Lori Gardi and a medium for light: a primer

Source gap (repeated from Step 1 and the arbiter). The folder holds 16 PDFs (2017-2023) by Lori(-Anne) Gardi, some written alone, most with Jeff Yee, one with Ionel Dinu. All of them have a text layer. In the Yee-Gardi papers, though, the numbered equations are images, so they are missing from the extracted text. After the quality check (QE), deep readers read those equation pages as images for six papers: "The Geometry of Particles" (all equations and figures), "The Geometry of Spacetime" v2.3 (PDF pp.5-33), "Particles of the Universe Meets Electric Universe" (pp.5-13), "Planck's Constant" V2 (pp.6-10), the CNPS 2018 paper (pp.4-9) and "PaperColour" (pp.5-12) (files in deep/). For this primer I also re-read the 2017 paper "A Medium for the Propagation of Light Revisited" in full [read], and the page of the Spacetime paper that gives the fine-structure formula [read: page image, PDF p.9, printed p.8]. Page numbers below are PDF pages; the printed page number is one less. The companion papers the authors cite are not in the folder, and nobody in this review read them: "The Relationship of the Mole and Charge", "The Relationship of the Fine Structure Constant and Pi", Distinti's and Yee's wave-constant papers, Gardi's "Calibrating the universe" (Physics Essays 29, 2016), and earlier versions of Yee's atomic-orbitals paper (only v6 was looked at, on the web, by a deep reader).

What changed after QE. The first version of this primer said the particle paper computes no masses. That was wrong: it gives a lepton energy formula and a table compared with measured values. Claim 6 is now scored on that formula (section 5, step 10, and the evidence table). The primer also now covers the papers' removal of Maxwell's displacement current (sections 2-4).

Tags: [read] from the documents; [web] with URL; [memory] from training, with confidence; [calc] computed in example.py (same folder); [arbiter] taken from arbiter.md.

---

1. The idea in one paragraph

Sound needs air, and water waves need water. Lori Gardi (her ResearchGate cover page lists Western University and a prostate-biopsy image-registration project [read, p.1 of the 2017 paper]) asks why light should be different. Her answer is that it is not. Empty space, she argues, is filled with a medium (an "aether"), and light is a wave in it, just as sound is a wave in a crystal. She first proposes that the medium is a lattice of bound electron-positron pairs. In later papers with Jeff Yee it becomes a lattice of tiny "granules" the size of the Planck length. From this picture the papers try to recover the speed of light, the strength of the electric force, the electron's mass, gravity and a dozen constants of nature. They also reread Planck's constant as "the energy of one cycle of light" and reinterpret the first gravitational-wave detection (2015) as an aether "earthquake". Some of the starting physics is perfectly real: wave speeds in solids really do come from stiffness and inertia, and electron-positron pairs really do carry 1.022 MeV of rest energy. The question this primer works through is whether the steps from those starting points to the conclusions hold. The main lesson is a skill you can use anywhere: telling a derivation (which can fail) from an identity (which cannot fail, and so cannot prove anything).

---

2. What you need to know first

Waves in a medium. A mechanical wave travels at a speed set by two properties of the medium. One is how hard it pushes back when deformed (stiffness). The other is how much mass must be moved (inertia). For a string, v = √(T/μ), where T is the tension and μ the mass per unit length. For sound in a fluid or solid, v = √(K/ρ), where K is the bulk modulus (the pressure needed for a given fractional squeeze, in pascals, Pa = N/m²) and ρ (rho) is the mass density in kg/m³. A solid also resists shear, so it carries two kinds of wave. Longitudinal or P (pressure) waves are faster. Transverse or S (shear) waves are slower. Read: Halliday, Resnick & Walker, Fundamentals of Physics, the chapters on waves and sound; for solids, Kittel, Introduction to Solid State Physics, the chapter on crystal binding and elastic constants.

How crystals hold together (ionic binding). In a salt crystal such as NaCl, every Na⁺ ion is pulled toward its Cl⁻ neighbours by the Coulomb force. When electron clouds start to overlap they push back with a steep repulsion. The energy needed to pull one ion pair out of the crystal into separate gas-phase ions is the lattice energy. For NaCl it is 787 kJ/mol, or 8.16 eV per pair [web https://en.wikipedia.org/wiki/Born%E2%80%93Land%C3%A9_equation]. The Born-Landé model writes the energy per pair as U(r) = −A/r + B/rⁿ. Here r is the distance between neighbouring ions. The first term is the Coulomb attraction summed over the whole lattice (A includes the "Madelung constant"). The second term is the repulsion, and the "Born exponent" n is typically 5 to 12. Read: Kittel, crystal binding chapter; any physical chemistry text (Atkins) on lattice enthalpy.

Electromagnetic waves and c. Maxwell's equations in empty space give waves that travel at c = 1/√(ε₀μ₀). Here ε₀ (the electric constant, 8.854 × 10⁻¹² F/m) sets how strong the electric field of a charge is, and μ₀ (the magnetic constant, 1.2566 × 10⁻⁶ N/A²) does the same for the magnetic field of a current. Nothing in that derivation needs a material: ε₀ and μ₀ are constants of the field equations, not the density and stiffness of a substance. Read: Griffiths, Introduction to Electrodynamics, Chapter 9 (electromagnetic waves).

Rest energy and pair annihilation. Einstein's E = mc² says a mass m at rest carries energy mc². An electron (mass m_e = 9.109 × 10⁻³¹ kg) has rest energy 0.511 MeV, where 1 MeV = 1.602 × 10⁻¹³ J. When an electron meets its antiparticle, the positron, the two can annihilate into two gamma-ray photons that together carry 2m_ec² = 1.022 MeV. Run backwards, a 1.022 MeV photon can make a pair near a nucleus. Read: any modern-physics text (Krane, Modern Physics; Taylor, Zafiratos & Dubson, Modern Physics for Scientists and Engineers), the chapters on relativity and on particles.

Units and identities. An identity is an equation that holds for every possible value of its symbols because of how the symbols are defined. Example: if you define the speed v = d/t, then "v·t = d" cannot be tested. It is true by construction. Physics constants are full of such links. One is the Planck units: l_P = √(ħG/c³) (length), m_P = √(ħc/G) (mass) and q_P = √(4πε₀ħc) (charge). They are built from G (Newton's gravitational constant), ħ = h/2π (the reduced Planck constant) and c. So any formula that "derives" G from l_P and m_P is reading the definition backwards. Read: the Wikipedia article "Planck units" is reliable for definitions; Taylor's An Introduction to Error Analysis for uncertainties.

The fine-structure constant α (alpha). α = e²/(4πε₀ħc) ≈ 1/137.036. Here e is the electron's charge. α is a pure number, the same in any system of units, and it measures the strength of electromagnetism. It is measured to a relative uncertainty of 1.6 × 10⁻¹⁰ [web https://physics.nist.gov/cgi-bin/cuu/Value?alph, via the contrarian; arbiter accepts]. No accepted theory predicts its value, so closed-form guesses for 137.036 have a long history [memory, medium].

Michelson-Morley and relativity. If light moved through a medium at rest in some frame, then on a moving Earth the round-trip time of light along two perpendicular arms would change as the apparatus was rotated. Michelson and Morley (1887) and many more precise successors found no such change [memory, high]. Einstein's special relativity (1905) takes this as a principle: no experiment can find a preferred rest frame. Lorentz's earlier "ether theory" keeps a medium but arranges its effects so they can never be detected. It predicts the same results as relativity, so experiment cannot tell the two apart [memory, high]. Read: Taylor & Wheeler, Spacetime Physics; Griffiths, Chapter 12.

Maxwell's displacement current. Ampère's law says a steady current I makes a magnetic field that circles it. In differential form it is ∇×H = J, where H is the magnetic field strength (A/m), J is the current density (A/m²), and ∇× is the curl (how much a field circulates around a point). Maxwell (1860s) saw that this cannot be complete. The divergence of any curl is zero, so ∇·(∇×H) = 0 would force ∇·J = 0 everywhere. But charge piling up on a capacitor plate has ∇·J ≠ 0. The fix is to add the displacement current ∂D/∂t, where D = ε₀E + P is the electric displacement field (C/m²), E the electric field and P the material's polarisation: ∇×H = J + ∂D/∂t. With Gauss's law ∇·D = ρ (ρ = charge density), this makes the continuity equation ∂ρ/∂t + ∇·J = 0 (charge is conserved) come out automatically. In empty space, where J = 0, the ∂D/∂t term is the only thing that lets a changing electric field make a magnetic field, and so it is what makes light possible [memory, high]. Read: Griffiths, Chapter 7 (section on Maxwell's equations) and Chapter 9.

Leptons, quarks and neutrino masses. The leptons are the electron (0.511 MeV), the muon (105.658 MeV [web https://en.wikipedia.org/wiki/Muon]) and the tau (1776.86 MeV [web https://en.wikipedia.org/wiki/Tau_(particle)]), each with a neutrino. (Particle physicists quote masses as rest energies, mc², in eV; 1 eV = 1.602 × 10⁻¹⁹ J.) Leptons show no internal structure in any experiment so far [memory, high]. The proton and neutron (938.3 and 939.6 MeV) are made of quarks, and most of their mass is field energy, not the quarks' own mass [memory, high]. Neutrinos have tiny masses that have not been measured, only bounded: the KATRIN experiment, which weighs the electron neutrino through the endpoint of tritium beta decay, finds m_ν < 0.45 eV at 90 % confidence [web https://arxiv.org/abs/2406.13516]. Older experiments gave only upper limits of about 2 eV (electron neutrino), 0.17-0.19 MeV (muon neutrino) and 15-18 MeV (tau neutrino) [memory, high]. An upper limit is not a measured value. Read: Griffiths, Introduction to Elementary Particles, Chapter 1 (historical introduction); Krane, Modern Physics, the chapter on elementary particles.

---

3. The textbook view

(a) Wave speed in an elastic medium.

v = √(K/ρ) (bulk or longitudinal waves; for a cubic crystal along a cube edge, v_L = √(C₁₁/ρ) and v_T = √(C₄₄/ρ))

• K = bulk modulus (Pa); ρ = density (kg/m³)
• C₁₁, C₄₄ = elastic constants of a cubic crystal (Pa). C₁₁ resists compression along an axis and C₄₄ resists shear.

Measured for NaCl [web https://www.crystran.com/optical-materials/sodium-chloride-nacl]: ρ = 2170 kg/m³, K = 24.42 GPa, C₁₁ = 48.5 GPa, C₄₄ = 12.61 GPa. These give speeds of 3355 m/s (bulk), 4728 m/s (longitudinal) and 2411 m/s (transverse) [calc].

(b) Lattice energy and stiffness are linked. Take the Born-Landé energy U(r) = −A/r + B/rⁿ and the rock-salt volume per ion pair, V = 2r³. The bulk modulus is K = V d²U/dV² at the equilibrium spacing r₀. Working it out (done symbolically in example.py) gives

K = (n−1)A / (18 r₀⁴), |U₀| = (A/r₀)(1 − 1/n)

and with ρ = m/(2r₀³), where m is the mass of one ion pair,

K/ρ = (n/9) · |U₀|/m [calc]

So "lattice energy per unit mass" is almost the same thing as "stiffness per unit density" for an ionic crystal, with the correction factor n/9. This is the honest physics behind Gardi's NaCl calculation.

(c) The speed of light. c = 1/√(ε₀μ₀) = 299 792 458 m/s (exact by definition of the metre since 1983).

(d) Rest energy of a pair. E_pair = 2mc². For electrons, 2m_ec² = 1.6374 × 10⁻¹³ J = 1.022 MeV. This is the energy released when the pair ceases to exist. It is not the energy holding a pair inside some structure.

(e) Photon energy. E = hf, where f is the frequency (cycles per second, Hz) and h = 6.626 × 10⁻³⁴ J·s. This is the energy of one photon, whatever the beam's power. A beam of power P carries P/(hf) photons per second.

(f) Constants that are definitions. The classical electron radius r_e = e²/(4πε₀m_ec²) = 2.818 × 10⁻¹⁵ m is defined so that the electric energy at distance r_e equals the electron's rest energy. Hence m_ec² = e²/(4πε₀r_e) holds by definition. Likewise l_Pc²/m_P = G exactly.

(g) Ampère-Maxwell law and charge conservation. ∇×H = J + ∂D/∂t. Taking the divergence of both sides and using ∇·D = ρ gives 0 = ∇·J + ∂ρ/∂t: charge is conserved. In vacuum (J = 0, ρ = 0, D = ε₀E, B = μ₀H) the curl equations combine into the wave equation ∇²E = μ₀ε₀ ∂²E/∂t², whose speed is c = 1/√(ε₀μ₀). Drop ∂D/∂t and both results disappear.

(h) Lepton masses are measured, not predicted. The Standard Model of particle physics takes the electron, muon and tau masses as inputs; no accepted theory predicts their ratios (m_μ/m_e = 206.77, m_τ/m_e = 3477) [memory, high]. So a formula that predicted them would be a real discovery, and a formula that only fits them must be judged by how many free choices it uses.

---

4. Where Gardi departs

---

5. Worked example: Gardi's sound-speed formula, from salt to the vacuum

We follow her own two-step argument (2017 paper, eqs. 4 and 5) and ask at each step whether physics or algebra is doing the work. Every number below is reproduced by example.py.

Step 1. Units. A is an energy in joules (kg·m²/s²) and B + C is a mass in kg, so A/(B+C) has units m²/s², and its square root is a speed. Note that B + C is a mass, though the paper calls it "mass per unit volume". No volume appears anywhere [read].

Step 2. Her NaCl number.

v = √(1.3052 × 10⁻¹⁸ J / (3.818 × 10⁻²⁶ kg + 5.720 × 10⁻²⁶ kg)) = √(1.3684 × 10⁷ m²/s²) = 3699 m/s

That matches her "3700 m/s" [calc]. With standard atomic masses (Na 22.990 u = 3.8175 × 10⁻²⁶ kg, Cl 35.453 u = 5.8871 × 10⁻²⁶ kg) the mass sum is 9.6046 × 10⁻²⁶ kg and v = 3667 m/s. Her chlorine mass is about 3 % low, which changes v by about 1 % [calc].

Step 3. What is A? 787 kJ/mol ÷ 6.022 × 10²³ mol⁻¹ = 1.307 × 10⁻¹⁸ J = 8.16 eV. That is her A to 0.1 %. So A is the NaCl lattice energy [calc, web value above]. (The paper ties it to an 8 eV ultraviolet effect reported by Simhony [read]. Whatever its story, the number is the lattice energy.)

Step 4. Is 3700 m/s "the speed of sound in NaCl"? Measured values (section 3a) are 3355 m/s bulk, 4728 m/s longitudinal and 2411 m/s transverse. NaCl has no single sound speed. 3700 m/s falls inside the measured range, about 10 % above the bulk value. So the paper's "which is in fact the speed of sound within the NaCl medium" [read] overstates a good order-of-magnitude estimate.

Step 5. Why does it come out so close? This is the part to be fair about. Section 3(b) showed that for an ionic crystal K/ρ = (n/9)·|U₀|/m exactly, within the Born-Landé model. Na⁺ has a neon-like electron shell (n = 7) and Cl⁻ an argon-like one (n = 9) [web, Born-Landé page], so n ≈ 8 and

√(K/ρ) ≈ √(8/9) × 3667 m/s = 3458 m/s

against a measured 3355 m/s [calc]. Running it backwards, the measured K implies n = 7.5. Gardi's shortcut works for salt because, for this kind of potential, energy per unit mass and stiffness per unit density differ only by the factor n/9, which happens to be near 1 [calc]. Her intuition that "binding energy sets stiffness" has a real basis here.

Step 6. Now the vacuum. She keeps the formula and puts in A = 1.637421 × 10⁻¹³ J and B + C = 1.821876582 × 10⁻³⁰ kg [read]:

v = √(1.637421 × 10⁻¹³ / 1.821876582 × 10⁻³⁰) = 2.997925 × 10⁸ m/s

But 2m_ec² = 2 × 9.1094 × 10⁻³¹ × (2.9979 × 10⁸)² = 1.637421 × 10⁻¹³ J. Her A is 2m_ec², so

v = √(2m_ec² / 2m_e) = √(c²) = c

whatever m_e is. Put in a muon pair (m = 1.884 × 10⁻²⁸ kg) or a proton pair (1.673 × 10⁻²⁷ kg) and you get exactly 299 792 458 m/s again [calc]. (Her rounded inputs give 299 792 457 m/s, not her printed "299792458.0"; the difference is rounding [calc].)

Step 7. Why step 6 is different from step 5. In salt, the result came out near the measured speed because of physics that was not fed in: the shape of the potential (the factor n/9). For that reason the salt formula could have failed. Had NaCl's Born exponent been 30, the estimate would have been off by a factor of nearly 2. In the vacuum step, c was fed in through A = 2m_ec², and the answer could not have been anything else. An argument that cannot fail cannot show that a medium exists. It also gives no density, no spacing and no stiffness: none of the things a medium has.

Step 8. What a real medium would have to supply. Suppose, as an assumption (the paper names no spacing), that the pairs sit one per cube whose side is the electron's Compton wavelength, λ_C = h/(m_ec) = 2.426 × 10⁻¹² m. Then

ρ = 2m_e/λ_C³ = 1.82 × 10⁻³⁰ kg / 1.428 × 10⁻³⁵ m³ = 1.3 × 10⁵ kg/m³ (over 100 times the density of water)

K = ρc² = 1.3 × 10⁵ × 9.0 × 10¹⁶ = 1.1 × 10²² Pa (about 10¹¹ times the bulk modulus of steel, about 1.6 × 10¹¹ Pa [memory, high])

[calc]. A medium like that would have to explain why planets move through it with no measurable drag and why light passing through it shows no frequency-dependent speed. Fermi's GRB 090510 measurement limits such dispersion to beyond the Planck scale [web https://arxiv.org/abs/0908.1832]. The paper does not take up these questions. This is where a real test of the e⁺e⁻ medium would start.

Step 9. Contrast: the one claim in this family that can fail. The Yee-Gardi Spacetime paper sets α = xy/(4πl² + πdl + πd²) with x = y = d = r_e and l = πr_e [read: page image, printed p.8]. The r_e² cancels:

α = 1/(4π³ + π² + π) = 1/137.036304

Measured: 1/α = 137.035999177. The relative difference is 2.22 × 10⁻⁶. The measurement's relative uncertainty is 1.6 × 10⁻¹⁰, so the formula misses by about 13,900 standard uncertainties [calc]. To 6 significant figures it looks perfect (0.00729734 vs 0.00729735), and that is how the paper reports it. To the 10 figures the experiment supplies, it is wrong. Unlike step 6, this formula could have been right. That is to its credit, and it is also why we can say it is wrong. Note too how many free choices it contains: which areas to add, l = πr_e chosen "by hand", equal lengths, three terms [arbiter, from the mathematician]. With that many choices, landing within 2 parts per million of a target is not surprising. The miss also spreads: the paper feeds this α into m_e, the Bohr radius, the Rydberg constant and others, which shifts them by 1 to 7 parts per million, hidden by printing only 4-5 digits [arbiter, QE revision].

Step 10. A second kind of check: fitting integers after the fact. "The Geometry of Particles" (Yee and Gardi, 2019) gives particle energies as

E(K) = E_ν · K⁵ · Σₙ₌₁..K [n³ − (n−1)³] / n⁴ [read: Eq. 4.10, page image p.14]

Here E_ν = 2.38925 eV is the energy of a "fundamental" neutrino, K is a whole number (the count of "wave centres" in the particle), and n counts shells; n³ − (n−1)³ is the number of new points when a cube of side n−1 grows to side n. example.py part G reproduces the paper's Table 5.1 [calc; masses from the web pages above]:

Three things to notice. 1. The electron is an input. The paper says the "refined value of 2.38925 eV is used ... resulting in the exact calculation of the electron's rest energy" [read: p.17]. So the electron row tests nothing. (The deep reader also found that the earlier "derivation" of E_ν divides m_ec² by a product of chosen factors and a constant fitted to r_e [deep read, Q1].) 2. The grid is dense. E grows roughly like K⁵, so neighbouring K values differ by a factor of about 1 + 5/K: 1.19 between K = 28 and 29, 1.046 between 110 and 111. Any mass at all lies within about √1.19 − 1 ≈ 9 % of some grid point near K = 28, and within about 2 % near K = 110 [calc]. Agreement of 1-10 % is therefore what you get by picking the nearest K for a random number. The paper picks K "by best fit" after the mass is known [read: p.14] and predicts no new particle. 3. The choice was not even the best fit. K = 29 is closer to the muon than K = 28. The paper keeps 28 because 8, 20, 28 and 50 are nuclear "magic numbers" [read: p.17]. So the magic-number pattern was partly put in by hand. (Its Fig. 4.2 also plots the muon at K = 28 but computes its point with 29⁴, which hides the 10 % miss [deep read, Q3].)

The general rule: count the free choices before counting the matches. One tuned constant plus one free integer per particle can "match" any list of masses to within the grid spacing. The one output that was not fitted, the 2.39 eV neutrino, is ruled out by direct measurement.

---

6. Evidence table

(Verdicts from arbiter.md; "TESTED / NEVER TESTED / DISMISSED" follows the contrarian's distinction.)

No source shows any of these claims being dismissed without testing or blocked from publication. The work has mostly been ignored rather than rebutted [arbiter; absence of evidence, not proof].

---

7. What a learner should take away

What is solid. - Wave speed really is √(stiffness/inertia), and her NaCl estimate is genuinely good physics in disguise. For an ionic crystal, lattice energy per unit mass ≈ stiffness per unit density, with the factor n/9 (section 5, step 5). It is a nice example of how a dimensional estimate can work for a real reason. - Units are conventions. Charge need not be a base dimension (Gaussian units), and whether a "cycle" or radian hides inside E = hf is a real point in metrology. Neither choice changes a measurement. - Steinmetz's duality between inductance and capacitance, and her matrix treatment of complex numbers, are correct. - She and Yee state falsifiable numbers (α, a_e, a neutrino energy, a minimum wavelength) and a falsification condition for the LIGO idea. That is more than most aether papers do, and it is why parts of the work can be checked at all.

What is wrong, and why the mistake is instructive. - Identity mistaken for derivation. √(2mc²/2m) = c, l_Pc²/m_P = G and m_e = μ₀e²/(4πr_e) cannot fail, because the answer goes in with the inputs. The test to use on any "derivation": could this have come out differently if the theory were false? If not, it is not evidence. The salt step passes that test; the vacuum step does not. - Six digits is not agreement. α = 1/(4π³ + π² + π) looks perfect at the precision the paper prints and fails at the precision of the measurement. Always compare with the experimental uncertainty, not with the number of digits you happen to show. - The Michelson-Morley argument misreads the experiment. The experiment does not "detect the medium". It checks whether light's speed depends on direction, which any medium with its own rest frame would cause. - Count the free choices before counting the matches. The particle formula hits the tau to 1.2 % and the muon to 10 %, but with E_ν tuned to the electron and K picked per particle, the spacing of the K grid alone guarantees hits of that size (section 5, step 10). Choosing K = 28 over the closer K = 29 to match a "magic number" shows how a pattern can be put in by hand. - Check a proposed structure against simple counting. A proton of four electrons and one positron has charge −3e, not +e. The same kind of counting (charge, spin, mass) is what ended the "electrons in the nucleus" model around 1930 [memory, medium-high]. - Maxwell's displacement current is not optional. The Planck papers' claim that ∂D/∂t "is" J fails two textbook checks: without it there are no electromagnetic waves in vacuum, and charge could not pile up on a capacitor (section 3g). Cancelling the ∂x in ∂Dₓ/∂x as if it were a fraction is the algebra slip behind it. - A medium has properties. A density, a spacing, a stiffness and a dispersion. A model that names none of them cannot be confronted with the strong limits that exist on each.

What is genuinely open. - Whether some undetectable medium (a Lorentz-type ether) underlies relativity cannot be decided by experiment as such theories are built. That is a question for philosophy of physics, not evidence for this model. - Why α ≈ 1/137 is unexplained by any accepted theory. Closed-form guesses are allowed; they must match to 10 digits. - The deep reads of the Particles and Spacetime papers found no prediction that was not an input, apart from the 2.7 × 10⁻¹⁷ m wavelength and the 2.39 eV neutrino, both of which fail [arbiter, QE revision]. Whether the companion papers not in the folder contain one is unknown. - Why the masses of the electron, muon and tau have the values they do is an open problem in mainstream physics too (section 3h). A formula that predicted an unmeasured mass in advance would deserve attention; none here does.

---

8. Check yourself

1. Gardi's formula gives c for an electron-positron pair. Without a calculator, what does it give for a pair of particles each with mass 1 kg, if A is their combined rest energy? What does that tell you?
2. Using the Born-Landé result K/ρ = (n/9)|U₀|/m, what Born exponent n would make Gardi's NaCl shortcut √(U₀/m) exactly equal to √(K/ρ)? Is NaCl's n above or below that?
3. A formula E(K) gives energies that grow roughly as K⁵. Near K = 50, by roughly what factor do E(50) and E(51) differ? So within about what percentage of some grid point does any mass near the tau's lie? What does that say about the paper's 1.2 % match to the tau?
4. A 1 mW red laser (f = 4.6 × 10¹⁴ Hz) runs for 1 s. What is its energy per cycle, and is it equal to h? Explain in one sentence why "h is the energy per cycle" cannot be a general rule.
5. GW150914 was 1.3 × 10⁹ light years away and the Fermi transient came 0.4 s after it. If both signals travelled through the same medium, what fractional speed difference does that imply? Compare it with the P/S speed ratio of rock (about 1.7).

---

9. Further reading

Mainstream - D. J. Griffiths, Introduction to Electrodynamics (4th ed.): Chapter 9 (electromagnetic waves) and Chapter 12 (electrodynamics and relativity). - C. Kittel, Introduction to Solid State Physics: the chapter on crystal binding and elastic constants (Madelung energy, bulk modulus, sound waves in cubic crystals). - E. F. Taylor and J. A. Wheeler, Spacetime Physics: the experimental basis of special relativity, including Michelson-Morley. - J. R. Taylor, An Introduction to Error Analysis: comparing a prediction with a measurement and its uncertainty. - NIST CODATA values: α https://physics.nist.gov/cgi-bin/cuu/Value?alph and the electron anomaly https://physics.nist.gov/cgi-bin/cuu/Value?ae. - Connaughton et al. 2016, Fermi GBM observations of GW150914: https://arxiv.org/abs/1602.03920; Savchenko et al. 2016, INTEGRAL: https://arxiv.org/abs/1602.04180; LIGO/Virgo/Fermi/INTEGRAL, GW170817 and GRB 170817A: https://arxiv.org/abs/1710.05834. - Event Horizon Telescope Collaboration 2019, M87 paper I: https://arxiv.org/abs/1906.11238. - KATRIN Collaboration, direct neutrino-mass measurement from 259 days of data (m_ν < 0.45 eV): https://arxiv.org/abs/2406.13516. - D. J. Griffiths, Introduction to Elementary Particles: Chapter 1 (historical introduction to the particles). - H. Kragh, "Magic number: a partial history of the fine-structure constant", Archive for History of Exact Sciences* (2003): the history of attempts to compute 137 [memory, medium on the bibliographic details].

Gardi's own key works (all in the project folder) - "A Medium for the Propagation of Light Revisited", Proceedings of the CNPS 10 (2017): the e⁺e⁻ medium, NaCl analogy, MM and LIGO argument. - J. Yee and L. Gardi, "The Geometry of Spacetime and the Unification of Forces" v2.3: granule lattice, α as a geometric ratio, "derived" constants. - J. Yee, L. Gardi et al., CNPS 2018 proceedings paper, and "The Relationship of Mass and Charge" v2: charge as wave amplitude. - "Planck's Constant and the Nature of Light" V2/V3 and "Modified Unit Analysis: A Real Life Example": h per cycle. - "theOMparticle: A Unified Model of the Universe" (2023): Planck units from h, α/2π as the anomalous moment, Z₀/4π. - J. Yee and L. Gardi, "The Geometry of Particles and the Explanation of Their Creation and Decay" v2 (2019): the K⁵ energy formula (Eq. 4.10, Table 5.1), the composite proton and the solar-neutrino decay trigger. - J. Yee and L. Gardi, "Particles of the Universe Meets Electric Universe" v2: the 42,127 N strong force and the "orbital force". - I. Dinu and L. Gardi, "Fundaments of a Theory of Aether" (the MyART file): electrigen. - "Light as an LC Circuit" and "Steinmetz Analogy: Dielectric and Magnetic"; "Demystifying Complex Numbers"; "PaperColour" (Mandelbrot set and black holes).

Cited by the papers, not in the folder - J. Yee et al., "Atomic Orbitals: Explained and Derived by Energy Wave Equations": the "hydrogen to calcium" fit (Yee is the author, not Gardi) [web https://vixra.org/abs/1708.0146].

---

Answers

1. Exactly c again: √(2 × 1 kg × c² / 2 kg) = c. Any mass gives c, so the formula cannot tell an electron-positron medium from a medium of anything else, or from no medium. It is E = mc² rearranged.
2. n = 9 makes n/9 = 1. NaCl's n is about 8 (7.5 from the measured bulk modulus), slightly below, so the shortcut overestimates the bulk sound speed by about √(9/8) ≈ 6 %, and by about 10 % against the measured K (3667 vs 3355 m/s).
3. (51/50)⁵ ≈ 1.104; the slowly growing sum adds very little, and the exact ratio from Eq. 4.10 is 1.105 [calc]. The worst case is a mass halfway (on a log scale) between grid points, about √1.105 − 1 ≈ 5 % from either one, so a match within 5 % is guaranteed for any mass near there. A 1.2 % match is somewhat better than the guaranteed worst case, but one such match, with K chosen after the fact and E_ν fitted to the electron, is weak evidence.
4. Energy per cycle = P/f = 10⁻³ J/s ÷ 4.6 × 10¹⁴ s⁻¹ = 2.2 × 10⁻¹⁸ J, which is about 3 × 10¹⁵ times h (6.6 × 10⁻³⁴). Energy per cycle depends on the beam's power, so it equals h only for the special beam that delivers one photon (hf) per second.
5. Travel time = 1.3 × 10⁹ yr × 3.156 × 10⁷ s/yr ≈ 4.1 × 10¹⁶ s; fractional difference ≈ 0.4/4.1 × 10¹⁶ ≈ 1 × 10⁻¹⁷. A seismic P/S ratio of 1.7 is a 70 % difference, so the "earthquake" analogy does not hold quantitatively; at best it would describe a medium whose two wave types differ in speed by one part in 10¹⁷.