Research: independently bind QQ16 consumer budget to the first F0 caller

research/qq16-consumer-budget-independent/PREDECESSORS.json

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+{
+  "audited_pr": 3908,
+  "head": "0d8880e38ed7c88d70eec6a48e5aa829d0fb22f3",
+  "author": "1337isnot1337",
+  "archive_sha256": "9ccaf27da52f96a5701d8d7f9e9934a64f8bbc15f0aa12bd45c7168785d6966d",
+  "files": [
+    {
+      "path": "research/qq16-consumer-cap-audit/PR3181-README.md.gz",
+      "sha256": "c9ebe8149ff8cc0dc4f67bc52fd6bef822ba99a8a803bbddba0eb99c032cedde",
+      "git_blob": "14779caff5262387a6fb42e284f94524b8c66550"
+    },
+    {
+      "path": "research/qq16-consumer-cap-audit/README.md",
+      "sha256": "b62dfba1a486ee0e28d799abe0c95f0a72e1942fd9ddcf2ed51c0ead780f579e",
+      "git_blob": "d4606ff57f84566b1fd586dd02a192d66d9da895"
+    },
+    {
+      "path": "research/qq16-consumer-cap-audit/SOURCES.json",
+      "sha256": "d48c7151a4553351d4f9de58eaa39f5660591fb3a84edc4aee3978bdefec2323",
+      "git_blob": "a7d38c77808c4f43c43db4c948af1d3a623f3f44"
+    },
+    {
+      "path": "research/qq16-consumer-cap-audit/TARGET.json",
+      "sha256": "b11729b374a14775c754ff561f52ab48562a4c12ea1e5303e3829ec15cebf1c3",
+      "git_blob": "750e72bdb5709a9b41993c12948729374faac38c"
+    },
+    {
+      "path": "research/qq16-consumer-cap-audit/TASK.json",
+      "sha256": "1c9c1ca540312c72ff27ffbfbb17e3306e4195da7f9ff254c508ada1efb3cc07",
+      "git_blob": "c869052a9e86d9314442c64c1213476fa7d6a867"
+    },
+    {
+      "path": "research/qq16-consumer-cap-audit/VERIFY.json",
+      "sha256": "021f6f76b07a0c96cbc6d870bdd8a44de61b0dc9d726b483731fe0417ab79544",
+      "git_blob": "37caf50e7cbcec733ddfde93f6a3a55b4096f91e"
+    },
+    {
+      "path": "research/qq16-consumer-cap-audit/verify.py",
+      "sha256": "30ce90d9d0a8d6bda4341ebfc26a2dbcd0fbf60170e50a010ccbad4ce54274a9",
+      "git_blob": "dd6c8b9f123e85da984edfc847bb40469d0cc555"
+    }
+  ],
+  "caller_pr": 3932,
+  "caller_head": "12c324c576821f78070627c5eeb684d03704e20e",
+  "credit": [
+    "Ali Romman and QQ16/native lineage",
+    "Douglas Colkitt and coverage-ceiling lineage",
+    "ywz65 PR2707",
+    "dcharb78 PR3181",
+    "Nick Petalas producer/decoder numerical lineage",
+    "Chris R / Clear1nsight with substantial OpenAI assistance"
+  ]
+}

research/qq16-consumer-budget-independent/PROOF.md

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+# A caller-bound audit of the QQ16 consumer budget
+
+Chris R / Clear1nsight, with substantial OpenAI assistance. Apache-2.0.
+This is an ordinary scoped proof and executable rational certificate, not
+human review, proof-kernel verification or supplier admission.
+
+## 1. Audited claim and independent source binding
+
+The audited PR3908 head is
+`0d8880e38ed7c88d70eec6a48e5aa829d0fb22f3`, by 1337isnot1337.
+It checks the consumer-budget branch of dcharb78's PR3181, not its
+dynamic-range, adjoint or grid theorems. The canonical baseline is
+`20ecf2891d80b5acf18d980b2164fb44ea7faa90`, priority revision 6.
+
+The original seven-file package is retained byte-for-byte in
+`vendor/PR3908.tar.gz`, with per-file Git blob/SHA256 identities in
+PREDECESSORS.json. Its original attribution and limitations are retained.
+The regression suite authenticates and replays that exact executable in a
+temporary source export. The independent calculation imports neither that
+executable nor `ceiling.py`.
+
+Unlike a check of reported profile dictionaries alone, the new checker
+expands the literal `programs.F_0` and `programs.B_1_2` in the pinned
+`research/joint-query/certificate.json.gz`. `retained_core` expands the named
+core word; `cover_sweep` and `private_sweep` contribute their ordered bodies.
+Each instruction is counted once per full-cover invocation, not multiplied
+by the number of cover owners. This matches the per-call native invoice.
+`climb` supplies its literal rank; `native_line` supplies its recipe rank.
+
+The resulting event counts are 26,099 and 27,019. Their positive child
+profiles are:
+
+| Program | Rank 1 | Rank 2 | Rank 3 | Rank 4 | Rank 5 | Rank mass |
+|---|---:|---:|---:|---:|---:|---:|
+| F0 | 2,420 | 418 | 432 | 142 | 82 | 5,530 |
+| B_1_2 | 2,436 | 418 | 432 | 142 | 82 | 5,546 |
+| Pair average | 2,428 | 418 | 432 | 142 | 82 | 5,538 |
+
+The same independent census checks all 37 native cases: each of the nine F
+cases has the F0 profile and each of the 28 B cases has the B_1_2 profile.
+This is a count check, not 37 complete operator-correctness replays.
+
+They agree with CONSERVED-QQ16.json, whose payload hash also binds the exact
+decompressed certificate. Independently counting the literal #2707 WORD.json
+gives 3,823 instructions, including 336 signed rank-one lines and no stage
+children. The source proof lowers each signed line using one positive
+rank-one child and explicit monomial wrappers. Its functional equivalence
+and physical realization are inherited, not re-proved by this census.
+
+The per-call profile with this decoder is therefore
+`{1:2596,2:418,3:432,4:142,5:82}`, of mass 5,706. PR3181 additionally reports
+eight rank-eight children per call. The main obstruction below drops all of
+them, so their origin or necessity is not a premise of that obstruction.
+The tables explicitly labelled as retaining rank eight condition on that
+additional reported profile component.
+
+## 2. Applicability to the audited first F0 caller
+
+The first positive child in the same literal F0 expansion is zero-based
+event 9894:
+
+```text
+["climb","A",546,0,4,1]
+```
+
+Event 9895 is `L546 -= L561` in frame 4. The canonical JSON hash of the
+preceding 9,894 events is
+`15115954461f2cb93e2266822d82bfac1ab8de7d460ff78fcf594d21c61d2b53`,
+exactly the prefix audited in PR3932, head
+`12c324c576821f78070627c5eeb684d03704e20e`.
+Thus our caller is inside the F0 program counted here. It is not a new
+producer profile to which an unrelated histogram has been assigned.
+
+The implication remains conditional: if this F0 and B_1_2 are used with the
+unchanged #2707 decoder and their listed paid recursive ranks, and the
+centre-64 coverage/halving law is retained, the budget fails at the selected
+target. Correct encoding or a faster implementation of a scalar/axis
+operation does not by itself delete those recursive charges.
+
+Conversely, PR3884's six-tape operation is a local bit-tape implementation,
+and PR3932 is an arithmetic caller audit. Neither supplies the complete
+recursive invoice. A proof that direct implementation actually *replaces*
+a listed recursive tariff, with all-depth costs and the complete consumer
+charged, would change this model and must be assessed separately. This
+obstruction does not prohibit that possibility or every physical caller.
+
+Changing only F/B frequencies cannot fix the screen if the decoder charge
+remains 168 rank-one children per call: the most favorable all-F mixture,
+still waiving rank eight, has moment in
+`[1.000090397821,1.000090398426]`. All B substitutions increase it.
+This does not establish the actual decoder frequency in a new global
+schedule. Different reuse/frequencies must be proved and priced; the
+pair-level audit is not an admission of an outer schedule.
+
+## 3. The conditional moment lower bound
+
+Fix `m=73`, `w=896/9`, `R=51`, one unpaid coordinate per family and centre
+charge `c`. The inherited coverage law fixes total paid rank mass
+
+```text
+T(c) = 72w + c.
+```
+
+At c=64 this is 7,232. For local profile n_r, let L=sum r*n_r and
+I=T(c)-L. The outer halving guard `2R^2<m^2` permits R=51 and excludes 52.
+For nonnegative a,
+
+```text
+b_a(r) = (r/73)^(1-a)
+b_a(r)/r = 73^(a-1) r^(-a)
+```
+
+is non-increasing in r. Consequently any remaining idle histogram with
+ranks at most 51 and mass I has cost at least `(I/51)b_a(51)`, even allowing
+fractional child counts. A necessary budget test is the favorable relaxation
+
+```text
+G(n,c,a) = [sum n_r b_a(r) + (I/51)b_a(51)] / w < 1.
+```
+
+Moving any paid rank-eight child mass to rank-51 idle mass decreases this
+relaxation. Omitting all eight such children is therefore favorable, not an
+extra charge. At the selected
+
+```text
+kappa = 836330098538459/500000000000000000
+a = kappa/(1-kappa) = 836330098538459/499163669901461541
+```
+
+the decoder-inclusive minimal profile has L=5,706 and I=1,526.
+The independent enclosure is
+
+```text
+1.000097677678 <= G <= 1.000097678284.
+```
+
+In particular G>100009/100000. Every admissible idle packing in this fixed
+model has moment above one before additional physical costs. Since the
+moment is increasing in a, this model cannot support the selected saving
+or a larger one. No claim about an exact cap root is needed.
+
+## 4. Independent exact interval construction
+
+PR3908 uses a positive atanh/log expansion and one-sided exponential sums.
+This checker instead encloses logarithms by integration. On [1,y],
+1<=y<=2, the function f(t)=1/t is convex. For a uniform partition into
+1,024 panels, the composite midpoint sum is a lower bound for its integral
+and the composite trapezoid sum is an upper bound. Every evaluation and
+sum is rational. Reduce x>=1 by powers of two and use
+`ln(x)=k*ln(2)+ln(x/2^k)`; add the corresponding enclosures.
+
+For 0<=l<=u<=1/2, the degree-six exponential polynomial at l is a lower
+bound. At u, the first omitted term is t7=u^7/7!. All later term ratios
+are at most u/8, so
+
+```text
+exp(u) <= sum_{j=0}^6 u^j/j! + t7/(1-u/8).
+```
+
+All used a*ln(73/r) intervals lie in this domain; the code checks it.
+Every interval is rounded outward to the dyadic grid 2^-80. Positive
+coefficients, mass and division by w preserve enclosure. The decimal
+endpoints in RESULT.json are rational outward bounds at 10^-12, not
+floating-point calculations. Full JSON equality checks every reported
+field and distinguishes booleans from integers.
+
+The retained PR3908 replay reproduces its output lower bound
+1.000097677879937. Its centre-free lower bound being below one is only a
+scope control, not evidence of a feasible supplier. Our new upper bounds
+below one certify passing *relaxations*, also not supplier admission.
+
+## 5. Exact decoder and centre tradeoffs
+
+Keep the producer average fixed and let d be the total number of additional
+rank-one decoder children across the two-call pair. Per-call decoder cost
+is d/2. Since each added rank-one child displaces one unit of rank-51 idle
+mass, G increases linearly in d with positive coefficient
+
+```text
+[b_a(1)-b_a(51)/51] / (2w).
+```
+
+This allows exact integer thresholds without a search for a root:
+
+| Centre charge per call | Eight rank-eight children | Last integer d passing relaxation | First d excluded |
+|---|---|---:|---:|
+| 64 | Dropped favorably | 121 | 122 |
+| 64 | Retained | 61 | 62 |
+| 63 (hypothetical changed law) | Retained | 363 | 364 |
+
+Odd d is permitted as a fractional per-call average; an even, equal split
+would restrict the passing ceilings to 120, 60 and 362. This generalization
+does not affect PR3908's actual even d=336.
+
+Thus a decoder-only replacement under centre-64 must remove at least 215
+of 336 calls even when all rank-eight work is waived, or at least 275 with
+that work retained. Merely removing two 112-call groups leaves d=112 and
+still fails the retained-rank-eight test. These are budget requirements,
+not decoder lower bounds or constructions.
+
+For the unchanged d=336 with rank eight retained, a hypothetical c=63 gives
+
+```text
+0.999987387153 <= G <= 0.999987387763.
+```
+
+Its available additional normalized moment is between 0.000012612237 and
+0.000012612847. Any complete replacement must fit strictly inside its
+actual remaining allowance. In the very special case where the only
+additional work is rank-one local children displacing idle mass, at most
+27 additional children per pair pass this relaxation; 28 are excluded.
+This is not a conversion rate for tape actions or physical overhead.
+
+## 6. Research decision and limits
+
+Pause expansion of the unchanged F0/B_1_2 + #2707 centre-64 recipe **as a
+route to the selected target**. Its optimistic bound already fails.
+Retain the Gaussian/caller results as useful local interface contributions.
+Do not infer a global QQ16 or multiplication impossibility statement.
+
+The next budget-admissible research question is a *changed consumer
+endpoint or native operator-specific identity*: can one identify real
+eliminated centre work, corresponding to at least one average mass unit
+per call in the aggregate coverage ledger, while retaining the required
+final output and avoiding repayment beyond the allowance above?
+One unit per call means two over this pair; a smaller saving can be
+assessed by the same affine c-dependence. No such identity is supplied.
+
+The first test must specify the actual downstream F17-native demand map,
+the producer output rows it reads, chronological owner/frame interval,
+arbitrary dirty state and return/restoration requirements. Compose that
+map with the literal decoder before proposing cancellation. A final-output
+quotient or identity specific to the fixed native operators is a declared
+changed premise; relabelling selected independent demands or storing the
+same first transcript is not one.
+
+Relevant adjacent restrictions are PR2391's native first-transcript
+counterexample, PR2461's owner-local repair floor, PR3120's selected
+copied-demand rank floor and pending PR3873's uniform scalar-oracle and
+selected coverage audits. We inspected their stated scopes, not re-proved
+those theorems here; none is silently asserted for a new QQ16 endpoint.
+Any proposed escape must name its difference from those contracts.
+
+If no named consumer supplies the saving, stop at this handoff. If the
+changed contract passes its complete budget, the next implementation slice
+is the already identified L546 child-plus-ADD with actual tape intervals,
+entry grid/range, monomials, movement/address/control and eventual return
+charged. Local bit counts cannot be inserted into this recursive moment
+without a justified transfer. Unknown costs remain unknown.
+
+The complete native/bridge algebra, decoder correctness, global coverage
+realization, huge all-depth payload execution, physical hosting/routing,
+precision/restoration admission, F17 consumer and selected full cold replay
+remain inherited or unrun. No frontier record is changed.