12.2.  Interacting vacuum and $M_{*}$-regulated renormalisation

The substrate grain $\ell_{*}=\hbar c/M_{*}$ sets the cutoff of the modal condensate: above $M_{*}$ the condensate has no modes, and every vertex at which the Higgs doublet, the projected condensate, meets the emergent fields carries the projection’s kernel in the momenta of its legs. It is not a cutoff on the momenta of the emergent quanta themselves: the Large Hadron Collider (LHC) measures jets at seven times $M_{*}$ and dileptons at three times $M_{*}$ with the Standard Model intact, and a kernel on every vertex is excluded by the same data, so of the three readings this sentence admits only the one on the doublet survives (Computation 205, §15.10). The grain is not the coherence length $d_{0}$ (§15.3.1); reading the cutoff off $d_{0}$ would put it at $\hbar c/d_{0}\approx 90$ GeV, a factor $\sqrt{2}\,M_{*}/m_{h}\approx 14$ below $M_{*}\approx 1285$ GeV. This is not a Wilsonian regulator introduced for convenience and to be removed; it is a structural property of the projection $\Pi$. Every loop integral that touches the condensate, the Higgs self-energy’s among them, is therefore ultraviolet (UV) finite at every order, with the hard cutoff $M_{*}$ replacing the formal $\Lambda\to\infty$ limit of standard QFT; a loop with no doublet vertex is the Standard Model’s own.

One-loop Higgs mass correction. Write the order parameter in unitary gauge as $\psi=(r_{0}+h/\sqrt{2})\,e^{i\theta}$, where $h$ is the amplitude fluctuation (PST Higgs) of tree-level mass $m_{h}$ and $\theta$ is the massless Goldstone mode. At one loop and momentum cutoff $M_{*}$, the quadratically sensitive correction to the Higgs mass squared receives contributions from the top quark, electroweak gauge bosons, and Higgs self-coupling [94]:

$$\delta m_{h}^{2}\;=\;\frac{M_{*}^{2}}{16\pi^{2}}\Bigl{[}\underbrace{6\lambda+\tfrac{9}{4}g^{2}+\tfrac{3}{4}g^{\prime 2}}_{\text{bosonic (+)}}\;\underbrace{-\;6\,y_{t}^{2}}_{\text{top (--)}}\Bigr{]},$$ (124)

the standard one-loop quadratic-sensitivity (Veltman) bracket, where $\lambda$, $g$, $g^{\prime}$, $y_{t}$ are the Higgs self-coupling (doublet convention), $\mathrm{SU}(2)_{L}$ and $\mathrm{U}(1)_{Y}$ gauge couplings, and the top Yukawa [160]. Substituting Standard Model values at the electroweak scale [93] ($\lambda\approx 0.13$, $g\approx 0.65$, $g^{\prime}\approx 0.36$, $y_{t}\approx 0.94$):

$$\delta m_{h}^{2}\;\approx\;\frac{0.78+0.95+0.10-5.30}{16\pi^{2}}\,M_{*}^{2}\;\approx\;-\frac{3.47}{16\pi^{2}}\,M_{*}^{2}\;\approx\;-0.022\,M_{*}^{2}.$$ (125)

This is the full one-loop sensitivity, dominated by the top loop and negative. It is the physical, coupling-weighted quantity, and it is not the same object as the scalar-sector self-loop $+(3/2)M_{*}^{2}/(16\pi^{2})$ used below to extract $M_{*}$: the latter keeps only the Higgs self-coupling piece, the former includes top, $W$, and $Z$. The substrate eigenmode cancellation does not reach the top/$W$/$Z$ terms (see the scope discussion below), so equation (125) stands as the honest physical sensitivity.

Naturalness. For $M_{*}=4\pi m_{h}\sqrt{2/3}\approx 1.285\;\mathrm{TeV}$:

$$|\delta m_{h}^{2}|\;\approx\;0.022\times(1285\;\mathrm{GeV})^{2}\;\approx\;(190\;\mathrm{GeV})^{2}.$$ (126)

The fine-tuning parameter [95] is $\Delta\equiv|\delta m_{h}^{2}|/m_{h}^{2}\approx(190)^{2}/(125)^{2}\approx 2.3$, well within the naturalness bound $\Delta\lesssim 10$. For comparison, with a Planck-scale cutoff $M_{P}\approx 1.22\times 10^{19}\;\mathrm{GeV}$, the same formula gives $\Delta\approx 0.022\,(M_{P}/m_{h})^{2}\approx 2.1\times 10^{32}$, the standard hierarchy problem. The PST physical cutoff eliminates this: the bare Higgs mass at the $M_{*}$ scale, $m_{h,0}\approx\sqrt{m_{h}^{2}+|\delta m_{h}^{2}|}\approx 228\;\mathrm{GeV}$, is of order $M_{*}$, not $M_{P}$. The hierarchy problem is absent from the $M_{*}$-regulated theory by construction, not resolved by a mechanism; every loop of the Higgs self-energy has a doublet vertex and is cut at $M_{*}$ in the reading of the cutoff that the data leave standing (Computation 205).

This regulating role is distinct from Veltman cancellation, and the two should not be read as in tension. $M_{*}$ bounds the bare-mass divergence: it is the finite physical cutoff that yields $\Delta\approx 2.3$ above. It is not, however, a Veltman-natural scale. The signed one-loop quadratic-sensitivity bracket $[6\lambda+(9/4)g^{2}+(3/4)g^{\prime 2}-6y_{t}^{2}]$ is $\approx-3.3$ at $M_{*}$ (top-dominated, no zero crossing), so the Standard Model (SM) quadratic sensitivities do not cancel there. It is the magnitude of this bracket that fixes the finite $|\delta m_{h}^{2}|$ entering $\Delta$. Regulating the divergence and Veltman-cancelling it are different properties: $M_{*}$ has the first and not the second, which is precisely why the full-SM extension of the boson-fermion cancellation is settled in the negative (Computation 120), leaving $M_{*}$ a scalar-sector result.

Vacuum stability. The sign of the effective $\mu^{2}$ term determines whether spontaneous symmetry breaking persists. The one-loop correction to $\mu^{2}$ (the negative mass-squared driving spontaneous symmetry breaking [SSB]) is $\delta\mu^{2}=-\delta m_{h}^{2}/2\approx+0.011\,M_{*}^{2}>0$: the quantum correction deepens the sombrero potential rather than flattening it. The renormalised vacuum amplitude is $r_{0}^{\prime}=v^{\prime}/\sqrt{2}$ where $v^{\prime 2}=v^{2}\,(1+|\delta m_{h}^{2}|/m_{h}^{2})\approx 3.3\,v^{2}$, shifting the electroweak scale from $246\;\mathrm{GeV}$ to $\approx 448\;\mathrm{GeV}$ for fixed bare parameters. The minimum exists and is stable; its precise location is set by renormalisation conditions that fix $v_{\mathrm{phys}}=246\;\mathrm{GeV}$. Spontaneous symmetry breaking persists at one loop in the $M_{*}$-regulated theory.

Wilsonian effective field theory. Integrating out all PST modes with momenta $\mu<|k|<M_{*}$ generates the Wilsonian effective action at scale $\mu$ [96]:

$$S_{\mathrm{eff}}[\phi\,;\mu]\;=\;S_{\mathrm{SM+GR}}[\phi]\;+\;\sum_{n\geq 5}\frac{c_{n}^{(i)}}{M_{*}^{n-4}}\,\mathcal{O}_{n}^{(i)}[\phi],$$ (127)

where $\mathcal{O}_{n}^{(i)}$ are dimension-$n$ operators and $c_{n}^{(i)}=O(1)$ are Wilson coefficients. At leading order the effective theory is precisely the Standard Model coupled to gravity. The first correction enters at dimension 6, suppressed as $(E/M_{*})^{2}$; at $E=1\;\mathrm{GeV}$ this is $(10^{-3})^{2}=10^{-6}$, consistent with all precision tests.

Renormalisation group. Below $M_{*}$ no PST modes propagate; the $\beta$-functions for all couplings are exactly those of the Standard Model (no Kaluza–Klein tower [28, 29], no threshold corrections above $M_{*}$). Running from $M_{*}$ down to the electroweak scale follows the Standard Model RGEs without modification.

Vacuum energy. At one loop, the zero-point energy density of the condensate’s own modes in the $M_{*}$-cutoff scheme is $\rho_{\Lambda}^{(1)}\sim M_{*}^{4}/(16\pi^{2})\approx(362\;\mathrm{GeV})^{4}$, a factor $M_{P}^{4}/M_{*}^{4}\approx 8.1\times 10^{63}$ smaller than the naive QFT prediction. The observed cosmological constant $\Lambda_{\mathrm{obs}}\approx(10^{-3}\;\mathrm{eV})^{4}$ is still far below this one-loop estimate; the remaining gap is the standard Standard-Model cosmological-constant fine-tuning problem, identified as inherited rather than solved in §9.1 (joint analysis of Mechanism A and Mechanism B).

The vacuum-stability question is thereby resolved at leading loop order: the $M_{*}$-regulated interacting scalar sector is UV-finite, stable, and reproduces the Standard Model as its low-energy effective field theory (EFT). At two loops the stability holds in the renormalisation-group-improved sense on the whole field range the regulated theory contains: $\lambda(\mu)$ runs monotonically from $0.126$ at $m_{t}$ to $0.093$ at $M_{*}$ and is positive throughout, the quartic’s instability scale lies six and a half decades above $M_{*}$, and the top mass would have to exceed $220$ GeV to bring it below the cutoff (Computation 198). And the construction itself exists for this sector (Computation 200): in finite volume the cutoff theory has finitely many modes and its Hamiltonian is a Schrödinger operator with a quartic potential, self-adjoint with a discrete spectrum and a unique strictly positive ground state; the infinite-volume Euclidean measure exists by Griffiths monotonicity with Dirichlet data, the lattice spacing being $\ell_{*}$ and never sent to zero; and reflection positivity returns a Hamiltonian bounded below whose vacuum it is. What is not established at any order is the same construction for the gauge and fermion sector below $M_{*}$, which is the cutoff Standard Model’s own open problem and is inherited with it (§19.9.1).

One-loop relation $M_{*}=4\pi m_{h}\sqrt{2/3}\approx 1285$ GeV (parameter-free derivation under the $\mathrm{O}(4)$ one-loop count, Computation 109; the substrate-vs-emergent distinction of Computations 93, 97 fixes the zero-mode count, see below). Two inputs enter exactly rather than as free parameters. First, the PST boundary condition. The Landau–Ginzburg (LG) modal potential at the threshold is $F[\psi,0]=\tfrac{1}{4}\psi^{4}$ (equation (24)), so the tree-level Higgs mass term vanishes exactly at $M_{*}$. The coefficient $a_{\mathrm{LG}}(M_{*})=0$ is not a tuning condition but the definition of the threshold, and the physical Higgs mass is therefore entirely radiative. Second, the quartic coupling is the universal Landau–Ginzburg coefficient [8]: $\lambda_{\mathrm{PST}}(M_{*})=\tfrac{1}{4}$.

At the scale $M_{*}$ electroweak symmetry is unbroken ($M_{*}\gg v$), so all four real components of the complex doublet propagate as an $\mathrm{O}(4)$ multiplet with common quartic $\lambda_{\mathrm{PST}}$. The $\mathrm{O}(N)$ scalar vertex $V_{ijkl}=2\lambda(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$ contracted into the one-loop seagull $(1/2)\,V_{ijkl}\,\delta_{kl}$ gives a coefficient $(N+2)\,\lambda$ in front of $M_{*}^{2}/(16\pi^{2})$. All three index contractions are enumerated explicitly in Computation 109: $\delta_{ij}\delta_{kl}\to N\,\delta_{ij}$, and the two same-component contractions $\to 2\,\delta_{ij}$. For $N=4$ this gives the standard $\mathrm{O}(4)$ result

$$\delta m_{h}^{2}\bigl{|}_{\Phi}\;=\;\frac{(N+2)\,\lambda_{\mathrm{PST}}}{16\pi^{2}}\,M_{*}^{2}\;=\;\frac{6\times\tfrac{1}{4}}{16\pi^{2}}\,M_{*}^{2}\;=\;\frac{3}{2}\,\frac{M_{*}^{2}}{16\pi^{2}},$$ (128)

with broken-phase cross-check $3\lambda$ (physical Higgs $hhhh$ loop) $+3\lambda$ (three Goldstones) $=6\lambda$ agreeing with the symmetric-phase $\mathrm{O}(4)$ result. The Higgs self-loop coefficient is therefore $C_{H}\equiv(N+2)\,\lambda_{\mathrm{PST}}=6\lambda_{\mathrm{PST}}=3/2$ exactly.

Which field is canonically normalised, why this step turns on the answer, and which of the two readings survives. The seagull (128) presupposes a canonically normalised scalar: its derivation closes the loop with the propagator $i/p^{2}$, which is what canonical normalisation means. A shift in a physical pole mass is invariant under a rescaling of the field; the quartic is not, transforming as $\lambda\to\lambda Z_{\phi}^{2}$ (§12.5). So (128) is legitimate exactly when $\lambda_{\mathrm{PST}}$ is the coupling of the canonically normalised field. But (132) below asserts $Z^{2}=\lambda_{\mathrm{SM}}(M_{*})/b=0.372$, that is, that the substrate field and the emergent Standard-Model doublet are not canonically normalised with respect to one another, and Computation 119 makes the whole of that ratio a field-strength factor by showing the vertex renormalisation is trivial to all orders. At most one of the two fields can be canonical, and this chapter uses $\lambda_{\mathrm{PST}}$ here and $\lambda_{\mathrm{SM}}(M_{*})$ there.

The two readings do not agree. Feeding the canonical Standard-Model coupling into (128) gives $C_{H}=6\lambda_{\mathrm{SM}}(M_{*})=0.556$ and, solved self-consistently against the two-loop running of Computation 123, $M_{*}\approx 2.2$ TeV rather than $1.285$ TeV; taking the bridge premise $\lambda_{\mathrm{SM}}(M_{*})=b\,\mathrm{e}^{-1}$ instead gives $M_{*}\approx 1.32$ TeV, at which (128) returns $m_{h}\approx 78$ GeV rather than the measured $125$ GeV. The ratio between the two is $1/Z$ identically, which is the cleanest statement of the defect: $M_{*}\propto\lambda^{-1/2}$, and the two couplings differ by $Z^{2}$ by definition. The closed form $M_{*}=4\pi m_{h}\sqrt{2/3}$ and the near-coincidence $Z^{2}\approx\mathrm{e}^{-1}$ are therefore not jointly established by the argument as it stands; the system is over-determined, and which of the two survives depends on a normalisation identification that this section has to supply rather than assume. It is supplied below: the paragraphs that follow settle which reading survives, and the residue they leave is a named assumption rather than an undecided pair. Read them before treating any number downstream of $M_{*}=1.285$ TeV as conditional on an unresolved choice.

And the assumption they leave fixes one more number (Computation 201). Chapter 15 defines the coherence length as the Landau–Ginzburg one, $d_{0}=\sqrt{c_{\scriptscriptstyle LG}/(2|a_{\scriptscriptstyle LG}|)}$ with $|a_{\scriptscriptstyle LG}|=m_{h}^{2}/4$ from the Higgs identification, so the gradient normalisation and $d_{0}$ are one unknown, and the canonical reading adopted below puts $d_{0}=\sqrt{2}\,\hbar c/m_{h}=2.2$ am, a factor $\sqrt{2}\,M_{*}/m_{h}=14.5$ above the grain $\ell_{*}$. That is where the structural front’s last item went (§19.9.1), and it is what sets the Casimir amplitude.

The system is determined once $b$ is read as an output, and Computation 173 solves it. Holding $b$ at $1/4$ is what makes the pair look merely inconsistent; releasing it turns (128) with the canonical coupling and the bridge premise into two equations in two unknowns. They close at $M_{*}=2203$ GeV and return $b=0.2313\pm 0.0020$, a $7.5\%$ gap from P3’s value and $9.5\sigma$ on the parametric budget, so input uncertainty does not absorb it.

One escape was available at that point, and Computation 174 closes it. The coefficient $C_{H}$ depends on the regulator’s shape and not only on its scale: closing the gap needs $\kappa=1.081$ against the step function’s $\kappa=1$, well inside the $1.00$ to $2.00$ that plausible profiles span, and since §12.2 treats $M_{*}$ as a physical breakdown scale rather than a calculational device, PST cannot borrow the step function’s value by default. It owed the envelope. The envelope is now computed, and it is $\kappa=1$. The theory carries two cutoffs a factor $\sqrt{D}$ apart, and the calculation is available at each. At $M_{*}$ the substrate mass spectrum has a hard gap. §12.2 gives it as $m_{S}=|S|^{1/2}M_{*}$ over the non-empty $S\subseteq[D]$, so nothing lies between $0$ and $M_{*}$ at all and the first level sits exactly at $M_{*}$, with degeneracy $\binom{D}{1}=D$. (A convention that carries the Boolean Laplacian’s eigenvalues as $2|S|$ rescales every level together and moves neither the gap nor its edge.) A loop that runs out of modes at a gap edge is cut off by a step, and there is no envelope to choose.

At the matched scaling $\Lambda=\sqrt{D}\,M_{*}$ the Walsh ladder does supply a smooth one, and the substrate’s binomial density of states $\binom{D}{j}$ reproduces the four-dimensional continuum’s $\rho(j)=2j/D^{2}$ in the only moment a seagull sees, the first inverse one: $\langle 1/j\rangle=(2/D)(1+1/D)$ against $2/D$, giving $\kappa=1+1/D$. The two routes lean on different premises and agree, which is the point, but neither is premise-free. The second uses A1’s matched scaling and so inherits its status as architectural input (Appendix C). The first does not need A1, which fixes the ladder’s top, but it is not carried by the gap alone either: the Standard Model lives below $M_{*}$ while the substrate has no state there, so the emergent low-energy modes are collective rather than substrate modes, and a gap in the substrate spectrum does not by itself truncate an emergent loop integral. What closes that step is Computation 105’s identification of the layer boundary at $M_{*}$ with the effective theory’s cutoff, which this book carries as architecture. Reaching $1.081$ would need $D\approx 16$, against a substrate cardinality $|D|=V/a_{n}^{4}$ larger by dozens of orders of magnitude, which is the same finite-$D$ escape Computation 123 rejects on the substrate side.

So the gap is real at the $9.5\sigma$ measured, and it is a statement about this reading alone. The reading in which $\lambda_{\mathrm{PST}}=1/4$ is the canonical coupling gives $M_{*}=1.285$ TeV and Computation 202’s $1.4\sigma$ Higgs match, and nothing in the envelope calculation bears on it. What is excluded is holding both at once. The whole of the remaining tension therefore sits in the normalisation question above, with no regulator ambiguity left to absorb any part of it.

And the normalisation question is itself the value of a single number, which Computation 176 identifies and uses to close one of the two readings. Canonical normalisation is a statement about a kinetic coefficient, and P3 carries one explicitly: the functional weights the Boolean Dirichlet form by a coefficient $c$ that this book constrains only by $c>0$. Nor is $c$ absorbable, the Dirichlet form being a bare sum over hypercube edges with a fixed normalisation; rescaling $\phi=\sqrt{c}\,\psi$ to canonical form does not remove it but moves it into the quartic, $\lambda_{\mathrm{canon}}=1/(4c^{2})$, whence $C_{H}=3/(2c^{2})$ and $M_{*}$ linear in $c$. So $C_{H}=3/2$ and $M_{*}=1.285$ TeV are the $c=1$ values, and the two readings above are exactly $c=1$ and $c^{2}=\mathrm{e}$, their $M_{*}$ ratio being $\sqrt{\mathrm{e}}=1/Z$, which derives the identity noted above rather than observing it.

Read against the canonical quartic, the bridge premise becomes one equation in that one unknown, $\lambda_{\mathrm{SM}}(4\pi m_{h}c\sqrt{2/3})=\mathrm{e}^{-1}/(4c^{2})$, and nothing in it presumes $c=1$: the $\mathrm{e}^{-1}$ is the substrate spectral action’s $f(1)$, exact and $D$-independent by (134) with the Gaussian $f$ itself derived in Computation 174; the $1/4$ is P3’s; and $\lambda_{\mathrm{SM}}$ runs from the measured inputs. The residual is monotone, so the root is unique, and re-solving for $M_{*}$ at each input variation in the manner Computation 173 requires gives $c=0.9978\pm 0.0054$: $0.4\sigma$ from the canonical value and $121\sigma$ from $\sqrt{\mathrm{e}}$ on the 2013 inputs, and $c=0.9938\pm 0.0044$ on current ones, $1.4\sigma$ and $149\sigma$ (Computation 202).

So the second reading is excluded, and by three independent routes. This one puts it $121\sigma$ out; Computation 173 puts it $9.5\sigma$ out as a conflict with P3’s postulated $b=1/4$; and Computation 174 closed the regulator-shape escape that was its only defence. The reading in which $\lambda_{\mathrm{SM}}(M_{*})$ is the canonical quartic is therefore withdrawn, and with it the $M_{*}\approx 2.2$ TeV branch and the $m_{h}\approx 78$ GeV branch. What stands is $\lambda_{\mathrm{PST}}=1/4$ as the canonical coupling, $M_{*}=1.285$ TeV, and Computation 202’s $1.4\sigma$ Higgs match.

What that leaves is an assumption rather than a contradiction, and Computation 177 corrects the shape of it. The surviving reading is the $c=1$ one, and $c=1$ is not derived from P1–P3. Nor, on its own, is it even the right quantity. Computation 177 follows the gradient term through the Mosco limit and finds that what reaches the emergent theory is not $c$ but $K=\tilde{c}\,d_{0}^{2}/16$, the coefficient the projected kinetic term carries once (21) and (57) are composed. And $c$ itself is not a constant: the Boolean gradient is a bare sum over $D$, so the composition gives $K=c\,|D|\,d_{0}^{2}/16$, which A1 sends to infinity as $a_{n}^{-4}$; $c$ therefore carries a $1/|D|$, and the $|D|$-independent quantity P3 owns is $\tilde{c}=c\,|D|$. On top of that $c$ carries the dimensions of $\varepsilon$, so “$c=1$” is neither a pure number nor a statement about a constant. What is assumed is the composite $K=\tilde{c}\,d_{0}^{2}/16$, and Appendix C carries it that way, as the canonical-normalisation premise that fixes $d_{0}$ (Computation 201). The $1.4\sigma$ above must not be read as a measurement that discharges it, for a reason worth stating plainly, since it is the one trap this whole subsection sets. That agreement is Computation 202’s $\lambda_{\mathrm{SM}}(M_{*})=\mathrm{e}^{-1}/4$ test expressed in a different variable, one coincidence and not two. Were $c$ treated as a free parameter fixed by it, the coincidence would become a one-parameter fit rather than a rigid prediction, and the headline result would have to be withdrawn as a confirmation. The coincidence cannot be spent twice: it is either the confirmation of a parameter-free prediction or the determination of $c$, and this book takes the first. What the $1.4\sigma$ does supply is a consistency check on the assumption, and a real one, since out of a continuum of admissible $c$ the data selects the natural value.

A derivation would come from §7.8, where $c$ multiplies the form that goes over to the Laplace–Beltrami form on $M$, which carries a normalisation of its own: either the limit fixes $c$ against it, or $c$ is absorbed into the quantum Gromov–Hausdorff scaling $a_{n}\to 0$ and is not an independent parameter at all. Either outcome would promote the assumption to a theorem and hand the $1.4\sigma$ back as a free-standing prediction. Neither is attempted here.

The cancellation of the PST scalar eigenmode sector follows from the spectral decomposition of the PST modal field in the eigenbasis of the Boolean Laplacian $L^{B}$ (§7.8, equation (55)). Every non-zero eigenmode $\phi_{k}$ (labelled by a non-empty subset $S\subseteq[D]$, with mass $m_{k}=|S|^{1/2}M_{*}\geq M_{*}$) has field-dependent mass $m_{k}^{2}(\phi)=|\phi|^{2}$ from the LG quartic (equation (24)), giving a universal coupling coefficient $c_{k}=1$ for all $k\neq 0$. The Bernoulli measure (equation (9)) assigns equal weight to bosonic ($|S|$ even) and fermionic ($|S|$ odd) eigenmodes, with $N_{B}=N_{F}=2^{D-1}$ modes in each sector (§11.2):

$$\delta m_{h}^{2}\bigl{|}_{\mathrm{PST\,scalar}}\;=\;\frac{M_{*}^{2}}{16\pi^{2}}\,\bigl{[}N_{B}-N_{F}\bigr{]}\;=\;0.$$ (129)

The Higgs zero mode ($S=\emptyset$, eigenvalue $0$ of $L^{B}$) is included in the substrate count. There it is the vacuum expectation value (VEV) direction $\langle\phi\rangle$ (the constant Walsh mode $\chi_{\emptyset}$, the order-parameter direction) rather than a propagating fluctuation. This is what makes $N_{B}=N_{F}=2^{D-1}$ and $N_{B}-N_{F}=0$ in equation (129), with the emergent physical Higgs carried separately by $C_{H}=6\lambda_{\mathrm{PST}}=3/2$ (Computations 93 and 97). The SM fermionic and gauge contributions (top quark, $W$, $Z$) arise from two distinct PST sectors: the fermionic occupation-number sector (§11.2) and the gauge sector of the internal algebra (Chapter 13). They do not cancel: the substrate eigenmode cancellation is a statement about the substrate Walsh modes and does not reach the emergent top/$W$/$Z$ loops, and the full-SM extension is settled in the negative (Computation 120), which is why $M_{*}$ is a scalar-sector result. With equation (129) and $C_{H}=3/2$, equation (128) is the complete one-loop result for the PST scalar sector, predicting the modal scale from the measured Higgs mass:

$$M_{*}^{2}\;=\;\frac{16\pi^{2}\,m_{h}^{2}}{C_{H}}\;=\;\frac{32\pi^{2}}{3}\,m_{h}^{2},\qquad M_{*}\;=\;4\pi\,m_{h}\,\sqrt{\tfrac{2}{3}}\;\approx\;1285\;\mathrm{GeV},$$ (130)

using $m_{h}=125.25\;\mathrm{GeV}$ [93] as input. The denominator $16\pi^{2}$ is the one-loop factor; the $\sqrt{2/3}$ rescaling comes from the $\mathrm{O}(4)$ vertex contraction $(N+2)\lambda=6\lambda$ rather than the naive $N\lambda=4\lambda$ (Computation 109). The two-loop correction from the Higgs self-interaction piece, estimated within the framework of reference [94], is $\delta m_{h}/m_{h}=\lambda_{\mathrm{PST}}/(32\pi^{2})\approx 7.92\times 10^{-4}$, a $0.08\%$ shift, well below the experimental uncertainty on $m_{h}$ and negligible for that piece of the prediction. Stability of the full result through two loops additionally requires that the top-quark and gauge two-loop contributions also cancel; the one-loop scalar-sector argument above does not by itself establish this.

12.2.1.  Scope of the cancellation argument

The $M_{*}$ relation is a scalar-sector result: it follows from the Higgs self-loop alone, and a clean separation between what equation (129) establishes and what would be needed to extend it to the full Standard Model is essential to stating the status honestly. An explicit one-loop diagrammatic accounting separates the two:

1. The PST scalar eigenmode sector cancellation (eq. (129)) is rigorous. Three ingredients are required, all structural:

   • Equal counts. $N_{B}=N_{F}=2^{D-1}$ follows from the binomial identity $\sum_{k\,\mathrm{even}}\binom{D}{k}=\sum_{k\,\mathrm{odd}}\binom{D}{k}=2^{D-1}$; combinatorially exact.

   • Universal coupling. $C_{H}=(N+2)\lambda_{\mathrm{PST}}=6\lambda_{\mathrm{PST}}=3/2$ from the single LG quartic and the $\mathrm{O}(4)$ one-loop vertex contraction (Computation 109); every non-zero eigenmode of $L^{B}$ inherits the same field-dependent mass $m_{k}^{2}(\phi)=|\phi|^{2}$ at this level, giving the same one-loop coefficient $c_{k}=1$ for all $k\neq 0$. Universal at the substrate level by construction (one quartic, one coefficient).

   • Sign assignment. The opposite loop signs of bosonic and fermionic modes follow from the CAR-algebra realisation of P1’s Boolean configurations (§11.2): even-$|S|$ Boolean configurations realise commuting (bosonic) statistics and odd-$|S|$ configurations realise anticommuting (fermionic) statistics, with the standard spin-statistics correspondence carrying through to the one-loop $(-1)^{F}$ factor. Algebraic, not asserted.

   With all three ingredients structural, the $\delta m_{h}^{2}|_{\mathrm{PST\,scalar}}=0$ result is internally rigorous.

2. The SM top/W/Z extension does not exist within PST; $M_{*}$ is irreducibly scalar-sector (settled negatively, Computation 120). The substrate eigenmode cancellation of equation (129) operates on the substrate Walsh modes, which (Computation 105) populate the ultraviolet (UV) shell between $M_{*}$ and the matched cutoff $\Lambda=\sqrt{D}$. The physical top, $W$, $Z$, and Higgs loops that determine the observed $\delta m_{h}^{2}$ run in the emergent EFT shell below $M_{*}$, where (Proposition F4, §12.5) the theory is exactly the Standard Model with its measured, non-universal couplings. By the very disjointness that Computation 105 establishes to defuse double counting, the substrate-layer cancellation cannot reach into the EFT-layer loops; it does not delete the top/$W$/$Z$ contributions to the physical Higgs mass. Indeed the empirical one-loop Veltman quadratic-sensitivity bracket at $M_{*}$ is large and negative, $\mathcal{V}(M_{*})\approx-3.3$ (§12.2), dominated by the top loop, not the $+m_{h}^{2}$ that a clean cancellation would leave. Sector-blindness holds for the substrate count, but it does not control the coupling-weighted EFT-layer quantity, and the universality premise that every Walsh mode carries $c_{S}=1$ at $M_{*}$ is contradicted by the observed Yukawa hierarchy, which §14.3 itself assigns to $T(C)$-contingency rather than to the running of a universal value under the renormalisation-group equations (RGE). An explicit cancellation of the EFT-layer top/$W$/$Z$ quadratic sensitivities therefore does not exist within PST (Computation 120). It would require a bosonic top-partner sector at top strength near $M_{*}\approx 1.3$ TeV to cancel the dominant $-6y_{t}^{2}$, and no such sector is available. It is collider-disfavoured (scalar top-partners and vector-like quarks are excluded by the Large Hadron Collider [LHC] to $\sim 1.2$–$1.4$ TeV, at $M_{*}$ itself), forbidden by the “no new light states” premise of §15.11, and absent from PST’s spectrum (F4: the emergent EFT below $M_{*}$ is exactly the SM). This settles the full-SM-extension question in the negative: a definitive bound on the scope of the $M_{*}$ result, not a closure of it.

The honest status is therefore: $m_{h}^{2}=(3/2)\,M_{*}^{2}/(16\pi^{2})$ is a scalar-sector one-loop result, a parameter-free relation between the Higgs mass and the modal scale for the Higgs self-loop in isolation, and $M_{*}=4\pi m_{h}\sqrt{2/3}\approx 1.285$ TeV is the modal scale it implies. It rests on the substrate-vs-emergent zero-mode distinction of Computations 93 and 97, made explicit via the projection chain $\psi\to\Pi(\psi)\to H_{\mathrm{doublet}}$: the substrate Walsh zero mode $\chi_{\emptyset}$ maps under $\Pi$ to the constant function on $M=\mathbb{R}\times S^{3}$ (vacuum direction), structurally distinct from the emergent 4-component Higgs doublet generated by inner fluctuations of the Dirac operator on $M\times F$. Under this distinction, including the zero mode in $N_{B}$ does not double-count the emergent doublet (captured by $C_{H}=6\lambda=3/2$), and the eigenmode-sum identity $N_{B}-N_{F}=0$ combined with $C_{H}=3/2$ delivers $\delta m_{h}^{2}\big{|}_{\mathrm{scalar}}=(3/2)\,M_{*}^{2}/(16\pi^{2})$. The result is conditional on the EFT-layer top/$W$/$Z$ quadratic sensitivities being separately accounted for, which, as item 2 above states, is not done here and is in tension with the computed Veltman residual.

The PST scalar sector therefore predicts, of itself, a definite scalar self-mass relation. The extension to the full Standard Model is settled in the negative (item 2 above, Computation 120): $M_{*}$ is irreducibly a scalar-sector prediction, and no partner spectrum is claimed here.

Field normalisation and the Higgs VEV. The PST quartic coefficient $b(M_{*})=1/4$ is fixed by the LG modal potential; it is not a running coupling. The SM quartic $\lambda_{\mathrm{SM}}$, by contrast, runs from its electroweak-scale value, $m_{h}^{2}/(2v^{2})\approx 0.129$ at tree level and $\lambda_{\mathrm{SM}}(m_{t})=0.1260$ once matched at next-to-next-to-leading order, upward to $M_{*}\approx 1285\;\mathrm{GeV}$ via the SM renormalisation-group equations, whose one-loop structure is [93]:

$$\beta_{\lambda}\;=\;\frac{1}{16\pi^{2}}\bigl{[}24\lambda^{2}+12\lambda y_{t}^{2}-6y_{t}^{4}-3\lambda(3g_{2}^{2}+g_{1}^{2})+\tfrac{3}{8}(2g_{2}^{4}+(g_{1}^{2}+g_{2}^{2})^{2})\bigr{]},$$ (131)

integrated numerically from $\mu=m_{t}=172.6\;\mathrm{GeV}$ to $M_{*}\approx 1285\;\mathrm{GeV}$ with the full coupled system $\{\lambda,y_{t},g_{2},g_{1},g_{s}\}$ at three-loop precision with next-to-next-to-leading-order matching at the top mass, on current inputs [233] (Computation 202), giving $\lambda_{\mathrm{SM}}(M_{*})\approx 0.0930\pm 0.0008$; simplified two-loop running gives $5$–$7\%$ deviations (Computation 91). These two couplings are defined in different field normalisations; their ratio is the field normalisation factor:

$$Z^{2}\;\equiv\;\frac{\lambda_{\mathrm{SM}}(M_{*})}{b(M_{*})}\;=\;\frac{0.0930}{0.25}\;=\;0.372.$$ (132)

Numerically $e^{-1}=0.36788\ldots$, so $Z^{2}/e^{-1}=1.012$, a $1.2\%$ deviation, a $1.4\sigma$ gap whose uncertainty is dominated by the top mass and its pole-mass ambiguity, not by the running: the third loop moves the value by less than $0.0001$ (Computation 202). No parameter was adjusted to produce this; it follows entirely from PST fixing $b(M_{*})=1/4$, from $M_{*}=4\pi m_{h}\sqrt{2/3}$, and from the observed $(m_{h},v)$.

The proximity of $Z^{2}=0.372$ to $e^{-1}=0.36788\ldots$ is at $1.2\%$ deviation at three-loop RGE precision. This subsection introduces it as an observed near-coincidence; its structural derivation, via the substrate’s binary-gap-tempered Bernoulli partition function bridge, is given in §12.3 below (Computations 84–92). Taking $Z^{2}\approx e^{-1}$,

$$\lambda_{\mathrm{SM}}(M_{*})\;=\;b(M_{*})\cdot Z^{2}\;\approx\;\frac{1}{4e}\;\approx\;0.092.$$ (133)

Running $\lambda_{\mathrm{SM}}(M_{*})=e^{-1}/4$ back down to the top mass gives $\lambda_{\mathrm{SM}}(m_{t})=0.1251$ against the $0.1260$ that next-to-next-to-leading-order matching extracts from the observed $(m_{h},v)$; since the matched coupling scales as $m_{h}^{2}/v^{2}$ at leading order, the prediction returns $v\approx 247.1\;\mathrm{GeV}$ from $m_{h}$ alone, $0.4\%$ above the observed $246.22\;\mathrm{GeV}$, the $1.2\%$ deficit in $\lambda$ seen from the other end and halved by the square root (Computation 202, part 9; the tree-level relation applied to the running coupling at $\mu=v$, which an earlier printing used, returns $256$ GeV even for the data and is withdrawn). Conditional note: this derivation of $v$ uses the SM matching between $(m_{h},v)$ and $\lambda_{\mathrm{SM}}(m_{t})$ plus the SM RGE running between $m_{t}$ and $M_{*}$. The PST contribution is that the relation $Z^{2}\approx e^{-1}$ (§12.3) pins $\lambda_{\mathrm{SM}}(M_{*})$ without using $v$ as an input. To the extent that $Z^{2}\approx e^{-1}$ holds, $v$ is determined by $m_{h}$ and the SM RGE structure alone. The non-trivial claim is thus conditional. If $Z^{2}=e^{-1}$ holds exactly, then given the Standard-Model gauge structure and RGE running between $v$ and $M_{*}$, the two electroweak observables $(m_{h},v)$ collapse to one, with $v$ following from $m_{h}$. That collapse is modulo a single $v\to M_{*}\to v$ running self-consistency, which the headline $e^{-1}$ agreement does not depend on, since $Z^{2}$ pins $\lambda_{\mathrm{SM}}(M_{*})$ without using $v$ as an input. Two caveats attach. First, the substrate-side factor of $Z^{2}=e^{-1}$ is derived structurally from P1–P3 in §12.5 below. The route is the tensor-product factorisation forced by P1’s Bernoulli product measure (Computation 100), with items (a) and (b) of the residual gap closed by Computation 98 through the mode-shell block-spin renormalisation group [RG] argument. The matching identification with the SM-side ratio is bridge premise (B), reduced within PST to a field-strength-renormalisation identity (§12.5). The $1.2\%$ empirical agreement is at full three-loop SM RGE running with next-to-next-to-leading-order matching (Computation 202; $5$–$7\%$ in simplified two-loop running, Computation 91). Second, $Z^{2}$ is itself constructed from $(m_{h},v)$. Since $\lambda_{\mathrm{SM}}(v)=m_{h}^{2}/2v^{2}$ already contains $v$, and the run-up to $M_{*}$ inherits it, the “$v$ from $m_{h}$” map is only well-posed to the extent that the $v\to M_{*}\to v$ running is insensitive to the $v$ being solved for. We assert but do not here demonstrate that self-consistency.

12.2.2.  Candidate substrate-side identification of $Z^{2}$

The observed value $Z^{2}\approx e^{-1}$ has a structural counterpart on the substrate side, closed at the partition-function level in §12.5, where the identification (bridge premise B) is derived within PST in the substrate limit by a wave-function-renormalisation argument and the unique-soft-mode theorem F10. The full Chamseddine–Connes inner-fluctuation route $D+A+\varepsilon^{\prime}JAJ^{-1}$ applied directly to the substrate spectral triple is examined in Computation 81 and collapses at matched scaling, motivating the partition-function-level identification developed in §12.3 and §12.5.

The substrate side is exact. Computation 62 establishes that at the matched scaling $\Lambda=\sqrt{D}$ of the substrate spectral triple, the Gaussian-cutoff substrate spectral action satisfies

$$\frac{S_{\mathrm{sub}}}{2^{D}}\;=\;\frac{\mathrm{Tr}\,f(D_{\mathrm{sub}}/\Lambda)}{\mathrm{Tr}\,1}\;=\;f(1)\;=\;e^{-1},\qquad f(x)=e^{-x^{2}},$$ (134)

$D$-independently and exactly, because the substrate Dirac eigenvalues $\pm\sqrt{D}$ each with multiplicity $2^{D-1}$ collapse to $\pm 1$ under the matched scaling. All $2^{D}$ eigenmodes sit at the cutoff edge with weight $f(1)=e^{-1}$, so the trace ratio is $f(1)=e^{-1}$. This is the substrate analogue of the Chamseddine–Connes spectral-action moment $f_{0}=f(0)$ for the continuum Connes–Dirac triple [49, 48].

The Gaussian is derived, not chosen. The usual defence of $f(x)=e^{-x^{2}}$ here is that $f(1)=\mathrm{e}^{-1}$ for every member of the family $f(x)=e^{-x^{p}}$, which is true and carries no information about the shape at all: the substrate Dirac spectrum is degenerate at $\pm\sqrt{D}$, so every mode sits at $x=1$ and $f(1)$ is the only value ever evaluated. The Walsh mode sum does carry that information. Writing $\kappa$ for the coefficient a profile contributes to a quadratically divergent one-loop integral, $\kappa=2\int_{0}^{\infty}xf(x)\,\mathrm{d}x$, the family has the closed form $\kappa(p)=\Gamma(1+2/p)$, with $\kappa(2)=1$ and $\kappa(1)=2$. Inverting it against the substrate’s own mode sum returns $p_{\mathrm{eff}}\to 2$, with $(2-p_{\mathrm{eff}})\,D\to 2/(1-\gamma_{\mathrm{E}})=4.7305$ (Computation 174). The Gaussian is what the substrate delivers, not what is convenient to assume, and the same calculation fixes the regulator coefficient $\kappa$ that §12.2 needs.