Tangent coverage, forms and horizontal operations

Tangent coverage, forms and horizontal operations

Let be the edge cochain (. At the identity readout, differentiating exp() and) = exp() gives

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p(φ, a) = a + d0φ,
j(u) = (0, u),
pj = I.

The kernel is exactly: this follows directly by solving) = 0. Oriented boundary cancellation gives = 0, hence. This kernel is infinitesimal frame redundancy; it is distinct from any subsequent physical gauge or harmonic restriction in the supplied construction.

Theorem A.1 (Operator and quadratic-form recovery). For a target operator on the specified finite cochain spaces, put. Then

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p2TZ = Tp1,
TZj1 = j2T.

Compositions are preserved:. A symmetric target form with matrix instead pulls back as. Its nullspace is(ker, and its restriction to the section is the original form.

Proof. Insert in the two intertwining identities and the composition. For the form, its value on isby the definition of the adjoint. Thus its radical consists precisely of for which = 0, since is onto. In particular every vertical vector is null. Taking

recovers the original form. The transported section inner product makes an isometry. It does not assert that is an isometry for an unrelated ambient inner product or that equals

For example the source stationary identity

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Qc = T ∗QfT + b(I −PT)∗(I −PT)

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pulls back to
p∗
cQcpc = T ∗
Zp∗
fQfpfTZ + b(pc −PpfTZ)∗(pc −PpfTZ).

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This follows by substitution and pfTZ = Tpc. Replacing the last readout difference by I −PZTZ
on the whole redundant space changes the form on vertical directions. Any positive lower bound
is retained only on the original restricted space or its isometric section, with the original norm and
restrictions.
For a differentiable finite-lattice curve let kx =  ̇rxr−1
x . Exact matrix differentiation gives

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 ̇Ue = r−1
x (  ̇Ae −kxAe + Aeky)ry.

Every target tangent lifts, for any differentiable frame curve, by

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 ̇Ae = rxher−1
y
+ kxAe −Aeky.

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Indeed differentiate Ae = rxUer−1
y ; this also proves group-tangent membership. At fixed target
input the two frame terms are still required. The flat target configuration at arbitrary frames is
Ae = rxr−1
y .
Let P be any actual tagged source predictor from fine links to coarse links on its stated domain.
With coarse frames s, define

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Bb = syP(π(r, A))bs−1
z ,
b : y →z.

Then)) identically. On a smooth branch, for coarse rates

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 ̇Bb = syDPU[h]bs−1
z
+ lyBb −Bblz.

Differentiating the coarse readout cancels the last two terms and recovers]. The proof uses the actual predictor, with its routes, minimizers, fallback tags and finite stencil. A multivalued target branch is lifted as a tagged relation; no new selection is introduced. Norm estimates transfer with the transported color-block norms on fixed-frame or horizontal derivatives. They do not bound arbitrary unbounded frame rates.