Price native helper reuse: even a free third H17 loses (no new κ)

research/homogeneous-chain-extraction/PROOF.md

@@ -205,3 +205,88 @@ At b=0 and t>0 the same expression is exact.
 Thus direct acquisition of each chain edge loses for every useful saving.
 Already-paid native observations or a changed joint provider would be needed.
 Their physical acquisition and restoration costs are **unknown** here.
+
+## Source-bound gate: repeating the native H9 helper loses even optimistically
+
+The actual one-port tall offer in Chris R / Clear1nsight's
+`research/one-terminal-h3-prefix/homogeneous/REPORT.md` has three original owners
+and a dependent copied prefix. Its third owner is the original H9 helper.
+Two native calls normally have different original helper owners. Reusing that
+owner through both calls changes the productive inventory; two H9 rewards
+cannot be retained for one owner.
+
+We now grant **more than has been constructed**: full112-port prefix capacity,
+exact homogeneous charts, free boundaries/precision/cleanup, and no added split
+cost. This is a budget relaxation, not a physical capacity or endpoint claim.
+Assume both calls retain the source paid profile P and all other original
+helper jobs. There are224 main owners and112 reused H9-helper owners in a
+17-axis embedding. The bare three-chain consumer supplies224 H17+112 H16.
+For a still more favorable test, grant even the third owner a free H17 output,
+whether or not the retained dependent observations can construct it.
+
+The pinned native receipt (source20bb93cd990d2ff683e4d0fc56b87a42cd5d3296,
+actual compressed word SHA2567679daa4ec6a40f56fc7f0202379a6f454310e94945aba4f48beaaaafb838d05)
+reports
+
+    P = {1:2201,2:412,3:510,4:127,5:40,8:8},
+    H = {4:28,7:7,8:47,9:326}.
+
+The receipt is authenticated byte-for-byte as an explicit replay input. This
+uses its reported native census; it is **not** a fresh native operator replay.
+The two-call paid profile has rank mass10654. Removing the two old H9 copies
+per reused owner gives the remaining helper targets2(H-112 H9). Thus
+
+    T_bare = 224 H17 + 112 H16 + 2(H-112 H9), rank mass10526,
+    T_bonus = 336 H17          + 2(H-112 H9), rank mass10638.
+
+Every owned prefix remains dependent and receives zero additional reward.
+The other original helpers are granted their full unchanged rewards.
+The actual positive split cost is also granted zero; adding it can only hurt.
+
+Here is an exact certificate for all real0<t<=1, without floating roots.
+For a profile V define K_V(z)=sum_r V_r min(r,z). For the bare case check
+
+    K_(2P)(z)-K_T_bare(z)-128 min(1,z) >= 0,
+
+and for the bonus case check
+
+    K_(2P)(z)-K_T_bonus(z)-16 min(1,z) >= 0.
+
+The respective exact residuals at all positive breakpoints are
+
+| z | Bare residual | Bonus residual |
+|---:|---:|---:|
+| 1 | 5540 | 5652 |
+| 2 | 6806 | 6918 |
+| 3 | 7248 | 7360 |
+| 4 | 6670 | 6782 |
+| 5 | 5894 | 6006 |
+| 7 | 4182 | 4294 |
+| 8 | 3340 | 3452 |
+| 9 | 2576 | 2688 |
+| 16 | 224 | 336 |
+| 17 | 0 | 0 |
+
+The bonus case is affine between9 and17, so16 is an optional additional
+point rather than a required knot. Both differences vanish at0 and are
+constant0 beyond17. Nonnegativity everywhere follows by piecewise affinity.
+For any increasing concave phi with phi(0)=0 on[0,17], write phi(x) as a
+nonnegative linear combination/integral of x and min(x,z). This follows
+from its decreasing nonnegative derivative, or from the corresponding
+piecewise-linear approximation. The cut inequalities therefore imply
+
+    sum_r (2P)_r r^t - sum_r (T_bare)_r r^t >= 128,
+    sum_r (2P)_r r^t - sum_r (T_bonus)_r r^t >= 16,
+
+using phi(x)=x^t and phi(1)=1. Both inequalities extend to t=0 by the
+right-hand limit, since all listed ranks are positive. Thus the entire
+saving range0<=C<1 loses even before any completion or outer charges.
+
+This stops **the stated full-capacity repeated-H9-owner policy**. It does not
+exclude a caller retaining both distinct helper jobs, a changed joint provider
+or productive census, additional independent owners, a different stage/axis
+count, broader physical address routing, or non-Gaussian recursive job types.
+It does not assert the hypothetical capacity112 exists. Building that capacity
+solely to feed this losing reuse policy is not justified. The actual remaining
+construction question is a complete positive distinct-owner or changed-provider
+caller, rather than another unpriced expansion of the same helper alias.

research/homogeneous-chain-extraction/README.md

@@ -55,9 +55,30 @@ expansion is warranted until a named changed provider has a complete favorable
 budget. The retained-center source has heterogeneous charts and incoming helper
 noise, as recorded in [Discussion4840](https://github.com/CrocSwap/integer-mult-bounds/discussions/4840).
 No native homogeneous offer is obtained from its scalar Jordan chains.
+The newer one-port H3 prefix supplies a distinct physical three-chain offer;
+its repeated-helper consumer is priced separately below.
 Acquisition, actual chart maps, dirty restoration and independent helper targets
 are unknown for the proposed longer chain. Physical movement/precision,
-completion, compiler, BIT and outer recovery remain unrun and unpriced.
+completion, compiler, BIT and outer recovery remain unrun and unpriced
+for a profitable distinct-owner or changed-provider caller.
+
+## A native repeated-helper policy is already too expensive
+
+The source-bound physical H3 prefix in Chris R / Clear1nsight's
+`one-terminal-h3-prefix` package uses an original H9 helper as the third
+chain owner. Repeating that owner across two calls replaces two independent
+H9 jobs; it does not retain both rewards. Granting112 prefixes, all chart and
+finite costs free, and the exact three-chain output still gives a moment
+loss of at least128. Even granting a free H17 third output at every port
+instead of H16 leaves a loss of at least16 for every0<=C<1.
+
+[PROOF.md](PROOF.md) gives the complete source profiles and exact
+piecewise-linear concave-moment certificate. The verifier authenticates the
+3772-byte native paid/helper receipt and checks both certificates plus three
+budget mutations. This is a fresh exact arithmetic gate using a pinned
+reported native census, not a fresh native replay or a112-prefix construction.
+**Stop this repeated-H9-owner expansion.** Retaining both distinct helper
+jobs or changing the provider/census is still a different open contract.
 
 ## Reproduce
 
@@ -89,12 +110,14 @@ claim, and explicit expansion gate. Upstream and frozen predecessors are unchang
 
 ## Credit and scope
 
-Cinna Davis / 1337isnot1337 supplies PR4694's coupled retained-center extension,
+Chris R / Clear1nsight supplies the actual H3-prefix offer and its phase-bound
+normalizers. Its frozen native source receipt is reused explicitly for the
+new all-C budget gate. Cinna Davis / 1337isnot1337 supplies PR4694's coupled retained-center extension,
 PR4698's helper normalization and conditional three-chain consumer, and the
 PR4440/4470 controller lineage. Retain Cao Yifan / intqwq, zxv, Jacob Alan
 Sussman, Ryanson Jonathan and all cited native/compiler/precision predecessors.
 Their actual native construction is not replayed or modified here. The new
 contribution is the all-length proof/circuits, restricted workspace bound,
-and explicit acquisition allowance. Prepared for Cinna Davis with substantial
+explicit acquisition allowance, and the source-bound repeated-helper budget gate. Prepared for Cinna Davis with substantial
 OpenAI Codex assistance. Apache-2.0. No independent human or proof-kernel
 acceptance is claimed.

research/homogeneous-chain-extraction/RESULT.json

@@ -310,6 +310,72 @@
     4,
     4
   ],
+  "native_reused_helper_budget": {
+    "source_receipt_sha256": "eec3b14241430b6c9f5de3cbf0e3d45a172d8071d2dccd2b59d56a1fbac3d428",
+    "two_call_paid": {
+      "1": 4402,
+      "2": 824,
+      "3": 1020,
+      "4": 254,
+      "5": 80,
+      "8": 16
+    },
+    "reused_H9_owners": 112,
+    "cases": [
+      {
+        "third_rank": 16,
+        "paid_minus_target_lower_bound": 128,
+        "targets": {
+          "4": 56,
+          "7": 14,
+          "8": 94,
+          "9": 428,
+          "16": 112,
+          "17": 224
+        },
+        "cut_residuals": {
+          "1": 5540,
+          "2": 6806,
+          "3": 7248,
+          "4": 6670,
+          "5": 5894,
+          "7": 4182,
+          "8": 3340,
+          "9": 2576,
+          "16": 224,
+          "17": 0
+        }
+      },
+      {
+        "third_rank": 17,
+        "paid_minus_target_lower_bound": 16,
+        "targets": {
+          "4": 56,
+          "7": 14,
+          "8": 94,
+          "9": 428,
+          "17": 336
+        },
+        "cut_residuals": {
+          "1": 5652,
+          "2": 6918,
+          "3": 7360,
+          "4": 6782,
+          "5": 6006,
+          "7": 4294,
+          "8": 3452,
+          "9": 2688,
+          "17": 0
+        }
+      }
+    ],
+    "budget_controls": [
+      "excess margin",
+      "double-counted aliased helper jobs",
+      "unpaid omitted centers"
+    ],
+    "scope": "Both hypothetical full-capacity policies lose for every 0<=C<1; no prefix capacity or native replay claimed"
+  },
   "proof": "PROOF.md",
   "native_embedding": "unknown",
   "selected_kappa_changed": false

research/homogeneous-chain-extraction/SOURCES.json

@@ -25,6 +25,24 @@
       "usage": "provenance",
       "git_blob": "7cf79600e5e46a3714e83fb04a694a1ddede065b",
       "sha256": "56dc7d458410637dd675e7807ec272f8e55381b4bfa1313ef5bb552e7488adc4"
+    },
+    {
+      "path": "research/one-terminal-h3-prefix/inputs/research/kappa-round3-warp-20261022/NATIVE-VALIDATION.json",
+      "usage": "input",
+      "git_blob": "d8a2e0a915edce68cfe7472b94bb87dfb9dc24c0",
+      "sha256": "eec3b14241430b6c9f5de3cbf0e3d45a172d8071d2dccd2b59d56a1fbac3d428"
+    },
+    {
+      "path": "research/one-terminal-h3-prefix/homogeneous/REPORT.md",
+      "usage": "provenance",
+      "git_blob": "89bc05d40ce08f664b573996a61cbc0aae273619",
+      "sha256": "f4d192ae906cf6d70b05ee134e806bfe0502ee4944a00024882bb042ac9fb2d3"
+    },
+    {
+      "path": "research/one-terminal-h3-prefix/README.md",
+      "usage": "provenance",
+      "git_blob": "d7bfa15070573221267e0c10391fc75f7ea1cc6d",
+      "sha256": "90a9602f073322436b6cf98cbc4a31f4ee6b6b28b4746f6c5e6115afbc217072"
     }
   ]
 }

research/homogeneous-chain-extraction/TASK.json

@@ -70,13 +70,13 @@
     ]
   },
   "scope_status": "changed",
-  "actual_scope": "Exact two-call capacity 2 floor((n-1)/2) of homogeneous query chains, with attaining odd/even circuits, restored-workspace upper bound in the scalar/global-P ring, and a complete conditional acquisition allowance.",
-  "scope_change_reason": "Replace the installed QQ16 demand with the conditional PR4698 chain interface; extend its proved three-field module before any physical synthesis. Native heterogeneity and helper contamination remain explicit. A bounded four-field middle-boundary test produced a weighted cyclic construction, extended algebraically to all even lengths in the same module; this closes the initially open even attaining case.",
+  "actual_scope": "Exact two-call capacity 2 floor((n-1)/2) of homogeneous query chains, with attaining odd/even circuits, restored-workspace upper bound in the scalar/global-P ring, and a complete conditional acquisition allowance. Source-bound complete optimistic all-C rejection of the known native repeated-H9-owner policy, even granting a third H17 output.",
+  "scope_change_reason": "Replace the installed QQ16 demand with the conditional PR4698 chain interface; extend its proved three-field module before any physical synthesis. Native heterogeneity and helper contamination remain explicit. A bounded four-field middle-boundary test produced a weighted cyclic construction, extended algebraically to all even lengths in the same module; this closes the initially open even attaining case. The next physical read/derive gate found the repeated-helper policy loses even at hypothetical full prefix capacity; an exact source-receipt-bound ledger is retained without constructing that capacity.",
   "bottleneck_link": "A cheap homogeneous offer could turn two paid calls into n-1 main queries, but direct unary acquisition loses. The exact E<(n-1)/2 allowance identifies the missing physical producer.",
   "budget_minutes": 60,
-  "stop_rule": "Complete the all-length proof and bounded replay; stop direct unary synthesis. Do not begin native expansion without a named complete positive provider invoice.",
+  "stop_rule": "Complete the all-length proof and bounded replay; stop direct unary synthesis. Do not begin native expansion without a named complete positive provider invoice. Do not expand repeated-H9-helper prefix capacity: even an extra free H17 third output misses the complete moment by at least16.",
   "outcome": "partial",
-  "evidence": "PROOF.md; deterministic RESULT.json with exact all-input lengths 2..32 and both parity circuits; registered verifier; no native provider or new kappa.",
+  "evidence": "PROOF.md; deterministic RESULT.json with exact all-input lengths 2..32 and both parity circuits; registered verifier; no native provider or new kappa. NATIVE-REUSE section: exact concave cuts prove margins128 and16 for the two hypothetical all112-port reuse policies.",
   "author": "Cinna Davis with substantial OpenAI Codex assistance",
   "ai_assistance": "Substantial OpenAI Codex assistance for algebra, implementation, proof review and publication; no delegated agents or independent human review.",
   "open_questions": [
@@ -119,5 +119,11 @@
       "local_notes_complete_at_head": true,
       "warning": "Title/body/local-note coverage is not exhaustive proof of novelty."
     }
+  },
+  "followup_checkpoint": {
+    "canonical_main": "a5df48c55ea806d7952b65acc9c76d878ff09b13",
+    "selected_kappa": "0.00302015",
+    "priority_revision": 7,
+    "reason": "PR5137 all-length operator package archived at f2d83999edccdad625ea686ff3f0d03b7d208965; this follow-up adds only the source-bound native reused-owner budget gates. Canonical onboarding-fixture fixes read before rebasing; original frozen task/source pins preserved."
   }
 }

research/homogeneous-chain-extraction/VERIFY.json

@@ -3,7 +3,9 @@
   "status": "ready",
   "python": "3.12",
   "requirements": [],
-  "inputs": [],
+  "inputs": [
+    "research/one-terminal-h3-prefix/inputs/research/kappa-round3-warp-20261022/NATIVE-VALIDATION.json"
+  ],
   "commands": [
     {
       "argv": [
@@ -12,7 +14,7 @@
         "{package}/verify.py"
       ],
       "timeout_seconds": 120,
-      "checks": "Exact cyclic-ring all-input identities for every n=2..32, both parity circuits and full boundary counts, thirteen active mutations, parity guards, 256 dirty-workspace rank fixtures and faithful tensor-width qualification. All-length statements proved in PROOF.md; no physical supplier or higher kappa."
+      "checks": "Exact cyclic-ring all-input identities for every n=2..32, both parity circuits and full boundary counts, thirteen active mutations, parity guards, 256 dirty-workspace rank fixtures and faithful tensor-width qualification. All-length statements proved in PROOF.md; no physical supplier or higher kappa. Pinned native paid/helper receipt; exact all-C concave-cut certificates for two reused-helper policies and three budget mutations."
     }
   ]
 }

research/homogeneous-chain-extraction/verify.py

@@ -4,13 +4,17 @@
 """
 
 import argparse
+from collections import Counter
+from hashlib import sha256
 import json
 from pathlib import Path
 
 ZERO = (0, 0, 0, 0)
 ONE = (1, 0, 0, 0)
 Q = (0, 1, 0, 0)
 P = (0, 0, 1, 0)
+NATIVE_RECEIPT = 'research/one-terminal-h3-prefix/inputs/research/kappa-round3-warp-20261022/NATIVE-VALIDATION.json'
+NATIVE_RECEIPT_SHA256 = 'eec3b14241430b6c9f5de3cbf0e3d45a172d8071d2dccd2b59d56a1fbac3d428'
 
 
 def need(condition, message):
@@ -230,6 +234,55 @@ def faithful_gaussian_check():
     return ranks
 
 
+def cut_certificate(paid, targets, margin):
+    knots = sorted(set(paid) | set(targets) | {1})
+    cuts = {str(z): sum(n*min(r, z) for r, n in paid.items())
+            - sum(n*min(r, z) for r, n in targets.items()) - margin*min(1, z)
+            for z in knots}
+    need(all(x >= 0 for x in cuts.values()), 'Nonnegative complete concave-moment cuts')
+    return cuts
+
+
+def native_reuse_budget():
+    """Read a pinned source receipt, not a fresh native operator replay."""
+    raw = (Path(__file__).resolve().parents[2] / NATIVE_RECEIPT).read_bytes()
+    need(sha256(raw).hexdigest() == NATIVE_RECEIPT_SHA256, 'Exact native paid/helper receipt')
+    source = json.loads(raw)
+    paid = {int(r): 2*n for r, n in source['paid'].items()}
+    helpers = {int(r): n for r, n in source['helpers'].items()}
+    need(helpers[9] >= 112, 'Sufficient H9 slots in the optimistic reused-owner policy')
+    remaining = {r: 2*(n-(112 if r == 9 else 0)) for r, n in helpers.items()}
+    cases = []
+    for third_rank, margin in ((16, 128), (17, 16)):
+        targets = Counter(remaining)
+        targets[17] += 224
+        targets[third_rank] += 112
+        cuts = cut_certificate(paid, targets, margin)
+        cases.append({'third_rank': third_rank, 'paid_minus_target_lower_bound': margin,
+                      'targets': {str(r): n for r, n in sorted(targets.items())},
+                      'cut_residuals': cuts})
+    optimistic = Counter(remaining)
+    optimistic[17] += 336
+    wrong = [
+        (paid, optimistic, 17),  # The claimed uniform margin is exactly too large.
+        (paid, optimistic + Counter({9: 224}), 16),  # Alias owners counted twice.
+        ({r: n for r, n in paid.items() if r != 8}, optimistic, 16),
+    ]
+    for bad_paid, bad_target, margin in wrong:
+        try:
+            cut_certificate(bad_paid, bad_target, margin)
+        except ValueError:
+            pass
+        else:
+            raise ValueError('False native reused-owner budget accepted')
+    return {'source_receipt_sha256': NATIVE_RECEIPT_SHA256,
+            'two_call_paid': {str(r): n for r, n in sorted(paid.items())},
+            'reused_H9_owners': 112, 'cases': cases,
+            'budget_controls': ['excess margin', 'double-counted aliased helper jobs',
+                                'unpaid omitted centers'],
+            'scope': 'Both hypothetical full-capacity policies lose for every 0<=C<1; no prefix capacity or native replay claimed'}
+
+
 def check():
     lengths = list(range(2, 33))
     rows = []
@@ -280,6 +333,7 @@ def check():
         'rejected_even_controller_mutations': even_mutations,
         'dirty_workspace_rank_fixtures': lower_bound_fixtures(),
         'tensor_evaluation_ranks_f1_through_f4': faithful_gaussian_check(),
+        'native_reused_helper_budget': native_reuse_budget(),
         'proof': 'PROOF.md',
         'native_embedding': 'unknown',
         'selected_kappa_changed': False,