[DreamingOfClouds]

Claim. Conditional κ = 0.00339993 (raw crossing 0.00339993013239573), up from 0.00337180047549672 of #6833 by +2.813e-5, +0.834%. Package research/p104-hosted-chain-root. Not claimed as a frontier change.

Conditional on the five framework assumptions of #6833 and on one more, A6, owned ascent, which is load-bearing: no ascent, no word. A6: an owned nested rise costs one paid H_r plus rank-0 adapters, with no credit, on arbitrary contents, factorwise at every width, with CLEAR at any frame under owned_erasure. The word has 23 owned rises (rank mass 56). Both verifiers, the rejection control and the direction-free cost argument are in PROOF-OWNED-ASCENT.md. The P112 native declares owned ascent (native_extensions.py:126, NATIVE-ADMISSION.md:249; #2479). The nearest text, notes/copied-centers-lemma.tex:58-62,71, covers a copy started by COPY; this word accumulates by ADD first, so A6 goes further than that text. Correction: an earlier version said the compiler text #4268 was unavailable; it is on main at research/lever-mixed-targets-2979. It does not condition its kernel count on direction (05-layers.tex:83), and allows paid jobs on correlated owned fields (FINITE-PRECOMPILATION.md:258-268). The P112 backward word does ADD first and then rises, 8 per call. No sentence decides how this word goes further (3 lifetimes read before the rise, 3 with contacts between rises, nonzero start frames, both directions in one word), so A6 stays a declared assumption.

What changes against #6833.

• Root: every one of the 18 stages hosts a pair of units, t = -1/2, N = n + 2, S = 656 + 48·18 = 1520. Exact, dyadic, max entry 3, irreducible.
• Native: the Conditional κ = 0.00330823 (+9e-8 over #6805): lift two helper inputs onto one shared carrier #6824 word run verbatim, preceded and followed by an exact contragredient mirror on fresh pads (R^-1 = F^T for any F), with 128 helpers shared. On each of the 104 ports X = x - y, Y = (x + y)/2, and every helper returns to the identity.
• The gain is a small difference of large terms: new = 2 × base + 29753.6 (hosted word) - 29741.0 (36 whole H8 idles per port), net +12.59 at the crossing.

Scope and caveats.

• κ rests on pricing the 36×104 hosted-stage idle jobs as whole paid H8 idles, which the line construction attains exactly; priced as eight rank-1 jobs each, κ would be 0.00298.
• Margin to the retained BIT pin 0.0034 is exactly 6.993e-8 at the tested price 0.00339993007, so nothing above 0.0034 is claimed.
• Predecessor literals are exceeded and declared: 172,166 events, public 1403, owned roles 51, 54 lifetimes, peak 13, coefficient height 9/8 bits, and the per-call check per_call < 10^7 (11,928,436). Root checker rules are relaxed for this family (N = n + 2, two units per stage), and the SPEC-level identities are re-proved in root_check.py.
• Raw total: 948,952,288 (literal) and 983,664,136 (actual per-call work) are below 10^9; the review's conservative per-call count gives 1,019,551,024, 2% above. The 10^9 is a parameter of the retained precision and tape contract, which no outer-ledger term reads, and that contract is not re-examined here.
• The endpoint literal changes: the hosted target replaces X = x, Y = x + y on the 104 main pairs.
• Direction-based lower bounds (P104 network-coded owned carriers: stops, shedding and pair merges do not lower owned paid 56 #6315 Lemma S, Prove confined two-clique P104 relays need eight paid units (no κ) #6229, Prove the exact reader-497 single-merge cache floor (no κ) #6257, Account the P104 invoice surplus as 48k − ports with private-descent mass, and fix the E8 structure of the 104 ports (no κ) #6831, the c >= 56 input of P104 surplus bound, hub rank law and form-capacity lemma with a hypothesis ledger (no kappa) #6881) are scoped to words whose owned registers only descend.
• Caller physical replay for HOST is included in caller_physical.py (contact phases, reverse cross term, whole-idle rank and axes, field map with aliasing control; swapped chronology and deleted idle rejected). It is vendored from the open Conditional κ = 0.00337180 (+6.36e-5, +1.92% over #6824): replace the supplier tree by a chain-nested root #6833 package, with one declared line changed for owned ascent.
• The 82 LAMBDA events, inherited from open Conditional κ = 0.00330720 (+3.000e-6, +0.09079% over #6716): two paid idle fragments unlock independent outputs #6726 and Audit the two-fragment caller’s physical contacts and complete invoice (no new κ) #6763, are checked algebraically at four characters and counted in ordinary work, but have no physical contact check; IDLE_ONE and LAMBDA are open lineage, not on main, and their admission is a review question.
• Not independently examined: an adjoint replay, and tensor width 3, where only widths 1 and 2 were run and width 3 follows from the ring replay. The physical replay uses one prime and one random input per run; separate runs on two primes agree. The inherited precision-tape and compiler assumptions are not discharged.

How to verify. python3.12 -B scripts/research_package.py --package research/p104-hosted-chain-root prints PASS_REGISTERED_RESEARCH_PACKAGE (about 75 s, standard library). It runs check.py (word replay, verifiers with rejection control, root, composition with wrong-gauge and wrong-pair controls, invoice with #6833 rebuilt by the same code, both interval engines, outer ledger), physical.py (all events as 512-component streams, tableau on all 104 ports), , ordinary.py (raw recount under all three counts) and caller_physical.py (caller replay).

Questions for review. Is A6 accepted for the native verifier? Are IDLE_ONE and LAMBDA, including whole H8 pricing of the hosted-stage idles, accepted by the compiler contract?

Credit: #6833, #6726 (zxv), #6716 (thu-nmrc), CubeJerry (#6805), Ali Romman (#6349, #6353, #6372), Jacob Alan Sussman (compiler), #6702 and #6824.

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