MTFT · Matter-03 · research lab

A Higgs branch exists.
The mass problem remains.

Follow three distinct gates: a classical vacuum, the charged modes surviving that vacuum, and the conversion from a holomorphic coupling to a mass. The controls expose the assumptions that separate them.

Paired arrowsClassical existence on the specified compact surface.
237 per charge signCharged classes remain; three families are not selected.
No selected scaleThe Higgs magnitude and kinetic metrics remain needed.
1

Release the reverse Higgs arrow

EXACT · CONDITIONAL CLASSICAL GEOMETRY

The full bundle has eight summands with powers (0,0,0,1,1,2,−3,−1). At the ample wall μ(ℒ) = 0, every summand has slope zero.

Canonical-valued Higgs maps between two line bundlesPaired down and up arrows connect the weak line and the B line. The remaining summands are spectators.ℒweak componentℒ⁻¹B componentHdHuMaps carry the canonical bundle KS twist

Paired nonzero branch: classical vacuum exists.

An odd exceptional valuation makes the active spectral polynomial irreducible. The stable active pair and slope-zero spectators give a polystable full bundle.

Φ₂ = [[0, Hd], [Hu, 0]]
det(λI − Φ₂) = λ² − s
ordD(s) = 3 + 2 + 2 = 7

Curve orders 3 and 2 contribute five zeros. Pulling back two top forms adds two exceptional Jacobian zeros. A square has even valuation, so s cannot be a square in the function field.

1.00
Spectral products(t) = 1.0000 s(1)
Scale selectionOPEN
Q = T₃ + YCommutes with both arrows
For each nonzero t, the existence argument applies. No potential selecting t was derived. The slider changes algebraic data; it does not evolve a computed metric.
Fixed assumptions and symmetry

S = Bl(P,O)(X₀(143) × 143a1), genus 13, P the rational width-one cusp ∞; ℒ = π*O(P) ⊗ O(−3D), no extra flat twists. A smooth compact projective surface and an ample slope-zero wall are required. Φ is KS-valued, as in the defect-free classical BHV bulk parent. Tanaka’s surface correspondence supplies metric existence. The paired branch leaves U(3) × U(1)⁴; axionic removal of extra Abelian factors and a unique massless photon remain separate gates.

2

Count what remains light

EXACT · CONDITIONAL MODE COMPLEX

The paired vacuum changes the complex End(V) → KSEnd(V). This panel compares the split background with the fixed nonzero pair of canonical orders 3 and 2; it is independent of the selector in panel 1.

Split Φ = 0 · each Q = ±1279 classes
279
Paired · each Q = ±1237 classes
Surviving classesConnection-map rankGauge-map rank 1
dim 𝕳¹ = 279 − 1 − rank(A)
rank(A) = 41 ⇒ dim 𝕳¹ = 237

An exact finite certificate computes the six connection-block ranks for the specified paired representatives. Changing the Higgs sections requires a new calculation.

For any nonzero aligned pair, the six block capacities still give a robust lower bound: at least 160 classes per charge sign survive. Positive kinetic normalization cannot reduce their number.

Arrow / spectatorBlock sizeCapacityRank

These are complex charged degree-one classes of the specified bulk deformation complex, not identified electrons or selected generations. The branch has far more modes than a three-family charged sector.
3

A flat ray does not select a scale

INPUT · ELECTROWEAK TOY REDUCTIONDIAGNOSTIC · PLOT

For supplied neutral amplitudes u, d and couplings g₂ = gY = 1, the illustrative quartic is

VD(u,d) = ¼(u² − d²)²
u = d ⇒ VD = 0 at every scale
The diagonal u = d has zero potential; off-diagonal points have positive potential.
1.00
1.00
Toy potential · arbitrary units0.000000
This finite electroweak example omits the extra U(1) factors and global surface equation. It is not the computed metric or potential of the paired branch above. In particular, equal algebraic arrow amplitudes do not assert equal physical norms.
4

Normalize a coupling before calling it a mass

INPUT · POSITIVE DIAGONAL GRAMS AND SCALEDIAGNOSTIC · NORMALIZED EXAMPLE

The known residue minor has rank three. It has not been identified with a three-family low-energy sector. Supply kinetic matrices to see why its rank cannot determine mass ratios.

Y₀ = [[1,0,0], [0,0,1], [0,1,0]]
CX†GXCX = I
Ycan = CLTY₀CE / √GH
M = (κvH/√2)Ycan

The transpose is ordinary: these are holomorphic field-coordinate slots. Physical Dirac masses are the singular values of M. The omitted overall parent coefficient κ and the Higgs scale must also be derived or supplied.

GL diagonal

GE diagonal

1.00
1.00

Sorted singular values · arbitrary units

Raw minor rank3
Normalized coupling rank3 for positive Grams
Displayed mass matrix rank3
Second / first0.125000
Third / first0.066667
Identity Grams give (1, 1, 1). The equally positive example GL = diag(1,4,25), GE = diag(1,9,16), GH = 1 gives (1, 1/8, 1/15) before the supplied scale. Neither set is a predicted particle spectrum.

Inputs still missing for a mass calculation

Required dataStatus
Selected Higgs expectation valueOPEN
Selected light-family subspacesOPEN
Full 18 × 44 × 10 Yukawa tensorOnly a slice known
Surface harmonic mode metricsOPEN
Parent kinetic / superpotential prefactorsOPEN

Curve Petersson or older HYM Grams are useful machinery, but cannot be substituted for the charged surface mode matrices of this construction.