The Planck Equivalence Grid

Systematic Enumeration of All Pairwise Relationships in the Equivalence Chain

T/T_P = f·t_P = m/m_P = l_P/l = E/E_P = p/p_P = F/F_P

Known Law - Historically discovered and named

Predicted Relationship - Structure tells us it must exist

↓ × →	Temperature (T)	Frequency (f)	Mass (m)	Length (l)	Energy (E)	Momentum (p)	Force (F)
Temperature (T)	—	T = f · kB/c² Temperature-Frequency [T] = [T⁻¹][ML²T⁻²K⁻¹][L⁻²T²] PREDICTED	T = m · c²/kB Temperature-Mass [K] = [M][L²T⁻²][ML²T⁻²K⁻¹]⁻¹ PREDICTED	T · l = hc/kB Thermal Length Scale [KL] = [ML²T⁻¹][LT⁻¹][ML²T⁻²K⁻¹]⁻¹ PREDICTED	E = kBT Boltzmann's Law [ML²T⁻²] = [ML²T⁻²K⁻¹][K] KNOWN	p = kBT/c Thermal Momentum [MLT⁻¹] = [ML²T⁻²K⁻¹][K][LT⁻¹]⁻¹ PREDICTED	F = kBT/lP Thermal Force Scale [MLT⁻²] = [ML²T⁻²K⁻¹][K][L]⁻¹ PREDICTED
Frequency (f)	f = T · c²/kB (Symmetric to T-f) PREDICTED	—	f = mc²/h Compton Frequency [T⁻¹] = [M][L²T⁻²][ML²T⁻¹]⁻¹ PREDICTED	f · l = c Wave Relation [T⁻¹][L] = [LT⁻¹] KNOWN (λf=c)	E = hf Planck-Einstein [ML²T⁻²] = [ML²T⁻¹][T⁻¹] KNOWN	p = hf/c Photon Momentum [MLT⁻¹] = [ML²T⁻¹][T⁻¹][LT⁻¹]⁻¹ PREDICTED	F = hf/lP Frequency Force [MLT⁻²] = [ML²T⁻¹][T⁻¹][L]⁻¹ PREDICTED
Mass (m)	m = T · kB/c² (Symmetric to T-m) PREDICTED	m = hf/c² (Symmetric to f-m) PREDICTED	—	m · l = h/c Compton Wavelength [M][L] = [ML²T⁻¹][LT⁻¹]⁻¹ KNOWN	E = mc² Mass-Energy [ML²T⁻²] = [M][L²T⁻²] KNOWN	p = mc Relativistic Momentum [MLT⁻¹] = [M][LT⁻¹] KNOWN	F = mc²/lP Mass Force Scale [MLT⁻²] = [M][L²T⁻²][L]⁻¹ PREDICTED
Length (l)	l = hc/(kBT) (Symmetric to T-l) PREDICTED	l = c/f Wavelength KNOWN	l = h/(mc) (Symmetric to m-l) KNOWN	—	E = hc/l Photon Energy (λ) [ML²T⁻²] = [ML²T⁻¹][LT⁻¹][L]⁻¹ PREDICTED	p = h/l de Broglie [MLT⁻¹] = [ML²T⁻¹][L]⁻¹ KNOWN	F = hc/l² Casimir-type Force [MLT⁻²] = [ML²T⁻¹][LT⁻¹][L²]⁻¹ PREDICTED
Energy (E)	E = kBT (Symmetric to T-E) KNOWN	E = hf (Symmetric to f-E) KNOWN	E = mc² (Symmetric to m-E) KNOWN	E = hc/l (Symmetric to l-E) PREDICTED	—	E = pc Photon E-p [ML²T⁻²] = [MLT⁻¹][LT⁻¹] KNOWN	F = E/lP Energy Force [MLT⁻²] = [ML²T⁻²][L]⁻¹ PREDICTED
Momentum (p)	p = kBT/c (Symmetric to T-p) PREDICTED	p = hf/c (Symmetric to f-p) PREDICTED	p = mc (Symmetric to m-p) KNOWN	p = h/l (Symmetric to l-p) KNOWN	E = pc (Symmetric to E-p) KNOWN	—	F = pc/lP Momentum Force [MLT⁻²] = [MLT⁻¹][LT⁻¹][L]⁻¹ PREDICTED
Force (F)	F = kBT/lP (Symmetric to T-F) PREDICTED	F = hf/lP (Symmetric to f-F) PREDICTED	F = mc²/lP (Symmetric to m-F) PREDICTED	F = hc/l² (Symmetric to l-F) PREDICTED	F = E/lP (Symmetric to E-F) PREDICTED	F = pc/lP (Symmetric to p-F) PREDICTED	—

Key Insights

• Out of 21 pairwise relationships, only ~8 are historically "known" and named. The rest have been sitting there for 126 years, waiting to be read off the structure.
• The grid is perfectly predictive: Every empty cell has its formula determined by the dimensional structure. We know what must be there before we look.
• The "predicted" relationships aren't speculative—they're mathematically necessary consequences of the equivalence chain. They may already be used implicitly in various contexts but haven't been recognized as fundamental.
• Several predictions match known physics:
  • f·l = c is just the wavelength-frequency relation λ = c/f
  • F = hc/l² has the same dimensional structure as the Casimir force
  • p = hf/c is the photon momentum (often written E/c with E=hf)
  • E = hc/l is photon energy from wavelength
• Some predicted relationships may represent new physics or ways of thinking about existing phenomena that haven't been explicitly formulated.
• The Force row/column is particularly interesting because force is underrepresented in fundamental relations—most entries here are genuinely predictive.
• Adding more axes expands the grid: With charge (10 axes → 45 relationships), angular momentum (11 axes → 55 relationships), etc.

The Algorithmic Power

For any empty cell at row R, column C:

1. Write the equivalence: R/R_P = C/C_P (or C_P/C if C uses inverse scaling)
2. Rearrange to solve for your target variable
3. The Planck unit ratios automatically give you the correct constant
4. The dimensional analysis tells you if it's right

This isn't discovery—it's extraction. The relationships are already there in the structure, waiting to be computed.