Make the positive-minimum return estimate uniform on the stated domain

.github/workflows/verify.yml

@@ -44,6 +44,8 @@ jobs:
         run: python tools/test_amplitude_identification.py --output amplitude-evidence/REPORT.json
       - name: Check reflection matching separately
         run: python tools/test_reflection_matching.py --output reflection-evidence/REPORT.json
+      - name: Check uniform positive-minimum estimates separately
+        run: python tools/test_uniform_minimum.py --output uniform-minimum-evidence/REPORT.json
       - name: Check preserved inputs and document links
         run: python verify.py --integrity-only
       - name: Run every scientific suite and retain comparisons
@@ -65,6 +67,7 @@ jobs:
             model-residual-evidence/
             amplitude-evidence/
             reflection-evidence/
+            uniform-minimum-evidence/
             tracked-source.zip
           if-no-files-found: warn
           retention-days: 14

README.md

@@ -86,6 +86,8 @@ The [physical-amplitude lemma chain](research/AMPLITUDE_IDENTIFICATION.md) now s
 an author-side retarded reconstruction, channel normalization and uniform tail comparison.
 The [reflection-matching note](research/REFLECTION_MATCHING.md) expands Gate B: explicit
 endpoint cancellation and exact-lattice positive/negative outer control at zero minimum.
+The [uniform-minimum note](research/UNIFORM_MINIMUM.md) expands Gate C: the full
+positive-minimum domain, including its narrow turning region and shifted negative tail.
 [Source labels](research/SOURCES.md) map the preserved notes' local reference numbers.
 [Assumptions](literature/ASSUMPTIONS.md) distinguishes ideal model premises from a joint
 apparatus, with a [primary-source register](literature/PHYSICAL_PRECEDENTS.md) and

STATUS.md

@@ -12,7 +12,7 @@ and history inputs remain byte-preserved.
 | Inverse-square-logarithmic return | Author-side matched-asymptotic derivation | Fixed local trap and order of limits |
 | Energy reflection equals physical return | Explicit Gate A lemma chain | Not independent proof review |
 | Leading reflection and outer matching | Explicit Gate B endpoint and exact-lattice estimates | Zero minimum; not independent proof review |
-| Positive-minimum joint law | Derived uniformly away from b=1 | No full crossover claim |
+| Positive-minimum joint law | Explicit Gate C uniform lemma chain | Fixed margin below b=1; no crossover claim |
 | Finite-volume family | Sufficient bound and finite examples | Not an optimized size requirement |
 | Fixed positive hopping anisotropy | Scope corollary with checked hypotheses | Not uniform at zero transverse hopping |
 
@@ -25,8 +25,12 @@ The new argument supplies that intermediate reasoning without changing the law.
 [Gate B](research/REFLECTION_MATCHING.md) now makes the endpoint-basis cancellation
 explicit, treats positive-side passivity without a chosen absorber, and bounds the
 entire negative tail with the exact positive spectral measure. No leading coefficient
-correction was found. These are author-side arguments at zero minimum; Gates C and D
-are not closed by this pass. The
+correction was found. [Gate C](research/UNIFORM_MINIMUM.md) now treats the entire
+positive-minimum domain at a fixed margin below b=1: its central perturbation is
+offset-independent, its passive denominator does not require a sign of Im(h), and
+its shifted exact negative-tail defect is dominated uniformly by the zero-minimum
+integral. The coefficient and domain are preserved. These are author-side arguments;
+Gate D remains separate. The
 [reader packet](research/READER_PACKET.md) remains a brief, not a received report.
 
 ## Prior art and physical premises
@@ -55,8 +59,8 @@ Full crossover, optimal volume and experimental superiority remain unclaimed.
 
 The preserved scientific suites contain **5 suites, 28 groups and 272 controls**.
 Fourteen infrastructure tests, five predecessor checks, six operational checks,
-seven model-residual checks, seven amplitude checks and six reflection checks are
-counted separately.
+seven model-residual checks, seven amplitude checks, six reflection checks and
+seven uniform-minimum checks are counted separately.
 Execution, exact byte reproduction and independent scientific review are distinct.
 
 The Gate A base was `5167d02b9504596e1c1b071ed425dea63dec62c6`. Its downloaded main
@@ -84,7 +88,13 @@ reclassify any previous failed, cancelled or non-byte-identical execution.
 
 ## Next bounded step
 
-Follow [CURRENT](work_orders/CURRENT.md): challenge uniformity of the existing
-positive-minimum law on the whole stated b domain, or record a concrete objection.
+[Gate C provenance](provenance/GATE_C_2026-10-04.json) records the recovered current
+main, source-tree identity and the new diagnostic evidence. The first seven-test
+uniform-minimum run passed; no new failed scientific attempt is omitted. The early
+base-source run overlapped addition of the new note, changing only its navigation
+link count; a frozen candidate and actual hosted reports are separate evidence.
+
+Follow [CURRENT](work_orders/CURRENT.md): scrutinize the finite-volume initial-state,
+periodic-seam and return-amplitude estimates without optimizing lattice size.
 Do not broaden the model to prolong development. A genuinely separate report
 remains absent. No manuscript, release or outside invitation is initiated here.

archive/GATE_C_DIAGNOSTIC_REFINEMENT.md

@@ -0,0 +1,56 @@
+# Gate C diagnostic refinement: keep the central equation off a singular wave chart
+
+**4 October 2026. Test implementation only; no physical equation or tolerance change.**
+The first seven-test Gate C candidate passed. An additional exact-turning-point probe
+then exposed an avoidable issue in its diagnostic RHS: it called `local_data(...)[0]`
+to obtain the regular scalar coefficient, but that helper also evaluated unused
+wave-basis quantities proportional to inverse powers of the coefficient. At a zero
+of the coefficient those unused quantities are singular.
+
+With NumPy floating-point warnings raised as exceptions, the actual probe was:
+
+```python
+with np.errstate(all='raise'):
+    local_data(-np.exp(-16.), 16., .5)[0]
+# FloatingPointError: divide by zero encountered in scalar divide
+```
+
+The central ODE coefficient itself is zero there and has no singularity. This is
+not a failure of an original 272-control suite, not a defect of the integrated
+central estimate, and not evidence for cutting out the turning region.
+
+The repair gives the central ODE a coefficient-only evaluator, including the exact
+origin limit. Wave data remain confined to their outer-basis use. A new assertion
+inside the existing central-remainder test checks both the exact turning point and
+the origin with floating warnings promoted to errors. The seven-test count is
+unchanged. No equation, integration tolerance, reference or protected source changed.
+
+The pre-refinement script, raw probe log and forward/reverse patch are retained in
+the conversation evidence package. The original candidate is also Git commit
+`52bc319675c2d77c894580059745f010a9fd2794`; its scientific report is not relabeled
+as failed. The exact diagnostic code change is:
+
+```diff
+--- a/tools/test_uniform_minimum.py
++++ b/tools/test_uniform_minimum.py
+@@ -22,0 +23,8 @@
++
++
++def local_coefficient(x: float, L: float, b: float):
++    """The central equation is regular even where a wave basis is singular."""
++    if x == 0:
++        return complex(-b)
++    D = complex(L-np.log(abs(x)), np.pi if x > 0 else 0.)
++    return L/D-b
+@@ -78,0 +87,3 @@
++        with np.errstate(all='raise'):
++            self.assertEqual(local_coefficient(-np.exp(-16.), 16., .5), 0j)
++            self.assertEqual(local_coefficient(0., 16., .5), -.5+0j)
+@@ -122 +133 @@
+-                    F = -b+0j if x == 0 else local_data(x, L, b)[0]
++                    F = local_coefficient(x, L, b)
+```
+
+The refined frozen tree and hosted runs require their own completed reports. Do not
+infer them from the initial pass or confuse this auxiliary probe with an interrupted
+scientific verification.

provenance/GATE_C_2026-10-04.json

@@ -0,0 +1,88 @@
+{
+  "date": "2026-10-04",
+  "repository": "GoGoKo699/Logarithmic-Threshold-Return",
+  "base_commit": "f00281f6fa9310091c7da402142f1f4e5560f272",
+  "base_tree": "db408710b9370ed31df0b397be6e294c19b938e1",
+  "scope": "Author-side Gate C: uniform positive-minimum matching at fixed delta>0; no changed law or independent review.",
+  "recovered_source": {
+    "run_id": 37229542484,
+    "artifact_id": 11313273793,
+    "sha256": "47f5b22eee51b0cbe192300e3cc855388bed7b35afb4fa2d1f0c82a1291d96b2",
+    "tracked_files": 57,
+    "source_tree_match": true,
+    "method": "Authenticated GitHub artifact after public clone failed DNS; prior local Gate B patch not applied because current upstream already has a different checked Gate B implementation."
+  },
+  "early_local_base_source_run": {
+    "status": "PASS",
+    "suite_count": 5,
+    "group_count": 28,
+    "case_count": 272,
+    "all_reference_bytes_identical": true,
+    "report_sha256": "83da0543ff11ec7d5a5a664f536acc0a3b233f725525381cc88ba8a6fa3e09e8",
+    "before_integrity": {
+      "status": "PASS",
+      "preserved_files": 27,
+      "local_links": 83,
+      "suites": 5
+    },
+    "after_integrity": {
+      "status": "PASS",
+      "preserved_files": 27,
+      "local_links": 85,
+      "suites": 5
+    },
+    "boundary": "Original scientific inputs unchanged. Addition of the new note overlapped this run, changing navigation links only; it is not labeled a frozen whole-tree run."
+  },
+  "existing_tests": {
+    "infrastructure": 14,
+    "predecessor": 5,
+    "operational": 6,
+    "model_residual": 7,
+    "amplitude_identification": 7,
+    "reflection_matching": 6,
+    "status": "PASS"
+  },
+  "new_checks": {
+    "count": 7,
+    "first_run": "PASS",
+    "report_sha256": "26f46c0806aed38d48ff200f1ee8c9b5f6ee590dac45dd17316118b4e198c314",
+    "original_272_members": false,
+    "sampled_sign_reversals": 169,
+    "scope": "Symbolic and finite integral/ODE diagnostics; not a supremum proof or long-time physical simulation.",
+    "initial_script_sha256": "02d7bc6223e585bd3e1762fee28751b3a5de5aec85fa600a8dfab709d87df681",
+    "refined_script_sha256": "7703afd291d29eb03bde1b4be1359f645d1d3644cd448809acfad229b4df52b1",
+    "refinement": "An extra exact-turning-point probe exposed unused singular wave-data evaluation in the central RHS. Coefficient-only evaluation and an exact-point assertion repair it without an equation or tolerance change. See archive/GATE_C_DIAGNOSTIC_REFINEMENT.md; first passing script/log/probe/patch retained."
+  },
+  "preservation": {
+    "mapped_inputs": 27,
+    "protected_equations_changed": false,
+    "references_changed": false,
+    "comparison_or_workload_policy_changed": false
+  },
+  "method_access": [
+    {
+      "source": "NIST DLMF 19.12",
+      "url": "https://dlmf.nist.gov/19.12",
+      "depth": "HTML convergent elliptic series 19.12.1–3 and complementary modulus conventions",
+      "boundary": "Inherited local expansion, not new primary physics attribution."
+    },
+    {
+      "source": "NIST DLMF 2.7(iii)",
+      "url": "https://dlmf.nist.gov/2.7#iii",
+      "depth": "HTML method context revisited",
+      "boundary": "No additional theorem imported; passive and shifted-defect bounds are derived explicitly."
+    }
+  ],
+  "operational_failures": [
+    "Public git clone failed: Could not resolve host github.com. Source recovered by connector; no scientific run interrupted.",
+    "A clean-baseline launch selected its source directory as output and was rejected before any scientific suite. The corrected retry uses fresh baseline-frozen paths; rejected logs are retained."
+  ],
+  "decision": {
+    "gate_c": "Explicit uniform author-side route; no coefficient/domain correction found.",
+    "gate_d": "Next bounded task.",
+    "separate_report": "Absent.",
+    "manuscript": "On hold.",
+    "outreach": "None."
+  },
+  "candidate_and_hosted": "Inspect frozen candidate and actual PR/main reports separately. Their success is not inferred from this baseline or stored prospectively here."
+}

research/CLAIMS.md

@@ -13,7 +13,7 @@ positive-minimum and finite-volume domain are supporting parts of that statement
 |---|---|---|---|
 | Local unitary source and bound-state return | CORE, model; ASYMPTOTIC Sections 2–4 | Suite 05 coordinate/spectral comparison, suite 06 time/energy comparison | Site occupation is not the return projector |
 | Leading logarithmic law and coefficient | ASYMPTOTIC Sections 5–7, read together with AUDIT | Suites 06–07 scalar formulations, physical-band and boundary checks | Numerical convergence is not the matching proof |
-| Positive-minimum law for b below one | ROUNDING Sections 2–4 | Suite 08 joint-coefficient and finite-time checks | No formula through b=1 or every fixed residual depth |
+| Positive-minimum law for b below one | ROUNDING Sections 2–4; UNIFORM_MINIMUM | Suite 08 joint-coefficient and finite-time checks | No formula through b=1 or every fixed residual depth |
 | A finite-system family with the same limit | ROUNDING Section 6 | Suite 08 finite torus/coordinate and locality controls | Sufficient size is not necessary or optimized size |
 | Fixed positive unequal hoppings | SPECTRAL_SCOPE Sections 2–4 | Suite 09 resolvent, causality and scalar checks | Not uniform as transverse hopping vanishes |
 
@@ -56,3 +56,7 @@ Full crossover, general trap universality, optimal volume, experimental superior
 and a joint apparatus are not claimed. The known terminology
 correction is in SPECTRAL_SCOPE: the lower-edge density is finite; the Green function
 is logarithmic. The original older wording is preserved, not silently rewritten.
+
+[Gate C](UNIFORM_MINIMUM.md) makes the positive-minimum quantifiers and shifted
+negative-tail domination explicit. Its seven finite diagnostics are separate from
+the preserved scientific suites and do not constitute an independent report.