Appendix C Claims Status

This appendix consolidates every substantive claim of the book into a single structured inventory. The chapters record the status of their results in context; this appendix is the consolidated, authoritative cross-cutting index. Each item is tagged with one of eight status categories defined below.

C.1.  Status categories

Postulate.

An ontological input; PST has exactly three (P1, P2, P3).

Modelling choice.

An item adopted within the framework where alternatives exist; documented and not derived from postulates alone.

Derived theorem.

Proved from P1–P3 with all conditions in place at proof-detail level.

Theorem under modelling choice.

Proved given a modelling choice; result depends on that choice.

Conditional theorem.

Proved under explicit assumptions (e.g. A1–A7 of §18.1).

Identification.

Structural correspondence between substrate objects and Standard-Model / general relativity (GR) / quantum field theory (QFT) counterparts.

T(C)-contingent.

Configuration content; not derivable from postulates alone (structural-scope theorem of §14.3).

Open.

Not settled either way: a derivation the book does not have, or a premise it carries without deriving. The register is §19.9.1 and §19.9; C.8 indexes it so that this appendix is complete.

C.1 Postulates

Postulate	Reference	Statement
P1, Property differentiation	§1.7, Chapter 2	The capacity for distinctions to exist; sole ontological primitive. Clause (iii), faithfulness (6), requires $\delta$ to record every difference the triple carries; it is a premise, not a theorem of (4) and (5) (Computation 184).
P2, Asymmetric vector tension	Ch. 3	Property differentiation carries a $V_{7}$-valued tension with $T(D)\neq 0$ generically.
P3, Modal sublimation	§4.1	$G_{2}$-invariant Landau–Ginzburg potential on $V_{7}$; sublimation instantiates the maximum-entropy ($\mu$-typical) configuration past threshold, supplying the equidistribution condition A3 and, the typicality reaching the content of $T$, its non-degeneracy (Computations 193, 195); instantiated geometry follows.

C.2 Modelling choices

Three rows left this table. The direction algebra $V_{7}=\mathrm{Im}(\mathbb{O})$, the minimal KO split and the maximal Dixon tensor were listed here as modelling choices in the first edition. They are not choices: the first and third are consequences of P1, a proper sub-object costing a selector P1 does not supply (Computation 168), and the second is not a selection at all, the admissible KO pair being unique (Computation 148). What remains below is a genuine choice; the $V_{7}$ decomposition behind the generation count sits in C.4 as a structural reading whose stated condition is withdrawn, not as a choice.

Choice	Reference	Alternatives & rationale
Gaussian projection kernel for Casimir $\xi$	Chapter 15, Computation 68	Six kernel families surveyed; Gaussian gives $\xi=90/\pi^{2}$; substrate’s binomial kernel converges to Gaussian (Computation 73).

C.3 Derived structural theorems (from P1–P3 at proof-detail level)

Here “proof-detail level” denotes the rigour of each derivation given its inputs, not unconditional proof: the spacetime chain below (Lorentzian signature, $n=4$, diffeomorphism invariance, the cosmological rows) rests on the equidistribution condition A3, the inequality $\eta_{K,R}<d_{0}^{2}/4$ of §18.1.6, supplied by P3’s maximum-entropy sublimation (equicoercivity A7 being a theorem given A1 to A4, with A3 so supplied), and the internal-algebra and gauge rows rest on the maximal $V_{7}$ direction algebra and the maximal Dixon tensor, which earlier editions carried as extremal modelling choices in C.2 and which are consequences of P1 rather than choices beside it (§1.7, Computation 168).

Theorem	Reference	Notes
Bernoulli measure $\mu=\bigotimes_{a}\mathrm{Bern}(\tfrac{1}{2})$ as the unique measure on $\mathcal{P}(D)$ invariant under every single-property complementation	§2.5	Derived from P1’s symmetry between the presence and absence of each distinction. Permutation invariance alone does not suffice: any weighting depending only on $|C|$ is permutation-invariant.
Modal threshold $\tau$ exists with specified value	§4.1	Bifurcation point of $F[\psi,\varepsilon]$ from P2 + P3.
Vacuum manifold $\mathcal{V}\cong S^{6}\subset V_{7}$ for $\varepsilon>0$	§4.1	$G_{2}$-orbit of vacuum direction.
Lorentzian signature $(+,-,-,-)$	Computation 2, §4.12	From hyperbolic well-posedness of Goldstone wave equation.
Diffeomorphism invariance	§4.11	From non-uniqueness of realisation map $\rho$.
Geroch time-orientability of emergent manifold	Ch. 5	P3-directed threshold + permutation invariance + Mosco convergence.
Cosmological principle	§16.2	From permutation invariance of $\mu$.
SM gauge group $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$	Chapter 13, Computation 6	Unimodular unitaries of $A_{F}$.
Internal algebra $A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$	§13.7, Computations 3, 19	From Cl(0,6) chain-algebra structure.
Born rule	§10.2	From Gleason’s theorem on $L^{2}(\mathcal{P}(D),\mu)$.
Fermionic excitations	§11.2	Boolean-CAR isomorphism on distinction primitive.
Spin-statistics structural	§11.3	Three closures, two of them conditional. Corrected at v32.82, when this row read “three closures” unconditionally against the section it cites, and completed at v32.123. The substrate-derived Dirac equation closes from P1–P3. CAR convergence holds once the relations are posed slice-wise, and the slice is supplied, a slab in the time function along $\Phi(\hat{\tau})$ normalised by the proper $3$-volume (Computations 186, 191, 192); W4 follows from it (Computation 185), its earlier argument withdrawn as vacuous and its substrate route closed (Computation 181). Those two hold given A1 to A4 with A3 supplied by P3 and are listed in C.5. The structural spin-statistics theorem itself is untouched, deriving statistics from Boolean exclusion without using microcausality.
Collapse rule (measurement postulate)	§10.3	Three closures and one condition: von Neumann measurement Hamiltonian, decoherence inheritance and threshold-crossing irreversibility close; agreement with the Born rule (§10.3.2.5) does not. It holds where the tension variable is uniform in its quantile, and that is a substantive constraint on the law of $T$ rather than a choice of scale: the comparison is cardinal, so Computation 140’s monotone-reparametrisation invariance does not reach it. Computation 145 shows the condition is satisfiable, with the tie floor falling as $|D|^{-1.5}$; satisfiable was not discharged there and is supplied almost surely by P3’s typicality clause (the non-degeneracy row of C.5, Computation 195). (98) does not deliver uniqueness and Born statistics at the same coupling, so the outcome separation is carried by the einselected pointer basis and the threshold supplies irreversibility.
Substrate-side factor $Z_{H}(\beta_{\mathrm{KO}})\to e^{-1}$ (substrate side of $Z^{2}\approx e^{-1}$)	§12.5, Computation 100	Tensor-product factorisation forced by P1’s Bernoulli product measure (substrate-side spectral-action cutoff identification), with the limiting value $\mathrm{e}^{-1}$ resting on the identifications $\beta_{\mathrm{KO}}=2$ and $\Lambda^{2}=D$ (C.6). The matching identification with the SM-side ratio $\lambda_{\mathrm{SM}}(M_{*})/b$ (bridge premise B) is reduced within PST to a field-strength-renormalisation identity (substrate side closed; SM-side matching open) (§12.5, Computations 110–116).
Cosmological-constant sign + equation of state $w=-1$; $\Lambda_{\mathrm{PST}}=8\pi G\rho_{\mathrm{inst}}/c^{2}$	§16.5	Steady-state non-dilution mechanism. The magnitude of $\Lambda_{\mathrm{obs}}$ is not fixed (C.7).
Weight profile $W(s)=-|s|$ of the pair sum, forced; $w=1/3$ for the balanced pair sum	§7.4.4, Computation 158	From (31), (30) and (40) having to agree on the trace pair by pair, with no free function; the trace of the sum is the summed $T-\tau_{\mathrm{null}}$ and vanishes in the large-window limit of a balanced cloud, so $w=1/3$ exactly there and for every configuration shape tried; a range cutoff at $d_{0}$ is needed for convergence and nothing is read off it. Out of balance the trace is the worldline sector’s kinetic scalar (Computation 159).
Dust is not in the pair sector; rest mass enters through the matter action	§7.4.6, Computation 159	The worldline sector is exactly kinetic, $w=\beta^{2}/3$; a photon contributes nothing from its own null worldline, which is why radiation is $1/3$; the worldline share stays at a few per cent for homogeneous matter because the window $|s|\lesssim d_{0}^{2}$ is a neighbourhood of the light cone, and the bond kernel cannot change that, so the projection is strictly conformal there up to that share.
Polarisation count, Stefan–Boltzmann constant and the Planck spectrum from the pair sum	§7.4.6, Computations 160, 161	Equilibrium radiation of the pair sum, inside the balanced semidefinite class; the photon’s dynamics were carried separately and closed by Computations 165, 166, 189.
$d^{-6}$ Casimir-correction power law	Chapter 15	Categorical theorem from parity + single scale $d_{0}$. Amplitude $\xi d_{0}^{2}$ with $d_{0}$ from the normalisation premise (Computation 201): exact in form, $10^{-19}$ of the force at $20$ nm.
Local gauge symmetry; minimal coupling	§13.3, §13.3.1	Minimal coupling $D_{\mu}$ follows from gauge covariance of the projected action (Computation 166). The gauged broken U(1) is massive (the $Z$); the photon is the unbroken $Q$, inherited from the SM’s unbroken-generator argument.
Canonical commutation relations $[\hat{x}^{\mu},\hat{p}_{\nu}]=i\hbar\,\delta^{\mu}{}_{\nu}$	§10.6	The Boolean gradient projects to the momentum operator, $\hat{p}_{\mu}=-i\hbar\,\Pi(\partial_{\mu}^{\mathrm{B}})$.
Angular momentum as Noether charge; $L\in\hbar\mathbb{Z}$	§9.4, §9.5	Noether charge of the would-be Goldstone $\mathrm{U}(1)$ on $\mathcal{V}\cong S^{1}$; the integer is $\pi_{1}$, the unit $\hbar$ is supplied by canonical quantisation, not derived. The winding number $n$ gives the discrete spectrum.
Uniqueness of $G_{\mu\nu}$ (Lovelock)	§8.6	Noether’s second theorem plus Lovelock, given diffeomorphism invariance and $n=4$ (both derived above).
Weak equivalence principle	§8.7	Single-source projection: all matter is the same projected tension, so $\nabla^{\mu}T_{\mu\nu}=0$ yields one geodesic law for every test body.
Chirality of $\mathrm{SU}(2)_{L}$	§14.4	Structural consequence of the threshold-direction parity argument.
Horizon and flatness problems dissolved	§16.3	From the Cosmological Principle theorem above plus a constant instantiation rate $j_{\tau}$; no inflationary epoch is invoked.

C.4 Theorems under modelling choice

Theorem	Reference	Modelling choice
$\xi=90/\pi^{2}\approx 9.12$ Casimir coefficient	Chapter 15, Computation 73	Gaussian projection kernel (substrate kernel converges to Gaussian at matched scaling, correction $-2/(3|D|)$).
$d_{0}\lesssim 2.3$ nm coherence-length bound	Chapter 15	Conditional on Gaussian kernel; factor-of-$\sim$3 spread across kernel families.
Three-generation count $N_{\mathrm{gen}}=3$ (structural reading)	§14.2	$V_{7}$-directional decomposition into three quaternionic sub-algebras through $\hat{\tau}$; the natural structural reading. Its stated condition is withdrawn, corrected at v32.82: this row rested on “the Dixon-synthesis block structure (Computation 48, 106)”, and Computation 106’s block-diagonality claim was withdrawn at v32.64. Computation 175 replaces it with a sharper statement, that orthogonality is not invariance and that $\hat{\tau}^{\perp}$ is the irreducible fundamental $\mathbf{3}$ by Schur, so no colour-invariant generation splitting exists inside it at all. What survives is the count: $\hat{\tau}^{\perp}$ is $3$-dimensional over $\mathbb{C}$, and that one fact is the source of both $N_{c}=3$ and $N_{\mathrm{gen}}=3$. PST derives one three, not two, and inherits the octonionic programme’s colour–generation question (Furey 2014) rather than closing it.
Colour–generation distinguishability, settled in the negative	§13.5, Computation 175	$\mathrm{SU}(3)_{c}=\mathrm{Stab}_{G_{2}}(\hat{\tau})$ acts on $\hat{\tau}^{\perp}$ as the irreducible $\mathbf{3}$, so by Schur no colour-invariant generation splitting of $\hat{\tau}^{\perp}$ exists: colour and generation are one three, not two. What PST inherits is the octonionic programme’s question of where generations live if not there; tree-level FCNC absence is not a structural theorem. Moved here from C.8 at v32.128.

C.5 Conditional theorems (explicit assumptions listed)

Theorem	Reference	Conditions
Mosco convergence $\mathcal{S}\to M=\mathbb{R}\times S^{3}$	§18.1	Seven conditions A1–A7. Equicoercivity A7 is a theorem given A1 to A4, its rate condition A3 supplied by P3 (§18.1.6); A3 is the rate hypothesis, its fluctuation half settled (McDiarmid, Computation 125) and its bias half a diffeomorphism-invariant equidistribution rate $\eta_{K,R}<d_{0}^{2}/4$ supplied by P3’s maximum-entropy sublimation (the realised configuration being $\mu$-typical, its cloud equidistributed at the rate A3 states) and corroborated on the i.i.d. surrogate (Computations 70, 83); the Mosco limit thus closes under A1–A7, A1 being architectural input rather than a substrate consequence, so it is not a theorem of P1–P3 alone. This is a typicality selection internal to P3, not a fourth postulate (no P4: the apparent bulk-density freedom is a flat-chart coordinate artefact, Computation 134).
Dirac-operator convergence (relaxed criterion)	§18.3, Computations 9, 65, 150, 213, 214	Relaxed-criterion closure delivered by the L_round/L_comm lemma chain; the uniform bound on $k\leq k_{D}$ is derived at its maximiser (the sharp value $3.6736$ an explicit algebraic number, Computation 214) and in its large-$N$ limit (closed form, Computation 213), scanned between; a proof over the whole regime remains its one quantitative refinement (existence of a finite bound there is analytic, as is its $D^{-1/4}$ mechanism; Ch. 19).
Slice-wise CAR limit and microcausality W4	§11.3, Computations 185, 186, 191, 192	A1 to A4 with A3’s rate supplied by P3, the standing of equicoercivity A7, and the Lorentzian signature of §4.12. No site is realised on a measure-zero slice, so the relations are posed on a slab normalised by the proper $3$-volume of the slice (Computation 186); the slab limit holds for every cloud whose discrepancy vanishes faster than the thickness, by Koksma–Hlawka, with no independence (Computation 191); the time function is cosmic time along $\Phi(\hat{\tau})$, its foliation forced by isotropy, its level sets the compact Cauchy surfaces, invariant under relabelling (Computation 192); and W4 is the corollary, $S$ being the causal propagator on a globally hyperbolic spacetime (Computation 185, [213, 214]). Moved here from C.8 at v32.123; the substrate-correlator route to W4 stays closed (Computation 181).
Orbital angular momentum, $L_{z}=nQ$	§9.5, Computation 194	The tensor from the Goldstone action and the integer from $\pi_{1}(S^{1})$ are derived, and $L_{z}=nQ$ is an identity for every axisymmetric configuration, $Q$ the Noether charge. The unit is conditional on the absolute normalisation of the Goldstone stiffness $r_{0}^{2}$, the canonical-normalisation premise of C.8: a derived unit is $r_{0}^{2}=\hbar c/(\kappa d_{0}^{2})$, with $r_{0}=v$ and the derived $d_{0}$ giving $\kappa=m_{h}^{2}/(2v^{2})=0.13$ (Computation 201; $1.2\times 10^{-19}$ at the Casimir bound, Computation 197), and $\hbar$ is inherited at (106). The far field is the global vortex’s, $L_{z}$ growing as $R^{3}$ and the energy logarithmically, and $\pi_{2}(S^{1})=0$ excludes the spherical defect. Moved here from C.8 at v32.125.
Non-degeneracy of $T$’s contingent content: Born agreement on the law of $T$, and non-balance	§10.3.2.5, §3.5, Computations 190, 193, 195	P3’s typicality clause read as reaching the contingent content of $T$ (§1.7), the footing A3 stands on. The graph half of Born agreement is a theorem (Computation 190); the two faces are one open dense condition, the magnitudes off the tie set and the directions off the balance set (Computation 193); the maximum-entropy law on the content has a density, so both faces hold almost surely under it, $\mu_{\pm}$ being its antipode (Computation 195). The reference measure is load-bearing: quantised directions would balance. Moved here from C.8 at v32.126.
Only the magnitude ordering of $T$ and $\hat{\tau}$ are load-bearing	§4.1, Computations 126, 140, 195	Established for every derivation but the Born comparison by the reads the body makes (Computation 140), and for that one cardinal read by the row above; the residual per-configuration content is contingent (§14.3) and a bounded underdetermination, not an open derivation. Moved here from C.8 at v32.126.
Stress-energy = projected tension; EFE as Seeley–DeWitt term	§7.8, Computation 5	Standard CC spectral-action machinery applied to substrate triple.
Newton’s $G$ as consistency relation among $(m_{h},M_{P},v)$	§8.5	Conditional on heat-kernel CC matching.
$m_{h}^{2}=(3/2)M_{*}^{2}/(16\pi^{2})$ scalar self-mass; $M_{*}=4\pi m_{h}\sqrt{2/3}\approx 1.285$ TeV	§12.2, Computations 93, 97, 109	Parameter-free one-loop result for the Higgs self-loop in isolation. The full-SM extension (separately accounting for the EFT-layer top/$W$/$Z$ quadratic sensitivities) does not exist within PST (Computation 120): the Veltman residual at $M_{*}$ is large and negative, eq. (125), with no zero crossing, so $M_{*}$ is irreducibly a scalar-sector result, not a full-theory prediction. Confronted with the LHC in Computation 205: survives in the Higgs-doublet reading of the cutoff, which is §15.2’s, and predicts a $23$ percent deficit in Higgs production above $650$ GeV of transverse momentum; the literal and the universal-kernel readings are excluded by the jet and dilepton spectra.
$Z^{2}\approx e^{-1}$ field-normalisation identity / Bridge premise (B)	§12.5, Computations 100, 110–116	Substrate-side factor $Z_{H}(\beta_{\mathrm{KO}})\to e^{-1}$ derived from P1–P3 (C.3, Computation 100). Matching identification with $\lambda_{\mathrm{SM}}(M_{*})/b$ (bridge premise B) is REDUCED to a field-strength-renormalisation identity, substrate side closed in the substrate limit: a wave-function (field-strength) renormalisation, multiplicative rather than the additive Wilsonian matching the measure-Jacobian route excludes (Computation 103), forced by P1’s product structure into the substrate-factor form (Computation 111), substrate-side value the embedding norm of the symmetric Higgs vacuum (Computation 114), with the total reduction supplied by a unique-soft-mode theorem (Computations 115, 116).
(the $Z^{2}$ row, continued)		Open, indexed in C.8 under Bridge premise (B): the matching of the substrate factor to the SM-measured coupling, resting on the emergence premise F4 (§12.6.2.1) plus the RGE test (Wilsonian threshold matching, quantified by a three-loop error budget as a rigid, $D$-independent prediction $\mathrm{e}^{-1}/4=0.09197$ consistent with the SM-run $0.0930\pm 0.0008$ at $1.4\sigma$ on current inputs, falsifiable as the top mass tightens, Computation 202; Computation 123’s $1.0\sigma$ rested on an anchor the literature’s formulas do not reproduce). Substrate-side residual: only the standard $O(1/D)$ caveat on the value $\mathrm{e}^{-1}$; the all-orders decoupling is closed exactly and non-perturbatively by sufficiency of the empirical mean (Computation 121), the Morse–Bott lemma (Computation 118) being its shadow, and F8’s linearity controls only that $O(1/D)$ (Computation 122).
Four-dimensional spacetime $n=4$	§1.5	From the minimal Connes-spectral-triple split given the KO-6 substrate; conditional on the spectral-triple framework assumption, which C.8 carries.
Weak-field limit: Poisson’s equation recovered	§8.2	Given the projected field equations (the Einstein-equation rows above); $G$ enters as the unit-conversion factor.
Friedmann equations; accelerating solution	§16.6	Given the projected field equations and the ongoing-sublimation source.
Conservation of $\Pi_{\mu\nu}$; the divergence-free ambiguity	§7.4.4, Computations 162, 163	Given the identification, exact on balanced clouds by (46), the traceless part of a conserved tensor being conserved when the trace is constant, verified on a Friedmann background with a massive scalar (Computation 163); without it, the residual is bounded and falls with the pair count (Computation 162). The Irving–Kirkwood ambiguity is restricted by tracelessness to a global harmonic function, which changes no integrated quantity.
The interacting vacuum of the regulated scalar sector	§12.2, Computations 198, 200	Conditional on the cutoff being kept, which is the theory’s own statement of it: in finite volume the Hamiltonian is a Schrödinger operator with a quartic potential, self-adjoint with discrete spectrum and a unique strictly positive ground state; the infinite-volume Euclidean measure exists by Griffiths monotonicity with Dirichlet data and the quartic’s uniform bound; reflection positivity returns a Hamiltonian bounded below. Stability at two loops (Computation 198) is an expansion about it. The gauge and fermion sector below $M_{*}$ is the cutoff Standard Model’s own existence problem (the weak-coupling infinite-volume limit, chiral fermions on a regulator), inherited rather than introduced. Moved here from C.8 at v32.131.
Coherence length $d_{0}=\sqrt{2}\,\hbar c/m_{h}=2.2$ am	Chapter 15, Computation 201	Conditional on canonical normalisation: the $\xi_{\psi}$ of (205), $\sqrt{c_{\scriptscriptstyle LG}/(2|a_{\scriptscriptstyle LG}|)}$ with $|a_{\scriptscriptstyle LG}|=m_{h}^{2}/4$, the frame (neutral or doublet, half-coefficient or not) fixing an order-one factor; $d_{0}/\ell_{*}=\sqrt{2}\,M_{*}/m_{h}=14.5$; nine orders below the Casimir bound; the Casimir correction $10^{-19}$ of the force at $20$ nm.
Gauge anomaly cancellation forces $N_{c}=3$	§13.8, Computation 136	Conditional on the Standard-Model hypercharge assignments. Explicitly not a derivation of anomaly cancellation from PST, but a consistency check: all six triangle coefficients vanish iff $N_{c}=3$, the dimension PST fixes independently as the $M_{3}(\mathbb{C})$ colour factor.

C.6 Identifications

Identification	Reference	Content
Substrate modal field = SM Higgs at different scale	§15.2	$\phi_{\mathrm{Higgs}}=\Pi(\psi)$, $m_{h}^{2}=4\varepsilon$.
Substrate Higgs Hamiltonian $H_{\mathrm{Higgs}}=\bar{X}$	Computation 89, §12.5	From additivity + matched scaling + uniformity; no free parameters.
Matched-scaling exponent $\beta_{\mathrm{KO}}=2$	Computation 100, 102	Spectral range of a single binary distinction, $2=1-(-1)$, forced by P1’s binary alphabet (equivalently the one-bit Clifford $\mathrm{Cl}(1,0)$ range). The KO Bott-periodicity value $\mathrm{KO}_{\mathrm{total}}\bmod 8=2$ shares the value but is not the load-bearing source.
Cutoff identification $\Lambda^{2}=D$ (the spectrum’s matched scale)	Computation 100	Substrate dimension as natural matched scale for Boolean Laplacian’s spectrum.
$\mathrm{U}(1)_{Y}$ hypercharges	§13.9	A determination, not a derivation: four structural facts fix all six hypercharges with no free parameter beyond the $\mathrm{U}(1)_{Y}$ normalisation, but the Gell-Mann–Nishijima relation and the empirical nucleon charges enter as inputs, not PST outputs.
Conformal part of $\Pi_{\mu\nu}$ = conformal part of $T^{\mathrm{(fields)}}_{\mu\nu}$	§7.5, Computations 158, 164, 196, 199	The conformal part of the pair sum reaches exactly the metric-traceless tensors obeying the null energy condition (Computation 199), so every classical field’s conformal part is reachable, static electric and magnetic fields included; on balanced clouds the conformal part is the pair sum itself (Computation 158); the magnitude is the state’s radiation content, the truncation spacing being comoving (Computation 196). An identification, not a derivation of $T^{\mathrm{(fields)}}_{\mu\nu}$ from the pair sum; the trace of an unbalanced sum, the worldline sector’s kinetic scalar (Computation 159), is not used. Carried in C.8 from v32.74 to v32.128.

C.7 T(C)-contingent (configuration content, not derivable from postulates alone)

Item	Reference	Status
Yukawa mass hierarchy (quarks + charged leptons)	§14.3, Computations 76–80	T(C)-contingent; structural-scope theorem (Computation 1) forces $S_{3}$ symmetry at leading order.
CKM mixing matrix	§14.3	T(C)-contingent; same structural-scope theorem.
Neutrino masses + PMNS matrix	§14.3	T(C)-contingent.
$\Lambda_{\mathrm{obs}}$ cosmological-constant magnitude	§19.9, Computations 82, 188	T(C)-contingent. Sign and equation of state $w=-1$ remain structural theorems; the theory’s own area-law and finiteness resources do not reach the magnitude (Computation 188).
Clustering of the tension structure, i.e. that a laboratory device exists at all	§19.9.2, Computation 156	T(C)-contingent. The compactness of a device is a corollary of (30), and being one is an open condition with sharp margin $m/2$; why $T$ clusters is configuration content.

C.8 Open (not settled either way)

This section indexes §19.9.1 and §19.9; those sections state each item in full and this table exists so that no substantive claim of the book is absent from this appendix.

Item	Reference	Status
Canonical normalisation of the emergent field (P3’s gradient term)	§12.2, Computations 173, 174, 176, 177, 197, 201	Carried as a named premise, not derived. Reading A of §12.2: $\lambda_{\mathrm{PST}}=1/4$ is the canonical quartic, the rival reading excluded at $121\sigma$ (Computation 176) and $9.5\sigma$ (Computation 173) with the regulator escape closed (Computation 174); the Mosco limit does not fix it (Computation 177) and the derived unit of angular momentum does not either (Computation 197). It fixes the coherence length, $d_{0}=\sqrt{2}\,\hbar c/m_{h}=2.2$ am (Computation 201), and with it the Casimir amplitude. The Higgs agreement ($1.4\sigma$ on current inputs, Computation 202) is its check, not its derivation. Was the last structural item until v32.132.
Bridge premise (B)	§19.7	Reduced, not closed: the SM-side matching rests on F4.
Framework premise F4	§12.5	Carried as an explicit unproved premise.
Matched-scaling condition A1	§18.1, Computations 179, 182	Architectural input, not a substrate consequence. The Mosco chain rests on it, and so, since Computation 182, does $\Pi_{\mu\nu}$’s magnitude: A1 is what names the realisation-point spacing on which that magnitude (C.6) rests.
Connes spectral-triple framework assumption	§1.5	Carried as a framework assumption; $n=4$ is enforced by it.
Residue: localising an apparatus in the property domain	§19.9.2, Computation 215	Not a derivation owed: an apparatus is inherited matter by necessity, and which clusters $T(C)$ has is configuration content (the clustering row of C.7).
Residue: the black-hole interior	§19.9.3, Computations 154, 216	Horizon, light and Wald entropy closed; the projected equations stop at $r_{\mathrm{break}}=37$ pm, where a point carries at most seven properties and the singularity has no substrate counterpart; what that region does, the theory does not say.
Residue: the uniform bound beyond $k=482$	§19.9.4, Computations 213, 214	The sharp value $3.6736$ is derived at its maximiser and the limit law holds for every $k\leq 482$ (the closed part is in the C.5 Dirac-convergence row); a proof over the whole regime and the growth of $D_{\infty}(k)$ beyond it are not supplied.