TauLib · API Book IV

TauLib.BookIV.Sectors.SpectralPage

TauLib.BookIV.Sectors.SpectralPage

Tensor-square derivation of 121/225 from the E₁ spectral page.

Registry Cross-References

  • [IV.D331] Tensor-Square Character Algebra — tensorModes

  • [IV.T133] EM Tensor Density Theorem — em_tensor_active_count, em_tensor_total

  • [IV.P179] E₁ Page Derivation of α-Coefficient — tensor_equals_sieve_times_correction

  • [IV.R388] OQ-A1 Status Update — RESOLVED

Mathematical Content

The boundary character algebra A_spec(L) has 15 modes (5 generators × 3 lobe configurations). The tensor square A_spec(L)^{⊗2} has 225 = 15² mode pairs. A mode pair (m₁, m₂) is jointly EM-active when both m₁ and m₂ are EM-active.

The EM-active count in the tensor square is 11² = 121 (since each factor contributes independently). The density 121/225 = (11/15)² is the coefficient in α = (121/225)·ι_τ⁴.

Physical interpretation: α is a coupling constant for emission-absorption. Each vertex (source, sink) contributes one factor of 11/15. The product (11/15)² = 121/225 is the joint probability that both endpoints of the EM propagator land on EM-active boundary modes.

Unification: The factorization (8/15)·(121/120) is a COROLLARY of the tensor-square density, not the other way around.

Ground Truth Sources

  • spectral_page_121_225_sprint.md: mathematical derivation

  • BoundaryFiltration.lean: structural EM-activity, twin prime residue

  • ModeCensus.lean: mode enumeration, 11/15 census

Tau.BookIV.Sectors.SpectralPage.tensorModes

source def Tau.BookIV.Sectors.SpectralPage.tensorModes :List (ModeCensus.BoundaryMode × ModeCensus.BoundaryMode)

[IV.D331] All mode pairs in A_spec(L)^{⊗2}. Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIV.Sectors.SpectralPage.emTensorActive

source def Tau.BookIV.Sectors.SpectralPage.emTensorActive :List (ModeCensus.BoundaryMode × ModeCensus.BoundaryMode)

EM-active tensor modes: both endpoints EM-active. Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIV.Sectors.SpectralPage.emTensorSilent

source def Tau.BookIV.Sectors.SpectralPage.emTensorSilent :List (ModeCensus.BoundaryMode × ModeCensus.BoundaryMode)

EM-silent tensor modes: at least one endpoint silent. Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIV.Sectors.SpectralPage.em_tensor_total

source theorem Tau.BookIV.Sectors.SpectralPage.em_tensor_total :tensorModes.length = 225

[IV.T133] Total tensor-square modes = 225 = 15².

Tau.BookIV.Sectors.SpectralPage.em_tensor_active_count

source theorem Tau.BookIV.Sectors.SpectralPage.em_tensor_active_count :emTensorActive.length = 121

[IV.T133] EM-active tensor modes = 121 = 11².

Tau.BookIV.Sectors.SpectralPage.em_tensor_silent_count

source theorem Tau.BookIV.Sectors.SpectralPage.em_tensor_silent_count :emTensorSilent.length = 104

Silent tensor modes = 104.

Tau.BookIV.Sectors.SpectralPage.tensor_partition

source theorem Tau.BookIV.Sectors.SpectralPage.tensor_partition :emTensorActive.length + emTensorSilent.length = tensorModes.length

Active + silent = total (consistency).

Tau.BookIV.Sectors.SpectralPage.density_is_square

source theorem Tau.BookIV.Sectors.SpectralPage.density_is_square :121 = 11 * 11 ∧ 225 = 15 * 15

121 = 11² and 225 = 15²: the tensor density IS a perfect square ratio.

Tau.BookIV.Sectors.SpectralPage.density_equals_square

source theorem Tau.BookIV.Sectors.SpectralPage.density_equals_square :121 * 15 * 15 = 11 * 11 * 225

The density 121/225 = (11/15)². Cross-multiplied form.

Tau.BookIV.Sectors.SpectralPage.tensor_equals_sieve_times_correction

source theorem Tau.BookIV.Sectors.SpectralPage.tensor_equals_sieve_times_correction :8 * 121 * 225 = 15 * 121 * 120

[IV.P179] Tensor square contains the sieve × correction factorization. (8/15)·(121/120) = 121/225. Cross-multiplied: 8 · 121 · 225 = 15 · 121 · 120.

Tau.BookIV.Sectors.SpectralPage.correction_cross_mult

source theorem Tau.BookIV.Sectors.SpectralPage.correction_cross_mult :121 * 15 * 120 = 8 * 225 * 121

The correction 121/120 is recovered from the tensor density and sieve. (121/225) / (8/15) = 121/120. Cross-multiplied (clearing denominators):