RESEARCH PROBE 003 — POLYHEDRAL CAVITY ENERGY CONCENTRATION LITERATURE
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Date: 2026-03-27
Status: COMPLETE — component physics known, specific synthesis is a gap
Origin: HPC-024 angular deficit sweep results
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PURPOSE
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Determine whether the HPC-024 finding — triangular-faced polyhedra
concentrate more energy at vertices than non-triangular-faced polyhedra
— is established, novel, or somewhere in between.

This is NOT a novelty claim. The TLT argument is parsimony: the same
{3} organizing principle appears in cavities, crystals, and cosmic
structure. The question is whether acoustics already knows this
(which validates the engine) and whether the cross-field connection
is already made (which determines what TLT adds).


WHAT IS ESTABLISHED (component physics)
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1. CORNER PRESSURE AMPLIFICATION (room acoustics):
   Trihedral corners (3 surfaces meeting) produce higher pressure than
   dihedral corners (2 surfaces). This is textbook studio acoustics.
   Bass trap placement guides rely on this.

2. MEIXNER EDGE CONDITION (1940s-present):
   EM fields near a conducting wedge have a singularity r^nu where
   the exponent nu depends on wedge angle. Sharper angles = stronger
   singularity. Well-established EM theory.

3. KELLER'S GEOMETRICAL THEORY OF DIFFRACTION (1962):
   Vertices and edges produce diffracted fields with coefficients
   that depend on geometry (cone angle, edge angle).

4. CONICAL TIP FIELD ENHANCEMENT:
   Field singularity at a conical tip depends on half-angle and solid
   angle. Smaller solid angle = stronger singularity. Basis of
   scanning probe microscopy, lightning rods.
   (Phys. Rev. B 74, 235442; COMSOL singularity literature)

5. ANGULAR DEFICIT AS DISCRETE CURVATURE:
   Well-established in discrete differential geometry. Descartes,
   Gauss-Bonnet for polyhedra. Values for all Platonic solids known.


WHAT IS NOT PUBLISHED (gaps)
------------------------------

1. NO SINGLE STUDY compares all 5 Platonic solids as acoustic or EM
   cavities. The closest is a comparative study of Platonic solid
   LOUDSPEAKERS (radiation outward, the inverse problem).

2. NO PAPER explicitly connects angular deficit (discrete Gaussian
   curvature) to acoustic/EM energy concentration at vertices. The
   discrete geometry community and the EM singularity community have
   not bridged this specific gap.

3. NO PAPER states "triangular-faced polyhedra concentrate more energy
   at vertices than non-triangular-faced polyhedra" as a general
   principle. The underlying physics (Meixner + angular deficit)
   would predict it, but nobody has published the synthesis.

4. Cavity resonator literature overwhelmingly covers rectangular,
   cylindrical, and spherical geometries. Icosahedral, octahedral,
   tetrahedral, and dodecahedral cavities are essentially unstudied
   as resonators.


ASSESSMENT
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The observation is "somewhere in between" — closer to novel synthesis
than established fact.

Component pieces are individually known:
  (a) sharper vertices concentrate more energy — KNOWN
  (b) angular deficit quantifies vertex sharpness — KNOWN
  (c) corners amplify acoustic pressure — KNOWN

The specific synthesis is NOT published:
  (d) ranking Platonic solids by vertex concentration — NOT DONE
  (e) attributing the ranking to face geometry — NOT DONE
  (f) connecting to {2,3} organizing principles — NOT DONE (TLT specific)

HPC-024 fills gap (d). The interpretation via (e) and (f) is where
TLT adds perspective — not new physics, but connecting known physics
across fields that haven't talked to each other.


TLT PARSIMONY CONTEXT
----------------------
The claim is NOT "we discovered that triangular cavities focus sound."
The claim IS: "the same {3} geometry that determines crystal archetypes,
material ductility, and conductor properties ALSO determines cavity
energy concentration — because it's one mechanism ({2,3} interference)
operating at every scale."

Acoustics engineers know about corner focusing.
Crystallographers know about coordination numbers.
Astrophysicists know about rotation curves.

None of them are talking to each other about why {3} appears in all
three problems. That's the silo problem TLT addresses.


REFERENCES
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  Meixner (1940s): edge condition for EM fields at conducting wedges
  Keller (1962): Geometrical Theory of Diffraction, JOSA 52(2), 116
  Phys. Rev. B 74, 235442: conical tip field enhancement
  COMSOL Blog: field singularities at sharp features
  Bass Traps 101 (arqen.com): practical corner pressure amplification
  Platonic solid loudspeaker comparison: ResearchGate/228939202
  Angular deficit: Wikipedia, nrich.maths.org/1381
  Geometry-invariant resonant cavities: Nature Comms, ncomms10989
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