61
| # | Description | Status | Sources |
|---|---|---|---|
| A. Inputs: imported and instance data; the frame’s multiplicities | |||
| A1p46001 | The lattice determinations of $m(0^{++})/\sig$ for $\SU(3)$: the $r_0$ route with two inputs, $3.70(6)=5.73(9)\times0.6462(30)$ and $3.63(10)=4.21(11)/1.160(6)$, and the direct continuum extrapolation $3.405(21)$ with the first scalar excitation $5.855(41)$, a spread of $9\,\%$; $T_c/\sig=0.6462(30)$ ($\SU(3)$), $0.7091(36)$ ($\SU(2)$); the $r_0$ masses of $0^{++}$, $2^{++}$, $0^{-+}$ and $0^{++*}$. | I | Rem. 2.6; [Hasenbusch & Necco 2004; Lucini et al. 2004; Morningstar & Peardon 1999; Athenodorou & Teper 2020] |
| A2p46002 | The colour factors measured at LEP: $C_A=2.89\pm0.21$ and $C_F=1.30\pm0.09$ from the four-jet angular correlations, $C_A/C_F=2.22\pm0.11$ from the jet multiplicities. | I | Rem. 2.6; [Kluth 2006] |
| A3p46003 | The strong coupling at two laboratory scales, $\alpha_s(M_Z)=0.1180(9)$ and $\alpha_s(m_\tau)=0.312(15)$, with the thresholds $M_Z$, $m_b$ and $m_\tau$. | I | Rem. 2.6; [Particle Data Group 2024] |
| A4p46004 | The one-loop coefficient $b_0=\frac{11}{3}C_A-\frac43T_Fn_f$ and, for the confrontation of P2 only, the two-loop coefficient $b_1=102-\frac{38}{3}n_f$. The paper imports the forms; the structure of the one-loop coefficient as a sum over the field species is the import A12, and its two rationals are the frame’s (C13). | I | |
| A5p46005 | The laboratory instances: the Subjects $\p=13$, $173$, $1549$ with their Carriers $\Om=233$, $30089$, $2408561$, and the sweep over the $21$ shells $\p=4\kap+1$ below $200$. The three Carriers are $5$ modulo $12$ (A18). The unitary frame’s centre has order three where $3$ divides $\p+1$, on the Subject $173$ and $12$ of the $21$ shells. | D | Def. A5 (def:lab), p. 4">2.1 |
| A6p46006 | Four results of the framework: time is the drive (a theorem), mass is winding rate with masslessness as drive-invariance, confinement is the wrap of $\Om$-hard quark residues into exact hadron residues, and the gauge sector has one $\Om$-hard scale. | I | |
| A7p46031 | Wilson’s lattice gauge theory: the plaquette action on the hypercubic lattice, its strong-coupling character expansion, and the area law $W(A)=c_3^{A}$ at strong coupling, whose chart reading is the tension $a^{2}\sigma=-\ln c_3$ in lattice units, within the expansion’s convergence. | I | |
| A8p46032 | Two results of the interactions paper: the character lift (near the identity the frame’s character equals the compact group’s, to the stated order; the lift’s parametrisation is imported as it stands and is not unitary as it stands, its repair the interactions paper’s) and the order $\gcdop(3,q+1)$ of the centre of $\SU(3,\F_q)$. | I | Prop. A8 (prop:fields), p. 6">2.3; [Akhtman & Voether 2026] |
| A9p46034 | The frame’s multiplicities, for $q$ a power of a prime $\p$ of the totality: $\mathrm{SL}_3(\F_{q^{2}})$ fixes the kernel modulo $\p$ of the matrix $D$, of dimension at least the compact count $C(a,b)$; for $\p>3$ and $q^{2}>a+b+\min(a,b)$ the pairing-and-determinant tensors span the vectors it fixes, of dimension $C(a,b)$; for $a+b+2\le\p$ the frame fixes the same vectors (to degree $9$ on Subjects above $11$). | T |   |
| A10p46035 | Casimir scaling of the string tension at intermediate distance, $\sigma_8/\sigma_3=C_A/C_F=\frac94$, the lattice’s observation; a bounding value of the tension parameter of C8. | I | Sec. 5; [Bali 2000] |
| A11p46039 | The $0^{++}$ channel of the strong-coupling expansion for $\SU(3)$: $am=-4\ln u-3u+9u^{2}-\frac{27}{2}u^{3}-7u^{4}-\frac{297}{2}u^{5}+\frac{858827}{10240}u^{6}+\frac{47641149}{71680}u^{7}$ in the chart’s fundamental plaquette coefficient $u$; its coefficients are rationals of the same multiplicities as $c_3$, not recomputed here. | I | |
| A12p46040 | The one-loop coefficient as a sum over the field species of a paramagnetic spin term and a diamagnetic charge term, $b_0=\sum(-1)^{2s}\bigl[(2s)^{2}-\frac13\bigr]T(R)$, one complex scalar, one Weyl fermion and one real vector each counting once. The form of the sum is imported; its two rationals are computed in C13. | I | Prop. C13 (prop:rationals), p. 26">4.4; [Hughes 1980; Nielsen 1981] |
| A13p46052 | Finite Schur–Weyl duality: over a finite field with more than $r$ elements the slot permutations span the commutant of $\mathrm{SL}(V)$ on $V^{\otimes r}$ (Benson–Doty, Cor. 4.4). For A9, C15. | I | |
| A14p46059 | Green’s theorem: for a finite group and a representation over a finite field, the sum of roots of unity that lift the eigenvalues is a virtual character (Green 1955); the central convolution by it, taken with the group’s mean, acts on the block of an irreducible character $\chi$ as $a_\chi/\chi(1)$ with multiplicity $\chi(1)^2$. For C19, C20. | I | |
| A15p46060 | The generic character table of $\SU(3,q)$ (Simpson and Frame 1973; CHEVIE): eight class types with centralizer orders and ten character families in the parameter $q$; verified at import by brute force at $q=2,3$ and by the orthogonality relations at nine $q$. For C16, C19, C20, C23, C25. | I | |
| A16p46069 | The stiffness of the connection (the interactions paper): the coefficient of the action, counted in action quanta $\hbar$ per link per chronon; by channel unity it is $1$ and the bare coupling is $\alpha=1/4\pi$, a canonical normalisation and not a measured coupling. Wilson’s coupling is $\beta=2N\stf$: $6$ on the colour frame at channel unity. The plaquette term is $\stf(\chi+\bar\chi)$. | I | |
| A17p46073 | The classical count of singlets: of the integer vectors of $\mathbf3^{\otimes a}\otimes\bar{\mathbf3}^{\otimes b}$ that the six matrix units $E_{st}$, $s\neq t$, annihilate, $C(a,b)$ are independent and no more; $C(a,b)$ is the constant term of the Weyl character polynomial, classically the multiplicity of the trivial representation of the compact group $\SU(3)$ (Weyl’s complete reducibility and character formula). For A9. | I | Prop. A9 (prop:mult), p. 6">2.4; [Humphreys 1972, Sections 6.3 and 24.3] |
| A18p46074 | The Carrier’s relation $\Om\equiv5\pmod{12}$: $3$ divides $\Om+1$, so $x^{2}+x+1$ has no root modulo $\Om$. It is a relation of the framework, imported from the interactions paper, which names it colour’s triality. The three laboratory Carriers satisfy it (C2). For B5. | I | Def. B5 (real:colour), p. 9">2.7; [Akhtman & Voether 2026] |
| A19p46075 | The searches for free quarks: the experiments have produced no evidence for a freely propagating quark (the review of the free quark searches of the Particle Data Group). For P4. | I | Prop. P4 (prop:free), p. 35">7.4; [Particle Data Group 2024] |
| B. The finite theory and its readings | |||
| B1p46007 | The residue of a state. A registration of a state by its frame is a line of the state’s space that every frame element maps to itself, and the residue of the state is the number of such lines, a count without a unit. The counts of the photon, the gluon and the hadrons are C1 and C3. P4 holds its falsifier. | D | Def. B1 (real:residue), p. 9">2.8; [Akhtman & Voether 2026] |
| B2p46008 | [chart] The chart’s scale is read on the drive: a registered scale is a nonzero element $w$ of the Subject, two scales differ by the power $n$ with $\gen^{n}=w_2/w_1$, and the chart’s $\ln(\mu_2/\mu_1)$ is read as $n\ln\gen$ within one window of the drive (a wrap adds $(\p-1)\ln\gen$). The registration $\mu\mapsto w$ is the quantum paper’s readout, not computed here. | D | Def. B2 (def:scale), p. 12">2.11; [Akhtman & Voether 2026] |
| B3p46009 | The finite theory is the transfer form of lattice gauge theory along the drive: a ring of plaquettes; the tally is the frame’s mean over the rungs of the amplitudes (B4), its paired weight registered once; the gap: the transfer tally’s second eigenvalue over its first. [chart]: counts of frame moves above $\Om\sim10^{123}$, the hypercubic lattice (A7), the wrap operator (C8), the scale (B2). | D | |
| B4p46055 | The spectral amplitude of a plaquette: roots of unity of orders $q^{2}-1$ and $q^{2}-q+1$ for two cycles, the units of $\F_{q^{2}}$ and the powers of a frame element $C$ with no eigenvalue there; $w(U)$ sums the three phases of the class of $U_s$; a configuration’s amplitude $\prod_\square w(U_\square)$, a tally’s paired weight $Z\bar Z$. B3 declares the configurations and the registered quantity. | D | Def. B4 (def:amplitude), p. 11">2.10; [L"ubeck 2020] |
| B5p46063 | Colour is the Carrier’s triality, the relation $\Om\equiv5\pmod{12}$ (A18). Every Subject reads colour in its unitary frame, $\SU(3)$ over its own extension. The type is declared from the relation: the cube roots of unity are not residues of the Carrier. P1 and P2 hold its falsifier, colour factors off those of a frame of degree three; no confrontation separates the unitary frame from the linear one. | D | Def. B5 (real:colour), p. 9">2.7; [Akhtman & Voether 2026] |
| B6p46070 | [chart] The chart’s cells. The one-loop law (A4) is a law of powers, $W(w_2)/W(w_1)=(w_1/w_2)^{b_0}$ with $W=e^{-8\pi^{2}\stf}$. A scale $n$ drive steps below channel unity (A16, B2) has $W$ higher by $\gen^{\,b_0n}$ and the stiffness $\stf_n=1-n\,b_0\ln\gen/8\pi^{2}$. For $b_0=11$, $\gen=2$ one step is $0.0966$, positive through $n=10$. The running is $\Om$-hard and the law an extrapolation. | D | Def. B6 (def:cells), p. 12">2.13; [Akhtman & Voether 2026] |
| C. The registrations, the sub-horizon tower and the tension | |||
| C1p46010 | The norm-one cycle has no shell eigenvalue. For every frame of the totality a norm-one element $a+b\sqrt\nu$ with $b\neq0$ has no eigenvalue in $\Fp$ (its discriminant $4\nu b^{2}$ is a nonsquare), so no line of the plane is mapped to itself by every element of the cycle; the split torus fixes exactly two lines. | T | |
| C2p46011 | The centre of the unitary frame: its order $\gcdop(3,q+1)$ is $3$ exactly when $q\equiv2\pmod3$. Among the Subjects $173$ has it, $13$ and $1549$ do not, and $12$ of $21$ shells below $200$ do; on $\SU(3,\F_2)$ it is found by enumeration. The three Carriers are $5$ modulo $12$ (A18). The Lean cell is the arithmetic of the order formula. | T | |
| C3p46012 | The residue series on the instance. On the $216$ elements of $\SU(3,\F_2)$ over $\F_4$ (no nontrivial character over $\F_4$) the adjoint action fixes no nonzero traceless matrix (gluon $0$), the determinant is the one invariant of $\Lambda^{3}$ (baryon $1$), and the commutants of the fundamental and the adjoint are one-dimensional (meson $1$, glueball $1$). | T | |
| C4p46013 | The Casimir rationals of the unitary frame from its torus exponents (C6): on the fundamental $(a,b,-a-b)$, on the adjoint their differences and two zeros; the second moments are $2$ and $12$ times $a^{2}+ab+b^{2}$, so $T_A/T_F=6$ and, with the chart’s normalisation $T_F=\frac12$, $C_F=\frac43$, $C_A=3$, $C_A/C_F=\frac94$; the three-quark singlet’s colour factor is $-8$. No lift enters. | T |  |
| C5p46014 | The window rational: $b_0=11-\frac23n_f$ takes the values $9$, $\frac{25}{3}$, $\frac{23}{3}$, $7$ for $n_f=3,4,5,6$, is positive for every $n_f\le16$ and negative for every $n_f$ from $17$ to $200$. | T | |
| C6p46015 | The drive exponent. On every shell of the totality the powers of the drive permute the nonzero elements: each winding rate has one exponent below $\p-1$, and exponents add modulo $\p-1$ — the discrete logarithm on the shell. The ledger file decides it on the Subjects and the sweep. | T | Thm. C6 (thm:exponent), p. 26">4.5; rests on A5 |
| C7p46016 | The tension’s coefficient to order $\stf^{7}$: the fundamental plaquette coefficient of $\SU(3)$ in the stiffness (A16) is $c_3=\stf/3+\stf^{2}/6-5\stf^{4}/72-\dots$ modulo $\stf^{8}$, every coefficient a rational built from the singlet multiplicities, the frame’s in the degrees used (A9); the computed terms sum to $\frac{23}{54}$ at channel unity. The instance $2T$ agrees with $\SU(2)$'s series through $\stf^{3}$, first differing at $\stf^{5}$. | T | |
| C8p46017 | [chart] The minimal wrap, the candidate operator for the mass gap: $\Delta=\lambda_{\mathrm{gb}}\sig$, $\lambda_{\mathrm{gb}}$ the lowest eigenvalue of the chart’s relativistic wrap operator $a|p|+\varkappa r$ with the kinetic multiplicity $a$ and the tension parameter $\varkappa$. Nothing in the operator is a frame object; the frame enters through $\varkappa$ (C9, A10). It must reproduce the determinations of A1; C12 and P3 record that it does not. | D | Def. C8 (def:wrap), p. 32">6.1; [Akhtman & Voether 2026] |
| C9p46030 | The adjoint wrap: the adjoint plaquette coefficient has $\stf^{1}$ coefficient $0$ and $\stf^{2}$ coefficient $\frac18=\frac98(\frac13)^{2}$, both identities holding on $\SU(3,\F_2)$ by enumeration; modulo $\stf^{6}$, $c_8=\frac98c_3^{2}(1-\stf/3+\stf^{2}/12+\stf^{3}/12)$: twice the fundamental string tension at leading order. | T | |
| C10p46037 | [chart] The short-range route of the chart fails: with the declared coupling $\alpha_s(\sig)=1.007$ the two-adjoint term has $C=C_A\alpha_s=3.02$, above the Herbst bounds $2/\pi$ and $4/\pi$, so the cutoff cannot be removed; with the coupling run to the bound state’s mass, fixed points $3.54$, $3.08$, $2.61$ at $r_c=1$, $\frac12$, $\frac14$ and none found at $\frac18$ (a scan of $149$ energies). | T | |
| C11p46071 | [chart] The plaquette channel of the chart fails: with A11’s series and $u$ the sum of $c_3$ through $\stf^{7}$ at B6’s cells, $m/\sig=6.9$, $6.2$, $5.6$, $5.1$ for $n=9,\dots,6$, orders $3,5,7$ within $1\,\%$; at $n=3$: $3.98$, $3.60$, $3.79$, the fifth alone in $3.4$–$3.7$; at channel unity, $u=\frac{23}{54}$, $2.95$, $0.45$, $2.82$. | T | |
| C12p46036 | [chart] The family of C8 against the lattice: $E_0=\sqrt{a\varkappa}\,2.232$ by the dilation symmetry, $2.232$, $3.157$, $3.348$, $3.157$, $4.464$, $4.735$ for $a\in\{1,2\}$, $\varkappa\in\{1,2,\frac94\}$; none inside any determination’s error; the nearest, $a=1$, $\varkappa=\frac94$, is $1.7\,\%$ below the direct value. | T | |
| C13p46041 | The two rationals of the one-loop coefficient are the frame’s: the spin term $(2s)^{2}=-2\,\mathrm{Tr}_V(J_{12}^{2})$ is $0$, $1$, $4$ on the scalar, the Weyl spinor and the vector; the charge term $\frac13$ is the leading Faulhaber coefficient of the scalar loop’s midpoint sum $3\sum_{k<N}(N-2k-1)^{2}=N(N^{2}-1)$, whose complete cycle $N=\p$ vanishes on the shell. With C4 the species sum of A12 is $3b_0=33-2n_f$ (C5). | T | |
| C14p46051 | The series’ window: the $\stf^{k}$ coefficient of $c_3$ uses the invariant counts of degree $\le k+1$, that of $c_8$ degree $\le k+2$, so the frame’s coefficients are built from the compact counts through $\stf^{\p-3}$, $\stf^{\p-4}$ by A9 and $\stf^{2\p-2}$, $\stf^{2\p-3}$ by C15; on the instance $M(4,1)=4$ against $3$ puts $c_3$'s $\stf^{4}$ at $0$ and $c_8$'s $\stf^{3}$ at $1/8$. | T | |
| C15p46053 | The frame’s window and first departure $d_*$: for odd $\p$ the frame’s invariants in $\mathbf3^{\otimes a}\otimes\bar{\mathbf3}^{\otimes b}$ are the compact ones through degree $2\p-1$ (the coordinate $\SU(2)$ subgroups, A13; the count, A9, at $\p=3$ computed), and the Frobenius tensor $v_\p$ at $(4\p-4,1)$ lies beyond them, so $2\p\le d_*\le4\p-3$: $[26,49]$ on $\F_{13}$, $[346,689]$ on $\F_{173}$; $d_*=5$, $9$ on $\F_2$, $\F_3$. | T | |
| C16p46056 | The amplitude’s finite properties: $U_s$ has three eigenvalues in $\F_{q^{2}}$, or none and an order dividing $q^{2}-q+1$; $w$ is a class function, $w(U^{-1})=\overline{w(U)}$, $w(zU)=\iota(z)w(U)$ for central $z$, and reduces to $\operatorname{Tr}U$; if $\p$ divides the order of $U$ then $U_s$ has eigenvalues $(\lambda,\lambda,\lambda^{-2})$, $\lambda^{q+1}=1$, so $w(U)=2\eta+\eta^{-2}$ and every unipotent has $w=3$; if $3\mid q+1$ then $\langle w\rangle=0$ (B3). | T | |
| C17p46057 | The amplitude on $\SU(3,\F_2)$ takes the values $3$, $3\zeta_3$, $3\zeta_3^{2}$ on $64$ elements each and $0$ on $24$; the moments $\frac1{216}\sum w^{a}\bar w^{b}$ at $(1,1)$, $(3,0)$, $(2,2)$ are $8$, $24$, $72$, not the invariant counts $1$, $1$, $2$ (C3); two involutions with $w=3$ have a product with $w=0$: $w$ is no character over $\Q(\zeta_3)$. | T | |
| C18p46058 | The transfer tally on the instance: on $\SU(3,\F_2)$ the step of one plaquette, $(Kf)(x)=\langle w(x^{-1}y)f(y)\rangle_y$ (B3), has the eigenvalues $\frac23$, $\frac16$, $0$, $-\frac13$ with multiplicities $9$, $36$, $144$, $27$ ($K(3K+1)(6K-1)(3K-2)=0$, $\operatorname{tr}K=3$); the trivial channel is annihilated, $K\mathbf1=0$ (C16), so the tally has no ratio to it; with the eigenvalue $-\frac13$ it is not a positive operator. | T | |
| C19p46061 | The transfer tally on the Subjects: for every prime power $q\ge3$, on each class type of A15, $w=\chi_{q^2-q+1}+\chi_{q^3+1}-\chi_{(q+1)^2(q-1)}$ (first members of the three families, A15), so the step $K$ (B3) has the eigenvalues $\frac1{q^2-q+1}$, $\frac1{q^3+1}$, $-\frac1{(q+1)^2(q-1)}$, $0$ with multiplicities the squared degrees; the frame’s mean of $w$ is $0$, moment $3$; $K$ not positive. | T | |
| C20p46062 | On C19’s frames the ring’s tally sums three characters’ terms $\chi(h_1)\chi(h_2^{-1})$ on the boundary classes; one plaquette multiplies each by its eigenvalue; the largest, one character’s, is $q+1$ times the next: the transfer tally’s gap $1/(q+1)$. Modulo $\p$ the eigenvalues are $1$ and at fixed classes the reduced tally is the same for every length; the paired weight reduces to a residue of the shell. | T | |
| C21p46064 | The spectrum in $2\pi=\p-1$, the shell’s phase cycle, certified by $2\pi\equiv-1$ and $\p$-hard: on the three Subjects $d_1=4\pi^{2}+2\pi+1$, $d_2=(2\pi+2)d_1$, $d_3=8\pi(\pi+1)^{2}$, and the gap of the transfer tally is $1/(2\pi+2)$, a pure number, one expression on every Subject, with the residue $1$ on the shell. [chart] $0.1207$ at the classical $\pi$. | T | Cor. C21 (cor:pi), p. 20">3.10; rests on C20 |
| C22p46065 | The product channels: where $3$ is invertible the frame fixes $A_n=\sum_k(-1)^{n-k}\binom nkM(k,k)$ dimensions of $\mathbf8^{\otimes n}$. On every Subject two mesons and three gluons have $d=2$ and $\p^{2}+1$ invariant lines, a meson with a glueball $d=3$, two glueballs $d=8$. On $\SU(3,\F_2)$ three gluons have $d=7$ and two glueballs $d=57$: the instance departs beyond degree $4$. | T | |
| C23p46066 | The tally with the same boundary links at every step, of the eigenvalues $g^{\pm j},1$ and $g^{\pm k},1$: the boundary classes of C20 move with the length $n$. On every Subject it reads $1+g^{(j+k)n}+g^{-(j+k)n}+g^{(j-k)n}+g^{-(j-k)n}$ at every regular length, with stated values at the others and the least period $(\p-1)/\gcdop(j,k,\p-1)$. Every exponent is a multiple of $\gcdop(j,k,\p-1)$; the three multipliers add none. | T | |
| C24p46067 | What a reading keeps of the window: the counts of C15 are equal as integers, and on $\F_{13}$ the pair $(1,13)$ has $2574=13\cdot198$ independent invariants and reads $0$, with $52$ of the $117$ nonzero counts. The coefficients of C14 below $\stf^{\p}$ have values in the Subject’s arithmetic; from $\stf^{\p}$ on they contain a factorial that reads $0$, and the paper states no reading. | T | |
| C25p46068 | Three rings of $n$ plaquettes on $n$ common links. For every prime power $q\ge3$ the tally sums $\chi_i(h_1)\chi_j(h_2)\chi_k(h_3)$ on the boundary classes with the invariant counts of the triples of characters; the mean of $w^{3}$ is $6$. On every Subject, on split classes of the eigenvalues $x^{\pm1},1$, $y^{\pm1},1$, $z^{\pm1},1$, it reads $1+N_{122}(c(x)c(y)+c(x)c(z)+c(y)c(z))$, $c(m)=m+m^{-1}$, $N_{122}=2$ or $4$. | T | |
| C26p46072 | The series on the shell: $d_R\,c_R=\sum_kh_k\stf^{k}/k!$ with integers $h_k$ ($0,1,1,0,-5,-15,21,413$ for $c_3$) from a recursion without division, on residues at every order. In the divided-power algebra on a Subject $f(0)=0$ gives $f^{\p}=0$, so a reading that respects sums and products sends $\stf$ to $0$; at channel unity on $\F_{13}$ the truncated sum reads $6$, the quotient of truncated sums $4$. | T | |
| X. What the construction explains | |||
| X1p46018 | Why the gluon is massless yet never seen alone: under B1 a gluon is drive-invariant (massless, A6) and its space has no frame-invariant line (C1, C3), so no registration singles it out. The row explicates the form of the confinement rule; it does not derive confinement. | T | |
| X2p46019 | Why only colour singlets are registered: under B1 each channel of the unitary frame, a space whose frame-fixed subspace is a line, has exactly one frame-invariant line, and the hadron paper’s series of such channels ends at the glueball (C3); products whose fixed subspace is larger are sums of channels, counted in C22. | T | |
| X3p46020 | [chart] Why the sector has one scale: the gauge sector has one $\Om$-hard scale, $\sig$ (Z1); every exact quantity of the sector is a sub-horizon ratio from the frame (C4–C7, C9); the ratio $\Delta/\sig$ is a quantity of the chart’s lattice, which the candidate operator (C8) does not supply (C12, P3). | D | Sec. 7 (X3) |
| X4p46021 | Why the colour factors take the values LEP measured: they are the second moments of the unitary frame’s torus exponents (C4), and the abelian alternative $C_A=0$, $C_F=1$ lies outside the measured brackets (A2). | T | |
| X5p46022 | [chart] Why the finite theory needs no ultraviolet renormalisation and the continuum chart does: every tally of the finite theory is a mean over a finite frame (B3), so nothing diverges; the finite matching of the substrate’s one scale to the chart’s scales is $\Om$-hard (Z1), and the chart’s counterterms are its account of that matching. | D | Sec. 7 (X5) |
| P. The paper’s four predictions, with their falsifiers and their present confrontations | |||
| P1p46023 | The colour factors of the strong interaction are $C_A=3$, $C_F=\frac43$, $C_A/C_F=\frac94$ (C4). Falsifier: a measured $C_A/C_F$ outside $\frac94$ beyond its error. Confrontation: LEP’s $2.89(21)$, $1.30(9)$, $2.22(11)$ (A2), each within one standard deviation. | T | |
| P2p46024 | [chart] Between laboratory scales $1/\alpha_s$ is linear in $\ln(\mu/\mu_0)$ with slope $b_0/2\pi$, $b_0$ the window rational (C5, A4). Falsifier: a measured $1/\alpha_s(\mu)$ off the linear law beyond the imported two-loop correction. Confrontation: one loop gives $\alpha_s(m_\tau)=0.279$ against $0.312(15)$, $2.2$ standard deviations below; with the imported $b_1$ the run gives $0.313$. | T | |
| P3p46043 | [chart] The radial ratio of the operator form: $E_1/E_0\approx1.49$ for every $a$ and $\varkappa$, a property of the form of C8 and of every state the one-body wrap describes. Falsifier: such a state’s first radial ratio measured off $1.49$. Confrontation: $1.54(11)$ (1999) agrees; $1.72(2)$ (2020) excludes $1.49$ by ten standard deviations under first-order error propagation at any correlation: the form fails for the glueball. | T | |
| P4p46076 | No state of residue $0$ is registered in isolation: on every Subject the triplet and the adjoint of the unitary frame have no invariant line (A9), so there is no free quark and no free gluon. Falsifier: a confirmed free particle that carries colour, such as a fractional electric charge. Confrontation: the searches for free quarks are negative (A19). | T | |
| Z. The $\Om$-hard horizon | |||
| Z1p46026 | The one scale of the gauge sector, $\sig$, which fixes $\LQCD$ up to a sub-horizon ratio, is $\Om$-hard: no bounded observer computes its absolute value, its ratio to the Planck energy. The paper reads its numeral from the lattice ($\sig\approx420$–$460$ MeV [import]). Every dimensionless quantity of the sector is a sub-horizon ratio. | Ω | |
Nothing matches that filter.
Status
- I
- Import. A standard result, a framework result, or a measured datum used here without reproof.
- D
- Definition. A naming, a declared dictionary, or a set-up move, carrying no empirical content of its own. A definition that names a physical term names the prediction rows that hold its falsifier. A [chart] definition is a declared continuum reading.
- T
- Theorem. Proved at the statement the source column names, which also lists the rows it rests on, and machine-verified by the named check. A Lean cell proves its arithmetic in the framework’s axiom-free core, and a leanML cell names a theorem on Mathlib as well.
- Ω
- $\Om$-hard. Decided by the totality; not closeable by a bounded observer; read from the chart.
Sources
- Gold: the python check that decides the predicate, at its line in the script (the source, rendered on this site); on hover the deciding family and the families that corroborate it, and the verdict of the last run (red when it fails). The script page's bar runs the predicate's cell of the package's notebook on Google Colab (one cell per predicate; the whole notebook from the header badge).
- The predicate as one Lean declaration, named by its accession key (
p46034) — the conjunction of the theorems that prove it — at its line in the rendered module, one mark per library that proves it, the core's first. Green: the core (lean/FrcCore) — the statement on the shell ring, by computation and induction, no Mathlib, no axioms: the finitist witness. Violet: Mathlib (lean/FrcMathlib) — the classical statement for every shell on Mathlib's hierarchy (its axioms,propext,Classical.choice,Quot.sound, on hover). Red: a proof the gate rejects (an unfinished proof,native_decide) — not a witness. A predicate proved in both libraries shows both marks; one mark, one proof. The module page's bar opens the module as one executable file in Lean Live (the core on plain Lean, no Mathlib; the file elaborates in the browser and prints every predicate's axioms at its end); the header's two Lean badges count the marks of each colour and open the two executables. - section
- The statement in the paper's body that carries the predicate: a proposition, a remark or a section.
- rests on
- The predicates it rests on, by label: realisations, definitions and imports of this ledger, predicates of other ledgers (
32:B3) and of the master (00:D15).