IDA consistent IC: dae_init_mode, battery hold, forcing t+, source currents - #3845

…acitors

Replace capacitive couplings with voltage-source constraints that hold
branch Δv and solve the augmented G/B algebraic system. Falls back to the
nano-step heuristic only on failure. Includes unit test for series C–R
current step and docs for mode 3.

Classify C structure: floating caps keep Δv battery stamps; diagonal mass
holds inductor current; one-sided C (OpAmp tau*vk') holds output voltage
and drops the dynamic equation. Add unit tests for op-amp continuous hold
and inductor current hold under dae_init_mode(3).

Hold continuous Vm and xc layer content via stiff springs on the resistive
(cj=0) matrix, then restore holds exactly (allowing algebraic common-mode
when outer xc is zero). Wire into Daspk battery init after LinearMechanism
projection and add a cable+extracellular smoke test.

Diagnostic dump of residual/state before discontinuity, after event,
and after re-init. Arm with dae_init_audit(level, t) for first reinit
at or after t; write to stdout or a file via dae_init_audit_file.

Panel A uses the continuous residual captured at integrator retreat
(interpolate after a step that may overshoot), with continuous play
synchronized to the event time—shared by Vector.play, NetCon, and
at_time—not a re-evaluation after the jump or the raw step endpoint.

Status, architecture, build/test notes, and a starting prompt for new
Grok sessions on hines-grok/ida-init in ~/neuron/nrnida.

Recover continuous y' from the mass equation at fixed projected y instead
of the dteps nano-step companion, so residual WRMS can reach zero on
stiff cases (e.g. SEClamp + tiny cm). Keep nano-step only as residual
fallback; classify algebraic vs near-singular residual eqs on failure;
suppress fallback when an IC audit is armed. Document and test.

Add nrn_vecplay_continuous_tplus matching VecPlayContinuous interpolate
geometry (outgoing slope at knots; linear extrapolation of the last two
points past the end). Expose VecPlayContinuous::forcing_tplus for later IC
use. Document forcing t+ info (1-jet) in cvode.rst and GROK-IDA-INIT;
unit tests for slope extraction oracles.

At each Daspk::init, sample all VecPlayContinuous instances for right-limit
u and classical u' (forcing t+ info / 1-jet), store on Daspk for A2, and
list them in the three-panel IC audit. Add ramp-play hoctest checking
u≈0.5 and u'≈0.5 after the t=1 jump onto the ramp segment.

After C*y'=b-Gy particular seed, adjust y' in null(C) so left-null
constraints Z'G y' = Z'b' hold, with b' from Vector.play slopes mapped
into LinearMechanism b. Fixes series C–R ramp IC (V1'=1, V2'=0.5 when
I=0.5, I'=0.5, R=C=1) without a nano-step.

Add series C–R integration tests for step, slope kink, post-end
extrapolation, flat hold, finitialize first-segment slope, and two-event
iramp-like sequence. Treat t=t0 as outgoing slope for forcing t+ (strict
hold only for t<t0) so finitialize ramps get the correct I'.

Add lm.dforce(callable, bdot) so non-play forcing can supply db/dt for
mode-3 free y' completion. Callable runs at IC time; optional FD of the
force callable when dforce is unset. Audit lists LM dforce/bdot/fd
entries. Tests: sinusoid analytic and FD fallback on series C–R.

Track mode-3 success/fallback and free-y' sources (play, dforce, FD).
Expose CVode.dae_init_stats() for print/reset/Vector fill; include in
statistics(). Audit notes list free-y' sources; fallback messages name
mode 3 vs nano-step explicitly.

Add piecewise-linear electrode stimulus (test-only) and mode-3 coverage for
IClamp/PWL istep, kink, jump-ramp, and E0 Section vs LM play parity. Update
GROK-IDA-INIT for (b) in progress and parked density dforce.

Re-ran test_ida_init_mode.py and isolated A1 ramp + A3 istep/kink suite
under mode 3; no code changes. Handoff notes WP2 done; next is WP3.

…i tests.

Coupled C y'=f seeding now puts electrode current into extracellular mass
(1-layer and multi-layer cumulative F), so mode 3 clears residual for
IClamp/PWL steps with extracellular. Add E1 Section/LM parity and Z0/Z1
LM end-ri() twins; cable zero-area end inject still may nano-step fallback.

Note hoctests::test_ida_source_current (PWLClamp via nrnivmodl hash) and
standalone run instructions; handoff points at C→A→B stack.

Add nrn_cable_battery_ic: hold CAP-list node voltages, free section-end
nodes so electrode at loc 0/1 can set absolute levels; soft-fail if the
cj=0 factor is singular. Pre-evaluate residual so BREAKPOINT/at_time sees
t+ before project. Extracellular Vm holds skip zero-area nodes. Tests
require path_mode=3 for end IClamp/PWL and Z0/Z1 Section+LM.

E2 xc=0 resistive xtral istep (path_mode 3); E3 multi-layer xc>0 istep;
E4 weak/strong xg istep with Vm hold. Completes WP3 topology matrix for
mid-compartment electrode sources.

Expand CVode.dae_init_mode documentation for electrode sources, at_time,
extracellular Vm vs vext, and play/dforce 1-jet use. Fix E2/E* tests to
treat seg.v as transmembrane Vm (vext may jump when xc=0). Handoff notes
nrndc1sim smoke: mode 3 at finitialize; transfer re_init often falls back.