Prove the complete fixed P104 R22 compiler tail envelope

research/p104-tail-feasible-envelope/ORIGINS.json

@@ -0,0 +1,18 @@
+{
+  "source_repository": "CrocSwap/integer-mult-bounds",
+  "source_head": "a1b338e5d9271b541f2a198f4fc19cefc565feed",
+  "files": [
+    {
+      "original_path": "research/p104-block-reframe-r22/RESULT.json",
+      "retained_path": "inputs/RESULT.json",
+      "git_blob": "9d47144bd99dc57d799345c940232eb26afdc33c",
+      "sha256": "062fb71b92ca620cbea4a4469458ede513e4dc1039bfa783e06ed11f51335a9e"
+    },
+    {
+      "original_path": "research/p104-block-reframe-r22/pricing.py",
+      "retained_path": "inputs/pricing.py",
+      "git_blob": "fe40a9cb39c087c3170ee7e79140f4884c8e1873",
+      "sha256": "2a8f0d74a16c8ad3b595e2100f8c074122ecd2c1babf66a4495b20ba9e651ce8"
+    }
+  ]
+}

research/p104-tail-feasible-envelope/PORTABILITY.json

@@ -0,0 +1,7 @@
+{
+  "readonly_files_unchanged": 13,
+  "optimized_git_free": true,
+  "empty_path": true,
+  "source_corruption_rejected": true,
+  "python": "Python 3.12.14"
+}

research/p104-tail-feasible-envelope/PROOF.md

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+# Fixed-contract feasibility proof
+
+Fix the retained PR5756 target and paid histograms `T_r,P_r`, saving
+`C=622627591/200000000000` and `t=1-C`. Put `T=sum T_r r^t` and
+`N=sum T_r=13390`. For integer cutoff `c>=1`, integer floor `b`, positive
+real offset `u` and rational decay `rho`, the two source tail gates are
+
+```
+g(c,u) = sum P_r r^(t+u) / (c^u T) < rho,
+d(c,b) = N b / (c^C T) < rho,
+rho^128 < 1/16.
+```
+
+The capacity condition `c^64 < b^63` implies `b>=c+1`. Let
+`R=2^(-1/32)`. A permitted rational rho exists if and only if both g and d
+are strictly below R: rationals are dense in the open interval between their
+maximum and R. Equality at R is excluded. This eliminates every decay choice,
+not just a grid. We first use the weaker condition b>=c+1 to exclude parameters;
+the sole remaining pair also passes the exact capacity condition.
+
+## Exclude all but one integer pair
+
+For positive u, g(c,u) decreases with c. For fixed c, g is a positive sum
+of exponentials in u and is strictly convex, since a paid rank differs from c.
+Its derivative is the same sum weighted by `log(r/c)`. A negative derivative
+at l and positive derivative at h locate its unique global minimum in (l,h).
+For x in this interval, convexity gives
+`g(x)>=g(l)+g'(l)(x-l)>=g(l)+g'(l)(h-l)` when g'(l)<0.
+The verifier uses a lower interval endpoint for both terms. The midpoint gives
+an upper bound on the minimum. Strict convexity extends this statement to all
+real u; no unsearched tail or sampled-grid premise remains.
+
+At c=15 the resulting lower bound is `0.983563652257169`, while
+`R<0.978572062087701`. Thus every positive u fails for c<=15.
+
+For real c>=1, `(c+1)/c^C` has derivative with sign `(1-C)c-C`, which is
+positive because `0<C<1/2`. Also d increases with b. At c=17,b=18 the
+lower bound is `0.999697814388247>R`. Thus every c>=17 and every permitted
+b fails. At c=16,b=18, d exceeds `0.999886507972867>R`, which excludes all
+larger floors. The only pair left is c=16,b=17. It passes `16^64<17^63`.
+
+## Full offset interval and optimum
+
+For c=16, g(0)>R, g(1/7)<R and g(1)>R. Strict convexity, together with the
+unique derivative zero, gives exactly two positive roots of g(u)=R and a
+single open feasible interval between them. The exact root brackets are in
+RESULT.json. The completion tail is constant in u and is below
+`0.944337257529930`. It is below even the global minimum of g, so it never
+binds in this interval. Thus these two roots specify the complete tail set.
+
+The derivative has opposite signs at the endpoints of the reported best-offset
+bracket. The convex minimum bound above gives the stated decay infimum.
+The infimum is not attained by rho because g<rho is strict. Positive rational
+u near the minimizer and rational rho above its value approach it arbitrarily
+closely. If u is restricted to rationals, intersect the same interval with Q;
+all exclusion and approximation statements remain valid.
+
+At u=1/5, g exceeds R, so no permitted rho can repair this offset. At u=1/10,
+`125/128<g<489/500`, and `(489/500)^128<1/16`; this checks the distinction
+between fixed-decay failure and full infeasibility. The source point also
+passes its reserve. Other compiler conditions are not inferred from these gates.
+
+## Arithmetic and limits
+
+The shared power engine encloses fractional powers with integer square roots
+and directed products on a dyadic grid. For arbitrary rational exponent e,
+write e=floor(e)+f with 0<=f<1 and multiply by the exact integer power.
+This is valid for negative exponents as well. Positive denominators permit
+outward interval division.
+
+For x in [1,2], put z=(x-1)/(x+1). The first80 terms of
+`2 sum z^(2j+1)/(2j+1)` are a lower bound for log x. The remaining positive
+tail is at most `2 z^161/(161(1-z^2))`. An integer rank is reduced to this
+range by an exact power of2. Subtraction gives a rigorous log(r/c) interval;
+all four endpoint products handle its possible negative sign. Root bisection
+proceeds only after strict opposite signs are proved. It fails on uncertainty.
+
+The infinite parameter coverage follows from these monotonicity and convexity
+arguments. The finite arithmetic controls do not establish the proof by
+extrapolation. The source invoice and gate formulas are pinned premises;
+this proof does not establish their native realization or a different compiler.

research/p104-tail-feasible-envelope/README.md

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+# Complete P104 R22 tail-parameter envelope
+
+For the fixed PR5756 invoice at saving `622627591/200000000000`, the existing
+128-block compiler tail gates admit exactly one integer cutoff and completion
+floor: **16 and 17**. The full positive offset interval is between two roots,
+with enclosures near **0.0901726680035 and 0.193311515488**. The unique best
+offset is in `(0.141270802754661, 0.141270802755571)`. Its best decay infimum
+is in `(0.975481788192714, 0.975481788192715)`.
+
+This is a scoped proof for the declared, complete invoice: 66,852 paid calls,
+13,390 targets and rank surplus 760. It is not a new κ or supplier admission.
+The source's prose count 12,958 is stale; this package uses its JSON and the
+complete inventory. [PROOF.md](PROOF.md) proves coverage of all integer cutoffs,
+all integer floors and the continuum of positive offsets. It permits every
+rational decay with `decay^128 < 1/16`, not only the source's `125/128`.
+
+The source point `(16,17,1/7,125/128)` passes. Offset `1/5` fails for every
+permitted decay. Offset `1/10` fails at the source decay but passes at `489/500`.
+Thus a failed fixed-decay test is not a proof that an offset is impossible.
+The optimized tail margin does not raise the fixed invoice's moment crossing.
+
+## Reproduce
+
+Use standard-library CPython 3.12 from the repository root:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/p104-tail-feasible-envelope
+python3.12 -I -O -B research/p104-tail-feasible-envelope/verify.py
+```
+
+The checker authenticates inputs, imports the existing directed-integer power
+engine and imports no source constructor or pricing engine. It uses exact
+rational logarithm bounds, strict derivative and root brackets, and a convex
+lower bound at the minimum. It includes 640 exact rational-power fixtures,
+80 separate Decimal power checks, 80 separate Decimal logarithm checks,
+missing-count rejections and passing/failing tail points. Decimal checks are
+corroboration; all feasibility certificates use rational bounds. `RESULT.json`
+uses outward rational rounding. No floating-point search is accepted as proof.
+
+`ORIGINS.json` binds the retained source bytes to the exact PR5756 Git blobs.
+`SOURCES.json` binds those bytes to this branch's retention commit and pins the
+power engine and its dependency. The direct replay is read-only and Git-free.
+The optional full-history provenance check needs history absent from this
+shallow checkout. Normal and sealed replay evidence is in `VALIDATION.md`.
+
+## Scope and credit
+
+Canonical [Discussion5773](https://github.com/CrocSwap/integer-mult-bounds/discussions/5773).
+Base `de5ec2e7a6074068452f45b1a191b5c72832df4a`, priority8. This changes the
+independent-moment card to a compiler parameter-existence question. The active
+R22 subspace optimizer has a distinct scope. The earlier adaptive-tail-entropy
+and adaptive-gate-existence results concern a different compiler contract.
+
+Credit Joel Pulikkan/GamingPuzzled's PR5756 construction, Ali Romman's PR5751,
+flyingviper's PR5718, and all retained P104, native and controller contributors.
+Credit Jacob Alan Sussman, zxv and Cao Yifan for the compiler gates; retain the
+existing directed-power and invoice-comparator credits. New proof and checker
+by Ali Romman with substantial OpenAI Codex assistance, without subagents.
+Apache-2.0. This is author-side verification, not independent human acceptance.
+
+The invoice's physical legality, native phases, chronology, precision, other
+compiler gates, outer transfer and five framework assumptions remain separate.
+No full source scientific replay, Lean or Docker run is claimed. The selected
+frontier is unchanged. This does not limit a different invoice or compiler.
+
+## Open next questions
+
+- Can a legal changed helper invoice retain a nonempty tail interval? Supply its
+  exact complete bill, then repeat these bounds. A legal construction is needed.
+- Can a new fully charged completion or block rule enlarge the envelope? State
+  its normalization, exit and rounding costs before optimizing its parameters.

research/p104-tail-feasible-envelope/RESULT.json

@@ -0,0 +1,60 @@
+{
+  "best_decay_infimum": {
+    "lower": "487740894096357/500000000000000",
+    "upper": "195096357638543/200000000000000"
+  },
+  "best_offset": {
+    "lower": "141270802754661/1000000000000000",
+    "upper": "141270802755571/1000000000000000"
+  },
+  "completion_at_16_17": {
+    "lower": "944337257529929/1000000000000000",
+    "upper": "94433725752993/100000000000000"
+  },
+  "controls": {
+    "decimal_logs": 80,
+    "decimal_powers": 80,
+    "exact_powers": 640,
+    "missing_count_rejections": 2
+  },
+  "cutoff15_minimizer": {
+    "lower": "22907539841617/200000000000000",
+    "upper": "28634424802249/250000000000000"
+  },
+  "cutoff15_minimum": {
+    "lower": "983563652257169/1000000000000000",
+    "upper": "98356365225717/100000000000000"
+  },
+  "cutoff17_completion_lower": {
+    "lower": "999697814388247/1000000000000000",
+    "upper": "124962226798531/125000000000000"
+  },
+  "floor18_completion_lower": {
+    "lower": "999886507972867/1000000000000000",
+    "upper": "249971626993217/250000000000000"
+  },
+  "maximum_decay": {
+    "lower": "9785720620877/10000000000000",
+    "upper": "978572062087701/1000000000000000"
+  },
+  "offset_interval_lower_root": {
+    "lower": "22543167000809/250000000000000",
+    "upper": "22543167000939/250000000000000"
+  },
+  "offset_interval_upper_root": {
+    "lower": "48327878871987/250000000000000",
+    "upper": "193311515488729/1000000000000000"
+  },
+  "offset_one_fifth_gamma": {
+    "lower": "979409001450321/1000000000000000",
+    "upper": "489704500725161/500000000000000"
+  },
+  "paid_calls": 66852,
+  "source_head": "a1b338e5d9271b541f2a198f4fc19cefc565feed",
+  "target_count": 13390,
+  "tested_saving": "622627591/200000000000",
+  "unique_integer_cutoff_floor": [
+    16,
+    17
+  ]
+}

research/p104-tail-feasible-envelope/SOURCES.json

@@ -0,0 +1,44 @@
+{
+  "schema_version": 1,
+  "base_commit": "de5ec2e7a6074068452f45b1a191b5c72832df4a",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "581f8d87008354eb5b869f7ed9de1317ad837ce8",
+      "sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "scripts/compare_recursive_invoices.py",
+      "usage": "input",
+      "git_blob": "fda2a852e9937e42fe40df0707694465ca7c1b48",
+      "sha256": "3930c22c11da7bc6042cfeb6e459ecb06e4478430037bee3469d034353855fc6"
+    },
+    {
+      "path": "scripts/evaluate_recursive_power.py",
+      "usage": "input",
+      "git_blob": "ca50f8021fc3238a58fd420383f4619a8d008ace",
+      "sha256": "52b20f41fd0bb55fa88dccd6015af2ce358917817b9506720f3e60ce681cc3ad"
+    },
+    {
+      "path": "research/p104-tail-feasible-envelope/inputs/RESULT.json",
+      "usage": "input",
+      "pin": "head:4d71a44c1cba1de9cf3c164bcaabb909f4440fc9:research/p104-tail-feasible-envelope/inputs/RESULT.json",
+      "git_blob": "9d47144bd99dc57d799345c940232eb26afdc33c",
+      "sha256": "062fb71b92ca620cbea4a4469458ede513e4dc1039bfa783e06ed11f51335a9e"
+    },
+    {
+      "path": "research/p104-tail-feasible-envelope/inputs/pricing.py",
+      "usage": "input",
+      "pin": "head:4d71a44c1cba1de9cf3c164bcaabb909f4440fc9:research/p104-tail-feasible-envelope/inputs/pricing.py",
+      "git_blob": "fe40a9cb39c087c3170ee7e79140f4884c8e1873",
+      "sha256": "2a8f0d74a16c8ad3b595e2100f8c074122ecd2c1babf66a4495b20ba9e651ce8"
+    }
+  ]
+}

research/p104-tail-feasible-envelope/TASK.json

@@ -0,0 +1,110 @@
+{
+  "schema_version": 1,
+  "task_id": "independent-moment",
+  "priority_revision": 8,
+  "base_commit": "de5ec2e7a6074068452f45b1a191b5c72832df4a",
+  "original_card": {
+    "id": "independent-moment",
+    "question": "Does a changed construction's serialized target/paid invoice independently support its claimed improvement?",
+    "starting_points": [
+      "docs/research/current-priorities.md",
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+      "docs/research/recursive-invoice-preflight.md"
+    ],
+    "methods": [
+      "Construct and independently check one source-bound instance of the stated first experiment.",
+      "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change."
+    ],
+    "acceptance_gate": "Demonstrate independence, source binding, positive cases and semantic failure controls. Name the existing checks and the missing coverage. More digits or another equivalent evaluation of an unchanged root is not by itself new verification coverage.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.",
+      "scope": "Source-bound orientation only, not a supplier replay or theorem proof."
+    },
+    "first_experiment": "Independently recount one changed emitted word and its complete caller, importing no constructor or pricing engine. Include target losses, owned lifetimes, native orientation and extra charges. Check existing #5633/#5708/#5717/#5719 coverage to avoid duplicate audits and preserve its exact head. Stop after the chosen missing implication; a correct root alone is not supplier admission.",
+    "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.",
+    "allowed_variations": [
+      "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes."
+    ],
+    "scope_change_triggers": [
+      "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.",
+      "Treating an archived proposal or local numerical pass as complete supplier admission."
+    ],
+    "overlap_terms": [
+      "invoice",
+      "signed moment",
+      "productive target",
+      "independent recount"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4088: mixed local/owner/phase ledger and moment reported.",
+        "evidence": "research/staged544-mixed-fourth-entry/README.md",
+        "remaining": "Independent binding of changed construction to paid histogram, distinct from full admission.",
+        "status": "author-checked",
+        "source_commit": "72d7efc181edb21806a878f174847c96dd410ab0",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4091: sharp pooled ceiling proposed for fixed THREE/FOUR words and inventory.",
+        "evidence": "research/staged-fourth-mixture-ceiling/README.md",
+        "remaining": "Validate dual, exhaustive pattern scope and exact source equivalence before applying to newer fills.",
+        "status": "author-checked",
+        "source_commit": "a29bdd8f56cd18b805917775cdec09ff935eddc4",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4067: paid (2,2) to (1,3) exchange and fixed OQ4 host reported.",
+        "evidence": "research/r437-two-window-carrier/README.md",
+        "remaining": "Independent full cost delta; remaining caller/precision/stock obligations are separate.",
+        "status": "author-checked",
+        "source_commit": "a95f11c52e3ca1e4c5171b3c20d8f4b0815aee2e",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.",
+        "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+        "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.",
+        "status": "reviewed",
+        "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137",
+        "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "All integer cutoff/floor and positive offset feasibility for the fixed PR5756 charged tail gates, with optimal decay.",
+  "scope_change_reason": "The exact compiler parameter set was missing. This resolves a parameter-existence question; it does not repeat native recount or the active R22 subspace optimizer.",
+  "bottleneck_link": "A favorable signed moment is insufficient if paid and completion tails cannot both be funded. The full feasible set directs the next helper change.",
+  "budget_minutes": 70,
+  "stop_rule": "Prove the complete fixed-contract parameter set; do not extend to changed invoices or compiler rules.",
+  "outcome": "completed",
+  "evidence": "PROOF.md and RESULT.json; registered verify.py with directed powers and derivative root bounds.",
+  "author": "Ali Romman with substantial OpenAI Codex assistance",
+  "ai_assistance": "Substantial OpenAI Codex assistance; no subagents; not independent human review.",
+  "open_questions": [
+    {
+      "question": "Can a changed helper invoice retain a nonempty tail interval?",
+      "why_it_matters": "New productive targets can increase the completion tail and remove compiler feasibility.",
+      "first_step": "Supply a source-bound complete new invoice and repeat the cutoff and convexity bounds.",
+      "blocked_by": "A legal changed native construction and full bill."
+    },
+    {
+      "question": "Can a different completion or block rule enlarge the envelope?",
+      "why_it_matters": "The result is exact only for this fixed 128-block contract.",
+      "first_step": "State a new fully charged compiler rule and prove its tail conditions.",
+      "blocked_by": "A new compiler proof, including the normalization and exit charges."
+    }
+  ],
+  "coordination": {
+    "canonical": "https://github.com/CrocSwap/integer-mult-bounds/discussions/5773",
+    "source_head": "a1b338e5d9271b541f2a198f4fc19cefc565feed",
+    "seed": "16582aeecf71f7a63a0a9bfc8454b4cc"
+  }
+}