Appendix B Computation Scripts, Computations 204 to 216

Computation 204 ·  §19.2.1 ·  [PROOF]
The $1/N$ cancellation of Computation 203 proved, and the $N^{-2}$ sawtooth in closed form.

Both shifts move the degree by $2m$, so the degree-dependent Bergman factors cancel and $\mathrm{ratio}_{m}(N)^{2}=m^{2}\prod_{i\leq m+1}(a^{*}+i)\prod_{i\leq m-1}(b^{*}+i)/\prod_{i\leq m}(x+i)^{2}$ with $x=s/2$, a finite product. Writing $a^{*}=(m+1)x/m+\delta$, $b^{*}=(m-1)x/m-\delta$, with $\delta\in((1-m)/(2m),(m+1)/(2m)]$ a function of $s\bmod 2m$, the logarithm’s expansion in $1/x$ is a finite Taylor expansion: the $\delta$ terms cancel at first order ($m\delta/x-m\delta/x$), the first-order sums are $m(m+2)/(2x)+m^{2}/(2x)=m(m+1)/x$ above and $m(m+1)/x$ below, and the second-order term is $-m(6m^{2}\delta^{2}-6m\delta+1)/(6(m^{2}-1)x^{2})$, equation (277). Verified symbolically (exact series, $\delta$ symbolic) for $m=2$ to $12$, the general-$m$ first-order identity checked as a closed sum, and the $N^{-2}$ coefficient verified in exact rational arithmetic at every residue of $s\bmod 2m$ for $m\leq 7$, the residual falling as $1/N$. The coefficient is $6(m\delta-\frac{1}{2})^{2}-\frac{1}{2}$ times $m/(6(m^{2}-1))$: negative when the lattice point sits at the continuum maximiser, $3m^{2}/2-1/2$ at the ends of $\delta$’s range, so the deficit changes sign along the sawtooth. Why it cancels: first-order stationarity at an interior maximum; an $O(1)$ displacement of $a^{*}$ leaves the $1/N$ term zero, a displacement proportional to $x$ moves the limit itself below $\mathrm{ratio}_{\mathrm{cf}}$, and at $m=1$ the maximiser is the corner $b^{*}=0$, where stationarity fails and $\mathrm{ratio}^{2}=4(N-1)/N$ keeps its $1/N$.

 

Computation 205 ·  §15.10 ·  [EMPIRICAL]
The $M_{*}$ confrontation: three readings of the modal cutoff against the LHC, two excluded by enormous margins, the third consistent and predicting.

The computation the book said had not been performed. Scope, v32.153: the first measurement in the bin R3 names, ATLAS’s boosted $H\to\tau\tau$ on Run 2 plus partial Run 3 [234], gives above $650$ GeV a Run 2 deficit and a combined upper limit of $4.5$ times the Standard Model at $95\%$ confidence, a limit and not the $15\%$ measurement that would exclude R3; the exposure stands untested. The readings: R1, a momentum cutoff above which no field mode exists; R2, the projection’s Gaussian kernel on every vertex; R3, the kernel on the Higgs doublet alone, every vertex with a doublet leg carrying it in the momenta of its legs, gauge and fermion vertices carrying none. The data, all ATLAS at $13$ TeV: inclusive jets to $p_{T}=3.5$ TeV and dijets to $9$ TeV with NNLO QCD [217]; dileptons above $2$ TeV, $25$ observed against $22$ expected, highest at $4.06$ TeV, contact-interaction limits of $32$ to $36$ TeV (constructive) and $26$ to $29$ TeV (destructive) [218]; $\kappa_{V}=1.035\pm 0.031$, $\kappa_{F}=0.95\pm 0.05$ [219]; $\mu_{HH}<2.4$ [220]; boosted $H\to b\bar{b}$ limits of $115$ fb above $450$ GeV and $9.6$ fb above $1$ TeV against $18.4$ and $0.13$ fb [221]; same-sign $WW$ at $2.92\pm 0.22\pm 0.19$ fb [222]. (1) R1 predicts zero events above $1.285$ TeV: excluded at probability zero to every decimal. (2) R2 multiplies the Drell–Yan yield at $2.2$ TeV by $0.053$, so $12.4$ expected dielectron events become $0.66$ against $19$ observed ($p=10^{-20}$); linearised, it is a contact interaction with $\Lambda_{\mathrm{eff}}=13$ to $16$ TeV by chirality against limits of $32$ to $36$ TeV, short by a factor $4$ to $7$ in $1/\Lambda^{2}$; the data require a universal kernel at $2.7\,M_{*}$ or above. (3) R3: coupling modifiers shift by $-0.5$ percent ($1.3\sigma$ and $0.9\sigma$ from the measurements); Higgs production suppressed by $\mathrm{e}^{-p_{T}^{2}/M_{*}^{2}}$, $0.88$, $0.77$, $0.55$ above $450$, $650$, $1000$ GeV, invisible at present precision; $\mu_{HH}=0.73$; with the whole doublet carrying the kernel the longitudinal cancellation is preserved, whereas a kernel on the physical $h$ alone breaks $|a_{0}|<1/2$ at $1.5$ TeV; every loop of the Higgs self-energy has a doublet vertex, so the naturalness result stands, and what is given up is finiteness of loops that never touch the condensate. (4) Predictions: a $23$ percent deficit above $650$ GeV and $45$ percent above $1$ TeV in the Higgs $p_{T}$ spectrum, $27$ percent in Higgs pairs, modifiers within half a percent of one. (5) Computation 10 revisited: its $M_{*}=1.573$ TeV predates the correction and its $O(1)$ Wilsonian reading is R2; its structural argument is R3. Tampers: a universal kernel at $3\,M_{*}$ passes the dilepton limits, so the exclusion is about the scale; under R3 a kernel at $M_{*}/2$ would put the $650$ GeV bin at $0.36$ of the Standard Model, which the measurement disfavours, so R3 is falsifiable.

 

Computation 206 ·  §19.2.1 ·  [NUMERICAL]
The two-monomial deficit to $N=15360$: no plateau, so the per-$k$ constant of Computations 157 and 203 is a value, not a limit.

Before an analytic route to the bridge’s $1/N$ constant is attempted, this file asks whether the number is a limit. Computation 157’s chain reduction with the weights vectorised, every chain scanned at $N\leq 3840$ and a refined window around the extrapolated extremal chain above it. (1) The deficit times $N$ rises monotonically through $N=15360$: $0.851$, $0.880$, $0.913$, $0.948$, $0.983$, $1.015$ at $k=2$ and $0.811$, $0.838$, $0.881$, $0.934$, $0.992$, $1.050$ at $k=6$ for $N=480$ to $15360$, increments per doubling of $0.03$ and $0.06$ that shrink only slowly; run offline to $N=61440$ the values are $1.072$ and $1.162$. The data do not decide between a limit approached logarithmically and a term of order $\log N/N$. (2) The commutator’s extremal chain moves, its start at $0.392\,N$ to $0.420\,N$ ($k=2$) and $0.208\,N$ to $0.222\,N$ ($k=6$), towards but not at the single-shift maximiser’s position; the tamper evaluating the norm on that maximiser’s chain falls short by a growing margin. (3) The symbol’s deficit moves the other way, $1.01$ to $0.46$ at $k=2$ from $N=480$ to $15360$, crossing the exact value, so the symbol route to a closed form is closed. (4) Every value computed lies below the certified $3.6736$ at $(6,12)$; the uniform bound holds on every point, and whether it holds as $N\to\infty$ for fixed $k$ is exactly what is left open, so the closure rate of Lemma 7 is established up to a possible logarithm. Settled by Computation 213: the value is a limit approached from below as $N^{-1/3}$, and the limit is in closed form.

 

Computation 207 ·  §20.6 ·  [SURROGATE]
The spectral dimension of the projected geometry: it rises between the grain and the coherence length, sits at four above $d_{0}$, and never reaches two.

The configuration-space walk of the Boolean Dirichlet form, one flip per unit of Mosco time, pushed to $M$ by the A4 displacement covariance, has an exact return probability at resolution $\epsilon$: a Binomial mixture over the number of odd-flipped bits of chi-square tails. Carlip’s $d_{s}=-2\,\mathrm{d}\log P/\mathrm{d}\log s$ is read off it for $|D|$ up to $65536$, with the cubic lattice $\mathbb{Z}^{4}$ at the same rate as the fixed-valency control. (1) The Mosco plateau is there: $|d_{s}-4|<0.1$ on a window of $s$ whose top end grows with $|D|$ ($25$ to $1690$ for $|D|=256$ to $65536$, the ratio per quadrupling tending to four). (2) At $\epsilon=d_{0}$, the resolution at which $\rho$ distinguishes points, the curve rises from zero to a maximum of $4.46$ and settles at four, an overshoot below the lattice’s own $4.87$: no rise. (3) At the grain $\epsilon=\ell_{*}=d_{0}/14.5$ it rises through four at $s=2.5$ to $22.9$ at $s=13$, a few coherence lengths, and settles at four by $s=30$; the height is a logarithm of the resolution, $8.4$ per $e$-fold, unbounded for exact return, where the bare hypercube’s $d_{s}=2s$ is recovered. The mechanism is the causal set one: a first step in any of $|D|$ directions makes the return probability fall exponentially until the diffusion has spread past the resolution. (4) The lattice’s overshoot is bounded and resolution-independent, so the rise is the unbounded-valency signature; reduction to two appears in neither model. Tamper: a deterministic number of flipped bits removes the not-moved term and the rise with it. Stands in for: the emergence-determined realisation, by Gaussian i.i.d. displacements with the A4 covariance; and $\mathbb{R}\times S^{3}$, by a flat Euclidean bulk.

 

Computation 208 ·  §18.1 ·  [SURROGATE]
The realised cloud as a sprinkling: PST’s discreteness passes the Lorentz-invariance theorem exactly when the realised positions are independent, which A3 must supply beyond the density.

Bombelli, Henson and Sorkin [225] prove that a Poisson sprinkling selects no frame and carries no finite-valency Lorentz-invariant graph. The theorem’s hypothesis is a Poisson process with the invariant volume as intensity; PST’s cloud has the invariant volume as density by A2 and A3, and $|\det g|$ is unchanged by the Wick rotation, so the Riemannian and Lorentzian processes coincide. What A2’s weak equidistribution does not fix is independence, and the file shows the gap on the i.i.d. surrogate in a $1{+}1$ diamond against a lattice of the same size. (1) Both pass A2 at the same rate. (2) The index of dispersion of cell counts separates them, $0.98$ against $0.00$. (3) The cloud’s counts in boosted blocks of equal volume keep their mean and dispersion at rapidities to $1.5$; the lattice’s dispersion moves from $0$ to $2.2$ with the boost. The cloud’s causal links fill rapidity flatly over $|\eta|<2$ on a range that widens with $N$; the lattice’s pile up along its own null directions. (4) The cloud’s links per element grow like $\log N$ ($4.0$ to $6.2$ for $N=250$ to $2000$), the finite-valency clause and the mechanism of Computation 207. (5) $|\det g|$ agrees in both signatures. Tamper: a lattice jittered by $0.3$ of its spacing passes A2 and still fails. So the theorem applies to PST exactly when the realised positions are independent: true by construction for the surrogate, and for the emergence-determined cloud one more thing P3’s typicality must deliver (Computation 132: no $T$-channel forces the placement). Scope, from Computation 211: the independence is what the maximum-entropy clause means once read as a law on the realised cloud, and the rate leaves it intact at the grain. Stands in for: the emergence-determined realisation, by the i.i.d. uniform cloud of Computation 70.

 

Computation 209 ·  §18.1 ·  [NUMERICAL]
The classical-scaling placement of the signed-tension metric has no Euclidean geometry, so the independence of realised positions cannot be tested on it.

The natural test of Computation 208’s condition is the one explicit placement the book has: classical scaling of eq. (30) on $\mu$-typical directions $v(a)$ i.i.d. on $S^{6}$, $T(\{a,b\})=|v(a)+v(b)|$, $\tau_{\mathrm{null}}$ a low quantile. With $T=\sqrt{2+2\,v_{a}\!\cdot\!v_{b}}$ the Gram form $B=-\tfrac{1}{2}Jd^{2}J$ is, at leading order, $-(\sqrt{2}/4)$ times the centred Gram matrix of the directions: negative of rank seven. (1) At $N=200$ to $1600$ the seven most negative eigenvalues of $B$ grow in proportion to $N$ and match that multiple of the direction Gram eigenvalues, the eighth an order of magnitude smaller. (2) The largest positive eigenvalue grows like $\sqrt{N}$: the Euclidean side is noise and no cloud emerges. (3) The index of dispersion read on the positive-side coordinates is $0.9$ and $1.2$ for i.i.d. directions and $1.1$ and $1.7$ for a design evened on the sphere until its nearest-neighbour spread is $0.03$ instead of $0.19$: the test cannot tell them apart. (4) Control: with the sign of $d^{2}$ flipped the seven eigenvalues turn positive, the positions become a function of each site’s own direction, and the evened design reads $0.58$ against $1.27$. Tamper: a cut above the bulk of the tensions reverses the signal, so the sign is set by which side of the cut the bulk sits on; eq. (30) puts it on the spacelike side, the negative case. The placement is the quantitative form of the remark in §4.12 that the Gram construction cannot supply the signature: it supplies seven negative directions, not one. The condition of Computation 208 stays P3’s, with this placement ruled out as its instrument.

 

Computation 210 ·  §20.6 ·  [NUMERICAL]
The spectral dimension on the book’s own bridged model: three between the truncation and the compact scale, zero below the truncation, no rise.

Computation 207 was computed on the surrogate. This file computes the same quantity on Computation 9’s bridge: Walsh weight $n$ mapped to degree $n$ on $S^{3}$, eigenvalue $n(n+2)$, the binomial multiplicity absorbed into the measure so that degree $n$ carries $(n+1)^{2}$, truncated at $k_{n}=(3\cdot 2^{D})^{1/3}$ where the mode counts match. (1) $|d_{s}-3|<0.15$ between about $5/k_{n}^{2}$ and the compact scale $0.075$ for $D=12$ to $24$ ($k_{n}=23$ to $369$), the lower end falling as $2^{-2D/3}$; the maximum is $2.96$ to $3.00$. (2) Below the truncation $d_{s}$ falls to zero: reduction, no rise, no two. (3) With the binomial multiplicity kept instead of reweighted, $d_{s}$ rises above three to $4.5$, $5.7$, $6.9$, $8.1$ for $D=12$ to $24$: the rise of Computation 207 is carried by the Walsh modes the bridge absorbs into the measure. Tamper: without the truncation the curve is the exact $S^{3}$ one. So on the book’s model the projected geometry’s spectral dimension is the Mosco value above $d_{0}$ and reduction below it; the rise between the grain and $d_{0}$ is seen by a probe of the substrate’s walk, not of the emergent geometry.

 

Computation 211 ·  §18.1 ·  [NUMERICAL]
Maximum entropy supplies the independence: the realised cloud is a sprinkling at the grain, and A3’s rate touches only the scales above the constraint.

Computation 208 named independence of the realised positions as one more thing P3’s typicality must deliver; Computation 209 showed the explicit placement cannot test it. This file tests the clause itself, read as a law on the realised cloud. (1) The theorem: among laws of $N$ labelled positions with the same one-point marginals the product law has the largest entropy, $H(X_{1},\ldots,X_{N})\leq\sum_{i}H(X_{i})$ with equality iff the positions are independent, so maximum entropy given the density is independence; verified exactly on $K=8$ cells and $N=16$ points over $p(n)\propto\mathrm{multinomial}(n)\,\mathrm{e}^{-J\sum_{k}n_{k}^{2}}$: the labelled-cloud entropy is $N\log K=33.27$ at $J=0$ and strictly below it on both sides, and the dispersion there is the binomial $1-1/K$. (2) The constrained cloud: the maximum-entropy law under A3’s rate is the product law conditioned on the constraint set; sampled in the $1{+}1$ diamond ($N=4000$, the eight lowest Fourier modes held to $|\overline{f}|\leq\varepsilon$ with $\varepsilon$ half and a tenth of the i.i.d. scale $1/\sqrt{N}$, thirty clouds each, greedy descent then Metropolis within the set), the index of dispersion is one to within $0.01$ at every cell size up to $0.05$ and $0.92$ to $1.02$ in boosted small cells at rapidities $0$ to $1.5$, and falls only towards the constraint’s half-wavelength (cell $0.5$: $0.75$, $0.36$, $0.24$; the first is the binomial $1-1/4$). The constraint makes the cloud hyperuniform at the Mosco scale and leaves it Poisson at the grain. (3) The causal links of a constrained cloud carry the sprinkling’s signature: $5.6$ links per element, rapidities flat to within a factor $1.2$ over $|\eta|<2$. Tampers: a jittered lattice passes the same constraint six times inside the tight $\varepsilon$ with dispersion $0.35$ at the grain and boost-dependent counts ($0.30$, $0.38$, $1.08$), so the rate alone gives neither independence nor the absence of a frame; and the tight conditioning moves $150$ of $4000$ points and still leaves the grain at one. What this changes: B.7 closes on P3’s footing; the fibres of $\rho$ (B.6) are a thinning of this cloud, which preserves the Poisson property.

 

Computation 212 ·  §19.9.2 ·  [NUMERICAL]
What a point carries and what a device is, on the theory’s own measure: fibres hold at most seven properties and are singletons almost surely, and no device of four or more properties is a substrate accident.

Two remarks shared a cause: B.6 left the fibres of $\rho$ unsaid, and the device accident rates of Computation 156 were computed against a uniform null because PST fixed no measure over configurations. Since Computation 141 and Computation 211 it does: directions i.i.d. under the $G_{2}$-invariant law. (1) A fibre is a clique of pairwise-null properties ($g=0$ means $T=\tau_{\mathrm{null}}$ exactly), and any $G_{2}$-invariant pair tension is a function of the angle alone, so when it is injective a fibre is a set of unit vectors in $V_{7}$ with one pairwise inner product $c_{0}$; its Gram matrix $(1-c_{0})I+c_{0}J$ has rank $k$ for $c_{0}>-1/(k-1)$ and does not embed in $\mathbb{R}^{7}$ beyond $k=7$. So a fibre holds at most seven properties when $c_{0}>-1/6$ and at most $1+1/|c_{0}|$ when the null angle is more obtuse (two at the $5\%$ quantile of the pair tensions, three at $20\%$, seven at the median); constructed at $k_{\max}$ for nine values of $c_{0}$ (residual $10^{-15}$) and failing at $k_{\max}+1$; in $\mathbb{R}^{8}$ eight, so the seven is $\dim V_{7}$. At the Rankin bound the clique’s net tension vanishes exactly: a maximal fibre is a balanced sub-configuration. (2) Under a continuous law on directions an exact coincidence $T=\tau_{\mathrm{null}}$ has probability zero: no pair lands on it in $220$ replicas up to $N=800$, and the nearest approach falls like $N^{-2}$, so fibres are singletons almost surely and A1 reads $|D_{n}|$ as the point count. (3) Computation 156’s device criterion on this measure, at two placements of $\tau_{\mathrm{null}}$, $100$ replicas of $60$ and $40$ of $200$: every nearest-neighbour pair qualifies, a triple in at most a few percent of replicas, nothing of four or more at either placement, where the uniform null gave $0.97$, $0.29$, $0.04$ at sizes three to five. Tampers: a planted compact device of four is found in $48$ of $50$ replicas (the two misses have a background point inside the planted separation), so the zeros are the measure’s and not the detector’s. What this changes: B.6 closes, fibres being singletons $\mu$-almost surely and at most seven in every case; the caveat on the accident rates closes the other way, the theory’s own measure being sharper than the null; what stays is Computation 156’s: why $T$ has clusters at all is $T(C)$-contingent, and the bound state’s dynamics is inherited.

 

Computation 213 ·  §19.2.1 ·  [NUMERICAL]
The two-monomial bridge constant is a limit after all: in closed form, approached from below as $N^{-1/3}$ through the Airy edge of the bridge’s own chain.

Computation 206 followed the deficit times $N$ to $15360$ and saw no plateau. This file finds the constant and why it was invisible. The deficit times $N$ is the sum of three terms, $W+Z_{C}-Z_{T}$ (an identity holding to $3\times 10^{-4}$ at every point computed): $W$ is the weights’ $1/N$ term, the two chains’ symbol maxima against $2\sqrt{k}$, and $Z_{C}$, $Z_{T}$ are the zero-point energies of the commutator’s and the bridge’s extremal chains below their symbols. (1) $W$ converges to a closed form from the $1/s$ expansion of the Bergman weight, $w^{2}=r^{p}(1-r)^{q}(1+e(p,q,r)/s)$ along a chain of degree $s$ and slope $r=a/(a+b)$, plus a drift term for the commutator because the hopping $o_{t}=v_{t}u_{t+1}$ couples neighbouring lattice points: $W_{\infty}=-\sqrt{k}\,(c_{C}+\gamma_{C}-c_{T})=0.2707$ at $k=2$, measured $0.2708$ from $N=7680$ on; without the drift term $0.129$. (2) The commutator’s symbol has an interior maximum in $r$ at $r^{*}=0.7143$ ($k=2$), so its extremal chain is the one through the corner $(r^{*},N)$, starting at $(2r^{*}-1)N=0.4286\,N$, the position Computation 206’s chain was drifting towards; along it the potential is quadratic, the eigenvector a harmonic ground state of width $\sim\sqrt{N}$, and $Z_{C}\to\sqrt{k}\,(2r^{*}-1)\sqrt{oS^{\prime\prime}/2}/S_{C}=0.9556$, confirmed to $O(1/N)$ by moving the truncation to $2N$ so the chain has no wall. With the wall the eigenvector keeps a distance $\sim\sqrt{N\log N}$ from it and the excess decays like $N^{-1/2}\sqrt{\log N}$ (ratios per doubling $0.734$, $0.732$ against the law’s $0.736$, $0.734$). (3) The bridge’s diagonal chain has $r=1/2$ fixed and a potential rising monotonically to the truncation, so its eigenvector is an Airy function pressed against the hard edge: $Z_{T}=A(k)\,N^{-1/3}$ with $A$ in closed form from the first zero of $\mathrm{Ai}$, $10.97$ at $k=2$ against the measured $11.26$ to $11.15$, falling towards it. This is the Airy edge, and it sits on the bridge’s own norm rather than on the commutator’s. So the deficit times $N$ is $D_{\infty}(k)-A(k)N^{-1/3}+E(N)$ with $D_{\infty}=W_{\infty}+Z_{C,\infty}$: $1.2263$ at $k=2$, $1.4738$ at $k=4$, $1.5235$ at $k=6$, approached from below, which is why no plateau was visible: at $N=61440$ the $k=2$ value $1.072$ is still $0.15$ short, and Computation 206’s increments are the Airy term’s decay minus the wall term’s, $0.093-0.064=0.029$ at $N=15360$. Over $k$: every $k$ to $1000$ closes with Computation 59’s $\alpha(k)$; $D_{\infty}\leq 2.465$ for $k\leq 80$ and below the certified $3.6736$ for every $k\leq 482$; its envelope grows like $k^{0.22}$, crossing $3.6736$ at $k=483$, where the asymptotic regime needs $N$ beyond $(A/D_{\infty})^{3}\sim 10^{8}$. Tampers: the drift term dropped, the Neumann Airy zero in place of the Dirichlet one (a coefficient $2.3$ times too small) and a half-well zero point (three times too large) are each rejected.

 

Computation 214 ·  §19.2.1 ·  [NUMERICAL]
The sharp value $3.6736$ in closed form: an algebraic number at the operative corner $(6,12)$, and why that corner is the maximum.

The one quantitative thread Computation 213 left: the sharp value sits in the small-headroom regime $N$ close to $2m$ that the asymptotic laws do not cover. Write $N=2m+h$; the lower shift acts on $a+b\leq h$ and the upper on $a+b\leq h-2$, so every chain has at most $\lfloor h/2\rfloor+1$ points and $G(k,2m+h)$ is an algebraic function of $(m,\alpha,h)$. At the operative corner $N=2k$ the headroom $h=2k-2m(k)$ grows with $k$; the only $k\leq 80$ with $h\leq 4$ are $k=2,3,4$ ($h=2$) and $5,6$ ($h=4$). (1) At $(6,12)$ both extremal chains are the diagonal $(0,0),(1,1),(2,2)$ (the commutator’s drift towards $0.43\,N$ is a large-$N$ effect), the weights are exact rationals, and the two $3\times 3$ matrices have the characteristic polynomials of (275); $\alpha(6)$ is the root in $(1,2)$ of a degree-$24$ integer polynomial obtained by eliminating $q$ and $\sqrt{1-q^{2}}$ from Lemma 5(c)’s closure $R_{4}(q)=2\sqrt{6}$, so $G(6,12)=12\,(2\sqrt{6}-\sqrt{L_{C}/L_{T}})=3.673556413812905\ldots$ is an explicit algebraic number, inside Computation 203’s enclosure and equal to the float evaluator to $10^{-13}$. (2) At fixed $(m,h)$ the map $\alpha\mapsto G$ is single-humped with the hump where the two monomials are comparable and its top rising with $m$ ($2.00$, $3.30$, $3.81$, $4.13$, $4.45$ at $h=4$ for $m=2$ to $6$); the integer lattice $(k,m(k),\alpha(k))$ samples those curves at few points, and $k=6$ is the only one with $h\leq 4$ and $\alpha$ on a hump ($3.6$ percent below the $(4,4)$ top), $k=5$ sharing that headroom with $\alpha=28.7$ far down the tail ($G=1.08$). Larger $m$ gives higher humps but larger $m$ means larger $k$ and larger headroom. (3) Rescanned over every chain for $k\leq 80$ and $N\leq 600$ (offline; the script rescans $k\leq 40$, $N\leq 200$): the maximum of $|G|$ is $3.6736$ at $(6,12)$, then $3.084$ at $(4,10)$, $2.862$ at $(9,18)$; with Computation 213’s limits (below $2.47$ for $k\leq 80$) the supremum over the operative regime at $k\leq 80$ is derived at its maximiser and scanned elsewhere. Tampers: the commutator on the chain through $(2,0)$ gives $5.58$; the other real closure root $29.85$ gives $-5.8$; $\alpha$ moved by $10^{-3}$ moves $G$ by $3\times 10^{-3}$. No analytic mechanism sets the value: it is the one point where a headroom of four meets a balanced bridge.

 

Computation 215 ·  §19.9.2 ·  [NUMERICAL]
The native sector cannot build an apparatus: Derrick’s scaling on P3’s functional, and the regulated lattice at PST’s own discreteness, three orders below pinning.

The apparatus question’s sharper half: could the theory’s own field, the order parameter of P3 with the functional (1), hold a localised object at all? (1) Derrick’s scaling: for a static localised configuration in three dimensions with $E_{\mathrm{pot}}\geq 0$, $\psi(\lambda x)$ has energy $E_{\mathrm{grad}}/\lambda+E_{\mathrm{pot}}/\lambda^{3}$, strictly decreasing in $\lambda$, so every localised configuration lowers its energy by shrinking and the only static finite-energy solution is the vacuum; evaluated on a dip, a bubble of the other vacuum and a ring. (2) On the regulated lattice (Computation 200) the shrinking stops at the spacing, so a localised object could survive only by pinning, which needs the discreteness $\kappa=(m\ell)^{2}$ large: PST’s is $(m_{h}/M_{*})^{2}=0.0095$ with the grain and exactly $2$ with the coherence length $d_{0}=\sqrt{2}\,\hbar c/m_{h}$. Gradient flow of the lattice energy at $\kappa=0.0095$ from four localised initial data (bubbles of radius $6$ and $12$, a Gaussian dip, a single flipped site) relaxes every one to the vacuum; by bisection a flipped site persists only for $\kappa>47.0$ and a four-grain bubble only for $\kappa>14.7$, which PST undercuts by three orders with the grain and by factors $24$ and $7$ with $d_{0}$. (3) The loophole named: the theory has oscillons, localised oscillations keeping half their core energy for $10^{3}$ to $10^{4}$ periods (a radial simulation to $t=1000/m$); in PST’s units a period is $1/m_{h}=5\times 10^{-27}$ s and the core a few $1/m_{h}$, a breather of the order parameter lasting $10^{-23}$ s, not an apparatus. Tampers: at twice the threshold the flipped site persists; with the potential’s sign flipped Derrick’s $E(\lambda)$ acquires a stationary point; a one-dimensional kink survives the same flow, so the theorem is dimensional. What this changes: “inherited” is forced, not chosen: an apparatus is a bound state of the inherited matter sector, realised in $T$ as a cluster that (30) reads as a device (Computation 156); which clusters $T(C)$ has stays the configuration’s content.

 

Computation 216 ·  §19.9.3 ·  [NUMERICAL]
The black-hole interior on the theory’s current lengths, and the central singularity dissolved by the fibre bound.

(A) Computation 154 evaluated the validity condition $(k\,d_{0})^{2}\ll 1$ on Schwarzschild with $d_{0}=2.3$ nm, the Casimir bound of its day, and the chapter printed those values after $d_{0}$ moved to $2.2$ am (Computation 201). Re-evaluated: $(k\,d_{0})^{2}=1.9\times 10^{-42}$ at a solar-mass horizon, $r_{\mathrm{break}}=37$ pm $=1.2\times 10^{-14}\,r_{s}$, enclosing $1.9\times 10^{-42}$ of the interior volume, a mass floor of $2.8\times 10^{9}$ kg; on the grain $\ell_{*}$: $9.4\times 10^{-45}$, $6.2$ pm, $1.9\times 10^{8}$ kg, the floor the chapter already had. Every conclusion of Computation 154 survives by wider margins; $d_{0}=2.3$ nm reproduces every printed number (tamper), so the discrepancy was the length alone. The uncovered region holds about $10^{25}$ grains. (B) The initial-singularity argument the book declined to transfer is made in a different form. A point is a fibre of $\rho$, a clique of pairwise-null properties holding at most seven (Computation 212), so the realised content of a collapsing body, $n\gg 7$ properties, cannot occupy one point: some pair fails to be null, and its proper distance $|T-\tau_{\mathrm{null}}|^{1/4}$ is the region’s diameter. How far from null that pair must be is a packing bound: Rankin (some pair has $\cos\geq-1/(n-1)$) and Welch (among $n>7$ unit vectors in $\mathbb{R}^{7}$ some pair has $|\cos|\geq\sqrt{(n-7)/(7(n-1))}$), evaluated with Computation 212’s exemplar tension at its two placements of $\tau_{\mathrm{null}}$ beside the best configurations a descent on the sphere finds. At the median placement seven properties can be exactly null (an orthonormal frame) and eight cannot (Welch’s $1/7$, attained by the simplex), the deviation growing with $n$ ($0.25$, $0.28$, $0.39$, $0.45$, $0.58$ at $n=10$, $14$, $20$, $30$, $50$); at the obtuse placement only two can, Rankin attained at $n=4$ and $7$. In the proper distance a region holding $n$ properties has diameter at least $0.56$ ($n=8$) to $0.80$ ($n=50$) at the median and $0.76$ to $1.05$ at the obtuse placement, in the exemplar’s units, and the single-pair distance saturates at $(2-\tau)^{1/4}$, the local character of (30). What this changes: the chapter’s numbers move to the current lengths; the singularity as a point carrying the collapsed content is excluded on the substrate side with the strength of the fibre result and no more; what replaces it, a region whose minimal diameter the bounds give in local units and whose dynamics the projected equations do not reach, the theory does not describe. The statistical entropy, the information question and the horizon’s status are untouched.