Appendix A Notation Conventions

The following symbols recur throughout the book. The “Where” column points to the place each is first introduced, a section such as §4.6, or a whole chapter where the symbol is the chapter’s subject.

A.1.  Substrate and primitives

Symbol	Meaning	Where
$(D,\delta)$	Discrete distinction substrate: a set $D$ with the distinction primitive $\delta$	Ch. 2
$|D|$	Substrate cardinality (number of distinction bits), generic letter $D$ also used by abuse for $|D|$ in dimensional analysis	Ch. 2
$\mathcal{P}(D)$	Boolean configuration space of subsets of $D$	Ch. 2
$a\in D$	Single distinction (Boolean bit)	Ch. 2
$C$	A precausal configuration, $C\subseteq D$	Ch. 2
$T(C)$	Asymmetric-tension functional on $\mathcal{P}(D)$ (distinct from the Dixon algebra $T=\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$, Ch. 13, and the oblique parameter $T$)	Ch. 2
$\mu$	Bernoulli product measure on $\mathcal{P}(D)$, $\bigotimes_{a}\mathrm{Bern}(1/2)$	Ch. 2
$V_{7}$	7-dimensional algebra carrying the substrate’s directional content, identified with $\mathrm{Im}(\mathbb{O})\cong\mathbb{R}^{7}$ via Hurwitz’s classification	Ch. 2, Ch. 13
$\hat{\tau}\in V_{7}$	Unit direction selected at modal sublimation; fixes the three quaternionic sub-algebras of $\mathbb{O}$ containing $\hat{\tau}$ that index the three matter generations	§1.7

A.2.  Modal threshold and order parameter

Symbol	Meaning	Where
$\tau$	Modal threshold value of the global tension $T(C)$; unsubscripted $\tau$ always means this one	§4.6
$\varepsilon=|T(C)|-\tau$	Excess tension (the control parameter)	Ch. 3
$\psi$	Modal order parameter ($V_{7}$-valued; the complex-scalar form is its 2-plane projection, cf. Ch. 3)	Ch. 3
$\psi^{*}=\pm\sqrt{2\varepsilon}$	Order parameter at the potential minimum	Ch. 4
$F[\psi,\varepsilon]$	Modal potential (Landau–Ginzburg form)	Ch. 4
$\mathcal{V}$	Vacuum manifold; for $\varepsilon>0$, $\mathcal{V}\cong S^{6}\subset V_{7}$ (a 6-sphere of directions); the 2-plane projection is the familiar ring of minima	Ch. 4
$r_{0}$	Vacuum radius, $r_{0}^{2}=\varepsilon/(2b)$	Ch. 4
$m_{0}$	PST elementary mass scale, $m_{0}=4\pi^{3}r_{0}^{2}d_{0}/(3c^{2})$; the form is derived, the magnitude inherits $d_{0}$ and $r_{0}$	§9.5

A.3.  Modal sublimation and projection

Symbol	Meaning	Where
$\mathcal{S}$	Precausal (Boolean / distinction) category; source of $\Phi$	Ch. 4
$\mathcal{G}$	Geometric (manifold / spectral-triple) category; target of $\Phi$	Ch. 4
$\Phi:\mathcal{S}\to\mathcal{G}$	Modal-sublimation functor (precausal $\to$ geometric)	Ch. 4
$\rho$	Realisation map from substrate to spacetime	Ch. 4
$\Pi$	Projection operator, substrate $\to$ four-dimensional field theory	Ch. 4

A.4.  Foundational object: the real spectral triple

Symbol	Meaning	Where
$M=\mathbb{R}\times S^{3}$	Four-dimensional Lorentzian spacetime factor (KO-dimension 4)	§1.5
$F$	Finite noncommutative internal factor (KO-dimension 6)	§1.5
$M\times F$	Foundational real spectral triple (total KO-dimension $\equiv 2\pmod{8}$)	§1.5
$(A,H,D,J,\gamma)$	Spectral-triple data: algebra, Hilbert space, Dirac operator, real structure, chirality grading	§1.5
$A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$	Internal algebra of $F$	§1.5
$\mathrm{Im}(\mathbb{O})\cong\mathbb{R}^{7}$	Imaginary octonions: the algebraic space on which $F$ is built	Ch. 13
$\mathrm{Cl}(0,6)$	Clifford algebra on $\mathrm{Im}(\mathbb{O})$	Ch. 13
$\overleftarrow{\mathbb{O}}\cong\mathrm{Cl}(0,6)\otimes\mathbb{C}\cong M_{8}(\mathbb{C})$	Octonionic left-action algebra on $\mathbb{O}$ (Furey)	Ch. 13
$\omega=L_{e_{1}}\cdots L_{e_{6}}$	Cl(0,6) chirality grading volume element (Spin(6)-invariant)	Ch. 13
$\gamma_{K}=i\omega$	Substrate parity grading; equals the Cl(0,6) chirality grading in the Jordan–Wigner Majorana representation (Computation 19)	Ch. 13
$f=(I+i\omega)/2$	Chirality projector on $H_{F}$; its canonical rank-1 refinement is Furey’s primitive idempotent	Ch. 13
KO-dimension	Connes mod-8 invariant of a real spectral triple; $4+6=10\equiv 2\pmod{8}$ for $M\times F$. The name is from real K-theory, $K$ for Grothendieck’s Klasse and $O$ for the orthogonal group a real bundle carries; it abbreviates no English phrase	§1.5

A.5.  Standard Model and gauge sector

Symbol	Meaning	Where
$\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$	Standard Model gauge group; unimodular unitaries of $A_{F}$	Ch. 13
$\mathrm{SU}(3)_{c}$	Colour sector; the $M_{3}(\mathbb{C})$ factor of $A_{F}$	Ch. 13
$\mathrm{SU}(2)_{L}$	Left-chiral weak sector; the $\mathbb{H}$ factor with the directed modal threshold selecting the left factor of $\mathrm{SO}(4)\cong\mathrm{SU}(2)_{L}\times\mathrm{SU}(2)_{R}$	Ch. 13
$N_{c}=3$	Colour multiplicity; the rank of $M_{3}(\mathbb{C})$	Ch. 13
$N_{\mathrm{gen}}=3$	Generation count; the three quaternionic sub-algebras of $\mathbb{O}$ containing $\hat{\tau}$ (§14.2)	Ch. 14
$D_{F}$	Yukawa / mass operator on $H_{F}$	Ch. 13
$Y$	$\mathrm{U}(1)_{Y}$ hypercharge	Ch. 13
$Z^{2}\approx e^{-1}$	Field-normalisation ratio: substrate-side factor $Z_{H}(\beta_{\mathrm{KO}})\to e^{-1}$ derived from P1–P3 (Computation 100); its matching to the SM-side ratio is bridge premise (B), whose standing the (B) row below records	§12.3

A.6.  Scales and constants

Symbol	Meaning	Where
$m_{h}$	Higgs mass ($\approx 125.25$ GeV)	Ch. 7
$v$	Higgs vacuum expectation value ($\approx 246$ GeV)	Ch. 7
$M_{*}=4\pi m_{h}\sqrt{2/3}$	Electroweak modal scale ($\approx 1.285$ TeV)	Ch. 7
$\ell_{*}=\hbar c/M_{*}$	Modal length scale ($\approx 1.535\times 10^{-19}$ m)	Ch. 12
$d_{0}$	Substrate coherence scale (Landau–Ginzburg coherence length)	Ch. 15
$\ell_{P}$	Planck length	Ch. 8
$M_{P}$	Planck mass	Ch. 7
$G$	Newton’s gravitational constant (consistency relation in PST)	Ch. 7
$\alpha_{G}\equiv G/d_{0}^{2}$	Gravitational-coupling coefficient (in $\hbar=c=1$)	Ch. 8
$\xi=90/\pi^{2}$	Casimir-correction coefficient (parameter-free under the Gaussian projection-kernel choice, dimensionless; the $d^{-6}$ exponent is kernel-independent; distinct from the coherence length $\xi_{\psi}=\xi_{\mathrm{BCS}}=d_{0}$)	Ch. 15
$\Lambda$	Cosmological constant (Ch. 16) and, by reuse of notation, the spectral-action cutoff $\mathrm{Tr}\,f(D/\Lambda)$ (Ch. 7). The two roles are kept apart by context.	Ch. 16, Ch. 7

A.7.  Substrate Dirac and Walsh sector (L_round closure)

Symbol	Meaning	Where
$D_{\mathrm{sub}}$	Substrate Dirac operator on $\bigotimes_{a}\mathbb{C}^{2}$; built by Jordan–Wigner from the $\chi_{a}^{\mathrm{Cliff}}$, satisfying $D_{\mathrm{sub}}^{2}=D\cdot I$	§18.3
$\chi_{S}$	Walsh mode indexed by $S\subseteq D$; eigenvector of all $\chi_{a}^{\mathrm{Cliff}}$	§18.3
$\chi_{a}^{\mathrm{Cliff}}$	Clifford-valued Walsh generator for bit $a\in D$ (Jordan–Wigner)	§18.3
$T_{\mathrm{tail}}$	Walsh-tail operator $\sum_{|S|>k_{D}}\chi_{S}$; complement of the bridge support	§18.3
$T_{\mathrm{tail}}^{+-}$	Off-diagonal block $P_{+}T_{\mathrm{tail}}P_{-}$ in the $D_{\mathrm{sub}}$ eigenbasis	§18.3
$\sigma_{1},\sigma_{2}$	Closed-form operator norms of $T_{\mathrm{tail}}^{+-}$ on $S_{D}$-isotypic components: $\sigma_{1}=2^{D-1}-D$ (trivial), $\sigma_{2}=D-2$ (standard rep)	§18.3
$\rho_{F}$	Fermionic-signed $S_{D}$ representation acting on Walsh modes by $\rho_{F}(g)|S\rangle=\mathrm{sgn}(g|_{S})\,|g(S)\rangle$; commutes with $D_{\mathrm{sub}}$	§18.3
$k_{D}$	Walsh truncation cutoff; the maximum $|S|$ retained in the bridge support	§18.3
$L_{\mathrm{round}}$	Round-trip Lip-norm criterion (substrate $\to$ Bergman $\to$ substrate)	§18.3
$L_{\mathrm{comm}}$	Commutator-norm Lip criterion: $\|[D_{\alpha},b_{C}(\chi_{S})\otimes I_{2}]\|/\|b_{C}(\chi_{S})\|\to 2\sqrt{|S|}$	§18.3
$L_{F}$	Fréchet smooth Lip-norm $L_{F}(\cdot)=\sup_{j}\|\mathrm{ad}_{D}^{j}(\cdot)\|/j!$	§18.3

A.8.  Bergman–Toeplitz sector and the refined bridge (L_comm closure)

Symbol	Meaning	Where
$B^{2}$	Open unit ball $\{(z_{0},z_{1})\in\mathbb{C}^{2}:|z_{0}|^{2}+|z_{1}|^{2}<1\}$	§18.3
$H^{2}_{\alpha}(B^{2})$	Weighted Bergman space on $B^{2}$ with weight $(\alpha+3)(1-|z|^{2})^{\alpha}/\pi^{2}$; canonical weight $\alpha=0$	§18.3
$H^{2}_{\alpha}(B^{2})^{(N)}$	Truncation to monomials $z_{0}^{a}z_{1}^{b}$ with $a+b\leq N$	§18.3
$e_{a,b}=z_{0}^{a}z_{1}^{b}/\|z_{0}^{a}z_{1}^{b}\|$	Orthonormal monomial basis of $H^{2}_{\alpha}(B^{2})$	§18.3
$w=z_{0}z_{1}$	Diagonal monomial; the bridge symbols are polynomials in $w$	§18.3
$T_{f}$	Holomorphic Toeplitz operator with symbol $f$: $T_{f}h=P(fh)$, $P$ = Bergman projection	§18.3
$T_{f_{k}}$	Refined 2-monomial bridge $T_{w^{m(k)}}+\alpha(k)\,T_{w^{m(k)+1}}$ at weight $k$	§18.3
$J_{+},J_{-},J_{z}$	SU(2) generators acting on holomorphic polynomials by $z_{0},z_{1}$ rotation	§18.3
$D_{\alpha}=J_{a}\otimes\sigma_{a}$	SU(2)-equivariant round-$S^{3}$ Dirac on $H^{2}_{\alpha}(B^{2})\otimes\mathbb{C}^{2}$	§18.3
$b_{C}(\chi_{S})=T_{f_{|S|}}$	Bridge map: substrate Walsh mode $\to$ Bergman Toeplitz with the weight-class symbol	§18.3
$B(\cdot)$	Berezin transform: $B(T)(z)=\langle k_{z},Tk_{z}\rangle/\|k_{z}\|^{2}$, $k_{z}$ = reproducing kernel	§18.3
$N$	Bergman truncation parameter; total-degree cutoff $a+b\leq N$	§18.3
$\sigma_{x},\sigma_{y},\sigma_{z}$	Pauli matrices on the $\mathbb{C}^{2}$ spinor factor	§18.3
$I_{2}$	$2\times 2$ identity on the spinor factor	§18.3

A.9.  Bridge parameters and closure-rate quantities (Lemmas 5–8)

Symbol	Meaning	Where
$m(k)$	Bridge monomial power for weight $k$; the unique integer with $m/2<\sqrt{k}$ in the appropriate $(m,m+1)$ range (Lemma 5(b))	§18.3
$\alpha(k)$	Closing coefficient from Lemma 5(c); pins the bulk ratio to $2\sqrt{k}$ exactly	§18.3
$\beta(k)$	Intermediate coefficient in the 3-monomial form of $f_{k}$ (where used; absent in the 2-monomial form)	§18.3
$q$	Parametric variable in the Lemma 5(c) closed form $\alpha(q)=2m(1-mq)/[(m+1)\sqrt{1-q^{2}}((m+1)q-1)]$	§18.3
$\mathrm{gap}(k,N)$	$\mathrm{ratio}(k,N)-2\sqrt{k}$; finite-$N$ deviation from the bulk ratio	§18.3
$c_{1}(k),c_{2}(k)$	Leading and subleading coefficients of the gap expansion $\mathrm{gap}(k,N)=c_{1}/N+c_{2}/N^{2}+O(1/N^{3})$; $c_{1}(1)=-1$ exactly (Lemma 7(a))	§18.3
$\gamma_{D}$	L_comm closure rate at substrate size $D$: $\gamma_{D}=\max_{k\leq k_{D}}|\mathrm{gap}(k,N(D))|$	§18.3

A.10.  Substrate symmetry, complement, and configuration spaces

Symbol	Meaning	Where
$v:D\to V_{7}$	P1 direction-content assignment: each distinction $a$ carries inherent $V_{7}$-valued direction $v(a)$; the third component of the substrate triple $(D,\delta,v)$, with $v(C)=\sum_{a\in C}v(a)$	Ch. 2
$\langle\cdot,\cdot\rangle$, $\times:V_{7}\times V_{7}\to V_{7}$	Octonionic inner and vector products on $V_{7}\cong\mathrm{Im}(\mathbb{O})$; the bilinear product satisfies $\langle u\times v,u\rangle=0$ and $|u\times v|^{2}=|u|^{2}|v|^{2}-\langle u,v\rangle^{2}$	Ch. 2
$\bar{C}=D\setminus C$	Configuration complement; the set of distinctions absent from $C$	Ch. 2
$\Delta T(C):=T(C)-T(\bar{C})$	Net configuration-complement tension imbalance, a nonzero $V_{7}$-vector central to complement-asymmetry and the Bernoulli $p=1/2$ derivation (distinct from the electroweak custodial $\Delta T$, below)	Ch. 3
$\mathrm{Aut}(D,\delta)$	Distinction-preserving bijections of $D$; the canonical measure is invariant under them	Ch. 2
$S_{D}$, $S_{D}\times\mathbb{Z}_{2}$	Full symmetric group on $D$ (driving $\rho$-equivariance); $S_{D}\times\mathbb{Z}_{2}$ adds complementation	Ch. 2, Ch. 6
$S_{3}$	Generation-permutation subgroup; forces the leading Yukawa degeneracy across the three generations	Ch. 13
$L^{2}(\mathcal{P}(D),\mu)$	PST quantum Hilbert space of square-integrable complex functionals on configuration space under $\mu$	Ch. 4
$W^{1,2}(\mathcal{P}(D),\mu)$	Boolean Sobolev space $\{\psi:\mathbb{E}_{\mu}|\nabla_{\mathcal{C}}\psi|^{2}<\infty\}$ on which the Dirichlet form lives	Ch. 4

A.11.  Order parameter and the Boolean gradient

Symbol	Meaning	Where
$\hat{\eta}\perp\hat{\tau}$	Unit direction perpendicular to $\hat{\tau}$ spanning the projection 2-plane in $V_{7}$	Ch. 3
$\mathbb{C}_{\hat{\tau},\hat{\eta}}$	The 2-plane $\subset V_{7}$ spanned by $\hat{\tau}$ and $\hat{\eta}$ onto which the $V_{7}$ Landau–Ginzburg potential projects	Ch. 9
$\psi_{\mathbb{C}}$	Complex-scalar order parameter $\psi_{\mathbb{C}}=\langle\psi,\hat{\tau}\rangle+i\langle\psi,\hat{\eta}\rangle$ recovering the textbook complex LG/Higgs slice	Ch. 9
$\hat{\psi}\in S^{6}$	Unit direction of the order parameter, $\hat{\psi}=\psi/|\psi|$ on the vacuum 6-sphere; modal sublimation locks it to $\hat{\tau}$	Ch. 3
$\nabla_{\mathcal{C}}\psi$, $|\nabla_{\mathcal{C}}\psi|^{2}$	$V_{7}$-valued Boolean Dirichlet gradient summed over single-bit-flip (Hamming) edges of $\mathcal{P}(D)$	Ch. 3
$\partial_{a}^{B}$	Per-site Boolean difference, $\partial_{a}^{B}\psi=\psi(C\cup\{a\})-\psi(C\setminus\{a\})$	Ch. 7
$C\triangle C^{\prime}$	Symmetric difference of configurations; $|C\triangle C^{\prime}|=1$ indexes the gradient edges	Ch. 4

A.12.  Mosco convergence and A7 reduction

Symbol	Meaning	Where
A1–A7	The seven explicit conditions for Mosco convergence; A7 (equicoercivity) is a theorem given A1, A2, A4 and A3, its equidistribution condition A3 supplied by P3’s maximum-entropy sublimation, while A1 is architectural input	§18.1
(240)	The uniform-rate hypothesis to which A7 reduces; the $K$-uniform Berry–Esseen rate $\eta_{K,R}<d_{0}^{2}/4$	§18.1.6
$\eta_{K,R}$	Rate constant in the uniform-rate hypothesis; the equidistribution defect the continuum geometry can absorb while still closing	§18.1.6
$C_{K}$	Equicoercivity (closure-margin) constant, $C_{K}=(d_{0}^{2}/4-\eta_{K,R})/(C_{P}(K)^{2}+1)$; $C_{K}>0$ is the condition that spacetime closes as a manifold	§18.1.6
$C_{P}(K)$	Poincaré constant of compact $K\subset(M,g)$ for test functions supported in $K$	§18.1.6
$D_{n}$, $\rho_{n}$	Sequence of finite site sets $|D_{n}|\to\infty$ with realisation maps $\rho_{n}:D_{n}\to M$	§18.1
$\mathcal{T}_{n}$, $\phi_{a}$	Delaunay triangulation of the realised cloud and its hat (Lagrange) basis functions	§18.2
$a_{n}$, $V_{M}$	Matched site spacing per scaling A1, $a_{n}\to 0$ with $|D_{n}|\,a_{n}^{4}\to V_{M}$ the Riemannian volume of $M$ (distinct from the fixed coherence length $d_{0}$)	§18.1
$E^{B}$ / $E^{B}_{n}$	Boolean Dirichlet form on $L^{2}(\mathcal{P}(D_{n}),\mu_{n})$; central object of the Mosco argument	§18.1
$L^{B}$ / $L^{B}_{n}$	Boolean Laplacian; generator of $E^{B}$	§18.1
$E^{\mathrm{Riem}}$, $\Delta_{g}$	Mosco limits: the Riemannian Dirichlet form and the Laplace–Beltrami operator on $(M,g)$	§18.1
$\Pi^{\dagger}$ / $\Pi_{n}^{\dagger}$	Adjoint pullback $C_{c}^{\infty}(M)\to L^{2}(\mathcal{P}(D_{n}),\mu_{n})$; comparison map in the Mosco recovery sequence	§18.1
$A_{n}(u)$	Empirical aggregate (exact pullback of $E^{B}_{n}$) whose mean deviation is the binding residual	§18.1.6
$\xi_{a}$	i.i.d. Gaussian per-site displacement with covariance $\sim d_{0}^{2}$	§18.2
$\mathcal{F}(K,R)$	Test-function class $\{u\in C_{c}^{\infty}(K):\|u\|_{H^{1}}\leq R\}$ on which the rate hypothesis is required uniform	§18.2
$c_{\mathrm{BE}}=0.4748$	Optimal classical Berry–Esseen constant (Shevtsova), quoted only for the i.i.d. surrogate central-limit comparison; the proof’s fluctuation channel is the McDiarmid concentration of Step 3	§18.2
$\rho_{3}$	Third absolute moment of the per-site displacement in the Berry–Esseen rate	§18.2

A.13.  Higgs naturalness and the $M_{*}$ one-loop derivation

Symbol	Meaning	Where
$\delta m_{h}^{2}$	One-loop quadratically-sensitive correction to the Higgs mass squared at cutoff $M_{*}$	§12.2
$\mathcal{V}(M_{*})$	Veltman quadratic-sensitivity bracket $[6\lambda+\tfrac{9}{4}g^{2}+\tfrac{3}{4}g^{\prime 2}-6y_{t}^{2}]\approx-3.3$ at $M_{*}$ (top-dominated, no zero crossing)	§12.2
$\Delta\equiv|\delta m_{h}^{2}|/m_{h}^{2}$	Fine-tuning (naturalness) parameter, $\approx 2.3$, well within $\Delta\lesssim 10$	§12.2
$C_{H}=6\lambda_{\mathrm{PST}}=\tfrac{3}{2}$	Higgs self-loop coefficient $(N+2)\lambda_{\mathrm{PST}}$, fixing $m_{h}^{2}=C_{H}M_{*}^{2}/(16\pi^{2})$	§12.2
$N_{B}=N_{F}=2^{D-1}$	Substrate boson and fermion mode counts; their equality cancels the leading $\delta m_{h}^{2}$ divergence	§12.2

A.14.  Partition-function / Chamseddine–Connes correspondence

Symbol	Meaning	Where
$\beta_{\mathrm{KO}}=2$	Binary-gap tempering exponent $2=1-(-1)$ (the spectral range of one binary distinction; the KO subscript is historical, not load-bearing)	§12.3
$Z_{H}(\beta)$	Tempered substrate partition function $Z(H_{\mathrm{Higgs}},\beta)=\mathbb{E}_{\mu}[\exp(-\beta H_{\mathrm{Higgs}})]\to e^{-1}$ at $\beta_{\mathrm{KO}}$	§12.5
$H_{\mathrm{Higgs}}(C)=\bar{X}$	Substrate Higgs Hamiltonian $=|C|/D$, forced by additivity, matched scaling, and uniformity (no free parameters)	§12.3
$\bar{X}(C)=|C|/D$	Empirical mean of bit-occupation over the substrate	§12.3
$B_{a}(C)$	Bit-occupation indicator for site $a$ in configuration $C$	§12.3
F1–F11	The eleven labelled statements of the partition-function-level CC correspondence; rostered in full, with pages, in the next block	§12.5
Bridge Premise (B)	$\lambda_{\mathrm{SM}}(M_{*})=\lambda_{\mathrm{PST}}^{\mathrm{bare}}\cdot Z(H_{\mathrm{Higgs}},\beta_{\mathrm{KO}})$; substrate side closed, the SM-side matching reduced to a field-strength-renormalisation identity (open)	§12.5
$\phi$ / $\Phi$	SM-side Higgs field $\phi=\Pi(\psi)$ and doublet $\Phi$ (distinct from the substrate $\psi$ and the functor $\Phi$)	Ch. 12
$\Phi^{\dagger}\Phi$	Higgs bilinear $=\langle\psi,\psi\rangle$ under the projection	§1.7
$\mu^{2}$	SM mass-squared parameter matched to PST excess tension $\varepsilon$	Ch. 12
$\lambda_{\mathrm{SM}}$	Running SM quartic, $\lambda_{\mathrm{SM}}(M_{*})=b\cdot Z^{2}\approx 0.092$; distinct from the substrate quartic $b=\lambda_{\mathrm{PST}}=1/4$	Ch. 12

A.15.  The framework statements F1–F11

The eleven statements of the partition-function Chamseddine–Connes framework (§12.5), which the text cites by label alone.

Label	Kind and statement	§
F1	Definition. The PST partition-function spectral triple	12.6.1.1
F2	Definition. The $\beta$-partition function $Z(O,\beta)=\mathbb{E}_{\mu}[\exp(-\beta O)]$ of a substrate observable	12.6.1.2
F3	Definition. The matched-scaling exponent $\beta=2$, the binary-distinction gap $2=1-(-1)$	12.6.1.3
F4	Proposition. The PST effective theory below $M_{*}$ is exactly the Standard Model. The emergence premise, and the one the closure ultimately leans on: the $\lambda_{\mathrm{SM}}(M_{*})$ prediction is parameter-free only if it holds	12.6.2.1
F5	Proposition. Wilsonian matching at $M_{*}$	12.6.2.2
F6	Proposition. The PST UV Higgs quartic $\lambda^{\mathrm{bare}}_{\mathrm{PST}}=b=1/4$	12.6.2.3
F7	Proposition. Substrate UV–IR scale separation at $\Lambda=\sqrt{D}$	12.6.2.4
F8	Theorem. The substrate Higgs Hamiltonian $H_{\mathrm{Higgs}}=\bar{X}$, forced by additivity, matched scaling and uniformity (Computation 89)	12.6.3.1
F9	Theorem. The binary-gap-tempered Bernoulli partition function $Z(H_{\mathrm{Higgs}},\beta_{\mathrm{KO}})\to\mathrm{e}^{-1}$, by the Cramér–Bernoulli asymptotic (Computation 88)	12.6.3.2
F10	Theorem. Unique soft mode; total reduction of the bridge premise	12.6.3.3
F11	Theorem. $Z^{2}\to\mathrm{e}^{-1}$ in the substrate	12.6.3.5

A.16.  Standard Model sector, extended, and the algebra synthesis

Symbol	Meaning	Where
$G_{2}=\mathrm{Aut}(\mathbb{O})$	Exceptional 14-dimensional automorphism group of the octonions; preserves the inner and vector products on $V_{7}$	Ch. 13
$\mathrm{Stab}_{G_{2}}(\hat{\tau})=\mathrm{SU}(3)_{c}$	Stabiliser of the threshold direction $\hat{\tau}$ in $G_{2}$; the source of colour $\mathrm{SU}(3)_{c}$ (Freudenthal–Tits)	Ch. 13
$A_{\mu}$, $F_{\mu\nu}$	U(1) connection, massive here since the U(1) acting on $\theta$ is broken (the $Z$, Computation 172); the photon is the unbroken combination, with $D_{\mu}\theta=\partial_{\mu}\theta-A_{\mu}$, and its field strength $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$	§13.3
$S_{\mathrm{Maxwell}}$	Gauge kinetic term, supplied by the spectral action’s inner fluctuations (Computation 6); its coupling is not fixed by the theory	§13.3
$\theta(x)$	Local Goldstone / vacuum phase assigned per spacetime point by $\Pi$, gauging the global U(1)	§13.3
$H=(H^{+},H^{0})^{T}$	SM Higgs doublet; the neutral component $H^{0}$ is identified with $\psi$ in the broken phase	§13.4
$\mathcal{V}_{\mathrm{EW}}$	Electroweak vacuum manifold $\{H:|H|^{2}=v^{2}/2\}\cong S^{3}\cong\mathrm{SU}(2)$	§13.4
$\mathrm{SU}(2)_{L}\times\mathrm{SU}(2)_{R}$	Isometry group of $\mathcal{V}_{\mathrm{EW}}$; $\mathrm{SU}(2)_{R}$ broken by the directed threshold crossing, leaving chiral $\mathrm{SU}(2)_{L}$	§13.4
$\lambda_{a}$, $f_{abc}$	Gell-Mann matrices and their structure constants in the octonionic colour derivation	Ch. 13
$C_{2}$	SU(3) quadratic Casimir $\sum_{a}(\mathrm{ad}\,\Lambda_{a})^{2}/4$; irrep discriminator in the $M_{8}(\mathbb{C})$ decomposition	Ch. 13
$\alpha_{k}$	Fermionic ladder operators $\tfrac{1}{2}(L_{2k-1}+iL_{2k})$ from octonionic left-multiplications, building one generation’s colour triplet	Ch. 13
$Q_{L}$, $u_{R}$, $d_{R}$, $L_{L}$, $e_{R}$, $\nu_{R}$	One-generation SM fermion multiplets: quark doublet, up/down singlets, lepton doublet, charged-lepton singlet, right-handed neutrino	Ch. 13
$\tilde{H}=i\sigma_{2}H^{*}$	Conjugate Higgs ($Y=-1$) for the up-type Yukawa coupling	Ch. 13
$c_{a},c_{a}^{\dagger}$, $n_{a}$	CAR creation/annihilation operators with $\{c_{a},c_{b}^{\dagger}\}=\delta_{ab}$ and occupation $n_{a}=c_{a}^{\dagger}c_{a}\in\{0,1\}$	Ch. 11
$N=\sum_{a}n_{a}$, $\mathcal{H}_{D}=\bigoplus_{k}\wedge^{k}\mathbb{C}^{|D|}$, $\mathcal{H}_{D}^{\pm}$	Total occupation grading the exterior-algebra Fock space $\mathcal{H}_{D}$ by $(-1)^{N}$ into bosonic/fermionic sectors $\mathcal{H}_{D}^{\pm}$	Ch. 11
$T=\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$	Dixon hyperspinor algebra (64-dimensional); the full electroweak setting (distinct from the tension functional $T$)	Ch. 13
$\mathrm{Cl}(1,3)\cong M_{2}(\mathbb{H})$	Local Lorentz Clifford algebra of $M$; its right-$\mathbb{H}$ action supplies the internal $\mathrm{SU}(2)$	Ch. 13
$\mathcal{S}$	Minimal left ideal $(\mathrm{Cl}(0,6)\otimes\mathbb{C})\,p\cong\mathbb{C}^{8}$, generated by $f$’s canonical rank-1 refinement $p$; one generation’s content	Ch. 13
$\mathcal{C}$	Furey chain algebra; its commutant on the minimal left ideal is $\mathbb{C}$ (the overall phase), and the algebra generated by $\mathrm{Stab}_{G_{2}}(e_{7})$ on the chiral half is $\mathbb{C}\oplus M_{3}(\mathbb{C})$	Ch. 13

A.17.  Spectral action, Seeley–DeWitt, and cutoffs

Symbol	Meaning	Where
$f_{0},f_{2},f_{4}$	Cutoff-function moments $f_{0}=f(0)$, $f_{2}=\int_{0}^{\infty}uf(u)\,du$, $f_{4}=\int_{0}^{\infty}u^{3}f(u)\,du$ of the spectral-action heat-kernel expansion	§7.8
$\kappa_{\mathrm{grav}}$	$O(1)$ scheme constant in $1/(16\pi G)=(f_{2}\Lambda^{2}/\pi^{2})\,\kappa_{\mathrm{grav}}\,N_{\mathrm{dof}}$	§7.8
$a_{1}=\tfrac{1}{6}$	Universal first Seeley–DeWitt coefficient, forcing the leading $R$ (Einstein–Hilbert) curvature term	§7.8
$\mathcal{A}_{2}(x)$	Second Seeley–DeWitt coefficient ($\tfrac{1}{180}(\dots)+\tfrac{1}{72}R^{2}$ curvature combination) in the gravitational spectral-action expansion	§7.8
$\Lambda_{\mathrm{UV}}=1/\ell_{*}$	Physical UV cutoff; a genuine breakdown scale of the continuum description, not a regulator	Ch. 13
$\rho_{\mathrm{A}}^{\mathrm{bare}}$	Bare Goldstone zero-point density $\Lambda_{\mathrm{UV}}^{4}/(16\pi^{2})+\text{counter-terms}$ at fixed UV regulator	§9.1
$\Lambda_{\mathrm{eff}}=M_{*}/\sqrt{c}$	Effective four-fermion / dimension-six operator scale ($\approx 30$ TeV), with $c$ the (loop-suppressed) $O(1)$ dimension-six Wilson coefficient	§15.11
$\Lambda=\sqrt{D}$	Matched-scaling substrate spectral-action cutoff, at which the substrate Dirac eigenvalues collapse to $\pm 1$	Ch. 12

A.18.  Quantum mechanics from the substrate

Symbol	Meaning	Where
$\hat{x}_{\mu}$	Position operator, $(\hat{x}_{\mu}\psi)(x)=x_{\mu}\,\psi(x)$ on $L^{2}(M,d^{4}x)$	§10.2
$\hat{p}_{\mu}=-i\hbar\,\partial_{\mu}$	Momentum operator; the strong limit of the projected Boolean gradient $\Pi(\partial_{\mu}^{B})$	§10.3
$P_{i}$	Born-rule orthogonal projectors onto outcome eigenspaces, $P_{i}P_{j}=\delta_{ij}P_{i}$, $\sum_{i}P_{i}=\mathbf{1}$	§10.2
$\mathcal{S}_{i}$	Measurement partition cell $\subseteq\mathcal{P}(D)$ whose projected wavefunction lies in eigenspace $\mathcal{H}_{i}$; the $\{\mathcal{S}_{i}\}$ partition $\mathcal{P}(D)$	§10.3
$\tau_{i}(C)$	Outcome-dependent modal threshold $\tau_{0}+\kappa(\lambda o_{i})^{2}$ crossed at measurement (distinct from the modal threshold $\tau$)	§10.3

A.19.  Cosmology and equation of state

Symbol	Meaning	Where
$\rho_{\mathrm{inst}}$	Effective energy density of ongoing instantiation, $\rho_{\mathrm{inst}}=j_{\tau}\,\Delta\mathcal{F}/c^{2}>0$; $\dot{\rho}_{\mathrm{inst}}=0$ (does not dilute)	§16.5
$w=-1$	Equation-of-state parameter: the instantiation reservoir’s pressure over its energy density, $P_{\mathrm{inst}}/\rho_{\mathrm{inst}}$, with $P_{\mathrm{inst}}=-\rho_{\mathrm{inst}}$ forced by covariant energy conservation at $\dot{\rho}_{\mathrm{inst}}=0$; a structural theorem (de Sitter, not anti-de Sitter)	§16.5
$\Delta\mathcal{F}$	Free-energy (tension) released per threshold crossing, $\mathcal{F}_{\mathrm{pre}}-\mathcal{F}|_{\mathcal{V}}>0$	§16.5
$j_{\tau}$	Instantiation current; threshold-crossing rate, constant by the Cosmological Principle theorem	§16.5
$\Lambda_{\mathrm{PST}}=8\pi G\rho_{\mathrm{inst}}/c^{2}$	Positive effective cosmological term contributed by ongoing sublimation	§16.5
$\Lambda_{\mathrm{obs}}$	Observed cosmological constant ($\approx 1.1\times 10^{-52}\,\mathrm{m}^{-2}$); its magnitude remains $T(C)$-contingent and open	§16.5

A.20.  Overloaded symbols (disambiguation)

Symbol	Meaning	Where
$\kappa$	Overloaded: scalar curvature (in $\Phi$); substrate-apparatus measurement coupling (collapse rule); and the spectral-action moment factor $\kappa_{\mathrm{grav}}$. Kept apart by context	Ch. 4, §10.3, §7.8
$\Delta T$	Overloaded: the substrate complement-tension $\Delta T(C)=T(C)-T(\bar{C})$ (a $V_{7}$-vector) versus the electroweak custodial / oblique $T$-parameter $\Delta T$ ($\Delta T=0$ structurally). Distinct objects	Ch. 3, Ch. 7
$\tau$ family	Three distinct thresholds, kept apart by decoration: bare $\tau$ is the modal threshold on the global tension $T(C)$; $\tau_{\mathrm{null}}$ is the null threshold on the pairwise tension $T(\{a,b\})$, the causal boundary supplying the Lorentzian signature; $\tau_{i}$ is the modal threshold shifted by the measurement coupling. $\hat{\tau}$ is a direction, not a value	§4.6, §4.10, §10.3
$d_{0}$ vs $\ell_{*}$	Two different lengths, ten orders apart at the bound and one under the canonical normalisation, and the one confusion this appendix most needs to prevent. $d_{0}\lesssim 2.3$ nm is the coherence length of the modal vacuum, an LG-dynamics quantity that sets the Casimir amplitude. $\ell_{*}=\hbar c/M_{*}\approx 1.5\times 10^{-19}$ m is the substrate grain, a momentum cutoff. The ultraviolet cutoff is $\Lambda_{\mathrm{UV}}=1/\ell_{*}$, never $1/d_{0}$; reading it off $d_{0}$ would put it at tens of electronvolts instead of $M_{*}\approx 1285$ GeV, and Computation 167 prices the difference at $(d_{0}/\ell_{*})^{6}\sim 10^{61}$ at the bound; under the canonical normalisation $d_{0}=2.2$ am and the ratio is $14.5$ (Computation 201).	§15.3.1
$\mathcal{V}$	Overloaded by an explicit convention: $\mathcal{V}\cong S^{6}\subset V_{7}$ is the full vacuum manifold, while $\mathcal{V}\cong S^{1}$ means its projection to the 2-plane slice, written $\mathcal{V}_{\hat{\eta}}\cong S^{1}$ where the distinction matters, as declared at §5. Appendix C uses both senses.	§5