Completed one-half endpoint routes for paper Step 4(A) #
The four signed endpoint calculations (DD, DE, ED, and EE) have
different dependent coordinate carriers. Their common numerical output is
much smaller: a completed residual prefix, a reachable complete-budget scale
on that prefix, its edge certificate, and the charge left by the first
exceptional incoming head (when there is one).
This file records exactly that common output. It deliberately does not put an endpoint density, an integral identity, or an a.e. majorization into the structure. Those facts are proved by the four branch lemmas only after the corresponding signed endpoint integrations have been performed.
If the first internal vertex survives the complete pairing reduction,
every head before an arbitrary retained terminal has a nonzero predecessor.
Consequently the literal stop trace is wholly ordinary. This is the direct
incoming half of the paper's DD/DE dichotomy.
Common numerical carrier left by any of the four paper endpoint routes
on one half. firstCharge is the scale-sized factor which precedes the
literal ordinary-J endpoint sacrifice; direct incoming routes may take it
to be their initial slot-zero budget.
- cases : R324PaperHalfEndpointCases
- final : R324WithinHalfResidualPrefix rho lam eps pairing
- terminalReachable : R324WithinHalfBudgetScaleReachable pairing rho C lam eps K A self.final.state self.terminalScale
- terminalCertificate : R324WithinHalfEdgeCertificate self.final.state self.terminalScale
- transportedMultiplier : ℂ
- firstCharge : ℝ
- multiplier_mul_firstCharge_le_terminal : ‖self.transportedMultiplier‖ * self.firstCharge ≤ self.terminalScale 0
Instances For
The endpoint cost remains the literal case-by-case cost until the two halves have been spliced.
Equations
- route.endpointSacrifice = Anderson4D.r324PaperHalfEndpointSacrifice eps route.cases
Instances For
Flatten a genuine completed alternating suffix into the common route
carrier. The branch lemma retains transport itself for its signed
identity; only its terminal numerical data are stored here.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Flatten a retained-outgoing terminal after its exact Fourier/primitive
collapse. The real witness is the caller's outgoing stop; unlike a
spurious identity transport, it records the actual paper route ending at
terminalPost.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Empty remaining schedule identifies the completed carrier with the canonical final active set.
Two completed one-half routes form the terminal carrier consumed by the nested-cross Step 3 estimate.
Equations
Instances For
Concatenate the two literal one-half case ledgers in the paper order
LI, LO, RI, RO.
Equations
Instances For
Exact two-half endpoint cost. No uniform eps^-8 enlargement occurs
in the common route layer.
The first exceptional incoming-head charge #
The following is the reusable numerical calculation previously repeated in the full/full endpoint proof. The complete budget and the signed analytic trace are synchronized on the same literal incoming stop. Therefore the ordinary primitive scale of that first head is paid by the after-head slot-zero budget, and the later alternating multiplier telescopes into the terminal slot-zero scale.
The exact analytic scale exposed by the first exceptional incoming primitive head.
Equations
- pack.firstExceptionalScale = Anderson4D.r324WithinHalfInternalEdgeScaleProduct pack.data.stopContext pack.data.stopScale * (C * lam) ^ (2 * Anderson4D.residualBlockOrder pack.data.terminal.2)
Instances For
The first exceptional head scale is already contained in the complete
after-head slot-zero budget. This is the paper's first-exceptional charge,
before applying the ordinary-J estimate and its literal eps^-2 cost.
End-to-end first-exceptional charge after the literal alternating suffix.
No ordinary-J sacrifice has yet been enlarged or made uniform.
Package a completed exceptional-incoming route once the paper's budget-synchronized alternating suffix has been constructed.
Equations
- One or more equations did not get rendered due to their size.