The autonomous echo arena: emission, reception and return as activated writes #

Category: 7-SigmaLayer (the record layer). BACKLOG #199 (out of #39; the author's third operational-geometry brief, #198–#205).

The activated write of #194 has one pointer, at the receiver. The echo puts a pointer at each station:

EchoArena = FieldArena 2 2 × (T² × T²), x = (p, (q_A, q_B)),

pointer A coupled to the occupancy of A, pointer B to the occupancy of B, each detuning the carrier by its own back-reaction (#198: H = J X + Δ_A n_A + Δ_B n_B). The dynamics is the one the time-independent joint Hamiltonian

𝓗 = ⟨p, J X p⟩ + g_A n_A(p) f_A(θ₂ᴬ) + g_B n_B(p) f_B(θ₂ᴮ)

prescribes: at fixed conserved angles the ray evolves by the two-site detuned exchange, pointer A turns at the rate −(2g_A/ħ) f_A'(θ₂ᴬ) · n_A(p) and pointer B at −(2g_B/ħ) f_B'(θ₂ᴮ) · n_B(p). Each pointer moves only while the carrier is at its station. For message 1 the carrier leaves A, arrives at B, and comes back to A on its own at the detuned round trip (twoDetuned_return), so pointer A is driven twice — once as the carrier leaves (the emission), once as it returns — and pointer B once (the reception). Read in thirds of a turn, pointer A's first arc is the emission record and its second arc the return record: two displacement arcs of one register at A, written by the carrier itself.

PointerCoupling; echoAmountA, echoAmountB, ★ echoFlow — the dynamics as data; echoFlow_zero, ★★ echoFlow_add — a one-parameter group (the group laws of both accumulated occupancies along the two-site flow); echoEvolve, echoEvolve_self, echoEvolve_comp, ★ echoEvolve_shift — the two-time law and autonomy;
★★ echoFlow_measurePreserving — Liouville: a skew product over the two conserved angles, assembled from three measure-preserving steps (measurePreserving_cshift_mid, the translation of one circle coordinate by a function of the others, used once per pointer; the unitary kick);
★★ echoEnergy_echoFlow — the joint energy is conserved; echoFlow_angles — both θ₂ conserved; ★★ hasDerivAt_echoAmountA, hasDerivAt_echoAmountB — the equations of motion;
echoFlow_msgPoint_zero — message 0 fixes both pointers; ★★★ echoAmountA_msgPoint_one, echoAmountB_msgPoint_one — the two pointer displacements in closed form: A turns by r_A (t − λ (ωt − sin ωt cos ωt)/(2ω)), B by r_B λ (ωt − sin ωt cos ωt)/(2ω), with λ = (J/Ω')² the transfer factor and ω = Ω'/ħ; ★ echo_return_ray — the carrier is back at A at the detuned round trip; senderDrift_hopTime, sender_rate_hopTime, receiver_rate_roundTrip — the plateaus: at the detuned hop time pointer A has turned by r_A τ (1 − λ/2) and moves at the rate r_A (1 − λ), the loss alone; at the round trip pointer B is stationary;
thirds, senderRead (pointer A read in thirds), receiverRead (pointer B read on #194's display arc); ★★★ emissionRecord, receptionRecord, returnRecord — the three writes as ActualOccurrences on the one arena (#191), sites {A}, {B}, {A}, from the start 0, with write intervals and windows derived from the occupancy integrals (emitStart, emitEnd, returnStart, returnEnd, senderWindow, and #194's writeStart/writeEnd/retention for B), under the explicit hypotheses: both pointers ready, both rates positive, the slowness conditions 3 r_A λ ≤ ω and 3 r_B λ ≤ 2ω;
★★ emission_return_disjoint — the return arc is disjoint from the emission arc, and the return record is written after the emission record; ★★ sender_returns, receiver_returns — the retention is finite for all three; ★★★ autonomous_echo — the package.

What is and is not claimed #

⚠️ Hamiltonian generation on ℂℙ³ × T² × T² is #202's second half. This file builds the dynamics the Hamiltonian prescribes and proves it is a two-time, measure-preserving, energy-conserving flow whose pointers obey the stated equations of motion and whose ray obeys Schrödinger's equation with the state-dependent generator (hasDerivAt_twoDetunedFlowVal).
The arc arithmetic uses the band |D_A(t) − r_A t (1 − λ/2)| ≤ r_A λ/(4ω) of the sender's displacement about its mean growth, and the record windows are where the band sits inside an arc. The plateaus are stated exactly but are not what the windows rest on, so the windows are valid for every λ ∈ (0, 1], the loss entering only through the band and the mean rate.
With λ < 1 pointer A never fully stops: its rate at the hop time is r_A (1 − λ). This is the detuning loss, read at A.
No clock is in this arena. The round-trip radar read entirely from A's records is #200.

References: specs/BACKLOG.md #199, #198, #200, #194, #191, #39; CV/TwoSiteDetuned.lean, RecordLayer/ActivatedWrite.lean (detuningAt, rateAt, pointerDrift, displayArc, writeStart, writeEnd, retention), RecordLayer/RecordOccurrence.lean.

The autonomous echo arena: emission, reception and return as activated writes #

What is and is not claimed #

The arena and the couplings #

A generic Liouville step: translating one circle coordinate #

The dynamics #

Continuity and Liouville #

The three steps of the echo flow #

Energy and the equations of motion #

The two messages #

The plateaus #

Bounds on the sender's drift #

The sender's write intervals #

The readouts #

Reading the pointers after a shift from the ready arc #

The three writes #

The package #