Standard Model Structure from
Stencil Geometry

Abstract

The Standard Model of particle physics contains approximately 25 free parameters: coupling constants, mixing angles, masses and CP-violating phases.

These values are measured experimentally and inserted by hand; the theory does not derive them from a deeper discrete substrate [1].

This paper proposes a structural classification of these parameters within the CAELIX framework. Parameters that depend on stencil topology, particle content and interaction combinatorics fall into the geometric class. Parameters that depend on runtime propagation burden fall into the computational class. This separation is the central claim of the paper.

For the geometric class, the framework uses a seven-voxel identity stencil together with a five-voxel propagation stencil on a lattice with state space \(\{-1,\,0,\,+1\}\), yielding a small set of countable structural integers. From these integers, candidate derivations are constructed here for nine non-mass Standard Model parameters: the weak mixing angle, the strong coupling constant, the strong CP angle, the three CKM mixing angles and the three PMNS mixing angles.

Within the framework, the strong CP angle \(\theta_{\mathrm{QCD}}\) is forced to zero by exact sign symmetry of the underlying alphabet.

Mass parameters are excluded from the geometric derivation on physical grounds. In the CAELIX picture, mass is not a label attached to a particle but the computational cost of propagating that particle's stencil through the lattice. Mass belongs to the computational tier, not the geometric one.

A small boundary class, including the fine structure constant and the Higgs vacuum expectation value, is identified as structurally important but not derived within the present geometric treatment.

The paper does not claim a completed proof of all Standard Model parameters. It claims that a single balanced-ternary stencil framework provides a unified route for separating geometric parameters from computational ones and for deriving the geometric class from countable lattice structure.

Reading guide

This is the fifth paper in the sequence.

The logical chain from “why does anything exist?” through “what is the minimal state space?” to “what constants does the algebra force?” and “what physics emerges on a lattice?” is established in:

1. On the Necessity of Existence
2. Balanced Ternary by Necessity
3. Constants from Balanced Ternary
4. Emergent Field Physics from Balanced-Ternary Microstates

This page picks up where the current experiment programme leaves off and is the next paper in the CAELIX sequence.
It takes the structural picture established so far and uses it as the foundation for the next suite of CAELIX experiments.

The question here is which Standard Model parameters should be expected to arise from lattice geometry alone and which should instead belong to the computational cost of propagating a stencil through the substrate. If a particle is a minimal lattice configuration, which of its measured properties follow from counting and which only emerge computationally when the lattice is run?

The structure is organised to build the argument from foundations:

Part I defines the stencil and its quantum numbers.

Part II separates the Standard Model parameters into geometric and computational categories. This classification is essential: the paper does not argue that all parameters should be derivable by counting alone.

Part III develops the modifier framework that corrects bare geometric values for particle-content overhead.

Part IV derives the geometric parameters treated within the scope of the present paper.

I. The Particle as Stencil

Intrinsic Properties And Their Information Content

Every elementary particle in the Standard Model is characterised here by five intrinsic properties relevant to the stencil description: mass, spin, electric charge, colour charge and chirality. Mass is categorically different from the others: it is continuous rather than discrete. The remaining four properties take values from finite sets and therefore define the particle’s discrete information content.

The purpose of this section is not to reproduce the full Standard Model taxonomy, but to ask a narrower question: how much discrete information must a minimal lattice object carry before it can represent a particle species at all?

Four of these properties are discrete and take values from finite sets:

Spin.
Elementary particles carry:
spin 0 (Higgs), \(\tfrac{1}{2}\) (fermions), or 1 (gauge bosons).
Three distinct values.

Electric charge.
Measured in units of \(e/3\), the values appearing across all elementary particles are: \(-1\), \(-\tfrac{2}{3}\), \(-\tfrac{1}{3}\), \(0\), \(+\tfrac{1}{3}\), \(+\tfrac{2}{3}\), \(+1\)
Seven distinct values.

Colour charge.
A fundamental particle is either colourless or carries one of three colour charges (conventionally red, green, blue).
Four distinct values.

Chirality.
The weak interaction distinguishes left-handed from right-handed fermions. Chirality is therefore an independent binary property, not recoverable from spin, charge or colour alone.
Two distinct values.

In a binary representation, encoding these four properties requires:

The importance of this count is architectural rather than digital. It shows that the particle problem is not one of continuous labels first and foremost, but of a small bundle of discrete quantum distinctions plus one categorically different quantity: mass. That is the split the rest of the paper exploits.

Mass Is Not A Label

Of these five properties, mass is categorically different from the other four. Spin, charge, colour and chirality are discrete and take values from small finite sets. Mass is a continuous parameter spanning many orders of magnitude with no evident quantisation.

In the CAELIX framework, this distinction has a structural explanation. The lattice is a sparse array of balanced-ternary voxels \(s(\mathbf{x}) \in \{-1, 0, +1\}\). The vacuum state is \(s = 0\).
A particle is a configuration of voxels flipped to \(\pm 1\).

Once such a stencil exists, the lattice must continually do work to sustain and propagate it: resolving neighbour mismatches, preserving identity under translation and carrying the pattern through time.

Mass is this computational overhead. It is not a label written on the particle; it is the cost of being that particle. Different stencil configurations create different propagation costs, producing different masses.

This is why 9 of the Standard Model’s 25 free parameters are fermion masses and a further 4 are neutrino masses or mass-squared differences. They are not geometric; they are computational. They do not belong to the same derivational class as angles, ratios or species-count structure, because they depend on the dynamics of propagation rather than on the topology of the stencil alone.

The Balanced-Ternary Stencil

The lattice does not speak in bits. Its native alphabet is \(\{-1, 0, +1\}\): balanced ternary.

A single balanced ternary digit (trit) can represent three states. The four discrete quantum properties require substantial balanced-ternary descriptive capacity. In a naive standalone ternary encoding this would amount to 6 trits, but the 3D identity stencil used here is not a register-by-register code: the quantum numbers are decoded from the composite motif itself. The encoding is therefore not arbitrary; it must be grounded in the lattice’s computational structure rather than imposed as an external coding scheme.

At this point a distinction is needed between two related but non-identical objects. The first is the identity stencil: the full 3D von Neumann neighbourhood, consisting of a centre voxel plus its six face-neighbours. This seven-voxel object is the minimal local motif from which particle identity is read. The second is the propagation stencil: the 2D five-cell cross used by the telegraph update at each local step. The propagation stencil is therefore a slice of the full identity stencil. Identity lives on the 3D motif; propagation samples it through local 2D crosses.

The centre trit encodes whether a non-vacuum stencil exists and, if so, which principal orientation class it occupies.

A stable topological defect in a 3D lattice is naturally a \((d-1)\)-dimensional object: a 2D surface-like defect embedded in 3D space. The three non-zero orientation classes are interpreted as the three principal plane families (XY, XZ, YZ) realised by the surrounding motif.

Centre \(= 0\) is the vacuum with respect to centred identity: no defect, no orientation, no particle is centred on that voxel. It does not by itself require the surrounding neighbourhood to be empty; nearby structure may still be present.

These assignments should not be read as a literal one-trit-to-one-quantum-number encoding table. A single trit cannot by itself carry all seven charge values or all four colour states. What the full 3D stencil provides is a minimal local scaffold from which those quantities are decoded.

Electric charge is identified with the normalised sum along a three-voxel axis:

so that the allowed values are exactly

The seven observed charge states therefore arise as an algebraic consequence of balanced ternary on a 3-cell line.

Colour is then not a trit value but an axis class of the defect support. A motif with non-trivial support concentrated preferentially along one axis belongs to the coloured triplet;
a motif with no preferred axis is colourless. In this way, fractional charge and colour appear together as consequences of axis-preferential structure, while integer charge and colourlessness arise from axis-symmetric structure.

The observed association of fractional electric charge with coloured states and integer charge with colourless states is therefore not an added rule in this framework but a geometric consequence of the same motif structure. A quark carries fractional charge because its defect is axis-asymmetric rather than axis-symmetric: one or two axes out of three contribute non-trivially to the normalised sum. A lepton carries integer charge because its defect is axis-symmetric: all three axes contribute together.

Colour and fractional charge are not independent labels here, but two descriptions of the same geometric property.

Quantum numbers are decoded from motif class, not from isolated single-cell labels.

The later derivations do not depend on a literal arm-to-property assignment, but on the counted structural constants extracted from the stencil and the particle inventory.

The particle is not something that lives on the lattice. The particle is a minimal lattice configuration.

Self-checking by construction.
Balanced ternary carries an intrinsic sign balance absent from ordinary unsigned binary encodings. Because the alphabet is centred on zero, local imbalance is directly visible as signed excess rather than requiring a separate parity bookkeeping layer. This does not mean every valid stencil must sum to zero globally. It means the representation itself makes imbalance explicit instead of hiding it behind external error-checking machinery. Every cell carries physics.

Matter–antimatter conjugation.
The sign flip \(-1 \leftrightarrow +1\) maps every particle to its antiparticle. No additional degree of freedom is needed; conjugation is a native operation of the alphabet.

The Structural Constants

The framework relies on a small set of countable integers: \(\{2, 3, 5, 7, 13, 17\}\).

These are not introduced as free fit parameters. They are treated as rigid structural counts arising from the balanced-ternary lattice and from the way Standard Model content occupies that carrying capacity.

The species capacity (13).
A 3×3×3 cube has exactly 13 distinct centre-crossing symmetry axes: 3 face-to-face, 6 edge-to-edge and 4 corner-to-corner. If species correspond to identity-bearing defect classes organised by these centre-crossing directions, then the cubic geometry itself supplies a 13-fold carrying capacity before any Standard Model inventory is counted. The observed Standard Model inventory of 12 fermions plus 1 scalar Higgs exactly saturates that geometric capacity. This does not by itself identify each axis with a specific Standard Model species, but it shows that the lattice geometry natively supplies a 13-fold identity capacity.

The colourless subset (7).
The integer 7 appears twice in the framework. First, as established above, a 3-voxel ternary axis permits exactly 7 normalised charge states. Second, of the 13 available species pathways, the Standard Model identifies exactly 7 colourless species: the 6 leptons plus the Higgs. The remaining 6 species are the coloured quarks. This alignment between the 7-state charge spectrum and the 7-member colourless subset is treated here as a structural resonance, not as a numerical coincidence.

The empirical type count (17).
Counting all distinct particle types presently resolved in the Standard Model gives 17: 12 fermion types plus 5 boson types (photon, W, Z, gluon, Higgs). In this paper that number is used conservatively as the current effective empirical inventory of fundamental structural roles, not as a claimed immutable lattice constant. It is therefore the most revisable integer in the set: if future experiment establishes additional fundamental types, the formulas using 17 must be recalculated rather than defended by ad hoc exception.

The boson split for colourless particles.
The 5 boson types are: photon, W, Z, gluon and Higgs, with W counted at the inventory level as the charged weak pair taken together. For the neutrino visibility split, however, the relevant count is not boson types but interaction channels.

A neutrino (which is both colourless and electrically neutral) can interact through three colourless channels: W⁺, W⁻ and Z (via weak isospin). It cannot interact through the photon channel (no electric charge) or the Higgs channel (mass coupling is computational, not geometric). This gives 3 visible channels and 2 invisible channels from the neutrino’s perspective.

Every formula in this paper is built exclusively from subsets of \(\{2, 3, 5, 7, 13, 17\}\). That restriction is deliberate. The claim is not that any small integer may be recruited after the fact, but that one fixed counted inventory is reused throughout.

II. Geometric versus Computational Parameters

Classification

The Standard Model’s approximately 25 free parameters do not all belong to the same ontological tier. In the CAELIX framework they divide into two categories: those that can plausibly arise from stencil topology and particle-content combinatorics alone and those that require the lattice to be run. This distinction is the organising principle of the paper.

Geometric parameters arise from stencil structure, species counting, symmetry class and interaction combinatorics. They are quantities for which a static counted architecture may reasonably matter: ratios, angles, visibility splits and hierarchy factors.

Computational parameters emerge from runtime burden, propagation lag or dynamic asymmetry. They are quantities whose value depends not just on what the stencil is, but on what it costs the lattice to sustain, move or distinguish that stencil during evolution.

This is a classification claim before it is a derivation claim. Not every Standard Model parameter should be recoverable by static counting; only one subset should be. The table below is therefore a classification reference for the broader CAELIX programme, while the derivations attempted in the present paper are restricted to the geometric class.

Boundary parameters.
Some quantities are not yet cleanly classifiable as either purely geometric or purely computational. The fine structure constant appears to depend on a geometric scaffold, but also on a scale-sensitive correction structure that is not closed within the scope of the present paper. The Higgs vacuum expectation value appears to sit near the interface between static geometric hierarchy and computational burden and its current residual does not justify assigning it cleanly to either class. These quantities are therefore placed in the boundary class: recognised as structurally important, but not derived in the present paper.

Note on the CP phases.
CP violation measures the asymmetry between a process and its conjugate. On a balanced-ternary lattice, conjugation (\(-1 \leftrightarrow +1\)) is an exact arithmetic symmetry at the alphabet level. Any non-zero CP phase must therefore arise from a dynamic asymmetry: a difference in runtime burden between a stencil and its sign-flipped conjugate, or from an irreducible imbalance in the particle-content seen by that stencil. The CKM CP phase is placed in the computational class for this reason.

Note on generations.
The electron, muon and tau are identical in every discrete quantum number. They differ only in mass. If mass is computational lag, then generations are not three independent labels written into the geometry, but three stable resonant modes of the same underlying stencil: ground state, first overtone, second overtone. This is a framework-level interpretation, not yet a derivation. Its role here is to explain why generation structure belongs naturally beside the computational class rather than inside the counted geometric inventory.

The practical consequence is simple: when a later section proposes a formula, the first question is not whether the number looks close, but whether the parameter belongs to the geometric class at all. If not, numerical agreement would not count as a success of the present programme.

III. The Modifier Framework

Bare Geometry And Particle-Content Corrections

Each candidate geometric parameter is written as a bare structural value plus a small correction, the modifier. The bare term is intended to capture what follows from stencil geometry alone. The modifier captures the finite particle-content overhead seen by that interaction channel. This section introduces the modifier framework as a disciplined working ansatz, not yet as a proved theorem.

Every modifier here shares the same anatomy:

The point of this shared anatomy is restraint. It prevents the later derivations from inventing a new correction law for every parameter. If the framework works, it must work with a very small modifier vocabulary.

Each factor is intended to represent a distinct layer of the lattice’s bookkeeping:

For reference, three modifier values arise in the broader CAELIX framework, one for each interaction domain:

These three values are not presented as an arbitrary menu. Within the present paper, the weak/CKM and leptonic/PMNS modifiers are the active reused correction families, while the electromagnetic modifier belongs to the broader CAELIX framework but is not derived further here. That reuse is part of the claim. If later derivations require a growing zoo of one-off corrections, the framework has failed.

The choice factor.
The leading factor distinguishes the nature of the interaction channel. For electromagnetic coupling, the counted response is split across the three charge-sign sectors \(\{-1, 0, +1\}\), giving a ternary choice factor of 3. For weak and colour interactions, the effective split is taken to be binary: emit or do not emit, couple or do not couple, colour or anticolour. The choice factor is therefore 2. This is one of the framework assumptions that must be judged by the consistency of the downstream derivations.

The universal backbone.
The product \(5 \times 13 \times 17 = 1105\) appears in the electromagnetic and weak modifiers. It gives the lattice’s full descriptive backbone for propagation bookkeeping: propagation-stencil voxels times species count times type count. Only the leading choice factor changes between these domains.

The colourless backbone.
Leptons are colourless. A colourless particle does not couple to the gluon channel, so the gluon type drops out of the bookkeeping. The type count reduces from 17 to \(17 - 1 = 16\)

The relevant species count becomes the number of lepton species: 6. This gives the colourless propagation backbone: \(5 \times 6 \times 16 = 480\). As with the universal backbone, this is a structural proposal whose value lies in repeated reuse rather than in isolated fit quality.

The colourless backbone is not intended as a second unrelated bookkeeping logic. It is the universal backbone restricted to the colourless interaction subspace. From that perspective, the change is a projection rather than a replacement: the gluon type is removed from the active type inventory and the active species inventory is restricted from the full 13-fold carrying capacity to the 6 lepton species. The backbone therefore changes by disciplined restriction of the accessible sector, not by invention of a new correction grammar.

Sign convention.
The sign of the modifier is not free. It is determined by whether the interaction’s accessible geometric phase space is reduced relative to the bare construction or enlarged beyond it. For electromagnetic coupling, the bare term represents ideal long-wavelength propagation in the continuum-isotropic limit, so resolvable particle content acts as an obstruction and the modifier is subtractive.

For quark mixing, colour-active structure opens additional configuration space beyond what the bare species/type ratios encode, so the CKM modifier is additive. For lepton mixing, the locked colour sector reduces the accessible configuration space relative to the bare overlap construction, so the PMNS modifier is subtractive. This sign logic remains a constrained geometric ansatz rather than a finished theorem.

A useful way to read the rest of the paper is therefore as a two-stage test. First ask whether the bare term is structurally plausible for the parameter in question. Then ask whether one of the three modifier families improves the result in a way that is consistent with the stated domain logic. If either stage fails, the derivation fails.

IV. Derivations

The Strong CP Angle: \(\theta_{\mathrm{QCD}} = 0\)

The problem.

The strong CP problem is one of the major unsolved puzzles in particle physics. The QCD Lagrangian permits a CP-violating term proportional to \(\theta_{\mathrm{QCD}}\). Experimentally, \(\theta_{\mathrm{QCD}} < 10^{-10}\) [1]. No mechanism within the Standard Model explains why this parameter is so extraordinarily close to zero. Proposed solutions include the Peccei–Quinn symmetry and the axion, neither of which has been experimentally confirmed [5,6].

The derivation.

\(\theta_{\mathrm{QCD}}\) parameterises the degree to which the strong interaction distinguishes a process from its CP-conjugate. In the present framework, the relevant question is whether the underlying alphabet permits such a distinction at the substrate level.

The balanced-ternary lattice has state space \(\{-1, 0, +1\}\). This alphabet is exactly symmetric under the sign conjugation \(-1 \leftrightarrow +1\). Within the CAELIX substrate picture, this is not an approximate symmetry, not a dynamical accident and not a fine-tuned cancellation. It is arithmetic. At the alphabet level, the lattice cannot distinguish between a configuration and its sign-flipped conjugate.

If the strong CP angle is sourced only by substrate-level sign asymmetry, then \(\theta_{\mathrm{QCD}}\) is forced to vanish in this framework. One would have to break the sign symmetry of the alphabet itself to make it otherwise.

The result.

(within the balanced-ternary sign-symmetric substrate picture)

Falsifiable consequence: an axion is not required by this framework. If a QCD axion is discovered as the mechanism that resolves strong CP violation, this derivation is falsified.

The Weinberg Angle: \(\sin^2\theta_W\)

The problem.

The weak mixing angle determines how the electromagnetic and weak forces are related after electroweak symmetry breaking. Its measured value is:
\(\sin^2\theta_W = 0.23122 \pm 0.00003\) [1]. The Standard Model does not derive this value from a deeper counted substrate.

The bare geometric value.

The Weinberg angle is the projection angle that decomposes electroweak coupling into the U(1) hypercharge and SU(2) weak-isospin sectors. Its bare value is taken to reflect a species-level split between ternary charge structure and the full counted species inventory. This is a candidate geometric identification, not an independent theorem.

The ternary radix is 3. The total species count is 13.
The resulting bare ratio is:

The modifier.

The electroweak split is treated here as a binary choice:
U(1) or SU(2). The choice factor is therefore 2. Using the universal backbone from Section III, the modifier is:

The modifier is additive: in this construction, the interaction’s accessible geometric phase space exceeds what the bare ratio alone encodes.

The result.

Measured: \(0.23122 \pm 0.00003\) [1]. Agreement within experimental uncertainty.

The bare ratio already sits in the correct electroweak neighbourhood and the reused weak-domain modifier then lifts it into direct agreement with the observed mixing scale.

The Strong Coupling Constant: \(\alpha_s\)

The problem.

The strong coupling constant at the Z boson mass scale is measured as \(\alpha_s(M_Z) = 0.1180 \pm 0.0009\) [1]. The Standard Model does not derive this value from a deeper counted substrate.

The bare geometric value.

The strong coupling describes colour interaction. Its bare value is identified with a binary colour-pairing structure measured against the total counted particle-type inventory. This is a candidate geometric identification, not yet an independent theorem. The relevant bare ratio is:

The modifier.

The colour–anticolour interaction is treated here as binary. Using the universal backbone from Section III, the modifier is:

The modifier is additive: in this construction, the interaction’s accessible geometric phase space exceeds what the bare ratio alone encodes.

The result.

Measured: \(0.1180 \pm 0.0009\) [1]. Agreement within experimental uncertainty.

The colour-pairing ratio \(2/17\) already lands in the correct QCD neighbourhood and the same weak-domain modifier used above then lifts it into the observed strong-coupling range without introducing a new correction law.

CKM Quark Mixing Angles

The Cabibbo–Kobayashi–Maskawa matrix describes the mixing between quark mass eigenstates and weak interaction eigenstates. It is parameterised by three angles and one CP-violating phase. The three angles are derived here; the CP phase is computational.

Physical picture.

Quarks occupy the full motif structure: spin, charge, colour and chirality are all active in the quark sector. Generations are harmonic overtones of the same underlying quark-sector motif. The mixing angles measure the overlap between these harmonics. On the full quark-sector motif, the fundamental overlap is determined by the ratio of the ternary radix to the species count: \(3/13\).

All CKM transitions are mediated by W boson emission or absorption, a binary process (emit or do not emit). The modifier is therefore \(1/2210 = 1/(2 \times 5 \times 13 \times 17)\) for all CKM angles.

CKM \(\theta_{12}\): The Cabibbo Angle

This subsection treats the Cabibbo angle as the fundamental first-to-second generation overlap on the full quark-sector motif. The bare value is identified with the radix-to-species overlap of the full quark-sector motif:

The sine requires the hypotenuse:

Applying the additive weak-domain modifier from Section III:

Measured: \(\sin\theta_{12} = 0.2253 \pm 0.0007\), \(\theta_{12} = 13.02^\circ\) [1].

The fundamental radix-to-species overlap of the full quark motif already lands close to the Cabibbo scale and the same reused weak-domain modifier then refines it into direct agreement with observation.

CKM \(\theta_{23}\): Higher Harmonic Coupling

This subsection treats the second-to-third generation coupling as a higher-harmonic overlap on the full quark-sector motif. Each adjacent generation step is modelled as a further structural suppression on that same motif. For \(\theta_{23}\), the coupling passes through both the species and type scales:

Applying the additive weak-domain modifier from Section III:

Measured: \(\sin\theta_{23} = 0.0415 \pm 0.0009\), \(\theta_{23} = 2.38^\circ\) [1].

The same higher-harmonic overlap logic and reused weak-domain modifier land near the observed second-to-third generation mixing scale.

CKM \(\theta_{13}\): The Generation Skip

This subsection treats the \(1\leftrightarrow 3\) direct coupling as a generation-skip overlap that bypasses the intermediate harmonic. Each generation gap is modelled as an additional species-scale suppression on the full quark-sector motif. Skipping two gaps costs \(13^2\), mediated through the full 5-voxel propagation stencil:

No additional external modifier is introduced here. The stencil factor (5) is already present inside the denominator. In this construction, the skip transition already carries part of the stencil overhead inside the overlap construction itself, so no separate external correction is added at this stage.

Compare the three CKM denominators:

Measured: \(\sin\theta_{13} = 0.00351 \pm 0.00013\), \(\theta_{13} = 0.201^\circ\) [1].

The generation-skip overlap construction lands on the observed \(1\leftrightarrow 3\) quark-mixing scale without an additional external modifier.

PMNS Lepton Mixing Angles

The Pontecorvo–Maki–Nakagawa–Sakata matrix describes neutrino mixing. Its angles are dramatically larger than the CKM angles. Conventional physics has no explanation for this difference.

Physical picture.

Leptons are colourless: the colour degree of freedom is locked to the symmetric class. This reduces the active motif structure from four quantum sectors to three. Reduced active motif complexity means the harmonic modes are more tightly packed, overlaps between generations are larger and mixing angles are near-maximal.

The CKM–PMNS asymmetry is not a mystery. It reflects the reduced number of active quantum sectors in the colourless lepton motif.

All PMNS corrections use the colourless modifier: \(1/960 = 1/(2 \times 5 \times 6 \times 16)\). The sign is subtractive: the locked colour sector reduces the accessible geometric phase space relative to the bare overlap construction.

PMNS \(\theta_{23}\): The Atmospheric Angle

This subsection treats the atmospheric angle as a colourless-overlap construction seen from the neutrino sector. Of the 13 counted species, 7 are colourless (6 leptons + Higgs) and 6 carry colour (the quarks). The bare angle is identified with the ratio of species the neutrino can couple to and those it cannot:

The hypotenuse:

Applying the subtractive colourless-domain modifier from Section III:

Measured: \(\sin\theta_{23} = 0.749^{+0.008}_{-0.010}\), \(\theta_{23} = 48.5^{+0.7}_{-0.9}\,^\circ\) [2].

The bare 7-to-6 colourless-versus-coloured species split already produces a near-maximal atmospheric angle and the reused colourless-domain modifier keeps the result in the correct atmospheric neighbourhood.

Why this angle is “nearly maximal” but not \(45^\circ\). Conventional physics describes \(\theta_{23}\) as “nearly maximal” without explaining why. The stencil construction suggests that it is not fundamentally tied to \(45^\circ\) at all, but is instead set by the ratio \(\arctan(7/6)\) before correction. It appears nearly maximal because 7 and 6 are close to each other and in this framework that closeness reflects the counted split between 6 coloured and 7 colourless species.

PMNS \(\theta_{12}\): The Solar Angle

This subsection treats the solar angle as a boson-visibility construction seen from the neutrino sector. The neutrino is both colourless and electrically neutral, so the relevant bare geometry is taken from which colourless boson channels it can and cannot see.

At the boson-type level there are 5 colourless boson types. For the neutrino visibility split, however, the relevant count is interaction channels: the neutrino can interact through 3 colourless channels (W⁺, W⁻, Z) and cannot interact through 2 (photon: no electric charge; Higgs: mass coupling is computational, not geometric). The ratio is:

where 2 is the count of invisible bosons and 3 is the count of visible bosons. The hypotenuse:

The appearance of \(\sqrt{13}\) as the hypotenuse of the boson-visibility triangle is presented here as a structural consistency check and secondarily as a numerical echo of the total species count 13.

Applying the subtractive colourless-domain modifier from Section III:

Measured: \(\sin^2\theta_{12} = 0.307^{+0.012}_{-0.011}\), \(\theta_{12} = 33.68^{+0.73}_{-0.70}\,^\circ\) [2].

The neutrino’s 3-visible-to-2-invisible boson-channel split already lands in the correct solar-mixing neighbourhood and the reused colourless-domain modifier then refines it into close agreement with the observed solar angle.

Note. The bare value \(\sin^2\theta_{12} = 4/13\) is close to the tribimaximal prediction of \(1/3\). Tribimaximal mixing was the leading theoretical prediction before \(\theta_{13}\) was measured to be non-zero in 2012 [3,4]. The stencil derivation recovers a similar value from a different structural argument, but crucially does not predict \(\theta_{13} = 0\).

PMNS \(\theta_{13}\): The Reactor Angle

This subsection treats the reactor angle as a cross-scale overlap between the boson-visibility structure of \(\theta_{12}\) and the full species inventory. The bare value is identified with that bridge between the boson scale and the species scale:

where 2 is the non-coupling colourless boson count and 13 is the total species count.

The hypotenuse:

Applying the subtractive colourless-domain modifier from Section III:

Measured: \(\theta_{13} = 8.52^{+0.11}_{-0.11}\,^\circ\) [2].

The cross-scale bridge between the boson-visibility count and the full species inventory already lands in the correct reactor-angle neighbourhood and the reused colourless-domain modifier preserves that scale without collapsing the angle back toward zero.

Note. This angle was measured to be non-zero in 2012 by the Daya Bay experiment [4]. The tribimaximal framework predicted it to be exactly zero, which was thereby falsified [3]. The stencil construction yields a definite non-zero value from counted structure.

V. Summary of Results

Complete Results

The table below summarises the nine candidate geometric derivations developed in this paper from the structural constants \(\{2, 3, 5, 7, 13, 17\}\):

For reference, the broader CAELIX framework uses the following modifier families to correct bare geometry for particle-content overhead:

In each factorisation below, the factor 5 denotes the propagation stencil size, not the full 7-voxel identity motif.

Discussion

Scope.

This paper develops nine candidate geometric derivations of Standard Model parameters. They are built from six countable integers taken from the identity/propagation stencil architecture. It does not derive mass values, which are computational rather than geometric in the CAELIX framework.

What is claimed.

Every formula in this paper is built from the same set of structural constants: \(\{2, 3, 5, 7, 13, 17\}\). Of these, 17 is treated explicitly as the current effective empirical inventory rather than as a lattice-intrinsic count on the same footing as the others. Each constant has an independent physical definition (radix, propagation stencil, species split, type count). The formulas are not selected from a large space of possibilities; they are developed from a single structural principle: a balanced-ternary lattice whose 7-voxel identity stencil and 5-voxel propagation stencil are applied consistently across all parameters.

What is not claimed.

This paper does not claim that the derivations constitute proofs in the mathematical sense. The identification of bare geometric values (which ratios correspond to which parameters) involves physical reasoning about which structural features are relevant to each interaction. The derivations are intended as falsifiable structural proposals, not as axiomatic identities.

The numerology objection.

Individual formulas involving small-integer ratios are vulnerable to the charge of numerology. The defence against this charge is not any single formula but the system: all nine candidate derivations are built from the same six integers, each with an independently motivated physical definition. A numerologist selects different constants for each target. This framework reuses the same counted constants across all targets rather than selecting a new inventory parameter by parameter.

Open questions.

The computational parameters remain only partially addressed: masses are not derived here and no computational CP phase is derived. Whether the stencil framework can constrain mass ratios, explain the observed generation count, or derive the CKM and PMNS CP phases from propagation dynamics is left for future work.

What remains to be worked out is not whether the 7-voxel identity stencil and the 5-voxel propagation stencil are connected, but how the full 3D identity motif projects into propagation dynamics, higher observables and possible additional quantum structure. The extra 3D degrees of freedom may therefore correspond not to a separate stencil ontology, but to motif features not yet resolved in the present propagation-level analysis.

Conclusion

The Standard Model’s geometric parameters are treated not as arbitrary inputs but as quantities constrained by the combinatorial structure of a balanced-ternary lattice with state space \(\{-1, 0, +1\}\), read through a 7-voxel identity stencil and propagated through a 5-voxel local update stencil.

The coupling constants appear as ratios of structural integers corrected by a small reused modifier whose leading factor depends on the interaction channel. The mixing angles are treated as overlaps built from counted species and boson-visibility structure. Within the balanced-ternary sign-symmetric substrate picture, the strong CP angle vanishes.

No numerical value is introduced as a fit parameter. No extra constants are recruited beyond the counted inventory stated at the outset. The claim is therefore one of constrained structural reuse rather than free numerical selection.

Positioning And Prior Art

The positioning claim of this paper is distinct: it does not begin from numerical coincidences. It develops candidate derivations from a single structural principle, namely a balanced-ternary lattice read through a 7-voxel identity stencil and a 5-voxel propagation stencil, applied uniformly across all parameters. The structural constants are not introduced as free choices; they are counted features of the lattice and the Standard Model’s particle content. The framework is falsifiable through its claim that a QCD axion is not required within the balanced-ternary sign-symmetric substrate picture.

Novelty Claim

The novelty claim is the unified derivational framework. It is not that any individual approximation is new, but that the paper develops nine candidate derivations from a balanced-ternary lattice read through a 7-voxel identity stencil and a 5-voxel propagation stencil, using six countable integers and a tightly constrained modifier framework. No free fit parameters are introduced and no additional continuous fit scale is imported into the derivations.

Appendix: Standard Model Particle Inventory

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