Mister-Meeseeks · Oct 27, 2026 · Oct 26, 2026
diff --git a/research/r437-unpooled-budget-obstruction/NOTICE b/research/r437-unpooled-budget-obstruction/NOTICE
@@ -0,0 +1,31 @@
+Retained source contributions and licenses:
+
+Chris R / Clear1nsight with substantial OpenAI Codex assistance, Apache-2.0:
+PR4067 two-window R437 carrier, complete fixed OQ4 host, paid profile and price.
+The source already establishes its word/geometry/dirty-state and local gain;
+this package does not claim those constructions or rederive the root.
+
+Retain ai-nanachi/PR3647 for R437 and helper464 deletion; Ali Romman/PR1043
+for SOURCE77-carrier deletion; Sagar Thakrar/PR3700 and PR2423 for OQ4 and
+independent checks; PR4005/Chris R for the source-quotient cut and boundary
+handoff; mightyOil/PR1281 and Rohan Arun/PR3959 for packing/pooled context;
+eumemic, DreamingOfClouds, DaysSky, Crow, chafreaky and retained notices for
+word, geometry, native, hosting and compiler lineage.
+The source host lowering, owner-invoice pattern and rational interval
+technique adapt the credited PR3647/3700 audit.
+
+zxv/PR3690 supplies the source-relative ordinary recurrence/transfer argument.
+unpaidmoon/PR4147 supplies the reverse-row coefficient follow-up and distinction
+between arithmetic envelope values and physical caller-bound admission.
+Retain Mihir Srivastava, JoyboyBrian, Krzysztof Wojt, commdata2338,
+Jose Manuel/jmbarrancoml, David Arias, Nick Petalas, ywz65 and the actual
+physical/caller/stock/pooling lineage in Discussion4682 and its linked packages.
+Current selected conditional P104/frontier5599 certificate and all inherited
+native/compiler/precision/tape/analytic authors retain their original credit.
+
+New fixed-unpooled current-budget implication, checker, independent integer
+oracle and scoped proof: ToikY with substantial OpenAI Codex assistance in
+separate primary, independent-oracle and proof-review contexts. Apache-2.0.
+Classical log/exp inequalities are not claimed as new general discoveries.
+No independent human or proof-kernel acceptance, physical theorem extension,
+full supplier admission or selected-frontier change is claimed.
diff --git a/research/r437-unpooled-budget-obstruction/ORACLE.json b/research/r437-unpooled-budget-obstruction/ORACLE.json
@@ -0,0 +1,107 @@
+{
+  "controls": {
+    "changed_profile_rank42_to_rank1": {
+      "bound": {
+        "A": "18673/18944",
+        "B": "4287625175886659/2380780979216640",
+        "base_denominator": 1818624,
+        "common_log_denominator": 335132457660,
+        "integer_cross_product_gap": -24934285326954060032,
+        "integer_lower_denominator": 3047399653397299200000,
+        "integer_lower_numerator": 3022465368070345139968,
+        "lower": "694500314354399159/700229699769600000",
+        "lower_minus_one": "-5729385415200841/700229699769600000",
+        "mass": 1792608,
+        "saving": "17/5000",
+        "slope_numerator": 1097632045026984704
+      },
+      "meaning": "Synthetic different-profile sign control only. L<1 is not a proof that its actual moment passes or that the changed jobs have a physical realization.",
+      "moved_paid_jobs": 384,
+      "total_paid_calls_unchanged": true
+    },
+    "rejections": {
+      "changed_source_byte": "Oracle exact source bytes: HOST-RESULT.json",
+      "different_normalization": "Oracle frozen width and normalization",
+      "different_profile": "Oracle frozen unpooled profile",
+      "negative_saving": "Oracle nonnegative saving",
+      "whole_kappa_as_supplier_target": "Oracle selected whole kappa is provenance, not a supplier target"
+    },
+    "zero_saving": {
+      "A": "18837/18944",
+      "B": "227656981418642299/126181391898481920",
+      "base_denominator": 1818624,
+      "common_log_denominator": 17762020255980,
+      "integer_cross_product_gap": -182451472069426560,
+      "integer_lower_denominator": 32302436326011371520,
+      "integer_lower_numerator": 32119984853941944960,
+      "lower": "18837/18944",
+      "lower_minus_one": "-107/18944",
+      "mass": 1808352,
+      "saving": "0",
+      "slope_numerator": 58280187243172428544
+    }
+  },
+  "inequalities": [
+    "For x>=1, ln(x)>=2(x-1)/(x+1): the derivative gap is (x-1)^2/[x(x+1)^2]>=0, and the gap vanishes at1.",
+    "For c>=0 and y=c*ln(x), exp(y)>=1+y; each paid weight is positive.",
+    "Therefore M0(c)>=L(c)=A+cB. B>0, so L(17/5000)>1 obstructs every c>=17/5000 in this retained moment-certification budget.",
+    "Under the declared necessary ordinary bridge a<=c, reaching a>=17/5000 would require such a rejected c. No physical bridge is extrapolated."
+  ],
+  "integer_certificates": {
+    "admitted_BIT": {
+      "A": "18837/18944",
+      "B": "227656981418642299/126181391898481920",
+      "base_denominator": 1818624,
+      "common_log_denominator": 17762020255980,
+      "integer_cross_product_gap": 78505822786798485248,
+      "integer_lower_denominator": 161512181630056857600000,
+      "integer_lower_numerator": 161590687452843656085248,
+      "lower": "37130213109568854799/37112174087788800000",
+      "lower_minus_one": "18039021780054799/37112174087788800000",
+      "mass": 1808352,
+      "saving": "17/5000",
+      "slope_numerator": 58280187243172428544
+    },
+    "ordinary_BIT": {
+      "A": "18837/18944",
+      "B": "227656981418642299/126181391898481920",
+      "base_denominator": 1818624,
+      "common_log_denominator": 17762020255980,
+      "integer_cross_product_gap": 44549682840639884280077593856,
+      "integer_lower_denominator": 80756090815028428800000000000000,
+      "integer_lower_numerator": 80800640497869068684280077593856,
+      "lower": "28693409267709186322542641191/28677589067836800000000000000",
+      "lower_minus_one": "15820199872386322542641191/28677589067836800000000000000",
+      "mass": 1808352,
+      "saving": "8590884599/2500000000000",
+      "slope_numerator": 58280187243172428544
+    }
+  },
+  "normalization": 28416,
+  "profile": {
+    "1": 380448,
+    "2": 400288,
+    "26": 32,
+    "3": 22464,
+    "4": 26592,
+    "42": 384,
+    "5": 29376,
+    "6": 17376,
+    "7": 26496
+  },
+  "scope": "Fixed unpooled invoice and retained moment-certification/contraction budget only; no runtime complexity lower bound or algorithmic impossibility. All omitted nonnegative charges are optimistic omissions, not real zero costs. The source ordinary proof fixes c=1/576 within c<1/512; it does not establish physical extension to 17/5000. Changed pooled invoices, native action, source words, compiler and full physical/stock implementation remain outside.",
+  "selected_stages": {
+    "admitted_BIT": "17/5000",
+    "ordinary_BIT": "8590884599/2500000000000",
+    "whole_kappa_provenance_only": "62009/20000000"
+  },
+  "source_sha256": {
+    "HOST-RESULT.json": "000837b2b7a1e9d7e4b6b657a055c6f5f3886c7ea343239f8620f79d5d7b2ae8",
+    "ORDINARY-TRANSFER-PROOF.md": "29ed43abaf5451a8514dff2233ba02f44c1d3839e72035ec5711d4edf060b8f9",
+    "SELECTED-CLAIM.json": "66fbb5b21f32f1c4ba544c20d5a44ebcdfd11bce2bfa2d609c656ae520d3b21e",
+    "SELECTED-RECORD.json": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5",
+    "host.py": "df01e61dc31a1cb6b2bf3549be1e2241f7a30829c2c98567f462271e3fc216ee"
+  },
+  "status": "PASS_R437_UNPOOLED_BUDGET_ORACLE",
+  "width": 64
+}
diff --git a/research/r437-unpooled-budget-obstruction/PROOF.md b/research/r437-unpooled-budget-obstruction/PROOF.md
@@ -0,0 +1,137 @@
+# A current-budget obstruction for one fixed unpooled moment contract
+
+This is an exact rational screen for the published PR #4067 unpooled OQ4 host,
+not a runtime lower bound or a universal R437 exclusion. It fixes the paid
+profile, productive normalization and moment-certification route below.
+All source files are the exact current-main bytes pinned in SOURCES.json;
+the predecessor host implementation is read as data, not imported or executed.
+
+## Source model and destination budget
+
+The frozen HOST-RESULT gives the new paid profile
+
+```
+{1:380448, 2:400288, 3:22464, 4:26592, 5:29376,
+ 6:17376, 7:26496, 26:32, 42:384}.
+```
+
+It has rank mass 1,808,352, dimension 64 and productive width W=28,416.
+These are supplied predecessor host results, not a new reconstruction of the
+288 calls or 125,856 hosted owner occurrences. The source `host.py:price`
+uses the moment `sum n_r*(r/64)^(1-c)/W`, after adding the nonnegative rank-one
+bad-event surcharge `32*64^2*E/10^16`, where E is the total paid count.
+
+Delete that surcharge and every other added cost as a favorable relaxation.
+For c>=0 define the fixed optimistic moment
+
+```
+M0(c) = sum_r [n_r*r/(64*28416)] * exp(c*log(64/r)).
+```
+
+The **budget-contract premise** is that the proposed retained admission route
+requires `M0(c)+D(c) <= 1`, where every added normalized charge D(c) is
+nonnegative. Some source contraction arguments require a strict inequality;
+failure even of this weak version suffices here. This premise is about that
+moment-certification method. An upper-bound recurrence with a failed moment
+does not prove a lower bound on actual runtime, exclude a different analysis,
+or make the underlying algorithm impossible.
+
+The pinned selected CLAIM distinguishes three quantities:
+
+```
+ordinary BIT saving = 8590884599/2500000000000,
+admitted BIT saving s = 17/5000,
+whole multiplication kappa = 62009/20000000.
+```
+
+The first is strictly larger than s. Thus s is a favorable ordinary-slot
+target for this screen; the smaller whole multiplication kappa is not the
+ordinary moment target. No selected frontier is changed.
+
+## Exact lower certificate
+
+For x>=1,
+
+```
+log(x) >= 2*(x-1)/(x+1).
+```
+
+For example, the difference vanishes at x=1 and its derivative is
+`(x-1)^2/[x*(x+1)^2] >= 0`. Also `exp(y) >= 1+y` for every real y.
+All supplied ranks satisfy `0<r<64`, and c>=0, so these inequalities give
+
+```
+M0(c) >= L(c),
+L(c) = sum_r [n_r*r/(64*28416)] * [1+2*c*(64-r)/(64+r)]
+     = A + B*c,
+A = 18837/18944,
+B = 227656981418642299/126181391898481920 > 0.
+```
+
+No numerical logarithm, exponential approximation or root search is needed.
+Exact rational evaluation at the freshly bound budget gives
+
+```
+L(17/5000) = 37130213109568854799/37112174087788800000,
+L(17/5000)-1 = 18039021780054799/37112174087788800000 > 0.
+```
+
+Consequently, for **every** c>=17/5000 and **every** nonnegative added charge
+D(c), `M0(c)+D(c) >= L(c) >= L(17/5000) > 1`. There is no feasible point
+with c in that range for the fixed budget contract. By contrast,
+`M0(0)=L(0)=18837/18944<1`; this is a positive-saving obstruction, not a claim
+that the zero-saving host fails.
+
+## Conditional ordinary-saving bridge
+
+The retained PR #3690 transfer text fixes `c=1/576` and sets ordinary saving
+`a=c*(1-c)`. Its physical estimates retain the historical restriction
+`17/10000<c<1/512`, plus the changed-program refinement, stock, history,
+primitive-cost and other framework premises. None is newly proved here.
+Its text does not by itself establish physical refinement of this changed
+PR #4067 host; only its algebraic saving law is reused as a declared bridge.
+
+For the budget screen, declare only the favorable bridge `0<=a<=c`.
+This follows from `a=c*(1-c)` in its nonnegative-saving range. If an attempt
+to fill the retained ordinary slot requires a>=s and uses this bridge, then
+c>=s, contradicting the fixed moment-contract bound above. The actual pinned
+ordinary saving is larger than s and therefore also cannot be reached by
+that unchanged contract and bridge.
+
+Allowing all c>=s deliberately relaxes the historical physical range to make
+the necessary algebraic test favorable. It does **not** extend PR #3690's
+physical transfer theorem to c=s or to any new parameter. Within its original
+range, c<1/512<s already prevents reaching this slot. The rational certificate
+shows that merely relaxing that range while retaining this invoice, moment
+gate and a<=c bridge still cannot certify the current target.
+
+## What this adds and what it leaves open
+
+PR #3668 already reports an earlier R437 invoice and historical c comparison;
+PR #3700 audits the earlier OQ4 invoice/root and its outer convention;
+PR #4067 certifies the local carrier improvement and charged fixed-host root.
+This package claims none of those constructions, roots or replay results as
+new. It supplies a short, source-bound implication for the revision-8 admitted
+ordinary budget: even the zero-surcharge relaxation of the unchanged unpooled
+invoice fails its retained moment gate at that budget and every larger c.
+Another root digit, a reduced nonnegative surcharge or completing stock work
+alone cannot change this particular budget decision.
+
+A changed paid profile, productive width, target allocation, moment analysis
+or transfer law is outside the proposition. In particular, the released
+[pooled allocation handoff](https://github.com/CrocSwap/integer-mult-bounds/pull/3967#issuecomment-6267387753)
+already reports a local-only allocation audit with a **different** paid
+profile. It does not yet give a complete pooled price or regenerated physical
+source catalogue. This result neither calls that allocation unperformed nor
+excludes it. Full native/ordinary compilation, caller integration, reverse
+precision charges, physical/stock costs and general R437 possibilities remain
+open. No whole multiplication kappa or supplier admission follows.
+
+Credit Chris R/Clear1nsight (#4067) for the carrier, complete fixed host and
+price; ai-nanachi (#3647) and Ali Romman (#1043) for the retained R437 word;
+Sagar Thakrar (#3700/#2423) for the OQ4/invoice lineage; zxv (#3690) for the
+ordinary transfer; unpaidmoon (#4147) for the reverse-envelope distinction;
+Rohan Arun and the retained pooling/native/compiler/physical/analytic authors.
+This scoped certificate/proof was prepared by ToikY with substantial OpenAI
+Codex assistance in separate primary, oracle and review contexts. Apache-2.0;
+retain NOTICE. No independent human or proof-kernel acceptance is claimed.
diff --git a/research/r437-unpooled-budget-obstruction/README.md b/research/r437-unpooled-budget-obstruction/README.md
@@ -0,0 +1,57 @@
+# Current-budget obstruction for the fixed unpooled R437 moment contract
+
+The published unpooled host of [PR #4067](https://github.com/CrocSwap/integer-mult-bounds/pull/4067), head a95f11c52e3ca1e4c5171b3c20d8f4b0815aee2e, cannot satisfy its retained moment-certification budget at the current admitted BIT target, even after every nonnegative surcharge is omitted optimistically. This is a statement about a fixed invoice and certification method. It is **not a runtime lower bound, a universal R437 exclusion, or a result about a changed pooled caller**. [Canonical claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/4682#discussioncomment-19205685).
+
+The exact host uses dimension64, productive width28416 and paid profile:
+
+```
+{1:380448,2:400288,3:22464,4:26592,5:29376,
+ 6:17376,7:26496,26:32,42:384}
+```
+
+The source supplies903456paid children and rank mass1808352. Its288calls and125856hosted owner occurrences are predecessor results, not newly rebuilt here. The source carrier changes each local (2,2) query pair to (1,3), retains rank mass, and adds3scalar ADDs per call,864overall. Its reverse coefficient1123650 is already reported in #4067/#4147; this package does not rederive it or certify its stock/precision envelope.
+
+## Exact result
+
+The live selected certificate distinguishes ordinary BIT saving8590884599/2500000000000, admitted BIT saving s=17/5000 and whole multiplication kappa62009/20000000. The admitted value is the weaker necessary supplier target; whole kappa is not the moment target.
+
+For the frozen profile let M0(c)=sum n_r*r/(64*28416)*exp(c*log(64/r)), with all extra charges omitted favorably. The elementary inequalities log(x)>=2(x-1)/(x+1) for x>=1 and exp(y)>=1+y give:
+
+```
+M0(c) >= L(c) = A+B*c,
+A = 18837/18944,
+B = 227656981418642299/126181391898481920 > 0,
+L(17/5000)-1 = 18039021780054799/37112174087788800000 > 0.
+```
+
+Thus **every c>=17/5000 fails M0(c)+D(c)<=1 for every nonnegative added normalized charge D(c)**. No numerical log, exponential or root search is used. The quantified implication is proved in PROOF.md; finite exact arithmetic authenticates its constants.
+
+The budget-contract premise is the retained contraction/moment gate requiring M<=1. A failed upper-bound recurrence test does not lower-bound actual runtime or exclude a different analysis. Under the separately declared ordinary bridge0<=a<=c, reaching a>=s would require one of these rejected c values. The historical #3690 text uses a=c(1-c) at c=1/576, with c<1/512 and physical refinement assumptions. Dropping that range is only a favorable mathematical relaxation. This package does not extend its physical theorem or establish physical refinement of the changed #4067 host.
+
+## Evidence and controls
+
+RESULT.json records each rational term, source/AST-site bindings, current budget stages and six controls. The separate ORACLE.json independently accumulates an integer common denominator and proves a positive cross-product gap78505822786798485248. Neither checker imports the other or executes predecessor code.
+
+The controls expose model dependence. At c=0, M0=L=18837/18944<1. Moving rank42jobs to rank1 preserves total paid calls and makes this lower-bound screen inconclusive, but the frozen-profile guard rejects the substitution; it is not a physically realized improvement or admission. Using whole kappa also loses this particular lower-bound rejection and is rejected as the wrong target role. Wrong productive normalization, source-byte drift and typed receipt aliases reject. The oracle additionally rejects negative saving. Default complete canonical receipt comparisons preserve bool/int/float distinctions.
+
+Source bytes are copied directly from current main and checked by raw SHA256/Git blob identity. The primary statically checks selected price AST sites; the complete source formula is also read and stated explicitly in the proof. This does not execute or independently replay the original host. All runtime inputs are package-local and no dependency installation is needed.
+
+## Reproduction
+
+Python3.12, standard library:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/r437-unpooled-budget-obstruction
+```
+
+Registered normal/optimized primary and separate optimized oracle use explicit guards. --emit is an explicit candidate mode on stdout and never rewrites a receipt. SOURCES.json pins main ancestry separately from the mathematical contract. Metadata, committed-byte/mode, export and cold checks are integrity evidence, not supplier admission. No native, source-word, pooled compiler, full selected replay or Lean build ran.
+
+## Open next questions
+
+A different invoice, productive normalization, target allocation, transfer law or moment analysis needs its own bound. The [released pooled allocation](https://github.com/CrocSwap/integer-mult-bounds/pull/3967#issuecomment-6267387753) already exists and changes the profile; its complete pooled price and regenerated physical catalogue remain open. This result does not call that work unperformed or exclude it. The #4682 stock/payload/caller handoffs remain separate.
+
+Freeze a complete changed pooled invoice and receiver contract before applying this screen. Native/compiler/stock/precision/tape/analytic obligations and the five framework assumptions remain open. No selected-frontier or kappa change follows; independent acceptance remains.
+
+## Credit
+
+Chris R/Clear1nsight supplied #4067's carrier, fixed host and price; unpaidmoon supplied #4147's reverse-envelope distinction. Retain ai-nanachi/#3647, Ali Romman/#1043, Sagar Thakrar/#3700/#2423, zxv/#3690, mightyOil/#1281, Rohan Arun/#3959 and all source/native/hosting/physical/compiler/analytic authors in NOTICE. #3668/#3700 already cover historical profiles/root comparisons; no such result is claimed as new here. The new contribution is this compact revision8 budget implication and independent exact certificate. ToikY with substantial OpenAI Codex assistance in separate primary/oracle/proof-review contexts. Classical inequalities are not new general discoveries. Apache-2.0.