Research note: staged544 width-unit prices, the exact staging break-even in the fixed-T360 ideal (no new κ)

[thakrarsagar]

Statement

This is a scoped exact result about PR3815's clock-free, fixed-bin ideal of the staged544 T360 template (PR3734). It makes no κ claim. It is not a construction and not a complete multiplication theorem.

Model, per owner replica:

• 9 calls of 408 helpers, with hosted widths {4:26,5:2,6:18,7:53,8:96,9:213};
• idle bins {8:672,16:448,32:448,48:448};
• local profile {1:2440,2:725,3:224,4:102,5:20,8:8};
• 896 live rank-73 families.

All prices are in F units per replica, evaluated at a_hi = 0.001730578437266246818741. That is the upper end of the staged544 ceiling bracket [...740, ...741], which reproduces PR3815. A "width unit" is one helper-call moved one width step.

1. Exact width-unit prices, within basis ranges.

   move	price per width unit	exact for up to (helper-calls per replica)
   9->8, 8->7, 7->6, 6->5	(h(48)-h(3)-5h(5))/20 = 0.987736781185353068089966	28 / 30 / 28 / 28
   5->4	h(1)-h(0) = 1	18
   4->3	h(5)-h(3)-1 = 0.991791369039984911721564	324/5 = 64.8

   Outside a range, the price is a lower bound for every further unit.

2. Staging screens. Under the observed per-call conservation (item 4), a literal staging must also change the local profile. The net change in F per width unit is (local saving) minus (price).

   • Documented signature {1:+3,2:+1,3:-2} (PR3605's 3H1+H2-2H3). It saves 0.991001081580029037511288 per unit, so the gain is:

     • +0.003264300394675969421322 for widths 6..9;
     • -0.008998918419970962488711 for 5->4;
     • -0.000790287459955874210276 for 4->3.

     All three observed units are 8->7 moves, so for other widths the signature is an assumption.

   • Hypothetical rank-1 deletion (a reference only; no realization is known). The gain is at most 0.012263218814646931910033 for 6..9, 0 for 5->4 and 0.008208630960015088278435 for 4->3.

3. General screen (*). By weak duality, any changed inventory over widths 1..9 with the same bins, and any changed local profile, satisfies F(a_hi) >= 9 sum n'_r h(r) + 9 sum pi_w s'_w + sum mu_c B_c - 896 h(73). For staged544 the right-hand side is 5.28e-20 > 0.

4. Per-call conservation (observed, not derived). Local paid rank minus hosted width is 1856 on both pinned (inventory, local) pairs. PR3605 states the same conservation for its staging.

5. Monotone-price lemma. A free width reduction with the local profile unchanged keeps the ideal ceiling below a_hi. This is a consistency statement about the model, not a staging closure, because item 4 makes such a move unrealizable.

6. PR3682 -> staged544 split at a_hi.

   • Width part: +8.889631030668177612809695088162, which is 9 x the price.
   • Local part: -8.919009734220261337601597266358.
   • Net: -0.029378703552083724791902178195, which is the documented-signature gain per call.

   The interaction term is identically 0 because F is additively separable in the two changes.

To first order (dF/da ~ 184132, floating point), one more width-6..9 staging in all nine calls with the documented signature raises the ideal ceiling by about 1.6e-7.

Milestone and decision informed

Milestone: a local module. It gives exact prices and screens in the ideal model.

Decision: expansion-gate point 4 for the staged-entrance family (canonical thread #3606), the engine of #3605 and #3734. It also partly answers PR3815's open question 1. This family is not a revision-6 lead.

• Under the documented signature, only width-6..9 staging pays. Over k such units in any mix, F(a_hi) >= 5.28e-20 - k x 0.003264300394675969421322.
• 5->4 staging cannot pay even with a hypothetical rank-1 deletion.
• Each candidate staging is decided by one exact inequality: its local saving must exceed the price.

Base, pins and reproduce

• Base: canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90. SOURCES.json pins certificates/selected-result.json and docs/research/agent-task-catalog.json.
• Predecessor inputs: taken from unmerged PR heads Stage helper544 after a returning producer excursion; improve conditional kappa #3734 fa041037, Research: certify the staged544 packing ceiling and reach it on PR3734's T360 clock (no new κ; same value as #3802) #3815 77ffbf28 and Consume retimed staged releases with smaller T360 stock #3682 431e6c60. They are recorded by sha256 in the README and in CERTIFICATE.json (predecessor_sources). The verifier does not byte-check them; this is a stated gap.

python3.12 -B scripts/research_package.py --package research/staged544-width-lever-screen
# or: python3 -B research/staged544-width-lever-screen/verify.py

The verifier is standard-library only, uses exact rationals and runs in about 6 s. Its output was identical under Python 3.12.15 and 3.14.4.

Expected output: a JSON report, then PASS_STAGED544_WIDTH_LEVER_SCREEN; the runner then prints PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed. The recorded output is evidence/VERIFY-OUTPUT.txt.

Certificates and negative controls

Dual certificates. Both are checked exactly on all 318,840 bin patterns with helper widths 1..9:

• the cheapest extension (pi_1..3 = pi_4), used by the lemma; a slack lemma also derives it from PR3815's 2,551-pattern dual;
• the tight extension, used for the 4->3 price.

An unbounded-knapsack recursion independently reproduces the minimum reduced cost. The 4->3 move is checked as an explicit four-column path: all its columns are tight, its net direction is e3-e4 with zero net change on every bin row, and its range is exactly 324/5.

Controls (each must fail; the verifier asserts that it does):

• pi_9 + 1e-30 makes the dual infeasible;
• the tight pi_3 + 1e-30 makes [3,9^5] infeasible;
• a non-monotone extension is rejected by the shared monotone check;
• one 8->9 widening per call drops the bound to <= 0;
• a +1 histogram change fails the shared paid_rank_ok check.

There is also one consistency check: the bound at a_lo is not positive.

Second, independent method: crosscheck_highs.py (float; numpy 2.5.4, scipy 1.18.1 HiGHS; not registered). It generates patterns separately, uses explicit reduction columns and bisects the root. It agrees with the exact unit prices to about 3e-12; see evidence/HIGHS.json.

Scope, assumptions, unresolved obligations

• The model is the clock-free fractional ideal with fixed idle bins and PR3815's payment rule.
• The per-call conservation and the staging signature are observed, not derived or proved forced.
• Prices are exact only within the stated ranges. mu gives subgradient bin prices only.
• All admission costs of a changed inventory are unknown, not zero, and unpriced: physical stock, coverage law and template checks, literal staging identities, and clock feasibility.
• dF/da and the ceiling conversions are floating point.

The package README lists three Open next questions:

1. Is the signature forced? Does PR3682's donor release survive in PR3734?
2. Can the conservation be derived?
3. What do bin-count changes offer?

Search record

This is a deterministic exact computation, not a search. The domain is every bin pattern with helper widths 1..9 in capacities 8, 16, 32 and 48: 318,840 patterns. The widths-4..9 subset (2,551 patterns) serves as a control. No seed and no stopping rule were involved.

Credit and what this adds

• PR3815 (ChaloupeMusic): the model, bins, payment rule, basis and basis dual (pi_4..9, mu), and the screening idea (its open question 1).
• PR3734 (dataisfire): the staged544 inventory and local profile.
• PR3682 (zxv): the predecessor inventory, the closed-form 8->7 width shadow and the staged-net calibration (price.py).
• PR3605 (RossRichardson): the staged-entrance mechanism, the per-helper delta 3H1+H2-2H3 and the rank-mass/width statement.
• Lineage runs through PR3795/PR3802 (hexapode).
• Method precedents: async505-inventory-wall (Rohan Arun), PR3630 (RossRichardson) and PR3638 (zxv).
• Concurrent: PR3836 (Matarener) on the idle alphabet, cited for the bin-template question.

What this PR adds:

• the dual read as width-unit prices: the common 6..9 price, 5->4 = 1 and an exact 4->3 price;
• the basis ranges;
• the widths-1..9 dual extensions, with the slack lemma;
• both staging screens;
• the conservation check;
• the exact PR3682 -> staged544 split.

This PR is not an audit or re-certification of any PR.

Claim: #3606 (comment)

Review provenance

Reviewed by four independent AI review passes in the same loop (reproducibility, mathematical audit, provenance, referee). Two of them (mathematical audit, referee) re-verified after the first revision; a second revision fixed only text and one verifier assertion. No human or formal-kernel verification is claimed.

Attribution

Prepared by Sagar Thakrar (@thakrarsagar) with substantial AI assistance from Claude (Anthropic, Claude Opus 5.5) via Claude Code, in an automated research loop. Apache-2.0.

🤖 Generated with Claude Code