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Prove exact signed idle completion for overlapping stage outputs (no κ) #6391

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110 changes: 110 additions & 0 deletions research/signed-overlap-idle-completion/PROOF.md
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# Signed whole-idle completion of an overlapping stage demand

This scoped proof about a determinant relaxation was produced by Nick Petalas.
It is not a construction of the proposed caller.

## Formal contract

Work over K = Q(q,A,B,C), with algebraically independent stage variables
A,B,C. There are six arbitrary independent main fields. Helpers are autonomous
outputs outside these six fields and cannot be mixed into them. Use each of
three full calls exactly once. The A call is A times a determinant one query
on a two-dimensional subspace, and identity on its complement. Define the B
and C calls likewise. Query orientations and chronology are unrestricted.

Any number of invertible maps over Q(q) are free in this relaxation, including
constant field mixing and determinant one thin queries. Their physical costs
are not declared zero. A paid whole-idle atom chooses a nonempty subset S of
{A,B,C}, a sign epsilon in {-1,+1}, and one field line. It multiplies that line
by (product of the stages in S)^epsilon. Constant conjugation is allowed.
Each stage has eight fresh axes. An atom costs H(8|S|), including an inverse
atom. There are no other stage-dependent operations. Every word is finite.

The desired stage multiset has two ABC outputs, two AB outputs and two BC
outputs. Central query factors may remain on the first pair, with
stage-independent determinant. The target determinant has stage degrees
(4,6,4). The three full calls have degrees (2,2,2). Each atom contributes
epsilon times the indicator of S. Constant mixing cannot change these degrees.
Every exact word therefore requires the signed idle balance

sum_atoms epsilon * 1_S = (2,4,2). (1)

This necessary condition does not assert that every satisfying inventory
has a legal query endpoint.

## Exact completion theorem

Let H(0)=0, with H positive on (0,24], nondecreasing and concave on [0,24]. Among all
finite signed inventories satisfying (1), the minimum cost is

2 H(24) + 2 H(8). (2)

Two positive ABC atoms and two positive B atoms attain this value in the
inventory relaxation. If H is strictly concave, this is the unique minimizing
signed subset multiset. Positions and conjugating maps are not classified.

Write a=H(8), b=H(16), c=H(24). Give the coordinates dual weights

y_A = y_C = (c-a)/2, y_B = a.

These are nonnegative. Each nonempty subset S satisfies y(S)<=H(8|S|).
For B this is equality. For A or C, c<=3a gives (c-a)/2<=a.
For AB or BC, midpoint concavity gives (a+c)/2<=b. For AC use c-a<=b:
the increment on [16,24] is at most the increment on [0,8], so c<=a+b.
For ABC there is equality at c.

A positive atom costs at least y(S). A negative atom costs at least -y(S),
since its cost is positive and y(S)>=0. Summing and using (1) gives
cost >= 2y_A+4y_B+2y_C = 2c+2a. The stated inventory attains this bound.
This proves (2) for arbitrary word length and arbitrary inverse calls.

Strict concavity gives c<3a, (a+c)/2<b and c<a+b. Only B and ABC are tight
positive subsets. Every negative atom has strictly positive dual slack.
Equation (1) forces exactly two of each tight subset, proving uniqueness.
Every power H(r)=r^s with 0<s<1 meets these conditions. No sampled exponent
or word-length cutoff enters this proof.

This is a three-coordinate application of the classical greedy dual for a
submodular set cost. The claimed new step is the signed caller completion
application and the following comparison.

## A strict improvement in the necessary budget

With one central axis and eight fresh axes per stage, the overlapping target
ranks are two 25 and four 17. Its optimal target-minus-idle reward is

R_overlap = 2H(25) + 4H(17) - 2H(24) - 2H(8).

A comparator with six ABC jobs has six rank-25 targets and deficit (4,4,4).
Its exact signed idle minimum is 4H(24). Uniform dual weights H(24)/3
certify this, since k H(24)/3<=H(8k) for k=1,2,3; four ABC atoms attain it.
Thus R_uniform = 6H(25) - 4H(24). For every strictly concave H on [0,25],

R_overlap - R_uniform
= 4H(17) + 2H(24) - 2H(8) - 4H(25) > 0. (3)

For an explicit proof, start with (8,8,25,25,25,25). Twice replace a pair
(8,25) by (16,17), then twice replace (16,25) by (17,24). The result is
(17,17,17,17,24,24). Every replacement preserves the sum and strictly
increases the sum of H by strict concavity. Both lists sum to 116.
For a linear tariff the rewards agree.

The comparison cancels identical full-call/helper and other charges between
two hypothetical contracts. It does not assume equal thin-query, ordinary,
storage or compiler costs. Any difference must be subtracted separately.
Neither six-field comparator is supplied as an operator word here. Equation
(3) is an improved optimistic budget, not a larger kappa or feasible supplier.

## Applicability and remaining obligations

The formal Laurent stage identity is an explicit hypothesis. A finite-order
Gaussian may satisfy additional relations. This degree condition is not an
impossibility theorem for every specialized native implementation.
Stage-dependent free maps, helper feedback, another rank/support relation,
changed native multiplicities or changed output contracts need new arguments.

An attaining inventory need not be placeable at legal chronological cuts.
The exact query coefficients, independent output map, native frames and
absolute phases, arbitrary helper preservation, owned erasure, thin-query
count, precision, stock, tape movement, compiler and outer costs remain open.
No selected-frontier or complete multiplication theorem follows.
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# Exact signed-idle bill for an overlapping stage contract

Scoped proof with exact finite corroboration. No new kappa is claimed.
This contribution was produced by Nick Petalas.

For six independent main fields with stage sets ABC, ABC, AB, AB, BC, BC,
and one native query-pair use of each stage, the determinant deficit is
(2,4,2). Allow arbitrary constant invertible mixing and forward or inverse
whole-idle calls on any nonempty stage subset. The exact idle-inventory
minimum, for every nondecreasing concave tariff H with H(0)=0, is

2H(24) + 2H(8).

For a strictly concave tariff the unique optimal signed subset multiset has
two positive ABC calls and two positive B calls. The proof covers all finite
inventories, with no search cutoff. It does not construct a chronological
caller with that inventory.

The corresponding target-minus-idle reward strictly exceeds the uniform
six-ABC completion reward for every power tariff H(r)=r^s, 0<s<1. The exact
positive difference is

4H(17) + 2H(24) - 2H(8) - 4H(25).

Four explicit balancing transfers prove this sign for the whole interval.
The comparison cancels identical full-call/helper charges. Thin queries,
ordinary work and other changed costs still need separate bills. This is a
strict improvement of an optimistic necessary budget, not a physical
construction, paid-profile improvement of the selected supplier or bound on
all possible native callers.

[PROOF.md](PROOF.md) specifies the formal Laurent-variable contract, proves
the determinant requirement, supplies an exact signed dual, classifies
optimal inventories and proves the reward comparison. A specialized
finite-order native action need not satisfy the formal identity hypothesis.

## Reproduce

From the repository root, Python 3.12 and the standard library suffice:

```sh
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/signed-overlap-idle-completion/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/signed-overlap-idle-completion/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/signed-overlap-idle-completion
```

Another host may substitute its CPython 3.12 executable. The scientific marker
is `PASS_SIGNED_OVERLAP_IDLE_COMPLETION`; the launcher adds
`PASS_REGISTERED_RESEARCH_PACKAGE`. Replays read and compare the generated
[RESULT.json](RESULT.json), including hashes of every source, proof and
metadata file in this package. Explicit `--write-receipt NEW_PATH` generates
an additional receipt and refuses an existing destination.

The verifier checks 42 signed subset/dual inequalities on the three extreme
nondecreasing concave cardinality tariffs. Those tariffs generate the full
three-point cone by nonnegative combinations. It separately checks 100 hinge
inequalities for the four balancing transfers, ten independent exact bounded
lattice shortest paths on five tariffs and two deficits, an attaining
inventory, the determinant degree debit and six rejecting controls. The
lattice search grants all fourteen forward/inverse subset moves. Its cube
[-4,4]^3 is finite corroboration only; the written dual proves the
unbounded-inventory claim. No predecessor executable or scientific suite is
run as this contribution.

The Python verifier has no recursion and uses explicit finite loops. The
shortest-path oracle has 729 vertices and fourteen directed moves per vertex;
at most 10,207 pushes and 10,935 pop attempts are allowed per run. Capacity
errors fail closed. Functions stay below sixty lines. Explicit exceptions
preserve acceptance under optimization. For Power of Ten rule 3, Python and
Fraction require runtime allocation; this bounded command has at most 729
distance entries, 729 settled entries and 10,207 queued entries. It owns no
persistent resource beyond optional exclusively created receipt output.
No stricter static analyzer configuration or production code is changed.

## Provenance and credit

Base main is `7c49fe32c55de02c2c873655c804b16e4c3fef0a`, priority revision 8.
The selected-source positive control passed. [TASK.json](TASK.json) records
an explicit change from constructing a complete caller to solving its signed
idle-inventory relaxation. [SOURCES.json](SOURCES.json) pins the catalogue,
selected record and predecessor proof as historical provenance; there are
no external scientific runtime inputs. Source authentication is not review
of predecessor mathematics.

[Ali Romman's released overlapping AB/BC proposal](https://github.com/CrocSwap/integer-mult-bounds/discussions/5593)
supplies the motivating changed demand. His [PR6348 proof](https://github.com/CrocSwap/integer-mult-bounds/blob/9894015d6f380206f990ad5fd14bc49bbd26b0d0/research/balanced-query-tree/PROOF.md)
supplies the full-query, eight-fresh-axis and whole-idle conventions. Cinna
Davis supplies the thin-query/controller interface. DreamingOfClouds and the
retained P104 helper, native, compiler and outer authors retain their credit.
Submodular greedy duality, determinant multiplicativity and concavity are
classical. The new result is this signed completion theorem and strict
comparison for the changed overlapping contract. The predecessor module,
native realization and tree are not reverified or optimized here.

The [advance claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19358970)
reserves this determinant-level question. Searches included live open PRs,
recent claims, local reservations, merged keyword results and current-main
README text. Sparse cached notes and unpublished work remain coverage limits.
The proof adds no literature-wide novelty claim.

Independent human and formal proof-kernel review are unclaimed. New files
are Apache-2.0 under the repository license.

## Open next questions

1. Can any six-field chronological word realize the overlapping query and
Gaussian outputs at the certified idle minimum? First solve the exact
Laurent endpoint with all six independent inputs, then price its thin
queries. The determinant inventory does not settle query coefficients.
2. If minimum-idle placement is impossible, what additional paid inventory
is necessary? Start with the four tight atoms and prove or falsify their
compatibility with three full-query stages. No placement obstruction is
supplied here.

Native frames, absolute phases, autonomous helper preservation, owned
erasure, ordinary operations, precision, stock, tape, compiler and outer
funding remain open. The five selected framework assumptions retain their
status. No selected cold replay, native execution, Lean, Docker or complete
rounded multiplication proof was run.
167 changes: 167 additions & 0 deletions research/signed-overlap-idle-completion/RESULT.json
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{
"concave_transfer_checks": 100,
"finite_shortest_paths": [
{
"optimum": "4",
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"rejecting_controls": 6,
"scope": "Scoped proof and finite corroboration; determinant relaxation only.",
"signed_dual_inequalities": 42,
"source_sha256": {
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},
"status": "PASS_SIGNED_OVERLAP_IDLE_COMPLETION"
}
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