Audit (no new kappa): the composed T488 profile is exactly PR3642's optimal packing vertex

[petalas]

What this adds

PR #3653's published paid profile for the T488 × retiming composition is
exactly PR #3642's published optimal fixed-capacity packing profile at
T = 488, coefficient for coefficient:

PROFILE_488 = 4392 * {1:2437, 2:724, 3:226, 4:102, 5:20, 8:8}
            +  488 * {1:234, 3:261/4, 5:57/4, 48:1531/4}

The second factor is PR #3642's ideal_idle_per_replica, the feasible primal
vertex of its 2,551-pattern relaxation dual. The certificate's own published
ideal_profile_at_T740 is reproduced by the same rule as a convention control,
so the identity is not an artifact of my reading of the model.

Because the local bill enters the moment additively and is fixed, the
moment-minimising packing does not depend on the bill. A construction already at
the vertex therefore stays optimal when the retiming changes the bill, and the
composition is at the vertex. Consequences, stated only at the strength the
cited certificate supports:

• The 2.4507952107748684e-8 root gap between PR Improve conditional T530 κ through two commuting release retimings #3642's literal T530 profile
  and its cap is closed exactly. That number is simultaneously PR Improve conditional T530 κ through two commuting release retimings #3642's
  published headroom and PR Compose retimed T488 suppliers: conditional kappa 0.00173041889976 #3653's claimed improvement over it: the whole gain
  is the fixed-inventory headroom PR Improve conditional T530 κ through two commuting release retimings #3642 had already certified, realised by
  PR Combine T37, T8 and T16: conditional κ > 0.00173009636352 #3633's T488 allocation.
• No packing inside the certificate's model (fixed capacities
  {8:672,16:448,32:448,48:448} per replica, widths
  {4:26,5:2,6:18,7:52,8:97,9:213}, retimed bill, dimension 73, live 896*T)
  can improve the composed κ. Further gain needs a changed word or bill,
  inventory, capacity set, or model — not more repacking of this one.
• At T = 488 the relaxation optimum is integer-attained (488 = 0 mod 4
  clears the quarters), so the vertex is a real child multiset.

This is a scoped ceiling statement. No new κ, no supplier admission, no
selected-frontier change. The optimality of the vertex is PR #3642's dual; I
re-executed its verifier on its pinned bytes and reproduced its receipt
(PASS_RETIMED_ALL_2551_FIXED_INVENTORY_PATTERNS, same bracket, same headroom),
but did not re-derive the dual.

Evidence

python3.12 -I -B research/t488-attains-retimed-packing-cap/verify.py
# PASS_T488_ATTAINS_RETIMED_PACKING_CAP checks=26 controls=7 identity_exact=true cap_attained=true no_new_kappa

26 checks, 7 controls, stdlib only, about 0.2 s: the exact identity; the T740
convention cross-check; rank mass 65088*488, live 896*488, physical
437757; the residual decomposition and per-replica idle mass 18873; capacity
feasibility; the moment bracket straddle at both published 10^-24 endpoints
with an 80-digit Decimal engine plus positive derivative and a one-grid-step
margin; the released 70-digit string; the headroom identity to one grid step.
Controls include the T530 literal residual (not at the vertex, two ways), the
pre-retiming bill, a wrong replica count, a one-unit rank-1 shift, swapped
endpoints, and T = 490 (fractional). RESULT.json is regenerated and compared
byte-for-byte by the verifier.

Local preflight, CPython 3.12.15 (the main checkout's
.venv/bin/python3.12):

• scripts/research_package.py --package research/t488-attains-retimed-packing-cap → PASS_REGISTERED_RESEARCH_PACKAGE outcome=completed
• scripts/check_research_commit.py --package research/t488-attains-retimed-packing-cap --check-source-history → PASS_RESEARCH_COMMIT_BYTES at c7386166f445ae47bbe04e87bc08d1ccf6ae4814, source-history commit 20ecf2891

The registered command runs in well under the CI timeout.

Exact sources (hash-recorded, not vendored; base 20ecf2891):

Source	Head	File	SHA-256
PR #3642	6568db75	packing-cap/RETIMED-PACKING-CAP.json	6ca9271b…4a6b0f3
PR #3642	6568db75	packing-cap/check_retimed_cap.py	63ae34f2…ec6a98e
PR #3653	607a09ff	retimed-t488-composition/PROOF.md	7a010afb…5fd04b

What is not established

• No word, operator, owner journal, native provider, admission stage or
  fresh-bootstrap wrapper is replayed. The certificate's scope excludes
  chronology and physical-family attainment; that remains PR Compose retimed T488 suppliers: conditional kappa 0.00173041889976 #3653's own
  owner-audit obligation.
• The dual optimality is inherited from PR Improve conditional T530 κ through two commuting release retimings #3642's certificate, re-executed but
  not re-derived. Re-deriving the 2,551-pattern dual independently is the first
  open question in TASK.json.
• The published headroom rational and the difference of the two published
  decimal floors differ by exactly one 10^-24 grid step; both are reported.

Complements, without duplicating, the independent arithmetic audit in #3740
(same family, different layer: that one reproduces profile → root → released
value; this one identifies the profile as the certificate's optimal vertex).

Credit

roger-sato's PR #3642 supplied the retiming, the fixed-capacity certificate, its
dual basis and the published headroom; rohanarun's PR #3653 supplied the
composed paid profile and released endpoint; hexapode's PR #3633 supplied the
T488 allocation that reaches the vertex; Ross Richardson's PR #3605 and #3630
supplied the staged suppliers and the exact dual basis; PR3531 supplied the
positive-binomial interval engine. The exact identity, the bill-additivity
observation, the T530 negative control and the seven controls here are new.

Produced by Nick Petalas with AI assistance; independent human and formal
proof-kernel review are unclaimed. Apache-2.0.