Polarizer Experiment

Malus’s Law from Graviton Mesh transverse-wave mechanics, and the single-photon layer above it.

Scope note. This chapter derives Malus’s Law — the cos²θ transmission rule for polarizers — within HGUT at the level of transverse-wave polarization geometry in the Graviton Mesh; Harrison Knot topology is not required at leading order. It then connects the result to the single-photon detection architecture of the Wave–Particle Duality chapter (occupation budget, first-passage resolution), so that the classical exposure law and the one-click event law occupy their correct, separate layers. The Bell-sector implications are bounded explicitly in the Bell scope section and are not closed by anything in this chapter.

The Puzzle, the Claim, and the Two-Layer Architecture

Malus’s Law is not in dispute. Since its empirical establishment in the 19th century, the transmission rule I(θ) = I₀cos²θ has been confirmed to extraordinary precision, and it is correctly derived within both classical wave optics and Quantum Mechanics. The mathematics is settled. What popular accounts of the three-polarizer experiment have long flagged with phrases like “quantum weirdness” is the absence of an agreed physical picture: why does nature obey this particular projection law, and why does a middle polarizer restore transmission through crossed filters?

The minimal instrumental formulation of Quantum Mechanics encodes the answer in the Born rule and stops there — deliberately. That is not a defect of the formalism; it is a choice about what a physical theory owes us, and interpretations of Quantum Mechanics divide on exactly this point. HGUT takes the realist side of that divide and supplies a candidate mechanism. The contribution of this chapter is therefore explicitly ontological, not mathematical: the equation is borrowed, the picture is new.

The chapter’s claim is organised as two layers, and keeping them apart is the discipline that earlier drafts lacked.

Layer 1 is what this chapter derives. Layer 2 is what stops this chapter from being a purely classical-wave replacement for quantum detection — the mistake would be to let the smooth projection story quietly imply that a photon can be half-detected. The two layers are assembled in Step 5.

Step 1: The Photon as a Transverse Excitation of the Graviton Mesh

The Medium and Its Wave Modes

The Graviton Mesh is characterized by an effective mass density ϱ and a shear modulus μ. The speed of transverse disturbances in the medium is

which HGUT identifies with the invariant speed of light: the photon’s propagation speed as a consequence of the elastic properties of the vacuum medium.

For a plane wave propagating in the +z direction, the transverse displacement field takes the form

where k = ω/c, φ is a phase, and A_x, A_y are real amplitudes along the two transverse directions. The displacement is entirely perpendicular to ẑ; a shear wave has no longitudinal component.

The Polarization Vector

The polarization state is encoded in the transverse amplitude vector

For a linearly polarized excitation whose axis makes angle φ with x̂,

Step 2: The Polarizer as an Anisotropic Region of the Medium

What a Polarizer Is Not

Three conventional descriptions obscure the mechanics in ways that matter for the HGUT framework.

At the field-projection layer, it is not an outcome-selecting measurement device. It imposes an anisotropic material constraint and sets the transmitted and absorbed exposure weights. A completed single-quantum absorption event belongs to the detector layer described in Step 5.

It is not an opaque object with a slit. The picture in which the polarizer is a separate entity acting on the field from outside is wrong in HGUT. The polarizer is a configuration of the same medium the field propagates through, with material structure that produces an anisotropic response.

It is not a filter in the information-theoretic sense. A polarizer does not “pass” or “reject” information. It physically reconfigures the field, transferring energy from one polarization component into the internal modes of the medium while the orthogonal component continues to propagate.

The Directional-Constraint Picture

The two axes are set by the material structure — aligned polymer chains in a Polaroid film, birefringent crystal axes, the geometry of a wire grid. In each case the medium in Ω_pol has an internal direction q̂ along which charges or other excitable degrees of freedom move freely, and a perpendicular direction p̂ along which they cannot. In HGUT terms, the filter material is itself a Harrison Knot configuration embedded in the Graviton Mesh: the constraint is imposed by structured medium on propagating medium, not by an external object on a field.

Writing the polarizer transmission axis as

the incoming field decomposes into components along the transmission and absorption axes:

where u_p = u · p̂ and u_q = u · q̂.

Mechanism: Anisotropic Coupling and Energy Transfer

The directional constraint is not a statement that one component “vanishes.” Energy is conserved. The mechanism is energy transfer from the field into the internal degrees of freedom of the polarizer medium. For a Polaroid-type polarizer, the component u_q aligned with the polymer chains drives oscillations of mobile charges along q̂, and field energy is transferred into internal motion:

Γ_q is large because the chains provide a continuum of low-frequency internal modes strongly coupled along q̂; Γ_p is small because the transmission direction lacks the corresponding modes. This is the HGUT reinterpretation of the standard electromagnetic account (fields driving currents along the conducting axis, dissipated ohmically); the two describe the same structure from complementary perspectives.

After passage through Ω_pol: the component u_q has had most of its energy transferred into internal modes, where it appears as heat or low-frequency lattice motion — it is absorbed by the medium; the component u_p propagates through essentially undisturbed.

For a polarizer of finite thickness, let t_p and t_q denote the complex transmission coefficients along the transmission and absorption axes. The outgoing amplitude is then

The strong anisotropy Γ_q ≫ Γ_p produces |t_q| ≪ |t_p|. In the ideal-polarizer limit,

so that

For a real polarizer the transmitted intensity is more generally

Malus’s Law is recovered in the ideal high-extinction limit.

At the many-photon or coherent-field level, the two terms above describe a literal partition of mean energy flux. At the single-photon level they must not be read as simultaneous fractional deposits. They are channel exposure weights: one complete occupation quantum is ultimately absorbed inside the polarizer or transmitted onward and later absorbed by a downstream detector. The mean fractions are continuous; the completed event is not.

Why a Polarizer Looks Dark

A linear polarizer in unpolarized light transmits roughly half the incident intensity: unpolarized light carries, on average, equal energy in the two orthogonal components; the polarizer transmits one and absorbs the other,

Real polarizers transmit less than the ideal value because of residual absorption, reflection, scattering, wavelength dependence, and finite extinction ratio. The visible darkening of a polarizer is the footprint of energy being absorbed by the medium: the suppressed component has not vanished — it has become heat in the polymer.

The Cape and the Bull

A polarizer can be visualized through the matador’s cape: the incoming field is not blocked but redirected — here, redirected into the medium’s internal modes rather than forward propagation. The analogy fails in one instructive respect: the bull and the cape are different objects, but in HGUT the field, the polarizer, and the surrounding vacuum are the same Graviton Mesh in different configurations. The cape is made of the same fabric as the bull. The “redirection” is a continuous reconfiguration of a single extended structure.

Step 3: The Projection Law and Malus’s Law

Amplitude Projection

The transmitted displacement field downstream of the ideal polarizer is

Substituting the expressions for A and p̂ above into the scalar projection A_out = A · p̂:

With θ ≡ φ − α the angle between the incoming polarization and the polarizer transmission axis:

This is vector geometry in the two-dimensional transverse plane of the Graviton Mesh — equivalently, the transmission/absorption decomposition above with u_p = |u| cos θ and u_q = |u| sin θ, the parallel component transmitted and the perpendicular component absorbed by the mechanism of Step 2.

Energy and the cos²θ Law

For a transverse elastic wave in a medium of density ϱ at speed c, the time-averaged energy density is quadratic in the displacement amplitude,

so that I ∝ A² — a universal property of linear wave mechanics in any elastic medium. Combining with the amplitude projection:

This is Malus’s Law from Graviton Mesh mechanics in four moves: the photon is a transverse shear wave with amplitude vector A; the polarizer projects A onto its transmission axis; the projected amplitude is A₀ cos θ; intensity is quadratic in amplitude. In Layer-1 language: |ψ_T,out|² = |ψ_T,in|² cos²θ. No quantum postulate enters at this layer; what the quantum layer adds is the subject of Step 5. [CLOSED — ideal anisotropic HGUT field model].

Step 4: The Three-Polarizer Experiment

Setup and Baseline

Three polarizers in sequence: the first at 0° (vertical, x̂), the final at 90° (horizontal, ŷ), and a middle polarizer inserted at angle θ₂ to the first.

Without the middle polarizer, the vertically polarized wave meets the horizontal filter at 90°:

Mechanically: the vertically oscillating shear wave has zero projection onto the horizontal axis. There is no component to transmit.

The Middle Polarizer Effect

Stage 1 → 2. The amplitude A₁ at 0° projects onto the middle axis at θ₂:

The wave emerging from the middle polarizer is polarized along θ₂: the filter has physically re-oriented the shear oscillation of the mesh. Its history of having oscillated at 0° is mechanically gone — the orthogonal mode was absorbed.

Stage 2 → 3. The wave at θ₂ meets the final polarizer at (90° − θ₂):

Combined:

At θ₂ = 45°: cos² = sin² = ½, so I₃ = I₁/4 = 25%. The middle polarizer restores a quarter of the intensity through otherwise crossed filters — because the 45° filter re-resolves the vertical mesh oscillation into a diagonal one, and the diagonal wave has non-zero projection sin 45° = 1/√2 onto the final axis. At the field layer this is re-orientation at each filter, not collapse.

The 22.5° Case and the 85% Figure

The 85% figure appearing in Bell-angle discussions is single-polarizer transmission at offset 22.5°. From the half-angle identity,

so

Not mysterious: it is the squared projection of the mesh shear amplitude at offset 22.5°. With the middle polarizer of the three-filter setup at θ₂ = 22.5°, the three-polarizer formula above gives I₃ = (I₁/4) sin²45° = I₁/8 ≈ 12.5%.

Step 5: The Single-Photon Layer — One Click, Not Fractional Clicks

Everything so far is Layer 1: a continuous field projecting onto a mechanical constraint. Left alone, that story invites a wrong conclusion — that a single photon at a polarizer should be partially transmitted, producing fractional detections. It never does. A single photon incident on a polarizer followed by detectors on the transmitted and absorbed channels produces exactly one event, on one channel, with probabilities cos²θ and sin²θ. This is where the architecture of the Wave–Particle Duality chapter takes over, and the two chapters lock together cleanly.

The division of labour is exactly the one established there:

• Layer 1 supplies the exposure weighting. The projection law splits the transverse exposure between the transmitted and absorbed channels in the ratio cos²θ : sin²θ. This is the classical, continuous, deterministic part — the likelihood map, not the event. [CLOSED — ideal anisotropic HGUT field model].
• Layer 2 supplies the event. The photon carries one occupation quantum (Integer B) — one unit of excitation, [a, a†] = 1, inherited from the superfluid base state. The exposed channels participate in a first-passage race whose rates are set by the local exposure; the first site to close its readiness gap ignites, claims the quantum, and depletion locks out every other site. One click; the losing channel is transiently perturbed but completes no irreversible transition. [CLOSED — HGUT architectural level].
• The seam between them is the open item. That the physical first-passage rate at a mesh site is proportional to the local exposure, r_i ∝ |ψ_T,i|², is the same rate-law seam identified in the Wave–Particle Duality chapter (its Seam 1); it is supplied, not yet derived from the spatial field equations. [OPEN].
• Spatial drainage. The spatial realization is also inherited rather than derived here. This chapter does not yet show, from the full field equations, how the occupation budget is extinguished in the losing channel quickly enough to prevent a second completed event while preserving the framework’s causal structure. That is the spatial-drainage seam of the Wave–Particle Duality chapter. [OPEN].

One consequence deserves emphasis because it corrects a sentence from earlier drafts. It is not true that “the entire sequence is deterministic.” The field-projection layer is deterministic; the single-trial outcome is not — it depends on the uncontrollable detector–mesh microstate (the readiness gap of the first-passage mechanism). What is deterministic is the mechanism; what is stochastic, over trials, is which site wins; and the trial-averaged frequencies reproduce Malus’s Law. Determinism at the field layer, Born-weighted statistics at the event layer: both statements are needed, and neither may borrow the other’s scope.

Mechanical Interpretation and the Ontological Point

In standard presentations, the insertion of a middle polarizer is described as “collapsing” the photon’s state, and the apparent strangeness enters with that word. In HGUT the field-layer sequence is: the first polarizer selects the 0° shear mode, and the mesh downstream physically oscillates vertically — a definite mechanical state of the medium; the middle filter’s structured lattice admits only the θ₂ mode, mechanically resolving the incoming oscillation, passing the parallel component and absorbing the orthogonal one; the re-oriented wave then presents projection sin θ₂ to the final filter. At the field layer this is local, mechanical re-orientation at each filter — and at the event layer, a single quantum is claimed once by the first-passage resolution of Step 5. The “paradox” was generated by the absence of a physical picture for the polarizer interaction, together with the conflation of the two layers.

The deeper point is ontological. In standard treatments the polarizer is one kind of thing (an object), the photon another (a particle or field quantum), the observer a third — and the measurement problem lives in the joints between them. In HGUT the trichotomy dissolves:

The same constraint logic unifies the standard puzzle apparatus, at the level of conceptual organisation (these are framings, not derivations): a polarizer is a directional constraint on the orientation of the field; a double slit is a positional constraint on the location of the field, with the downstream interference pattern the continuous propagation of one constrained field rather than the self-interference of a particle; a Bell apparatus is a pair of directional constraints applied to a single extended, correlated field configuration — with the essential caveat of the next section.

Scope Boundary: The Polarizer Problem vs. the Bell Problem

The derivation above closes the polarizer problem at the field layer, and Step 5 connects it to single-photon detection. It is important to state with equal clarity what it does not accomplish.

The 22.5° angle appears in Bell discussions because polarization-entangled photon pairs measured at relative analyzer angles produce joint statistics that violate Bell inequalities. Bell’s Theorem concerns joint measurement statistics across spatially separated detectors, not single-photon transmission fractions — the fact that cos²(22.5°) ≈ 85.4% follows from Malus’s Law is not what is at stake. What is at stake is the two-photon correlation function. For polarization-entangled pairs the quantum prediction carries the doubled angular dependence,

with the sign fixed by the entangled state and the outcome convention (for example, +cos 2(α − β) for the state (|HH⟩ + |VV⟩)/√2 with ±1 assigned to transmission/absorption); the frequently quoted −cos(α − β) is the spin-½ singlet form, not the photon-polarization form. Bell’s question is whether any locally realistic model reproduces this correlation; experiment answers that the inequality is violated.

The mechanical picture of this chapter does not resolve that. Two boundaries must be drawn precisely.

First, a shared medium is not automatically a Bell escape. A continuous elastic sea supplying each photon’s polarization as a pre-set local variable — or, more generally, any model whose effects on the two separated detectors factorize into local response functions — satisfies Bell’s assumptions and is bound by the inequality. The Layer-1 mechanics of this chapter is exactly such a local structure and therefore cannot be the whole Bell story. Anything in earlier drafts suggesting the constraint picture alone feeds a derived Bell correlation was an overstatement and is withdrawn here.

Second, the HGUT route to the Bell sector is a named, separate program: the Route 5 detector-as-knot / linked-corridor construction, in which the outcome is a collective reconfiguration of the source–detector–mesh system — one extended field configuration in a topologically linked sector, with two regions of constrained medium imposing geometric requirements on parts of the same structure — rather than a local field readout. That program is documented in the Bell-sector chapters, its status is adjudicated there, and nothing in the present chapter pre-pays it. At this chapter’s level the Bell problem is [OPEN]; the linked-corridor construction is the [CANDIDATE] mechanism class. This is the same fork identified in the Wave–Particle Duality chapter (its Seam 3): a single occupation quantum distributed across spacelike-separated detection regions already raises the drainage and exclusivity problem. The entangled-pair Bell sector treated in this program is stronger: it contains two occupation quanta in one joint entangled state and must reproduce correlations that no factorized local hidden-variable model can supply.

Status and Conclusion

Within HGUT, the three-polarizer experiment has a complete mechanical account at the field layer and an architecturally closed account at the single-photon event layer.

A photon is a transverse shear excitation of the Graviton Mesh carrying a polarization vector that physically describes the medium’s oscillation direction. A polarizer is an anisotropic region of the same medium: one transverse component couples strongly into internal modes and is absorbed as heat; the orthogonal component propagates. The transmitted amplitude is A₀ cos θ by vector projection; the transmitted intensity is I₀ cos²θ because wave energy is quadratic in amplitude. For the three-filter arrangement, I₃ = (I₁/4) sin²(2θ₂): 25% at 45°, 12.5% at 22.5°, with single-polarizer transmission cos²(22.5°) ≈ 85.4%. The middle filter restores transmission by physically re-orienting the shear oscillation — re-orientation, not collapse, at the field layer. At the event layer, a single photon yields one click because one occupation quantum is claimed once through the first-passage resolution, with the cos²θ/sin²θ exposure fractions setting the rates.