Scale and Boundary Candidates in CAELIX
Working Note on the Fine Structure Constant, the Higgs VEV
and Observation-Cost Running
Abstract
This note sets out the present CAELIX architecture for two low-energy boundary candidates: the fine structure constant and the Higgs vacuum expectation value. Neither quantity is treated here as a closed derivation of the full scale problem. Their importance is that both sit near the interface between geometric scaffold, computational burden and unresolved scale architecture.
The central claim is that CAELIX requires two distinct but structurally twinned meanings of scale. One is a burden ladder, associated with the computation required to sustain and propagate a motif. The other is a resolution ladder, associated with how deeply a probe forces internal structure into view. The fine structure constant is then treated as the place where those ladders meet. The scaffold \(4\pi^3 + \pi^2 + \pi\) is assigned to the long-wavelength side of the resolution ladder, while the modifier \(1/(3\times5\times13\times17)\) is assigned to the burden ladder.
The leading running hypothesis is observation-cost running: a broad low-energy plateau followed by threshold penetration once probe resolution begins to enter charged burden structure. On this view, the running of \(\alpha\) is not simple drift of the low-energy modifier. It is the macroscopic signature of additional computation forced by deeper observation of charged lattice patterns. The Higgs VEV belongs in the same discussion because it appears to land on the burden side of the same hierarchy, even though no finished closure is claimed here.
Reading guide
The logical chain from existence, to state space, to constants, to emergent field behaviour and then to geometric Standard Model parameters, is established in:
- On the Necessity of Existence
- Balanced Ternary by Necessity
- Constants from Balanced Ternary
- Emergent Field Physics from Balanced-Ternary Microstates
- Standard Model Structure from Stencil Geometry
This note is best read as a boundary-layer companion to Paper 5 rather than as a standalone derivation. The earlier papers establish why a balanced-ternary substrate is being used at all, why a small set of constants are treated as structurally forced and how lattice microphysics can support effective field behaviour. Paper 5 then pushes that programme into geometric Standard Model structure but leaves the fine structure constant and the Higgs vacuum expectation value in a more awkward class.
The present note picks up that awkward class directly.
It does not try to replace the geometric work. It asks how quantities near the electroweak and low-energy electromagnetic boundary should be read when pure geometric scaffold no longer seems sufficient on its own. The main proposal is that CAELIX needs two linked meanings of scale at once: one associated with burden and one associated with resolution.
A reader who already accepts the general CAELIX programme can therefore enter this note quite quickly.
A reader who does not should treat it as a working architectural extension rather than as a finished closure of either the fine structure constant or Higgs-scale problem.
Introduction
The present note concerns scale, but not in the usual single-axis sense. In CAELIX, “high” and “low” energy do not yet behave like one clean ladder with one clean interpretation. A low-energy numerical fit on its own does not settle the matter, because a quantity can sit near the correct magnitude while still mixing together different structural roles.
The fine structure constant is the sharpest place to confront this. A compact low-energy candidate exists and remains structurally interesting, but its existence alone does not explain why the expression should have the form it does, why the modifier is subtractive, or why the quantity runs under deeper probing. The low-energy fit is therefore only the start of the problem. It is not the whole of the scale architecture.
The Higgs vacuum expectation value raises the same issue from the opposite direction. It is not dimensionless, yet it appears to sit near a repeated 243-suppression from the Planck scale. That makes it useful here as a second boundary candidate. The point is not that the hierarchy problem is solved by one tidy ratio. The point is that the electroweak scale appears to land where a burden-style ladder would place it, while the fine structure constant appears to land where burden and resolution meet.
The note therefore adopts a split architecture. Scale is treated in two structurally twinned senses: burden and resolution. One concerns what it costs the lattice to sustain and propagate a motif. The other concerns how deeply a probe forces that motif into view. A single ladder is not enough. The fine structure constant is then read as the first quantity that appears to sit directly at the junction of the two.
The argument proceeds in stages. Section 2 sets out the two low-energy boundary candidates and explains why they belong in the same discussion without pretending that they are the same kind of observable. Section 3 defines the burden ladder and the resolution ladder. Section 4 then treats the fine structure constant as a junction quantity whose scaffold and modifier belong to different sides of the architecture. Section 5 gives the present running hypothesis: a broad low-energy plateau followed by threshold penetration and observation-cost running once deeper charged burden structure is forced into resolution. The appendices then record the 243-layer scale map and isolate the 26/27 boundary issue as a structured open question rather than a settled cosmological claim.
Boundary candidates at low energy
Fine structure constant
The low-energy fine structure candidate used throughout this note is
Numerically this gives
which lies very near the current measured low-energy value,
In this note that expression is treated as a boundary candidate, not as a closed derivation of electromagnetism across all scales.
Its importance here is not numerical closeness alone. The expression already appears internally split. The polynomial part
reads naturally as a long-wavelength continuum scaffold for the electromagnetic channel. The correction
reads very differently. It is not continuum geometry. It is a counted denominator built from the ternary radix, propagation stencil, species inventory and type inventory.
That split is the key interpretive move of the present note. The fine structure constant is not treated as one object with one meaning. Its leading scaffold and its subtractive modifier need not belong to the same scale logic. Once that distinction is made, the low-energy candidate stops looking like one isolated fit and starts looking like the first place where different parts of the CAELIX scale architecture meet.
The coefficient pattern inside the scaffold also deserves to remain visible. The form
contains the counted pattern \((4,1,1)\). In the present reading that is not decorative. The leading 4 is associated with the four non-centre propagation channels of the local cross, while the two unit coefficients mark one defect-surface tier and one propagation-axis tier. That reading is not yet a forward derivation from first principles. It is a disciplined identification of the scaffold structure that later sections will reinterpret more explicitly as belonging to the resolution side of the architecture.
Higgs vacuum expectation value
The second low-energy boundary candidate is the Higgs vacuum expectation value,
Using the unreduced Planck mass, this gives
against the observed value
The numerical residual is modest. The deeper significance is placement. The quantity lands near a repeated suppression by the local state-space factor
which is the natural combinatorial count of one five-voxel propagation slice with three balanced-ternary states per voxel.
Within the present note this does not close the electroweak hierarchy problem. It does something narrower and more useful for the scale argument. It places the Higgs VEV on a burden-style ladder anchored at the Planck scale and descended by repeated local state-space dilution.
In that reading the electroweak scale is not approached as a primarily geometric quantity. It is approached as the result of repeated computational suppression across a finite ternary substrate.
That is enough to make the Higgs VEV relevant to the same boundary discussion as the fine structure constant. One quantity is dimensionless and one is dimensionful. One appears partly geometric and one strongly hierarchical. Yet both seem to sit near the same unresolved interface between geometric scaffold, computational burden and scale selection.
Why these quantities belong together
The point of pairing the fine structure constant with the Higgs VEV is not that they are the same kind of observable. They are not. The point is that both appear to expose the same architectural fault line.
The Higgs VEV looks like a burden-side quantity. It is naturally written as repeated suppression from a top anchor. The fine structure constant looks mixed. Its leading scaffold is continuum-like, but its low-energy correction is counted and combinatoric. Put differently, the Higgs VEV suggests that repeated state-space burden matters for scale, while the fine structure constant suggests that even a continuum electromagnetic regime still carries a finite bookkeeping tax from the same substrate.
Taken together, the two candidates show why CAELIX cannot speak of scale with a single ladder. The Higgs VEV sits naturally on the burden side of the architecture. The fine structure constant appears to sit at the junction where burden and resolution meet. Read together, they expose the boundary problem that the rest of this note is trying to resolve.
Two meanings of scale in CAELIX
The central distinction is between a burden ladder and a resolution ladder.
The burden ladder is written as
This ladder tracks motif cost. It is about the computation needed to sustain, propagate or stabilise a pattern on the lattice. A deeper or richer motif generally requires more bookkeeping. A lower effective scale can therefore arise because the lattice spends more of its budget on maintaining structure rather than on free propagation.
This distinction matters because CAELIX does not treat particles as primitive objects with labels attached. A particle is better read as a stable lattice pattern, or motif, sustained on a balanced-ternary substrate. Mass then becomes the computation required to sustain and propagate that pattern. Once that move is made, the burden ladder stops looking like an abstract hierarchy map and starts looking like the natural language for persistence cost.
The resolution ladder is written as
This ladder tracks probe depth. It is about what wavelength or momentum transfer is required before a coarse effective description can no longer hide finer internal structure. The issue here is not what a motif costs to exist, but what a measurement costs to resolve.
That second point is decisive. A pattern may be stably carried at one burden cost while remaining only partially visible to a given probe. Shorter wavelength does not merely inspect a pre-existing interior for free. It forces additional internal structure into view. The same motif can therefore have one burden cost for existence and a further observation cost under deeper inspection.
These two ladders are structurally twinned because they are built from the same radix factor. That common backbone should not be mistaken for identity. The burden ladder answers the question: what must the lattice pay to carry a motif? The resolution ladder answers the question: when does a probe begin to see inside it? One concerns persistence and propagation. The other concerns access and penetration.
If only one ladder is used, the fine structure problem collapses into a false choice. Either everything becomes geometry, which leaves the subtractive modifier unexplained, or everything becomes burden, which leaves the polynomial scaffold unexplained. The note therefore treats scale in CAELIX as bifurcated but coupled. Burden and resolution are not rival descriptions. They are different aspects of one architecture.
The fine structure constant as a junction quantity
The working thesis of this note is that the low-energy fine structure candidate is a junction quantity,
That claim is narrower than a full derivation and stronger than a numerical fit. It says that the two pieces of the expression belong to different parts of the same architecture: the scaffold to the long-wavelength side of the resolution ladder, and the modifier to the burden ladder. The low-energy fine structure constant is then the point at which those two descriptions visibly meet.
The scaffold
is assigned to the resolution side because it is written in continuum language. At low energy the electromagnetic channel is observed in a regime where long-wavelength isotropy is already an excellent effective description. The expression is therefore not treated here as a fixed geometric shape. It is treated more usefully as a dimensional transformation law: a transport scaffold whose dominant contribution appears at cubic order, with reduced quadratic and linear trace terms as the same structure descends through effective dimensional tiers. This is why the scaffold can remain geometrically smooth without implying that the underlying substrate itself is continuous.
The counted pattern
matters because it stops the scaffold from floating free as a decorative polynomial in \(\pi\). In the present reading, the leading 4 is associated with the four non-centre propagation channels of the local cross, while the two unit coefficients mark one defect-surface tier and one propagation-axis tier. That is enough to anchor the scaffold structurally, but not enough to count as a completed forward derivation. The argument is therefore one of disciplined identification rather than proof of uniqueness.
The modifier
belongs to the burden side for the opposite reason. It is not generated by continuum isotropy. It is a finite inventory count built from radix, propagation stencil, species capacity and type inventory. In this note that denominator is interpreted as the full low-energy bookkeeping burden relevant to the electromagnetic channel. One possible refinement, worth keeping visible in a working note, is that the final factor may itself decompose into a conjugation-aware split \(17 = 14 + 3\), with fourteen conjugation-capable motif classes and three self-conjugate neutral classes. The photon does not sample empty geometry but geometry carrying a finite counted inventory of ternary states, propagation rules and channel content.
This also clarifies why the modifier is subtractive. The low-energy coupling is not treated as pure geometry with a small decorative offset. It is treated as continuum transport reduced by a finite computational tax. The lattice pays to resolve the electromagnetic channel through a bounded substrate and that payment appears as a downward correction from the bare scaffold.
The fine structure constant is therefore not treated here as “geometry plus correction” in any loose sense. It is treated as a junction quantity in the strict architectural sense: a long-wavelength scaffold from the resolution ladder combined with a counted burden term inherited from the substrate. That is the reason this quantity is central to the note. It is the first place where the split between burden and resolution becomes explicit inside one low-energy expression.
Threshold penetration and observation-cost running
The leading running hypothesis adopted here has three parts: a broad low-energy plateau, threshold penetration and observation-cost running.
At low energy the probe wavelength is coarse relative to the internal charged burden structure of the motif. In that regime the continuum scaffold remains a good effective description and the subtractive burden term behaves as a stable low-energy correction. The low-energy fine structure constant is therefore not treated as the value of one isolated resolution rung. It is treated as the plateau value of an infrared regime across which the probe remains too coarse to force deeper charged structure into explicit participation.
This point matters because the running of \(\alpha\) cannot be explained here as simple variation of the low-energy modifier. If a subtractive modifier merely shrank away, \(\alpha^{-1}\) would rise toward the bare scaffold. Experiment points in the opposite direction. That route is therefore closed.
The modifier marks the existence of burden, but it is not by itself the engine of the running.
The proposed trigger is threshold penetration. Running begins when probe resolution starts to enter charged burden structure itself. At that point the lattice is no longer asked merely to transmit the dressed exterior of a motif. It is forced to compute more of the motif’s internal structure explicitly. The measured coupling then changes because the observational act has crossed a burden threshold.
This is the sense in which the phrase observation-cost running is used. The probe is not a passive spectator. It imposes a cost. A shorter wavelength forces finer structural participation from the substrate. Time-like and space-like probing need not activate that burden in the same way: time-like processes can open new channel content, while space-like scattering can instead force deeper approach into the existing charged dressing of a motif without spawning real offspring. The measured running of the coupling is interpreted as the macroscopic signature of that added computation.
The stronger ontological reading follows directly. Particles are not primitive objects with labels attached. They are stable lattice patterns. Their mass is the computation required to sustain and propagate those patterns. Deeper probing forces deeper computation of the same patterns. Running is then the extra computation cost of resolving the pattern more deeply under observation.
This picture also helps explain why collider-scale running can appear smooth even if the substrate law is discrete. The underlying threshold structure may be discrete, but experiments observe a macroscopic envelope built from enormous ensembles, finite detector bandwidth and channel-dependent reconstruction. A discrete lattice law can therefore support a smooth effective running curve without contradiction.
The note stops short of claiming a finished running law. It does not yet derive exact threshold energies, activation rules or channel weights. What it does claim is that the broad infrared plateau, the later onset of running and the stronger CAELIX reading of particles as patterns fit more naturally into a threshold-penetration picture than into any simple drift of the low-energy modifier.
Present status of the hypothesis
The note treats the fine structure constant and the Higgs VEV as low-energy boundary candidates inside CAELIX. It argues that both matter because both sit near the same unresolved interface between geometric scaffold, computational burden and scale selection. It also argues that one scale ladder is not enough. Burden and resolution must be kept distinct if either quantity is to be read coherently.
Within that split architecture, the low-energy fine structure candidate is interpreted as a junction quantity. The scaffold belongs to the long-wavelength side of the resolution ladder. The modifier belongs to the burden ladder. The running problem is then approached through threshold penetration and observation-cost running rather than through any simple drift of the static low-energy modifier.
That is the positive claim of the note. It is architectural rather than final-form. The aim is to stabilise the present CAELIX reading of the problem, not to pretend that the whole renormalisation question has been closed.
The note does not claim a full forward derivation of the coefficient pattern \((4,1,1)\). It does not claim a complete theory of renormalisation. It does not claim that the Higgs hierarchy problem is solved merely because a \(243^7\) suppression lands near the observed electroweak scale. It does not yet derive the threshold locations at which the running of \(\alpha\) should change regime, nor the exact observation-cost law that would reproduce the measured running curve. The experimental appendices added at the end of the note are therefore reference guides rather than claimed derivations.
Those open questions are substantial, but they are now sharply posed. The architecture developed here is strong enough to separate what appears geometric from what appears computational and to mark where the observational act itself must enter the scale problem. That is enough for the present working note. It turns a loose cluster of numerical hints into a more disciplined research programme.
A practical consequence of that architecture is that CAELIX admits two complementary routes toward the same scale problem: top-down experimental probing from within the observed world, and bottom-up computational ascent from lawful substrate motifs. The strongest future test is not the absolute ceiling of either route taken in isolation, but the first regime in which their independently reached structures begin to overlap against existing experimental anchors.
Appendix: Scale map with radix-243 spacing
For the purposes of this note, the resolution ladder is written as
with
The cosmological ceiling is taken here to be the observable-universe radius,
rather than the diameter. The distinction between proper-distance and comoving horizon conventions is small at this level of precision but worth noting near a layer boundary.
On the convention used here, the visible universe lies within Layer 26, while the scale atlas is extended one further rung to a theoretical Layer 27.
The table is included as a scale atlas rather than as a finished dynamical map. Its role is to fix the numerical ladder used in the present note and to show where familiar physical bands sit relative to the 243-based spacing.
This appendix fixes a numerical resolution ladder only. It does not prove that every layer is dynamically distinct, nor that every familiar physical threshold sits exactly on a layer boundary. Its role is narrower: to establish the scale atlas against which later claims about interiors, boundaries, transitions and possible alignments can be checked.
The table should therefore not be read as assigning the electromagnetic modifier to one single layer. In the present note, the modifier belongs to the full low-energy continuum regime rather than to a unique band edge. What the atlas provides is the scale landscape within which different measurements of that effective low-energy quantity are actually carried out.
Visible-horizon placement and the 26/27 layer question
The 26/27 issue is treated here as an open structural question rather than as a finished cosmological claim.
Layer 26 is the first layer whose upper wavelength exceeds the observed horizon radius,
That is why the visible universe is said here to lie within Layer 26. Extending one further rung gives
which is recorded in the scale atlas in the short form:
Theoretical global-domain layer beyond the visible horizon; indirect curvature/topology constraints only.
The issue is not whether the ladder itself can be fractional. It cannot. The issue is which integer ladder the present cosmological ceiling is truncating. Under the current convention, the observable universe occupies only part of the highest presently visible rung. That leaves at least three structured possibilities open: the present convention is already the correct one and Layer 26 is the highest physically relevant occupied band, the true global closure lies above current observation and requires a full Layer 27, or the correct base scale for the effective resolution ladder is not the raw Planck length but a higher continuum-onset scale \(\ell_0\).
No stronger claim is made in this note. The question is kept visible because a discrete computational ontology makes a hard outer boundary structurally natural in a way that continuum ontology does not, but the present note does not claim that the global closure is known.
Low-energy fine structure constant measurements
The tables below record a small chronological guide to the main experimental routes used to determine the low-energy fine structure constant. They are not intended as a complete bibliography. Their purpose is to separate direct precision determinations of the low-energy constant from later higher-energy running measurements.
These entries are grouped together because they aim at the low-energy constant itself rather than at a running or fitted high-scale effective coupling. Tables C1 and C2 separate the experiment inventory from the reported outcomes so that chronology, method and interpretation do not have to compete for space in one overfilled layout.
The low-energy landscape is not made of identical measurements repeated in different laboratories. The free-electron \(g-2\) route and the atom-recoil route access \(\alpha\) through different observables and different theoretical pathways, even though both land on the same infrared quantity \(\alpha(0)\). That agreement is therefore part of the significance. It shows that distinct low-energy access routes converge on the same constant within very tight uncertainty.
The final CODATA row is included for a different reason. It is not an independent experiment but the adjusted metrological recommendation formed from multiple inputs. It belongs in this appendix because the present note uses the recommended low-energy value as its numerical anchor, but it is listed separately in order to keep the distinction between primary determinations and adjusted consensus values visible.
Older routes such as quantum Hall metrology, helium fine structure and related spectroscopy remain historically important cross-checks, but the modern precision landscape is dominated by the free-electron \(g-2\) route and atom-recoil interferometry.
Table C1. Low-energy experiment inventory
| ID | Year | Method | Physical system | Laboratory / collaboration | Regime |
|---|---|---|---|---|---|
| L1 | 2008 | Free-electron magnetic moment with QED inversion | Single electron in a Penning trap | Harvard / Gabrielse programme | Low-energy plateau |
| L2 | 2011 | Atom recoil interferometry | \(^{87}\mathrm{Rb}\) | Laboratoire Kastler Brossel | Low-energy plateau |
| L3 | 2018 | Atom recoil interferometry | \(^{133}\mathrm{Cs}\) | Berkeley / Müller group | Low-energy plateau |
| L4 | 2020 | Atom recoil interferometry | \(^{87}\mathrm{Rb}\) | Paris / CNRS-led collaboration | Low-energy plateau |
| L5 | 2022 | CODATA least-squares adjustment | Multi-input metrological adjustment | CODATA / NIST | Low-energy plateau |
Table C2. Low-energy results and significance
| ID | Scale | Measured quantity | Type | Reported result / significance | Ref. |
|---|---|---|---|---|---|
| L1 | \(q^2 \approx 0\) | Electron anomaly \(a_e\) | Indirect | Landmark free-electron route; one of the dominant modern low-energy determinations of \(\alpha\) | [4] |
| L2 | \(q^2 \approx 0\) | \(h/m_{\mathrm{Rb}}\) | Indirect | First high-precision rubidium recoil determination in the modern sequence | [5] |
| L3 | \(q^2 \approx 0\) | \(h/m_{\mathrm{Cs}}\) | Indirect | Reported \(\alpha^{-1} = 137.035999046(27)\) | [6] |
| L4 | \(q^2 \approx 0\) | \(h/m_{\mathrm{Rb}}\) | Indirect | Reported \(\alpha^{-1} = 137.035999206(11)\); improved rubidium result | [7] |
| L5 | \(q^2 \approx 0\) | CODATA least-squares adjustment | Adjusted | Recommended 2022 value \(\alpha^{-1} = 137.035999177(21)\); consensus adjustment from multiple inputs, not an independent experiment | [10] |
Higher-energy probes of the electromagnetic coupling
The tables below record a separate chronological guide to higher-energy probes of the electromagnetic coupling. These entries do not all determine the same quantity in the same renormalisation sense. Some are direct running measurements, while others are precision-fit inputs or determinations of effective high-scale quantities. They are listed separately in order to keep the low-energy constant problem distinct from the running problem.
These entries are grouped separately because they probe the scale dependence of the effective electromagnetic coupling rather than the low-energy constant alone. Tables D1, D2 and D3 divide that landscape into experiment identity, interpretive role and reported numerical output, because the higher-energy literature does not report one uniform kind of quantity.
That distinction matters. Some rows report direct running measurements, some report fitted effective couplings, some report differential running quantities and some report bridge terms such as \(\Delta\alpha_{\mathrm{had}}(M_Z^2)\) that are used to construct the high-scale effective coupling rather than to replace it. The tables are therefore separated so that unlike quantities are not forced into one misleading column structure.
The higher-energy appendix should not be read as a second low-energy table with larger numbers. It records a mixed measurement landscape in which time-like data, space-like running, global electroweak fits and hadronic-vacuum-polarisation reconstructions all contribute to the practical story. That is why only some rows naturally admit an \(\alpha^{-1}(q^2)\)-style number, while others are more honestly represented by differential quantities, fit parameters or bridge terms.
This distinction is directly relevant to the argument of the note. The empirical landscape above the infrared plateau is not accessed through one clean probe family. It is stitched together from several experimental and theoretical routes, and those routes begin to separate most clearly once hadronic vacuum polarisation enters the problem. The appendix is included to make that mixed access structure visible before any stronger CAELIX interpretation is attempted.
Table D1. Higher-energy experiment inventory
| ID | Year | Energy / regime | Process | Experiment / collider | Regime |
|---|---|---|---|---|---|
| H1 | 1997 | TRISTAN electroweak energies | \(e^+e^- \to \mu^+\mu^-\) and related ratios | TOPAZ / TRISTAN | Higher-energy running |
| H2 | 1998 | Space-like GeV\(^2\) regime | Large-angle Bhabha scattering | VENUS / TRISTAN | Higher-energy running |
| H3 | 2003 | LEP2 energies | \(e^+e^- \to f\bar f\) | OPAL / LEP2 | Higher-energy running |
| H4 | 2005 | Space-like running regime | Small-angle Bhabha scattering | OPAL / LEP | Higher-energy running |
| H5 | 2005 | High-\(Q^2\) LEP regime | Large-angle Bhabha scattering | L3 / LEP | Higher-energy running |
| H6 | 2006 | \(Z\)-pole electroweak regime | Global electroweak fits | LEP Electroweak Working Group / SLD | Higher-energy effective coupling |
| H7 | 2020 | High-scale vacuum-polarisation bridge | Dispersive hadronic input analyses | Davier et al. / global data compilations | Higher-energy effective coupling |
| H8 | 2022 - 2025 | Space-like HVP evaluations | Lattice QCD vacuum polarisation | Mainz / CLS lattice programme | Higher-energy effective coupling |
Table D2. Higher-energy quantities and significance
| ID | Scale | Measured quantity | Type | Reported result / significance | Ref. |
|---|---|---|---|---|---|
| H1 | Time-like, \(q^2 \sim\) \(3338\,\mathrm{GeV}^2\) | Cross-section ratios at electroweak energies | Direct | Early direct electroweak-scale running measurement from TRISTAN-era data | [12] |
| H2 | Space-like, GeV\(^2\) range | Angular dependence of large-angle Bhabha scattering | Direct | Early space-like probe of running in the GeV\(^2\) regime | [13] |
| H3 | LEP2 energies | Cross-sections and asymmetries | Fitted | Determination of \(\alpha_{\mathrm{em}}\) at LEP2 energies through fermion-pair production fits | [8] |
| H4 | Space-like, \(1.81 < -t\) \(< 6.07\,\mathrm{GeV}^2\) | Angular dependence of small-angle Bhabha scattering | Direct | Landmark direct measurement of the running of \(\alpha\) in the space-like region | [9] |
| H5 | Space-like, \(1800 < -Q^2\) \(< 21600\,\mathrm{GeV}^2\) | Large-angle Bhabha scattering cross-sections | Direct | High-\(Q^2\) validation of QED running at LEP scales | [14] |
| H6 | \(q^2 \approx M_Z^2\) | Precision electroweak observables | Fitted | Standard reference framework for effective high-scale electromagnetic coupling inputs near \(M_Z\) | [11] |
| H7 | High-scale vacuum-polarisation bridge | \(\Delta\alpha_{\mathrm{had}}(M_Z^2)\) and related inputs | Derived | Data-driven bridge from hadronic cross-sections to the effective \(Z\)-pole coupling | [15] |
| H8 | Space-like, few-GeV\(^2\) to \(M_Z^2\) bridge | Hadronic vacuum polarisation from lattice QCD | Computed | First-principles lattice route; relevant because of the reported tension with data-driven HVP evaluations | [16] |
Table D3. Higher-energy reported numerical outputs
| ID | Quantity reported | Numerical result | Note |
|---|---|---|---|
| H1 | \(\alpha^{-1}(q^2)\) | \(128.5 \pm 1.8 \pm 0.7\) | Early direct TRISTAN-scale running result; see H1 in Table D2 for the time-like electroweak-scale regime |
| H2 | Running sensitivity | No single canonical quoted value used here | Included as an early space-like running probe rather than as a precision anchor; see H2 in Table D2 for the scale domain |
| H3 | \(\alpha_{\mathrm{em}}\) at LEP2 energies | Fit-based determination; no single scalar value quoted here | Better read as a fermion-pair fit regime than as one headline number; see H3 in Table D2 for the LEP2 setting |
| H4 | \(\Delta\alpha(-6.07)\) \(-\,\Delta\alpha(-1.81)\) | \((440 \pm 58\) \(\pm 43 \pm 30) \times 10^{-5}\) | Landmark differential running measurement from small-angle Bhabha scattering; see H4 in Table D2 for the space-like interval |
| H5 | Running parameter \(C\) | \(1.05 \pm 0.07 \pm 0.14\) | Validates QED running and excludes a constant-coupling picture; see H5 in Table D2 for the high-\(Q^2\) range |
| H6 | Effective \(\alpha^{-1}(M_Z^2)\) | \(\approx 128.95(5)\) | Global-fit electroweak anchor rather than a direct scattering number; see H6 in Table D2 for the \(Z\)-pole regime |
| H7 | \(\Delta\alpha_{\mathrm{had}}(M_Z^2)\) | \((276.0 \pm 1.0) \times 10^{-4}\) | Data-driven hadronic contribution used in effective \(Z\)-pole coupling work; see H7 in Table D2 for interpretive context |
| H8 | Space-like HVP tension window | Tension reported in the few-GeV\(^2\) region, not one universal scalar summary | Included as a first-principles lattice route and comparison pressure point against data-driven HVP evaluations; see H8 in Table D2 for the bridge regime |
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