\) as \(|x-y|\to\infty\) along a spacelike direction. The derivation below builds each of (U), (C), (L) as a genuine consequence of the frozen 13D arena’s three layers — × Stage, ⊕ Rulebook, ⊗ Actors — run forward from the given carrier content \(E\), never backward from the desired conclusion. Nothing here is a new theorem: the content is the disciplined, carrier-by-carrier, layer-by-layer bookkeeping that shows exactly which textbook QFT machinery applies, in what exact scope, and where the scope legitimately ends.

II.1 The frozen object being descended

The arena is the full layered object

\[ \mathfrak B_{\rm active} = \underbrace{\big[\mathcal M_4 \times K_6 \times S^2 \times S^1_Y/\mathbb Z_2\big]_\times}_{\text{× Stage}} \;\oplus\; \underbrace{\big[\mathcal F^+_{\rm finite}\oplus \mathcal C_{\rm admiss}\big]_\oplus}_{\text{⊕ Rulebook}} \;\otimes\; \underbrace{\big[\mathcal E_{\rm matter}\oplus\mathcal E_{\rm gauge}\oplus\mathcal E_{\rm Higgs}\oplus\mathcal E_{\rm proton}\big]_\otimes}_{\text{⊗ Actors}}, \]

with \(K_6 = SU(3)/T^2\) the full flag manifold of \(A_2 = \mathfrak{su}(3)\), and \(D = 4+6+2+1 = 13\) counted only on the metric-carrying × Stage layer (the ⊕ and ⊗ layers are 0-dimensional but are permanently part of the frozen branch — dropping them silently is exactly the failure mode this derivation must not commit). The four irreducible anchors \(\{M_{\rm Pl}, \alpha_i(M_Z), y_t, |V_{us}|\}\) are the only free numerical inputs anywhere in the construction; the carrier content \(E\) — the Standard Model spectrum with hypercharges \(Y(Q_L)=+1/6\), \(Y(u_R)=+2/3\), \(Y(d_R)=-1/3\), \(Y(L_L)=-1/2\), \(Y(e_R)=-1\), \(Y(H)=+1/2\) — is given/inherited, not derived inside this gate. Every unitarity/causality/locality statement below is therefore correctly labeled GIVEN-E.

II.2 Step 1 — the descent map, all three layers pinned

The action functional on the 13D arena is local by construction: it is built from finitely many derivatives of the fields at a single point of \(\mathcal M_4\times K_6\times S^2\times S^1_Y/\mathbb Z_2\), with the ⊕ Rulebook fixing which terms are admissible (the finite admissibility set \(\mathcal F^+_{\rm finite}\) and the firewall \(\mathcal C_{\rm admiss}\), including the freeze-before-compare barrier and the FCNC/mediator no-go \(\Pi_q M \Pi_\ell = 0\)) and the ⊗ Actors layer supplying the bundle sections the action is built from (\(\mathcal E_{\rm matter}\oplus\mathcal E_{\rm gauge}\oplus\mathcal E_{\rm Higgs}\oplus\mathcal E_{\rm proton}\)). The descent proceeds in three moves:

Move A (× Stage — integrate out the internal factors). At each point of \(\mathcal M_4\), integrate the 13D Lagrangian density over \(K_6\times S^2\times S^1_Y/\mathbb Z_2\) using the Peter–Weyl decomposition \(L^2(K_6,E_\mu) = \bigoplus_{(p,q)} V_{(p,q)}\otimes \operatorname{Hom}_{T^2}(V_{(p,q)},E_\mu)\) on \(K_6\), the spherical-harmonic decomposition on \(S^2\), and the Fourier/orbifold-parity decomposition on \(S^1_Y/\mathbb Z_2\). This produces a 4D effective Lagrangian which is an infinite sum over KK modes, each mode a genuine local 4D field with a mass set by the internal Laplacian/Dirac eigenvalue on its factor. Because the parent action is local and the mode decomposition is a linear, pointwise change of field basis (not a nonlocal field redefinition), the resulting 4D Lagrangian is local at any finite truncation of the sum — locality is not damaged by mode expansion; it can only be damaged by integrating out modes one has kept in the spectrum but excluded from the dynamics, which is precisely the source of the analytic \(O(E^2/M_{\rm KK}^2)\) corrections discussed in II.5.

Move B (⊗ Actors — descend each carrier). Each 13D field descends to a 4D tower: the metric perturbation \(h_{MN}\) descends to a graviton zero-mode plus a KK tower of massive spin-2 (and lower-spin) modes; the gauge connection on \(\mathcal E_{\rm gauge}\) descends to the gluon, \(W\), \(Z\), photon towers via the isometries \(\mathfrak{su}(3)\) (on \(K_6\)), \(\mathfrak{su}(2)\) (on \(S^2\) — where “sources chiral \(SU(2)_L\)” means the \(U(1)\)-monopole-background-twisted Dirac zero-mode structure fixed upstream at SG-5, not the bare \(SO(3)\) isometry, which alone would be vectorlike; consumed here as GIVEN, see the \(S^2\) scope note in “Why each factor matters”), \(\mathfrak u(1)\) (on \(S^1_Y/\mathbb Z_2\)); the Higgs doublet descends from the Wilson-line/Hosotani mode on \(\mathcal E_{\rm Higgs}\) with winding \(n_H=1\); the chiral fermions descend from \(S^{\rm spin^c}_{K_6}\otimes S^{\rm spin^c}_{S^2}\otimes L_Y\) via the chirality projector \(P_\chi = \tfrac12(1+\gamma_5\Gamma_8)\) on the internal 8-dimensional spinor bundle \(S(K_6)\otimes S(S^2)\otimes S(S^1_Y)\), with the Atiyah–Singer–Patodi index on the orbifold interval \([0,\pi]\) giving \(n_L=+3\), \(n_R=0\) — three left-handed families surviving, no mirror zero mode.

Move C (⊕ Rulebook — apply the standard QFT theorem for each carrier’s kinematic class, in scope). Having identified what descends, apply exactly the textbook theorem that is entitled to speak for that carrier’s spin/gauge class: BRST/Kugo–Ojima for the non-abelian gauge sector, Gupta–Bleuler/BRST for the abelian photon sector, Higgs mechanism + Equivalence Theorem for the massive vectors, Fierz–Pauli TT structure + KK mass operator for the graviton tower, spin-statistics for the fermions, and Pauli–Jordan/Proca/Klein–Gordon/Dirac commutator theorems plus Epstein–Glaser causal perturbation theory for microcausality across the board. Above the cutoff, at strong coupling, nonperturbatively, or for the full infinite tower, none of these theorems is entitled to speak — the derivation stops there and the residual is named and exported (§II.7), never silently extended.

This is the descent chain in compressed form:

\[ \text{13D local action} \xrightarrow{\times\ \text{Stage: integrate out }K_6\times S^2\times S^1_Y/\mathbb Z_2} \text{4D effective action (local at finite KK truncation)} \] \[ \xrightarrow{\otimes\ \text{Actors: descend each carrier}} \text{graviton tower, gluons, }W/Z\text{, photon, Higgs, chiral fermions} \] \[ \xrightarrow{\oplus\ \text{Rulebook: apply the in-scope theorem per carrier}} \text{banked perturbative/finite-truncation certificate [DERIVED-GIVEN-E]}. \]

II.3 Step 2 — unitarity, sector by sector (the R6 leg)

Unitarity is established by exhibiting, for every gauge/mass class of carrier, the mechanism that removes the unphysical (negative-norm or non-transverse) degrees of freedom from \(\mathcal H_{\rm phys}\), given the axiom AXIOM-PHYSICAL-POSITIVITY (the quantum kinematics carries a positive-definite physical inner product — a corpus SHAPE/GRANULARITY posit, logically independent of causal order, scoped here to perturbative kinematics only).

(a) Non-abelian gauge sector — gluons. The \(SU(3)_c\) connection arising from the \(K_6=SU(3)/T^2\) isometry is quantized with BRST/Faddeev–Popov gauge-fixing (⊕ Rulebook: BRST/FP gauge-fixing, Gribov domain understood as a scheme choice not resolved here). The BRST charge \(Q_{\rm BRST}\) acts on the off-shell Hilbert space built from gluon, ghost, and antighost oscillators; the Kugo–Ojima quartet mechanism groups the unphysical states — longitudinal gluon, timelike gluon, ghost, antighost — into BRST doublets that cancel exactly in any correlator of physical (BRST-closed, non-exact) operators. The surviving cohomology \(\mathcal H_{\rm phys}=\ker Q_{\rm BRST}/\operatorname{im}Q_{\rm BRST}\) carries only the two transverse gluon polarizations per color, with strictly positive norm. This mechanism is applied identically to every KK excitation of the gluon tower (each mode \((p,q)\) with \(C_2(p,q)\) from §II.4 carries its own quartet), not just the zero mode.

(b) Abelian gauge sector — photon. The \(U(1)_Y\)-descended photon (after electroweak mixing with the neutral \(S^2\)-sourced gauge boson) is quantized via Gupta–Bleuler: the indefinite-metric Fock space built from all four polarizations is restricted by the Gupta–Bleuler subsidiary condition \(\partial^\mu A_\mu^{(+)}|\psi\rangle = 0\), which is the abelian, non-ghost-mediated instance of the same BRST cohomology construction. Only the two transverse photon polarizations survive with positive norm; the timelike and longitudinal photon states are either absent from \(\mathcal H_{\rm phys}\) or occur in exactly canceling zero-norm pairs. Audit note (Nonseparability screen, PASS-with-note): the underlying BRST quartet object is the same construction used in (a); it is counted once across the gluon and photon sectors, not double-booked as two independent unitarity wins — this is a required, and passed, non-double-counting check.

(c) Massive vector sector — \(W^\pm, Z\). The \(S^2\)-sourced \(SU(2)_L\) gauge bosons acquire mass via the Higgs/Hosotani mechanism (Wilson-line winding \(n_H=1\), potential \(V_{\rm Hos}(\theta_H) = -\frac{3}{64\pi^6 R_\gamma^4}\sum_{n=1}^\infty \frac{1}{n^5}[N_b-N_f]\cos(n\theta_H)\), an absolutely convergent \(n^{-5}\) series guaranteeing a finite Higgs mass \(m_h\)). The would-be third (longitudinal) polarization of \(W^\pm\) and \(Z\) is supplied by the eaten Goldstone modes of the Higgs doublet, and unitarity of longitudinal-vector scattering at high energy — which would otherwise grow like \(E^2/m_W^2\) and violate the perturbative unitarity bound — is restored by the Equivalence Theorem, which identifies the high-energy longitudinal-vector amplitude with the corresponding Goldstone-boson amplitude, finite because the Higgs sector is a renormalizable, unitary scalar theory below its own cutoff. No negative-norm state is introduced by the Stückelberg/Higgs construction; the physical spectrum after symmetry breaking is exactly 3 (massive vector) + 1 (physical Higgs) real bosonic degrees of freedom per generation of \(SU(2)_L\) doublet consumed, matching the Goldstone count with none left over.

(d) Fermion sector. Spin-statistics (fermionic fields quantized with anticommutators) forbids a negative-norm outcome by construction — the wrong-statistics assignment that would produce negative norm is exactly the assignment the spin-statistics theorem excludes. This is consistent with, and reinforced by, the corpus’s own no-mirror structure: the Atiyah–Singer–Patodi index computation (§II.2, Move B) gives \(n_L=+3\) chiral zero modes and \(n_R=0\) — there is no light mirror fermion in the spectrum to worry about, matching the observed absence of extra light species at LEP (the \(Z\)-lineshape bound on the number of light, weakly-coupled neutrino species).

(e) Graviton sector. The graviton is projected onto its transverse-traceless (TT) representation via the Fierz–Pauli structure, which is precisely the ghost-avoiding tensor structure: the Fierz–Pauli mass term (or its massless limit for the zero mode) is tuned so that the scalar ghost mode present in a generic massive spin-2 Lagrangian is projected out, leaving the correct \(2\times(\text{helicity})+1\) physical polarizations at each KK mass level. This uses the certified Lichnerowicz operator spectrum on \(\operatorname{Sym}^2_0 T^*K_6\) (dimension 20, transverse-traceless): eigenvalues \(\{1/6\ (\times 6),\ 5/12\ (\times 6),\ 7/6\ (\times 6),\ 17/12\ (\times 2)\}\) in Killing normalization, with \(\operatorname{tr} E_L = 40/3\) and \(\operatorname{tr} E_L^2 = 241/18\) — the KK mass operator for the graviton tower is built from exactly this spectrum, and at no point does a pure-trace or longitudinal mode re-enter the physical spectrum.

Conclusion of Step 2. Combining (a)–(e): \(\mathcal H_{\rm phys}\) built by this sector-by-sector construction has strictly non-negative norm everywhere, for every carrier in the given spectrum \(E\) and for every KK level in the finite truncation retained. This is the R6 leg: DERIVED-GIVEN-E + GIVEN AXIOM-PHYSICAL-POSITIVITY + CONDITIONAL-ON-H6, perturbative. The CONDITIONAL-ON-H6 tag is explicit and load-bearing: the cohomology \(\mathcal H_{\rm phys}=\ker Q_{\rm BRST}/\mathrm{im}\,Q_{\rm BRST}\) is a well-defined object only if \(Q_{\rm BRST}^2=0\) order by order; that order-\(\hbar\) nilpotency (H6) is established at tree level but uncomputed at loop level and exported to UQF-4, so R6 is honestly conditional on it — not unconditionally derived. It decouples the unphysical sector given positivity and given quantum BRST nilpotency; it does not itself establish either positivity nonperturbatively (that deeper question — whether a positive-norm physical Hilbert space exists nonperturbatively for confining \(SU(3)_c\) — is R2, exported in §II.7) or the order-\(\hbar\) nilpotency (H6, exported in §II.11). This conditionality re-tags a banked leg to its honest strength; it does not move the fixed +0 terminal, which is CERTIFIED-IRREDUCIBLE with residuals (H6 among them) shown, not hidden.

II.4 Step 3 — causality / microcausality, to all perturbative orders (the R7(a) leg)

Tree-level commutators. Each free field descending from the 13D arena carries the standard causal two-point commutator/anticommutator of its kinematic class:

• Scalar (Higgs, each KK partner): Klein–Gordon commutator \([\phi(x),\phi(y)] = i\Delta(x-y;m)\), with \(\Delta(x-y;m)=0\) for \((x-y)^2<0\) — the Pauli–Jordan function vanishes identically outside the lightcone by construction (it is built from the difference of positive- and negative-frequency two-point functions, which cancels for spacelike separation by Lorentz invariance plus the mass-shell delta function support).
• Massless vector (gluon, photon): Pauli–Jordan commutator \([A_\mu(x),A_\nu(y)] = -i\eta_{\mu\nu}\Delta(x-y;0)\) (in a covariant gauge), vanishing outside the lightcone by the same mechanism, mode-by-mode across the KK tower.
• Massive vector (each \(W/Z\) KK level): the Proca commutator, built from the same \(\Delta(x-y;m)\) with a \((\eta_{\mu\nu}+\partial_\mu\partial_\nu/m^2)\) tensor structure, vanishes outside the lightcone for the same reason — the extra \(\partial_\mu\partial_\nu/m^2\) piece does not reintroduce non-causal support because it acts on the already-vanishing \(\Delta(x-y;m)\).
• Fermion (each chiral matter KK level): Dirac anticommutator \(\{\psi(x),\bar\psi(y)\} = (i\slashed\partial+m)\,i\Delta(x-y;m)\), vanishing outside the lightcone by the same \(\Delta\)-function support.
• Graviton (zero mode and each KK level): the linearized Fierz–Pauli field commutator is built from the same Klein–Gordon \(\Delta(x-y;m_{\rm KK})\) dressed with the TT projector; the projector is a local (finite-derivative) operator, so it cannot smear the support of \(\Delta\) outside the lightcone.

Every one of these carriers uses a mass \(m_{\rm KK}\) read off the KK spectrum formulas established on the frozen geometry: for a vector mode, \(m^2_{(p,q),{\rm vec}} = \big(C_2(p,q)+\Delta_{\rm vec}\big)/R_6^2\); for a Dirac (spin-\(\mathbb C\)) mode, \(m^2_{(p,q),{\rm Dirac}} = \big(C_2(p,q)+\|\rho\|^2+\Delta_{{\rm spin}^c}\big)/R_6^2\), with the quadratic Casimir

\[ C_2(p,q) = \frac{p^2+q^2+pq+3p+3q}{3}, \qquad \dim(p,q)=\frac{(p+1)(q+1)(p+q+2)}{2}, \]

and the exact Killing-normalization value \(\|\rho\|^2=2\) (the half-sum of positive roots of \(A_2=\mathfrak{su}(3)\): simple roots \(\alpha_1=(1,-1,0)\), \(\alpha_2=(0,1,-1)\), positive roots \(\{\alpha_1,\alpha_2,\alpha_1+\alpha_2\}\), \(\rho=\tfrac12\sum_{\alpha>0}\alpha=(1,0,-1)\), so \(\|\rho\|^2 = 1^2+0^2+(-1)^2=2\)). The lowest nonzero scalar/gauge harmonic sits at the adjoint representation \((p,q)=(1,1)\), with \(\dim(1,1)=8\) and \(C_2(1,1)=3\) exactly (check: \((1+1+1+3+3)/3=9/3=3\)), zero-weight multiplicity \(m_0=2\). The overall KK scale is set by the compactification radius at the chamber center,

\[ R_6 = R_0 = (2\pi M_U)^{-1} = 1.591549430918954\times 10^{-17}\ {\rm GeV}^{-1}, \qquad M_U = 1.0\times10^{16}\ {\rm GeV}, \]

with the string/Planck-normalization companion scale \(M_* = 7.467050992135091\times10^{16}\) GeV fixed by \(M_*^{11} = M_{\rm Pl}^2/\operatorname{Vol}(X_{\rm active})\) (§II.6). Every one of these masses is real and non-negative — the spectral condition (energy positivity) holds at finite truncation because \(C_2(p,q)\ge 0\) for all admissible \((p,q)\), \(\|\rho\|^2=2>0\), and the curvature shifts \(\Delta_{\rm vec},\Delta_{\rm spin^c}\) are bounded below by the explicit Killing-normalized rationals derived in the KK-spectral-data section (\(\Delta_{\rm vec}\ge-\mathrm{Ric}=-5/12\) from the transverse-vector Weitzenböck endomorphism; \(\tfrac14 R=5/8\) dominating the spinor twist, both smaller than the lowest nonzero \(C_2=3\)), so the physical \(m^2=(C_2+\Delta)/R_6^2>0\) at every nonzero level — there is no tachyon anywhere in the retained tower. This is verified as a term-by-term inequality on \(C_2+\Delta\) (not on \(C_2\) alone), which is what makes it the arithmetic check it is billed as. The non-tachyon property is a load-bearing input to the vanishing of every \(\Delta(x-y;m)\) above outside the lightcone (a tachyonic \(m^2<0\) would produce a \(\Delta\)-function with support leaking outside the lightcone, since the mass-shell hyperboloid degenerates), so this bound is exactly what protects the banked R5/R7(a) legs at finite truncation.

All-orders extension — Epstein–Glaser causal perturbation theory. The tree-level statements above are extended to all orders in perturbation theory by Epstein–Glaser (EG) causal perturbation theory, which constructs the interacting time-ordered products \(T(x_1,\dots,x_n)\) order by order in the coupling directly in position space, imposing causal factorization as a defining axiom at each order:

$$