A complicated-looking orbit becomes two circles.
IDWT starts with A = cos(Lθ)eⁱθ and B = sin(Lθ)eⁱθ, pairing the four coordinates.
Combine them as (A+iB)/√2 and (A−iB)/√2. The sine and cosine merge into pure rotations at rates L+1 and −(L−1).
Result: two circles of radius 1/√2, with squared radii adding to 1.
https://idwt-v2.github.io/papers/petals-tubes-and-rings-20261111.pdf
Tilt a pattern. Move its visible crossing.
IDWT works it out for a free line centred a distance h along a hidden spatial direction:
1. Along the line, x₄ = h + τ cos φ and x₁ = τ sin φ.
2. Our slice has x₄ = 0, so τ = −h/cos φ.
3. Substitute: x₁ = −h tan φ.
The crossing position follows directly from the tilt.
https://idwt-v2.github.io/papers/free-orientation-20261111.pdf
Why does one orbit draw 3 petals, another 8?
IDWT derives it from the projected path r = cos(Lθ).
• Tips occur at Lθ = πj: 2L tip visits per turn.
• For odd L, the second half retraces the first.
• For even L, it adds distinct petals.
Result: L petals for odd L, 2L for even L.
https://idwt-v2.github.io/papers/petals-tubes-and-rings-20261111.pdf
How IDWT maps a muon decay onto an edge:
1. The assigned muon pattern has 35 edge points.
2. Electron and neutrino-3 corners cover 13 and 22.
3. 13 + 22 = 35: no gap or overlap.
Exact geometry for supplied assignments, not a full decay mechanism. The antineutrino and decay rate are not derived by this edge count.
https://idwt-v2.github.io/papers/particle-decay-as-rearrangement-of-simplex-patterns-20261110.pdf
Balance can be unstable.
In IDWT’s fixed-shape scale test, gradient energy resists shrinking while attraction favors it. Their balance can sit on an energy hill, not in a valley: a slight change of radius lowers the energy either way.
Exact for this trial energy—not a sustained particle solution.
An orbit is not a position.
IDWT’s orbit review exposes a missing variable: where each electron is along its path. Let those phases vary, and the repulsion problem changes. Five points on a sphere in four spatial directions can all be equally far apart—a regular simplex.
Exact geometry; a physical shell arrangement is not yet derived.
https://idwt-v2.github.io/research/notes/20261110-case-orbit-geometry-and-molecular-shape/
A ladder can grow while its percentage gaps shrink.
In IDWT’s counting rule, the next rung adds a fraction d/n of the current count. At depth 6: rung 6 to 7 doubles the count; rung 12 to 13 adds only 50%.
Exact combinatorics. Treating these counts as particle-mass multipliers remains a physical hypothesis.
A narrowing family reaches a limit.
In one exact IDWT field family, width is a=(10−d)/(2M). Hold positive M fixed: going from 6 to 8 bound directions halves the width.
At depth 10, width collapses and the required coupling diverges: no regular solution of this form survives. A limit of this family, not a ban on ten-dimensional fields.
https://idwt-v2.github.io/research/notes/20261110-case-simplex-shapes-and-branch/
A finite partial integral can hide an infinite total.
For one proposed IDWT response weight, integrating over three unresolved directions gives a finite result. Integrating over all six does not: the extra volume outruns the fading tail.
Exact for this branch and weight—not a derived detector probability law.
https://idwt-v2.github.io/research/notes/20261110-case-detector-weights-contact-and-fields/
An energy ladder can hide inside a fixed total.
In one IDWT orbit family, the potential-energy share is (1−1/L²)/2. Change L: the share changes, but total energy stays the same at fixed drift and radius.
Exact within that family. Emitted hydrogen lines require differences in total energy; that step remains unproved.
https://idwt-v2.github.io/research/notes/20261110-case-shell-counts-and-rydberg-squares/
A multiple bond as shared timing.
IDWT proposes that electrons in a multiple bond follow equal orbits around the same nuclei, staying apart through different phases—not by taking different-sized paths.
How many phase slots fit depends on electron extent along the orbit relative to its length. That capacity has not been computed.
A physical hypothesis, illustrated schematically.
https://idwt-v2.github.io/research/notes/20261110-case-orbit-geometry-and-molecular-shape
A steady gap between moving circles.
In four spatial directions, two equal circular paths can keep the same separation as points travel around them at matching speed and sense. Their planes must have equal tilts in both paired directions.
IDWT explores this exact geometry for shells. Whether it works for the proposed noncircular electron paths remains open.
https://idwt-v2.github.io/research/notes/20261110-case-orbit-geometry-and-molecular-shape/
A ripple can average to zero and still leave an energy trace.
For density ρ=1+0.8 cos φ, the cycle average is 1. But the average of ρ^(10/9) is greater than 1: a nonlinear tally remembers the ripple.
IDWT uses this exact convexity result in a frozen-background energy test. It does not prove dynamical alignment.
https://idwt-v2.github.io/research/notes/20261110-case-block-pattern-and-six-plane/
A detector need not turn twice the density into twice the response.
IDWT explores the weight w=(1+u)^(1/9)−1, where u is added density divided by ambient density. For tiny u, w≈u/9; at larger u, the response bends below that straight line.
Exact mathematics for an assumed detector coupling—not a derived probability law.
https://idwt-v2.github.io/research/notes/20261110-case-detector-weights-contact-and-fields/
A mass scale can disappear from a test.
IDWT assigns three neutrino mass-state counts: 2,002; 11,628; 65,780. If each mass is its count times one common scale, the ratio of squared mass differences is ≈0.03035, whatever that scale is.
Exact conditional arithmetic. The assignments are inputs; this does not determine absolute masses.
![Algebraic scaling diagram: amplitude 1× gives squared-magnitude integral 1×; amplitude 2× gives integral 4×. Formula N[2Ψ]=4N[Ψ]. N is the integral of squared field magnitude, not automatically an integer or a probability. Scaling alone does not establish a second field solution. Bars show a calculated ratio, not measurements.](https://glommer.org/uploads/71/124529-thumb.jpg)
Field strength is not a head count.
Double a field’s amplitude everywhere: its integrated squared magnitude quadruples. That integral can vary continuously.
IDWT distinguishes this classical field norm from an integer portion count and from a position-probability law. The scaling is exact algebra; it does not show that the rescaled field solves the same equation.
Turn the phase dial. The cancellation stays.
In one IDWT radial trial, two chosen seeds have opposite angular-charge means. Give them equal weight: the mean is exactly zero for every relative phase, because the integrated cross terms vanish.
Exact for this trial—not a demonstrated conserved electric source or neutrino state.
13 = 10 + 3. The counts do not simply add.
Vandermonde’s identity splits IDWT’s assigned electron count, 18,564, into seven pieces—not just the two proposed partner counts.
Exact arithmetic; the particle-index pairing was chosen after inspection. It is an exploratory pattern, not a particle-combination law.
Same seed spin. Different placement.
In one IDWT depth-five trial, a rotation plane wholly in bound directions gives angular momentum per field norm ½. A plane crossing a bound and a free direction gives ½−w/5, where w is the lower-component norm share.
Exact for this trial—not a measured particle spin.
A square hiding in angular patterns.
Degree-2 harmonics in four spatial directions have 9 independent modes. Under rotations of three observed directions, they split into 1 + 3 + 5—the s, p and d groups.
IDWT gives this standard mathematics a physical shell interpretation. The count is exact; electron paths and shell-filling dynamics are not derived.
https://idwt-v2.github.io/research/notes/20261110-case-shell-counts-and-rydberg-squares/