The Companion Theorem Programme

17.1 Purpose

The preceding chapters select the mathematical objects, physical mechanisms and inter-sector maps of Shadow Theory. Nine remaining theorem programmes carry the deeper work. They are stated here so that each companion paper begins from a fixed construction rather than redesigning the theory. The present chapter neither replaces those proofs nor treats their absence as a defect in the architectural monograph.

17.2 R1 — Minimal stable source closure

Definitions. Let $\mathfrak G_{\rm src}$ be the target-neutral family of finite source cells, including the initial cell $\mathbf0_{\rm src}$, their source faces, and cells for the finite source-local incidence types in the fixed source alphabet. For each noninitial cell $C$, let $F(C)$ be the source model generated by its incidence, dependency, marking, lineage and intrinsic boundary equations, and define

$$\mathcal W_0 =\operatorname{IsoSat} \{F(C):C\in\mathfrak G_{\rm src},\ C\ne\mathbf0_{\rm src}\}.$$

The walking marked dependency is one member of $\mathcal W_0$, not the whole witness basis. If $\partial C$ is the colimit of the proper source faces of $C$, a generator-relative attachment has the form

$$\begin{array}{ccc} \partial C&\longrightarrow&C\\ \downarrow&&\downarrow\\ X&\longrightarrow&X\cup_{\partial C}C, \end{array}$$

where the attaching map is strong and mark-reflecting. It adds only the interior generators and relations of $C$ along the displayed source face. Let $\mathfrak L_{\rm src}$ be the complete lattice of isomorphism-saturated classes of models of the marked source signature, ordered by inclusion, and define

$$\mathbb L_{\rm src}(A) =\operatorname{IsoSat}\!\left( \mathcal W_0\cup A\cup\mathscr O_1^{\rm NFE}(A) \right)$$

where $\mathscr O_1^{\rm NFE}$ contains only mark-preserving generated images, generator-relative attachments, generated retracts and stabilized generated colimits. The admitted class is

$$\operatorname{Ob}\mathsf S_{\rm adm}=\mu\mathbb L_{\rm src}.$$

Target and conclusion. Prove that $\mathbb L_{\rm src}$ is monotone, that its transfinite iteration from $\varnothing$ stabilizes, and that the resulting nonempty least fixed point is closed under precisely the stated generated operations while preserving the marked deposition structure and declared source invariants. Prove in particular that generator-relative attachments preserve the source axioms and that the admitted class is independent, up to strong source isomorphism, of a compatible refinement of the finite cell presentation.

Hypotheses. The source signature, source-local alphabet, finite cell family, source-face maps and witness class are well defined; the family contains a noninitial marked cell; generator-relative pushouts exist on their declared strong mark-reflecting attachments; the permitted generated images, attachments, retracts and stabilized colimits remain in the source category; source morphisms preserve the marked structure; isomorphism saturation is small in the chosen universe.

Lemma route. (i) construct the target-neutral finite cell family and show that the walking marked dependency is one witness among its noninitial cells; (ii) prove that source faces and generator-relative attachments preserve dependency, marked support, strict pairs, lineage and intrinsic boundaries; (iii) prove compatibility of cell presentations under common refinement and presentation independence up to strong source isomorphism; (iv) prove preservation under the remaining generated operations; (v) prove monotonicity and inflationarity; (vi) identify the closure ordinal of the transfinite chain; (vii) prove fixed-point minimality; (viii) prove closure of strong source morphisms and invariants.

Dependencies, limits, and comparisons. R1 has no physical-sector dependency. Concrete finite source families provide the first examples; comparison is by source invariants and closure rank, not by agreement with a desired Tier-1 output.

Companion-paper boundary. The companion proof must establish the preservation lemmas, nontrivial concrete families and their richness. R1 does not assert that any admitted source has a successful physical readout.

17.3 R2 — Realization and compatible Tier-1 projection

Definitions. For $X\in\mathsf S_{\rm adm}$, realizations lie in $\operatorname{RealPrep}_{\Omega}[X]$. Each sector relation $\mathscr K_s(Y)\subseteq\mathsf O_s^{\rm raw}\times\mathsf D_s$ retains a candidate and its discrepancy. The raw coupled family is $\mathcal Z_Y^{\rm raw}$, and redescription gives $\widetilde{\mathcal Z}_Y$. For each sector, fix its discrepancy $\delta_s$ and tolerance $\varepsilon_s\ge0$, and define the physical restriction

$$\mathsf P_s(\varepsilon_s) =\{p_s\in\mathsf O_s^{\rm raw}: \delta_s(p_s)\le\varepsilon_s\}.$$

The sectorwise partial physical relation is

$$\Gamma_{\rm phys}^{\rm sect}: \widetilde{\mathcal Z}_Y \dashrightarrow \prod_{s\in V}\mathsf P_s(\varepsilon_s),$$

with exact sector theory at $\boldsymbol\varepsilon=\mathbf0$. A fixed coarse map $C_{\mathfrak c}$ is attached only after the relevant sectorwise tests and induces the kernel equivalence $\sim_{\mathfrak c}$. The sectorwise relation descends to the coarse quotient only when its domain and output are proved constant on every coarse fibre. Pairwise comparison maps are defined conditionally on overlaps of successful sector domains. Their simultaneous coherent equalizer and its nonemptiness are the subject of R9, not an assumption or conclusion of R2.

Target and conclusion. Prove conditionally that raw construction precedes physical filtering, that redescription and coarse observation are natural and logically distinct, that the successful sector restrictions and pairwise interface maps are well defined on their stated domains, and that unsuccessful raw branches remain outside those restrictions rather than being removed from the construction. When a physical map is asserted on a coarse quotient, prove its constancy on the corresponding coarse fibres. R2 makes no claim that a common globally compatible fibre is nonempty.

Hypotheses. R1; a prepared realization for the branch under study; well-defined raw sector relations, physical restrictions, redescription and coarse-observation maps; and pairwise comparison maps on their declared common domains.

Lemma route. (i) prove realization preparation is source-local; (ii) define each raw sector relation on its realization domain and verify its candidate and discrepancy spaces; (iii) construct the sectorwise restrictions $\mathsf P_s(\varepsilon_s)$ and prove redescription invariance; (iv) prove that $\ker C_{\mathfrak c}$ is an equivalence relation and respects protocol refinement; (v) prove the conditions under which each sectorwise restriction descends through that quotient; (vi) prove naturality and commutation for each conditional pairwise interface on its common successful domain.

Dependencies, limits, and comparisons. R2 depends on R1. Its comparison tests are the ordinary quantum, flat-QFT, classical-field and other Tier-1 limits of individual successful branches. Source lineage remains inverse-problem information and is not inserted into the physical output.

Companion-paper boundary. Establish realization support and the conditional descent and interface theorems for specified source classes. Global fixed points and nonempty simultaneous compatibility belong to R9. Physical success is not made automatic by restricting the range in advance.

17.4 R3 — Objective records, metric reconstruction, and gravity

Definitions. Fix a realization $Y$, network $\mathcal N$ and gravitational boundary data $B_Y$. The archive used in this branch must satisfy

$$\mathcal A_Y^{\rm obj}\in \operatorname{ObjArch}_Y(g,\omega,\mathcal I,u;\mathcal N), \qquad (R,J)\in\operatorname{Emb}_g(\mathcal A_Y^{\rm obj}).$$

The same operational QFT on $g$, normal instruments $\mathcal I$, selector coordinate $u$, actual-deposition kernels and objectivity tests must regenerate $\mathcal A_Y^{\rm obj}$; an arbitrary pre-existing archive cannot be inserted into the gravitational branch. The embedding $(R,J)$ first supplies the unrestricted covariant kernel $\Gamma_{\rm arch,CTP}^{\rm full}$. At a fixed scale and in a fixed EFT operator basis, let

$$\mathfrak m_R =\bigl(\mu,\mathcal B_{\rm op}, \Pi_M,\Pi_{\rm QFT},\Pi_R,\Pi_{MR}\bigr)$$

be the matching datum, with mutually orthogonal projections whose sum is the identity. The terms already carried by the matter and QFT actions are matched through $\Pi_M$ and $\Pi_{\rm QFT}$, while

$$\Gamma_{R,\rm CTP} =\Pi_R\Gamma_{\rm arch,CTP}^{\rm full}, \qquad \Gamma_{MR,\rm CTP} =\Pi_{MR}\Gamma_{\rm arch,CTP}^{\rm full}$$

are the record-exclusive and genuinely mixed remainders. The scheme $\mathfrak m_R$ is fixed before $\operatorname{Sol}_G(\cdot;\mathfrak m_R)$ is evaluated. From this same-branch archive, form $\kappa^{\rm inf}$, the cycle-safe quotient $\mathsf P_Y$, the strict order $\prec_D$, the relational distance $d_{\rm rel}$, and a cofinal refinement sequence $(\mathfrak P_{Y,n})_n$.

A candidate $C$ is metric-comparable for archive $I$ only through the construction of Chapter 7, equations (7.16)–(7.23). On a Lorentzian branch its normalized comparison region $A_C$ must be compact, causally convex and causally distinguishing, with finite continuous time separation $\tau_C$, and

$$\begin{aligned} d_{\rm str}^{C}(p,q) &=\sup_{r\in A_C} \left| \tau_C(p,r)+\tau_C(r,p) -\tau_C(q,r)-\tau_C(r,q) \right|,\\ d_C^{\rm cmp}(x,y)&=d_H^{\,d_{\rm str}^{C}}(B_x,B_y). \end{aligned}$$

The same normalized $\widehat g_{C,I}$ induces

$$x\prec_Cy\Longleftrightarrow B_x\times B_y\subset I^+_{\widehat g_{C,I}}, \qquad \mu_C=(q_C)_*\overline{\operatorname{Vol}}_{|\widehat g_{C,I}|}, \qquad n_C=\dim A_C.$$

For a bandwidth $\epsilon_C$ fixed before reconstruction, its Gaussian cell weights and normalized graph Laplacian are

$$\begin{gathered} W_{xy}=\exp\!\left[-(d_C^{\rm cmp}(x,y)/\epsilon_C)^2\right] \quad(x\ne y), \qquad W_{xx}=0,\\ D_x=\sum_yW_{xy}, \qquad L_C=\mathbf1-D^{-1/2}WD^{-1/2}. \end{gathered}$$

Weights at infinite distance are zero, and isolated cells are assigned zero blocks. Hence $L_C$ is self-adjoint and positive semidefinite with a positivity-preserving heat semigroup. A Riemannian candidate uses the normalized geodesic distance and carries no chronological order. The response family $\operatorname{DepSig}_C$ is calculated from the QFT and matched record dynamics on that same geometry. Failure of compactness, causal convexity, causal distinction, continuity or finiteness of $\tau_C$, finite positive volume, cell support or assignment, fixed bandwidth, dimension, or any same-geometry datum leaves the candidate raw and assigns $+\infty$ to every affected discrepancy.

Write $\mathbf C=(I,C,\phi_C)$ for a normalized metric-comparable candidate. On declared common refinements, the directed primitive is

$$\delta_{\rm ref}(\mathbf C,\mathbf C') =\inf_{R\in\mathfrak R(\mathbf C,\mathbf C')} \max\!\left\{ D_{\rm ord}^{R},D_{\rm vol}^{R},D_{\rm nerve}^{R}, D_{\rm dim}^{R},D_{\rm context}^{R},D_{\rm refine}^{R} \right\},$$

where every discrepancy is recomputed after transport to the common refinement. Set

$$\delta_{\rm sym}(\mathbf C,\mathbf C') =\max\!\left\{ \delta_{\rm ref}(\mathbf C,\mathbf C'), \delta_{\rm ref}(\mathbf C',\mathbf C) \right\},$$

and define the extended path pseudometric by the capped edge cost

$$d_{\rm ref}(\mathbf C,\mathbf C') =\inf_{\substack{m\ge0,\ \mathbf C_0=\mathbf C,\ \mathbf C_m=\mathbf C'\\ \mathbf C_1,\ldots,\mathbf C_{m-1}}} \sum_{j=1}^{m} \min\!\left\{1, \delta_{\rm sym}(\mathbf C_{j-1},\mathbf C_j) \right\}.$$

The refinement uniformity is generated by $\mathcal U_\eta=\{(\mathbf C,\mathbf C'): d_{\rm ref}(\mathbf C,\mathbf C')\lt \eta\}$, $\eta>0$. These are the four constructions (7.30)–(7.33). The comparison transports must satisfy identity and composition on every common refinement. No unproved one-step discrepancy is itself called a distance. Metric candidates form the set

$$\mathcal G_Y=\operatorname{Rec}_{\rm met}((\mathfrak P_{Y,n})_n),$$

then let $\pi_{\rm sc}$ be the quotient from metrics modulo diffeomorphism to metrics modulo diffeomorphism and positive overall scale. Fix, independently of the observable used for comparison, a normalization functional $\operatorname{Norm}_Y$, a section $s_Y$, and an admissible scale family $\mathcal L_{{\rm dim},Y}\subset\mathbb R_{>0}$ such that

$$\pi_{\rm sc}\circ s_Y=\mathrm{id}_{\mathcal G_Y}, \qquad \operatorname{Norm}_Y(s_Y(\gamma))=1, \qquad \operatorname{Norm}_Y([M,c^2g]) =c\,\operatorname{Norm}_Y([M,g]).$$

Writing $s_Y(\gamma)=[M,g_Y(\gamma)]_{\rm Diff}$, form

$$\mathcal M_{{\rm dim},Y} = \{[M,\ell^2g_Y(\gamma)]_{\rm Diff}: \gamma\in\mathcal G_Y,\ \ell\in\mathcal L_{{\rm dim},Y}\}.$$

The simultaneous branch consists of tuples satisfying

$$\begin{gathered} \mathcal A_Y^{\rm obj}\in \operatorname{ObjArch}_Y(g,\omega,\mathcal I,u;\mathcal N), \qquad (R,J)\in\operatorname{Emb}_g(\mathcal A_Y^{\rm obj}),\\ \mathcal M_{{\rm dim},Y}\in \operatorname{DimCal}\!\left( \operatorname{Rec}_{\rm met} (\operatorname{PreGeom}(\mathcal A_Y^{\rm obj})) \right),\\ \mathfrak m_R\text{ is fixed before the gravitational solution,}\\ \Gamma_{R,\rm CTP}=\Pi_R\Gamma_{\rm arch,CTP}^{\rm full},\qquad \Gamma_{MR,\rm CTP}=\Pi_{MR}\Gamma_{\rm arch,CTP}^{\rm full},\\ (g,\Phi)\in \operatorname{Sol}_G (\mathcal M_{{\rm dim},Y},\omega, \mathcal A_Y^{\rm obj},B_Y;\mathfrak m_R). \end{gathered}$$

Thus the archive that constrains geometry is regenerated by the same $g,\omega,\mathcal I,u$ whose state-dependent stress and record functional source that geometry. If the normalized section or scale family is absent, no dimensional gravitational branch is defined.

Target and conclusion. Prove existence and stability conditions for this simultaneous QFT–record–geometry system rather than accepting an arbitrary archive input. Classify its reconstruction as empty, a single metric-equivalence class, a non-singleton family controlled relative to a predeclared observable vector covering every claimed observable, or an uncontrolled non-singleton family that remains insufficient for a unique physical prediction. Control for lensing alone is not global control. For every class admitted to gravitational evolution, prove that the resulting solutions satisfy the action-derived field equation, record influence functional, boundary data and Ward/Bianchi identities. No representative is selected from a non-singleton family by notation alone.

Hypotheses. R2 and R7; an operational QFT completion on each candidate metric with an admitted state, normal instruments and selector preparation; well-defined actual-deposition and objectivity maps; a covariant archive-embedding relation $\operatorname{Emb}_g$ whose record currents are fixed by deposition and persistence; an unrestricted archive kernel $\Gamma_{\rm arch,CTP}^{\rm full}$; a fixed EFT matching datum $\mathfrak m_R$ whose projections separate already-owned matter and QFT terms from record-exclusive and genuinely mixed remainders before gravitational solution; measurable admissible perturbations; redescription-invariant influence; a cycle-safe quotient; positive support measure; a cofinal refinement family with transports obeying identity and composition; finite primitive discrepancies on declared comparison pairs and the induced path-pseudometric uniformity; bounded reconstruction discrepancies; a normalized scale section and nonempty independently fixed dimensional-scale family; a predeclared observable vector or family for every claimed degeneracy-control statement; admissible action and boundary data.

Lemma route. (i) regenerate the archive from the same instruments, selector, actual-deposition and objectivity chain used on the candidate geometry; (ii) construct $(R,J)\in\operatorname{Emb}_g(\mathcal A_Y^{\rm obj})$ and prove covariance of $\Gamma_{\rm arch,CTP}^{\rm full}$; (iii) fix $\mathfrak m_R$, match the $M$ and QFT projections into their existing coefficients, and prove that $\Gamma_{R,\rm CTP}=\Pi_R\Gamma_{\rm arch,CTP}^{\rm full}$ and $\Gamma_{MR,\rm CTP}=\Pi_{MR}\Gamma_{\rm arch,CTP}^{\rm full}$ are the only archive-induced terms added to the total action; (iv) establish intervention-kernel measurability and invariance; (v) prove the quotient order and $c_{xy}\ge1$ with the extended-distance properties; (vi) prove transport identity/composition, that $d_{\rm ref}$ is the extended path pseudometric induced by $\delta_{\rm sym}$, and that its entourages define the refinement uniformity; (vii) prove stability of spectral, volume and causal data under that uniformity; (viii) establish compactness and the stated reconstruction alternatives; (ix) prove every degeneracy-control claim relative to its predeclared observable family and propagate the full family for all other observables; (x) calibrate every admitted scale class without using the test observable; (xi) vary the effective action and prove conservation/exchange compatibility; (xii) propagate every surviving dimensional metric through $\operatorname{Sol}_G(\cdot;\mathfrak m_R)$; (xiii) prove simultaneous existence and stability under source-local, record, matching-scale and state perturbations.

Dependencies, limits, and comparisons. R3 depends on R2, R7 and the Chapter 5 QFT construction. It must recover the finite diamond, flat-space, semiclassical and ordinary-GR limits. Reconstructed distances, redshifts and lensing observables provide comparisons with Tier-1 data.

Companion-paper boundary. Prove same-branch nonemptiness and stability, continuum convergence, controlled uniqueness for nontrivial regenerated archives, and global gravitational existence. Numerical higher-curvature coefficients remain a separate calculation.

17.5 R4 — Standard Model source selection

Definitions. Let $\mathfrak B_M(Y)$ be the target-neutral class of gauge-matter candidates compatible with the source invariant signature $I_M(Y)$, the declared global quotient, representations and intertwiners, and the full local/global anomaly condition $\mathcal C_{\rm anom}^{\rm full}$. Let $\mathcal C_Y$ be the source-complexity functional and

$$\operatorname{Sel}_M(Y) =\operatorname*{arg\,min}_{b\in\mathfrak B_M(Y)}\mathcal C_Y(b).$$

Target and conclusion. Determine whether $\operatorname{Sel}_M(Y)$ is nonempty and selects the Standard Model global group, chiral representations, anomaly-free hypercharges, Higgs/Yukawa intertwiners and generation structure uniquely or up to a declared physical degeneracy.

Hypotheses. R1 and R2; an exhaustive candidate filtration independent of the observed answer; a source-invariant comparison map; local, global, BV/QME, quotient-descent, line-operator and spin consistency conditions.

Lemma route. (i) prove candidate-filtration exhaustiveness; (ii) establish representation descent to the global quotient; (iii) impose local and global anomaly cancellation; (iv) recover electroweak breaking and the invariant Yukawa tensors; (v) prove source-invariant matching; (vi) prove stability of the minimizing equivalence class as the filtration expands.

Dependencies, limits, and comparisons. R4 depends on R1 and R2. The recovered branch is compared with observed gauge representations, charges, CKM/PMNS structure, neutrino alternatives and strong-CP constraints, while recovery of known Tier-1 physics remains distinct from source selection.

Companion-paper boundary. Prove nonemptiness, uniqueness or classified degeneracy of $\operatorname{Sel}_M(Y)$. Numerical parameter values belong to R5.

17.6 R5 — Parameter fixation and renormalization

Definitions. Parameter fixation is staged. For a selected matter branch $b_M$, $\operatorname{FixParam}_{M}(b_M)$ is the solution set of its source boundary relations, renormalization-group equations and threshold conditions at a declared scheme and scale. Conditional gravitational parameters are then treated on an admitted R3 branch. Cosmology-dependent parameters are not assumed before R6: after the background and perturbation solution exists, they enter a joint consistency set $\operatorname{FixParam}_{C}(b_M,g,C)$. Parameters at every stage are classified as source-derived, constrained, independently calibrated or unresolved.

Target and conclusion. Determine the existence, identifiability and scheme-equivalence class of the staged solution sets, and classify which dimensionless couplings, masses, mixings, gravitational quantities and cosmological parameters follow from source data rather than from the observables later used to test them. No cosmological parameter may be used as an upstream hypothesis for the R6 solution that is later claimed to determine it.

Hypotheses. R2 and R4 for the matter/QFT stage; R3 for conditional gravitational parameters; a specified renormalization scheme and scale; beta functions and threshold maps; source-invariant boundary relations; identifiable observables with uncertainties. The cosmology-dependent stage additionally assumes an R6 solution but is not an input to its own construction.

Lemma route. (i) establish local existence of RG trajectories; (ii) prove matching consistency at every threshold; (iii) prove scheme covariance of physical observables; (iv) analyze the rank and degeneracy of the inverse map; (v) propagate source, theoretical and measurement uncertainties; (vi) test stability and naturalness without using them as substitutes for derivation.

Dependencies, limits, and comparisons. The pre-cosmological part of R5 depends on R2 and R4, with gravitational parameters conditional on R3. Its cosmological classification follows, rather than precedes, the R6 solution. Comparison uses held-out dimensionless relations and observables not used to determine the boundary data. Standard effective-field-theory running must be recovered on the relevant scale intervals.

Companion-paper boundary. Perform the numerical matching and classify irreducible degeneracy. A fitted parameter is not relabelled as source-derived.

17.7 R6 — Cosmology, vacuum response, and the dark sector

R6 contains two ordered subtargets: the cosmological construction $R6_C$ and the temporal-arrow lemma suite $R6_{\rm arr}$. Together they constitute the single R6 residue; $R6_{\rm arr}$ is not a tenth theorem programme.

Definitions. A cosmological branch consists of a solved Lorentzian geometry, the exact-once matter/QFT/record/dark stress decomposition, a parameter branch, initial data, the FLRW background, gauge-invariant scalar/vector/tensor perturbations, thermal history and transfer functions. The EFT matching datum $\mathfrak m_R$ is fixed before $\operatorname{Sol}_G(\cdot;\mathfrak m_R)$; record stress is admitted only through $T_{\mu\nu}^{R}$, obtained from $\Gamma_{R,\rm CTP}=\Pi_R\Gamma_{\rm arch,CTP}^{\rm full}$. It vanishes when archive variables merely reparameterize matter/QFT degrees of freedom, and $\chi_{MR}T_{\mu\nu}^{MR}$ is included only for a genuinely independent mixed interaction. The ordinary and Henneaux–Teitelboim gravitational actions are mutually exclusive branches.

$R6_C$ target and conclusion. Prove existence of a branch satisfying the background constraints and scalar/vector/tensor evolution, a consistent thermal history, stable dark-sector dynamics and the stated vacuum-response condition, with every action and stress contribution counted once.

Hypotheses. R3 and R4, together with the pre-cosmological matter/QFT and conditional-gravity parameter branches from R5; admissible initial and boundary data; one selected gravitational branch (ordinary or Henneaux–Teitelboim); the fixed pre-solution matching datum $\mathfrak m_R$ and its record-exclusive and genuinely mixed projections; well-defined matter and dark stress tensors; stable perturbation kinetic and gradient terms.

Lemma route. (i) prove background existence and constraint propagation; (ii) derive the gauge-invariant SVT system from the same field equation; (iii) establish well-posed species and kinetic evolution; (iv) prove transfer-function existence; (v) join reheating or its alternative through BBN, recombination and structure formation; (vi) prove stability of the dark and vacuum branch; (vii) derive CMB, lensing, growth and gravitational-wave observables.

Dependencies, limits, and comparisons. R6 depends on R3, R4 and the pre-cosmological part of R5. Its solution then supplies the cosmology-dependent input returned to the final R5 classification. It must recover standard radiation-, matter- and $\Lambda$-dominated regimes where appropriate. Its principal comparisons are CMB spectra, growth, lensing, large-scale structure and gravitational waves.

Companion-paper boundary. Prove viable branch existence and compute spectra for fixed source and parameter data. Dark-sector microphysics and the numerical value of $\Lambda_{\rm br}$ are not fixed by architectural notation.

17.7.1 Temporal-arrow lemma suite within R6

Definitions. From the objective-event carrier $(\mathsf P_Y,\precsim_D)$, form the down-set archives $\mathcal A_Y^{\rm obj}(p)$ and canonical retention maps $\iota_{pq}$. Define $\preceq_A^{\mathsf P}$ first on $\mathsf P_Y$ by dependency, injective archive retention and nondecreasing record-count entropy. Let

$$\Theta_A:(\mathsf P_Y,\precsim_D)\hookrightarrow (\mathcal T_Y,\preceq_t)$$

be an order embedding compatible with the low gravitational-entropy boundary. The temporal arrow is the image of $\preceq_A^{\mathsf P}$ under $\Theta_A$; retarded Green support is imposed only after the causal Lorentzian branch exists.

Target and conclusion. Prove that the objective-event dependency relation descends through physical redescription, that the retention system is functorial under refinement, that $\preceq_A^{\mathsf P}$ and its strict part are well defined, and that $\Theta_A$ transports them to a temporal order compatible with the low boundary and retarded propagation. No general monotonicity statement for Shannon, von Neumann, thermodynamic or gravitational entropy follows merely from record-count growth.

Hypotheses. R3, $R6_C$ and R7; objective event formation; a cycle-safe physical quotient; injective retention on persistence classes; monotone eligibility; a stated low-boundary condition; a globally hyperbolic or Green-hyperbolic solved branch for the retarded-propagation statement.

Lemma route. (i) prove descent of dependency and archives; (ii) prove the partial-order and retention-functor laws; (iii) establish conditional record-count monotonicity; (iv) construct the order embedding and prove well-defined temporal descent; (v) establish the low-boundary orientation; (vi) prove retarded support from the solved causal branch and boundary conditions.

Dependencies, limits, and comparisons. The subtarget $R6_{\rm arr}$ follows R3, $R6_C$ and R7 and is completed before R8a. It is compared with objective archive growth, cosmological low-boundary data and observed retarded response. A local memory may forget without violating objective archive retention.

Companion-paper boundary. Establish the low-boundary mechanism and generalized entropy results for concrete realizations. R9 later asks whether this arrow branch is simultaneously compatible with all other sectors.

17.8 R7 — Contextual selection and single-record realization

Definitions. Let $(\mathsf U_*,\Sigma_*,\mu_*)$ be the realization-supported selector space prepared independently of freely chosen settings. For a complete admissible context $(v,h)$, the measurable map $\operatorname{Sel}_{v,h}$ selects one instrument outcome. The selected record then enters $\operatorname{Dep}^{\rm act}$ and $\operatorname{ObjPers}$; selection, deposition and persistence remain distinct.

Target and conclusion. Construct the selector so that its pushforward equals the Born measure for every admissible context, its joint blocks reproduce entangled correlations, and its histories agree with the normal quantum instruments while preserving operational no-signalling.

Hypotheses. R2; normal instruments on the operational detector algebra; measurable selector variables; a covariant context code; consistent subsystem composition.

Lemma route. (i) prove selector measurability; (ii) establish the single-context pushforward; (iii) prove consistency of conditional and cylinder probabilities; (iv) prove composition for spacelike and entangled subsystems; (v) establish repeatability under repeatable instruments; (vi) prove covariance and setting-independent local marginals; (vii) prove compatibility with actual deposition and objective persistence, including null outcomes.

Dependencies, limits, and comparisons. R7 depends on R2 and the Chapter 5 detector-algebra construction. It must recover Born frequencies, sequential histories, entangled correlations and no-signalling in their standard regimes.

Companion-paper boundary. Derive or constrain the source dynamics of the selector variables and test concrete selector models. Context dependence is permitted; superluminal signalling is not.

17.9 R8 — Structural observer readout and qualitative finality

Definitions. For observer candidate $o$, use the common history

$$\mathcal H_{Y,o} = \operatorname{ObjHist}_{Y,o} \times_{\mathcal T_Y} \operatorname{ArrowHist}_{Y,o},$$

ordered windows $W_{Y,o}(t)$, the set-valued integration relation $\operatorname{Int}_{Y,o}$, self-index $I_{\rm self,Y}$, unity $\mathcal U_{Y,o}$, differentiation structure $\mathcal Q_{Y,o}$, and report map $\mathcal K_{\rm rep,Y}$.

R8a target and conclusion. Prove that, under the stated accessibility, invariance and stability hypotheses, $\mathfrak O_{Y,o}(t)$ is nonempty and supports ordered integration, self-indexing, unity, qualitative differentiation, report coupling and explicit fragmentation conditions.

R8a hypotheses. $R6_{\rm arr}$ and R7; an objective history and compatible arrow history; a stable subsystem boundary; intervention-defined internal access; ordered windows; a fixed unity normalization and threshold; redescription invariance; an individual-structure metric $d_{\rm 1P}$; and a metric family of physical perturbations with declared algebraic transports or correspondences.

R8a lemma route. (i) prove existence of the history pullback; (ii) prove access and set-valued integration are well defined using the physical algebra embeddings; (iii) establish self-index normalization without an external identity label; (iv) prove unity and differentiation invariance; (v) establish report coupling without defining the internal structure by its report; (vi) construct the cross-branch transports and prove the $d_H^{\rm 1P}$ Hausdorff stability bound or the stated fragmentation alternatives.

R8b target. Determine whether and under what additional independently stated principles the structural package is sufficient for consciousness and qualitative finality.

Dependencies, limits, and comparisons. R8a depends on $R6_{\rm arr}$ and R7. It is compared through temporal-window stability, controlled fragmentation and report correlations. Disjoint systems and unstable or redescription-dependent integrations are counterexamples, not successful observer branches.

Companion-paper boundary. R8a is a mathematical-physics problem posed by Chapter 15. R8b remains a distinct foundational problem; the monograph does not identify structural observer readout with a completed derivation of consciousness.

17.10 R9 — Integrated compatibility and formal closure

Definitions. Let $\mathsf{Comp}_{T1}=\operatorname{Eq}^{\rm coh}(\Delta_L,\Delta_R)$ be the coherent equalizer of the sector-output product under the two interface maps. A global branch is a common realization together with sector solutions satisfying the compatibility conditions of Chapter 16 and every required path and loop coherence condition.

Target and conclusion. Prove that the compatible fibre over at least one admitted source is nonempty and contains a Tier-1 readout satisfying the quantum, QFT, record, geometric, gravitational, matter, cosmological, temporal and observer equations simultaneously.

Hypotheses. The relevant existence conclusions of R1–R7 and R8a, including both R6 subtargets; nonempty pairwise comparison spaces; path and loop coherence; suitable compactness or fixed-point hypotheses for mutually coupled QFT–geometry, cosmology and record–arrow–observer sectors. R8b is independent of physical-sector compatibility and is not a hypothesis of R9.

Lemma route. (i) prove each local interface square commutes; (ii) establish consistency on overlapping triples; (iii) prove path independence and loop coherence; (iv) solve the coupled QFT–geometry, cosmology and record–arrow–observer correspondences; (v) prove nonempty intersection of the compatible fibres; (vi) establish the principal correspondence limits and observable maps on the resulting branch.

Dependencies, limits, and comparisons. R9 depends on the existence conclusions of R1–R7 and R8a, including $R6_{\rm arr}$. It does not depend on R8b. It is tested by simultaneous recovery of quantum, QFT, GR, Standard Model and cosmological limits and by the source-to-observable routes of Chapter 16.

Companion-paper boundary. Complete the global existence and compatibility proof. R9 is not discharged by the internal consistency of the architectural monograph alone.

17.11 Dependency of the theorem programme

The principal dependencies are

$$\begin{aligned} R1&\to R2,\\ R2&\to(R4,R7),\\ (R2,R7,\mathrm{QFT})&\to R3,\\ (R2,R4)&\to R5_M,\\ (R3,R4,R5_M)&\to R6_C,\\ R6_C&\to R5_C,\\ (R3,R6_C,R7)&\to R6_{\rm arr},\\ (R6_{\rm arr},R7)&\to R8a,\\ R8a+\text{additional foundational principles}&\to R8b,\\ (R1,R2,R3,R4,R5,R6,R7,R8a)&\to R9,\\ R6&=(R6_C,R6_{\rm arr}). \end{aligned}$$

These arrows order the research programme; they are not substitutes for proofs.

The physical comparisons associated with the programmes are summarized once: