Angle-Free Note on the Repeated Appearance of 2/3

Reading Position

This page has been rebuilt from the local PDF. The document is about 4,518 extracted words across 16 pages. Its central move is diagnostic: it does not claim that every appearance of 2/3 has one origin. It separates the mechanisms by asking how each one responds to a formal deformation D = 3 + epsilon.

Read this after the recursive closure and Hopfion/topology notes, because it is a discipline page for number coincidences. It teaches the site how to say: "these routes agree at the undeformed point, but their deformation response is different, so they are not automatically the same mechanism."

Document Shape

The note starts with notation, rewrites Koide as a projector equality, generalises the projector statement to D components, compares three routes to 2/3, computes epsilon expansions, defines a Koide residual diagnostic, evaluates physical charged leptons, checks FFGFT/T0 inputs, and ends with a route table.

Central Principle

At D = 3, several algebraic paths pass through 2/3. Under D = 3 + epsilon, they usually separate. That separation is not a nuisance; it is the test. The deformation reveals whether the claim is about equal projector weights, a complement-of-step rule, a midpoint, a rank split, an FFGFT exponent ladder, or recursive closure.

At D = 3, many roads meet at 2/3; at D = 3 + epsilon, the roads separate.

Formula Spine

These formulas are the live algebraic spine of the page. They are intentionally angle-free.

x_i = sqrt(m_i)

u = (1, 1, 1)^T

Q = (x^T x) / (u^T x)^2

P = u u^T / (u^T u) = u u^T / 3

P_perp = I - P

A = x^T P x,    B = x^T P_perp x

Q = (A + B) / (3 A)

Q = 2/3  <=>  A = B

P_D = u_D u_D^T / D,    P_{D,perp} = I - P_D

Q_D = (A_D + B_D) / (D A_D)

equal-projector route: Q_proj(D) = 2 / D

complement route: p_comp(D) = 1 - 1 / D

midpoint route: Q_mid(D) = (1/2)(1 + 1/D)

Delta_K = 3 Q - 2 = B/A - 1

Delta_K^(D) = D Q_D - 2 = B_D/A_D - 1

recursive closure: S = (1 - r)L + rS

resolved complement: 1 - r

rank split: rank(P_+)/dim(V) = 8/24 = 1/3, rank(P_-)/dim(V) = 16/24 = 2/3

FFGFT generator: m_i = r_i xi^{p_i} v

FFGFT-to-Koide map: (r_i, p_i, xi) -> x_i = sqrt(r_i) xi^{p_i/2} -> (A, B) -> Q

Epsilon Discriminator

The clean discriminator is the first derivative at the undeformed point. The three simple routes meet at 2/3 when epsilon is zero, but they respond differently as epsilon changes.

Numerical Diagnostics

The physical charged-lepton vector is extremely close to equal projector weight but not exactly equal at the central values used in the note. The diagnostic residual is small and is dominated experimentally by the tau mass uncertainty.

physical charged leptons: Q ~= 0.6666644634

physical residual: Delta_K ~= -6.6097565e-6

For the quoted FFGFT/T0 rational inputs, the note obtains a nearby but distinct Koide value. That number is not treated as an exact derivation of Koide. It is a proximity diagnostic for the mass-generator route.

FFGFT quoted input: xi = 4/30000 = 1/7500

FFGFT coefficients: r_e = 4/3, r_mu = 16/5, r_tau = 25/9

FFGFT exponents: p_e = 3/2, p_mu = 1, p_tau = 2/3

Q_FFGFT ~= 0.6677418577

Q_FFGFT - 2/3 ~= 0.0010751910

Route Separation

The document's mature conclusion is that repeated numbers need maps, not excitement. Koide projector equality, FFGFT exponent architecture, rank-split mediation, and recursive closure can all touch 2/3, but each starts from a different primitive object.

Koide projector form

Primitive object: square-root mass vector. The 2/3 mechanism is equal democratic and residual quadratic weight.

FFGFT/T0

Primitive object: xi exponent ladder and rational coefficients. The tau exponent is read as a complement-of-step route at exact topological D = 3.

Rank-split mediation

Primitive object: finite mediation space with involution and projectors. The 2/3 mechanism is a rank ratio.

Recursive closure

Primitive object: a contractive recursive whole. The 2/3 mechanism is the complement of a 1/3 unresolved remainder.

Claim Discipline

The safe reading is: this is a diagnostic note, not a proof that all 2/3 appearances share one source. Its purpose is to prevent number-chaining by demanding the primitive object, map, residual, and deformation response.

• Do not treat agreement at D = 3 as proof of shared mechanism.
• Do not mix topological D with FFGFT fractal D_f unless the page explicitly says so.
• Do not call FFGFT an exact Koide derivation unless the equal-projector condition is forced, not merely approached.
• Require a map before identifying projector, complement, rank, FFGFT, and recursive-closure routes.

Terms To Carry Forward

Angle-free Koide

A projector statement about equal quadratic weight, avoiding trigonometric parametrisation.

Democratic projector

The projector onto the all-ones direction in component space.

Residual projector

The complement of the democratic projector.

Koide diagnostic

Delta_K = 3Q - 2 = B/A - 1, measuring distance from equal projector weight.

Epsilon response

The deformation derivative that identifies which 2/3 route is being claimed.

Koide proximity

Nearness to the Koide surface without claiming exact closure.

Recommended Reading Move

Read sections 3 through 7 first, because they give the projector algebra and the residual diagnostic. Then read sections 12 through 20 for the FFGFT and cross-framework comparisons. Keep the conclusion in view: reproducing 2/3 is not enough; a mechanism must specify its route.