[DreamingOfClouds]

The same R480 word as #503, priced at nine stages instead of eleven. The word does not change. It is #493's R489
with its nine totals on dead registers (#503). Pricing it in the nine-stage whole-residual hosted pipeline gives:

	stages	bit c
#493 (R489)	11 (m = 78, 8 pooled replicas)	0.001565442026981656908596
#503 (R480)	11	0.001574753352953808650916
this PR (R480)	9 (m = 64, 32 replicas)	396861776523137034277/250000000000000000000000 = 0.001587447106092548137108 (+1.4057% over #493, +0.806% over #503)

κ does not change, because the side that binds is the complex side. With #484's complex b (draft; Lemmas A and G) κ =
662515748707749/500000000000000000 = 0.001325031497415498. With #471's it is 0.001303530097714685, and with #458's
0.001295523322678044. The bit gain is headroom.

Why nine stages win here. On the eleven-stage schedule the even-stage rank-7 idle children cannot host any width-8
residual, so they are paid in full. With only 480 helpers, nine stages host every residual whole: 138,240 = 9 × 32 ×
480, with 0 banks and maximum child 8. The smaller window m = 64 then prices lower. R489 at nine stages through the
same chain is 0.001576648413160438658485 (+0.716% over #493). On R715/R740, eleven stages were best.

What is in research/e8-r480-nine-stage/

• verify.py. Parent: Bit word R480: totals on dead registers, c = 0.001574753352953808650916 (+0.595% over #493) #503's verifier (--light, or full with Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493's verify.sh). Then:

  • nine-stage formal columns: columns_k, KPAIRS = 4; 1,440 columns; 18/18 controls; payload 102 bits < 104;
  • nine-stage template: 288 invocations, F2 column replay;
  • parent audit: 17 mutations rejected;
  • Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442 hosting, rediscovered exactly: all residuals whole, 0 banks;
  • independent hosted audit;
  • nine-stage bit invoice, both moment engines, and Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493's outer loop: 47 strict inequalities and controls with
    Complex supplier b ≈ 1.327·10⁻³ (+4.7%) on whole-residual hosting: shared-donor kernel and entrance gauges with paid residual wrappers (conditional on two lemmas) #484, Conditional kappa = 0.001274548075945587: full-nine-axis kernel normalization and same-owner spectator borrowing #471 and Conditional κ = 0.001274032589159993: complex kernel with paid residual adapters #458.

• vendor/: the nine-stage pipeline, byte-identical to PR Conditional κ = 0.001283479303751122: E8 bit word R715 (carrier lower bound + F2 overlap cancellation) with #458's complex supplier (+0.711% over #460) #463 at cdf0c1c (git blob ids in SOURCE.json,
  recomputed). It is not on main; it is in open PRs Conditional κ = 0.001283479303751122: E8 bit word R715 (carrier lower bound + F2 overlap cancellation) with #458's complex supplier (+0.711% over #460) #463 and Conditional κ = 0.001347744547714835: eleven-stage cross-copy hosting with target mixing (+1.278% over #448) #450. The lineage is Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443, Conditional κ = 0.001246073505487942: R740 replicated kernel with idle-data hosting #444, Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442 and Compressed nine-stage E8 windows and whole-block banking for a conditional κ = 9.98243154231047e-4 > 2^-10 bound #386.

• code/assemble9.py: Conditional κ = 0.001283479303751122: E8 bit word R715 (carrier lower bound + F2 overlap cancellation) with #458's complex supplier (+0.711% over #460) #463's nine-stage invoice verbatim, plus Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493's per-supplier loop verbatim.
  code/adapt.py: two listed edits to audit_parent_k.py.

• code/findings/: the checks for findings (a) and (b) below (flowlp2.py, levels.py, piv0.py), run by verify.py.

Findings on the R489 family (knowledge)

Checked by verify.py:

• (a) An LP floor of 480 for R480's skeleton. The model is a min-flow LP on the frame lattice of one 4-frame. Its
  dual is a max-weight antichain. Node bounds come from the junk-dimension lemma: a cross cell needs ≥ 3 registers
  and a matching cell ≥ 4. The optimum is 16 helpers per 4-frame, so 480 (8 without the junk bounds). It assumes 3
  branches per source, 2 transverse quads per cell, and matching-cell sources that feed the block frame. R480
  attains every node bound. Each of its 480 helpers stands at exactly one cell and one D6 frame, so levels 2 and 6
  are tight. verify.py checks the LP optimum given these node bounds, and the tight levels on R480's own data. The junk-dimension lemma and the flow model itself (that every word of the family is such a flow) are argued here, not machine-checked.
• (b) Kernel pivots on R480 and R489 are exactly 0 for coordinate and root-line entrances: all 120 line classes
  have nullity 0. All residuals stay width 8.

Exploration outside the package, not run by verify.py:

• (c) The helper ledger is at its floors: 240 carriers at the 4-design bound, and 240 further registers at the
  per-cell junk count, with every shared symbol at its Dilworth width. The copy-removal methods built on the
  κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370-lattice words (read-then-climb, XOR fusion, Bcw inlining) have nothing to act on.

• (d) Idle fusion has no legal glue:

  • an X→Y bridge add mixes frames at rank 1 in Conditional κ = 0.001263365464781156: eleven-stage R740 hosting with complete row bound #445's geometry (X wires contain the line q, Y wires do not);
  • with X→X/Y→Y shears only, a Gröbner test finds no glue at K = 2..5.

  Checks that compare only pair levels miss the X/Y frame difference. Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493's own audit hard-codes X→X/Y→Y, so it is
  safe.

Scope and review points (also in the README)

• The emitter is the per-replica nine-stage one (Conditional κ = 0.001246073505487942: R740 replicated kernel with idle-data hosting #444/Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442 lineage, 32 replicas in groups of 8), not Conditional κ = 0.001292099559442278: R715 shared kernel, target mixing and whole hosting #472's pooled
  eleven-stage emitter. Its finite prefix is Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443's per-stage product bound (193 bits), not Conditional κ = 0.001292099559442278: R715 shared kernel, target mixing and whole hosting #472's chronological
  bound. The prefix enters only the cutoffs.

• One audit adaptation. The template control "omit a residual route factor" needs an isometry that has a route
  factor. R480's helpers all enter at frame zero, so no isometry has one, and the selector raised StopIteration. The
  control is vacuous then and is skipped exactly in that case. The other 17 controls run.

• Mutable sources at nine stages. Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493 grounds source-register writes below the full frame in Conditional κ = 0.001292099559442278: R715 shared kernel, target mixing and whole hosting #472's PROOF ("Shared
  physical labels"). The nine-stage address map is Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443/Conditional κ = 0.001246073505487942: R740 replicated kernel with idle-data hosting #444's, whose proof does not discuss source destinations.
  What carries over:

  • κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425's four word-level hypotheses, checked on R480 by Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493's interface.py in the parent run;
  • the same local invocation and Compressed nine-stage E8 windows and whole-block banking for a conditional κ = 9.98243154231047e-4 > 2^-10 bound #386 frame operators;
  • the exact F2 replay of every template column, including the source ports' data families.

  The realisation sentence for Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443's address map is stated, not re-proved. Please treat it as a listed conditional
  interface, as Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493 does.

• Everything else is inherited from Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493 and Bit word R480: totals on dead registers, c = 0.001574753352953808650916 (+0.595% over #493) #503 unchanged. Complex savings are cited. No unconditional theorem is
  claimed.

Credits

eumemic (#493), #503 (R480). Nine-stage pipeline: chafreaky (#443, #444, with OpenAI Codex), DaysSky (#442, #381),
Zhihao Chen (#386), DreamingOfClouds (#315; the #450/#463 pipeline adaptations), Dylan Crow (#370).

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Integrated on main in 9449300, with original commit history and individual credit retained.

Your nine-stage R480 supplier is integrated and now supplies the binding bit rate in the selected composition with icekylinx’s #535: κ = 0.001584931111785787. The fresh replay covers 1,440 formal columns, 138,240 whole hosted residuals, physical audit/mutations, finite invoice and both exact moments. The unchanged #493 foundation was reused with matching source hashes.

Maintainer review and exact scope · Contributor record. The multiplication claim remains conditional on the inherited all-size/realization/analytic interfaces; these checks do not establish a complete unconditional theorem.

Thank you for contributing this work.