Prove conditional uniform R437 COPY contract and complete relative cost

research/r437-uniform-copy-reduction/.gitattributes

@@ -0,0 +1,2 @@
+inputs/** -text -whitespace
+R437-* -text -whitespace

research/r437-uniform-copy-reduction/CALLBACK-PROBE.json

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+{
+  "status": "PASS_ENCODED_R437_COPY_COEFFICIENT_PROBE",
+  "q": 3,
+  "atom_precision": 2,
+  "actual_R437_center": 475,
+  "actual_R437_frame": 515,
+  "copied_projector_rank": 7,
+  "synthetic_ancestor_rank": 26,
+  "root_width": 4096,
+  "parent_width": 1664,
+  "child_width": 182,
+  "child_is_nonterminal": true,
+  "generated_coefficients": [
+    {
+      "coordinate": 0,
+      "value": 5
+    },
+    {
+      "coordinate": 26,
+      "value": 8
+    },
+    {
+      "coordinate": 64,
+      "value": 2
+    },
+    {
+      "coordinate": 181,
+      "value": 2
+    }
+  ],
+  "child_metadata": [
+    0,
+    1,
+    7,
+    0,
+    0,
+    0
+  ],
+  "controls": [
+    {
+      "control": "foreign_current_source",
+      "rejected_by": "selected good current site"
+    },
+    {
+      "control": "wrong_ancestor_rank",
+      "rejected_by": "actual selected rank"
+    },
+    {
+      "control": "wrong_current_prefix_code",
+      "rejected_by": "original global raw-field code"
+    }
+  ],
+  "runtime_host_numerical_decoding": false,
+  "scope": "Authenticated reused numerical components, actual COPY projector and explicit finite new source record, synthetic ancestor; no full R437 catalogue, payload, stock, physical CALL/RETURN or all-size cost proof"
+}

research/r437-uniform-copy-reduction/COORDINATION.json

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+{
+  "checked_utc": "2026-10-19T06:18:00Z",
+  "coverage": "Partial targeted live PR/title/body and selected proof-file review, not novelty coverage",
+  "canonical_main": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "final_checkpoint": "Canonical main fetched unchanged; revision-6 priorities and selected frontier unchanged; no reroll on explicit assignment",
+  "predecessors": [
+    {
+      "pr": 3690,
+      "head": "03c141ef1fd372cf42951488767537e55675c072",
+      "source": "36ff441479ff9c7e70d83f0becccd44da50effb8",
+      "credit": "zxv",
+      "scope": "Finite actual R437 catalogue/event adapter; rank34 numerical callback and controlled 256->136 CALL/RETURN; uniform physical refinement and full ordinary/native/router cohort explicitly open",
+      "relationship": "Substantive predecessor. Current a24f9ff/OQ4 binding remains to refresh; our unchanged-selected-Catalogue guard does not refute this adapter"
+    },
+    {
+      "pr": 3562,
+      "head": "abb93be8b9ceca07a2b17f3a97db7bb6b074271d",
+      "credit": "Jose Manuel",
+      "scope": "Conditional ordinary retuning c=1/577; finite controlled returns; named physical premises"
+    },
+    {
+      "pr": 3724,
+      "head": "946d03dd6b9cc794190876fe3f0f89b0ebc01c59",
+      "credit": "Sagar Thakrar",
+      "scope": "Independent arithmetic review under inherited physical premises"
+    },
+    {
+      "pr": 3945,
+      "head": "91f81d131252db704f945891da1ea54e29d46b8b",
+      "scope": "Distinct staged544 terminal-fill conditional exponent; not this local COPY proof; no frontier change"
+    }
+  ],
+  "pr3936": {
+    "head": "e54c649865d737fe08edd4312a234cd186317db4",
+    "scientific_commit": "125f5afad680ef128e386fa58e76eef5fadaa5d6",
+    "ci_run": 39205477701,
+    "plan_job": 117907253747,
+    "replay_job": 117907773499,
+    "status": "success",
+    "delta": "Four-line P112 workflow-only repair; scientific package unchanged"
+  },
+  "claim": "https://github.com/CrocSwap/integer-mult-bounds/pull/3936#issuecomment-6260449978",
+  "outcome": "Partial: conditional local caller and full relative cost proof; no uniform physical supplier admission"
+}

research/r437-uniform-copy-reduction/MINOR-BOUND.json

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+{
+  "status": "PASS_RANK7_LOW_CLASS_MINOR_BOUND",
+  "generic_minor_numerator_degree": 64,
+  "product_degree": 448,
+  "conditional_fraction": "min(1,448*(q-1)/(q*(q-2)))",
+  "simple_bound": "delta_7 < 896/q for q>=3; delta_7 <449/q for q>=450",
+  "source_admissible_uniform_upper": "449/1208925819614629174706176",
+  "fallback_children_per_bad_class_upper": 49911,
+  "source_admissible_fallback_multiplier_upper": "22410039/1208925819614629174706176",
+  "inherited_epsilon": "1/10000000000000000",
+  "union_over_paid_labels": false,
+  "full_high_lifts_covered": true,
+  "coverage": "Written polynomial and conditional-caller proof; finite audits are supporting evidence, not proof of physical refinement.",
+  "complete_small_GL_tables": [
+    {
+      "dimension": 2,
+      "rank": 1,
+      "q": 3,
+      "GL": 48,
+      "bad": 20,
+      "fraction": "5/12",
+      "numerator_checks": 48
+    },
+    {
+      "dimension": 2,
+      "rank": 1,
+      "q": 5,
+      "GL": 480,
+      "bad": 144,
+      "fraction": "3/10",
+      "numerator_checks": 480
+    },
+    {
+      "dimension": 2,
+      "rank": 1,
+      "q": 7,
+      "GL": 2016,
+      "bad": 468,
+      "fraction": "13/56",
+      "numerator_checks": 2016
+    },
+    {
+      "dimension": 2,
+      "rank": 1,
+      "q": 17,
+      "GL": 78336,
+      "bad": 8448,
+      "fraction": "11/102",
+      "numerator_checks": 78336
+    },
+    {
+      "dimension": 3,
+      "rank": 2,
+      "q": 3,
+      "GL": 11232,
+      "bad": 6528,
+      "fraction": "68/117",
+      "numerator_checks": 22464
+    }
+  ],
+  "sign_mutation_separated": true,
+  "checked_obligations": [
+    "conditioning on GL explicitly charged",
+    "fallback children not free"
+  ]
+}

research/r437-uniform-copy-reduction/ORIGINAL-CALLER.md

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+# The original rank-seven caller: uniform semantics and a conditional full bill
+
+This is the route relevant to the inherited R437 tariff. The alternate all-class
+conjugacy in PROOF.md is deliberately kept separate: it pays additional ordinary
+interchanges and cannot inherit this bill. Neither result independently closes
+the published C0/B1 physical refinement, generated-source, or numerical reserve
+arguments. This file proves the rank-seven specialization **under those explicitly
+retained contracts**, including all bad classes; it does not assert their premises.
+
+## 1. The unchanged source-level primitive is applicable to arbitrary R437 dirt
+
+PR3936 binds the complete capture map, all 659 arbitrary input functions and
+the transported ERASE guards at R437 head a24f9ff. The nine rank-seven frame
+projectors, their rational root rows and their frame lists are unchanged from
+the selected PR589 word. The new word changes the functions occupying some
+centres; that does not change a callee quantified over **all** Boolean functions.
+The new data bindings and source codes still need authentication. Equality of
+projectors is not authority to replace one literal source record by another.
+
+For each fixed physical prefix, the actual requested residual is the weighted
+partial interchange D_H(gPg^-1) defined in PROOF.md. Address matrix equality
+implies equality of its pullback on every Boolean function, without independence
+or clean-helper assumptions. Clone the complete current centre, call that
+residual on the owned clone, keep its owned result unchanged during the literal
+read interval, then clear it at the transported equality guard. This gives
+source preservation and exact private output, conditional on a completed
+whole-packet residual with the stated source, spectator and cleanup contracts.
+
+In the actual pinned compiler (`macro_code.py`, COPY branch), the first copy
+commits the staged centre to `copy_source`; `residual_begin` initializes owned
+`current`; `residual_end` transfers the result back to `copy_source`; the final
+MOVE transfers it to the private role. `Machine.commit` preserves a permanent
+source and clears a returned temporary Lease only after copying it. ERASE
+clears the private role. This correspondence explains the clone/commit bill;
+it is source inspection, not an executed R437-specific physical integration.
+
+## 2. All high lifts have the same good/fallback classification
+
+Fix q, r=7 and the rank-seven split basis B_P over F_q. Right multiplication
+g0 -> G=g0 B_P is a bijection on GL_m(F_q). Its projector is
+
+    Q=G E_r G^-1,  E_r=diag(I_r,0), m=64.
+
+The source's `compile_specification` tests the successive leading pivots of Q.
+Equivalently it tests the leading principal minors Delta_k, k=1,...,r. In
+`good_form`, columns of Q are reversed before northeast elimination; its
+northeast pivot products are the same leading principal minors of Q, up to
+the fixed reversal convention. Every nonzero low pivot lifts to a unit in
+Z/q^w. Thus a good g0 stays good for **every** high lift at every precision.
+
+Incoming weights form a diagonal unit matrix H. Each leading principal block
+of H Q H^-1 is conjugate by its restricted diagonal H_I, so its determinant
+equals Delta_k. Therefore the classification is independent of the actual
+ancestor units and of the body/dirty payload. No new bad event is introduced
+per column or per high prefix. There is no probabilistic failure of the
+algorithm: the construction traverses a fixed complete cover and uses a
+completed fallback on every bad class.
+
+## 3. A rank-specific complete low-class count
+
+For I={0,...,k-1}, Cauchy–Binet and Jacobi's complementary-minor identity give
+
+    det(G) Delta_k =
+      sum_(J subset {0,...,r-1}, |J|=k)
+       (-1)^(sum I+sum J) det G[I,J] det G[I^c,J^c] = F_k(G).
+
+Every term of F_k has total degree k+(m-k)=m. At G=I, F_k=1. Hence F_k is
+a nonzero polynomial over **every** residue field, including q dividing r.
+The product F=product_(k=1)^r F_k is nonzero and has degree at most rm=448.
+This proof uses complementary minors of one invertible matrix, rather than
+assigning the full degree of each entry of G^-1 independently.
+
+A nonzero polynomial of total degree d in independent uniform field entries
+vanishes on at most a d/q fraction. One proof inducts on the number of
+variables: separate the highest power of the final variable, charge zeros of
+its nonzero leading coefficient by induction, then use the univariate root
+bound on the remaining fibres. The resulting degree contributions add to d.
+This bound is harmlessly trivial if d/q>=1.
+
+For the uniform full matrix distribution,
+
+    Pr(GL_m) = product_(i=1)^m (1-q^-i)
+             >= 1-sum_(i=1)^m q^-i > 1-1/(q-1).
+
+Conditioning explicitly on invertibility gives
+
+    delta_7 < min(1,448*(q-1)/(q*(q-2)))
+            < 896/q                    (q>=3),
+    delta_7 < 449/q                     (q>=450).           (A)
+
+The first expression can be used as a non-strict upper bound at the capped
+value one; strictness of the uncapped fraction is immaterial. For the admitted
+q>2^80, delta_7 <449/2^80<10^-16. The denominator 2^80 here is a monotone
+lower bound on q, **not** a claimed prime. This is a deterministic fraction
+of low classes; every low class has exactly the same H high-lift multiplicity.
+It is not a union over COPY labels, all paid requests, columns, or prefixes.
+
+The exact finite check independently evaluates Jacobi numerators with an
+adjugate over complete small GL domains, including 78,336 matrices in GL2(F17)
+and 11,232 in GL3(F3). A missing Jacobi sign is detected. These audits support
+the displayed polynomial proof; they do not prove its quantified statement.
+
+## 4. Good and fallback children, with every physical call paid
+
+Retain the source's completed normal-form identities and primitive contracts.
+A good residual has forward ordered adapters, one weighted reversed child
+of width 7f, then backward adapters. All current high fields remain frozen
+prefixes and all unused rows/spectators stay complete. The child is outside
+the prefix loop. Its [B',G',S'] and compatible fresh-row divisor are the ones
+in PROOF.md section 4. Width 7f<n gives the proper-width induction step;
+below the cutoff the source executes the completed eight-primitive leaf.
+
+The original good plan emits 420 head, 420 tail and 4096 cross ADD commands:
+the first two counts are sum_(k=0)^6(63-k). All 4936 commands have source
+physically before target. Its two sweeps therefore pay **9872*f** complete
+ordered ADD passes, even if some prepared coefficients are zero. They do not
+need the ordinary reverse adapters of the alternate construction.
+
+For a bad class the invertible 128-dimensional residual has a unit-pivot
+Gaussian fallback over the same local ring. The retained finite control has
+at most 127 row exchanges, 254 initial pair/unit child commands, and 16256
+child commands introduced to reverse above-diagonal shears. Each complete
+pair command has one main child and at most two same-bank half resets.
+The source omits zero-width halves and preserves the middle atom at odd f.
+Therefore a safe full bound is
+
+    J_bad = 3*(127+254+16256) = 49911
+          < 32*64^2 = 131072                              (B)
+
+coarse child calls per bad low class. Every one has width at most f; the
+half resets have width floor(f/2). Preparations, ordered scans, signs,
+canonical commits and restoration are additional local work. The terminal
+sign is not a recursive child and is included in that work. No child or
+private buffer disappears merely because it carries a low/rank tag.
+
+The fallback also has at most 16256 ordered ADD commands and one sign
+command per column, hence at most **16257*f** scalar passes, before its child
+and preparation costs. `source_audit.py` authenticates and executes the
+unmodified original pure plan definitions on actual ambient-64 good and bad
+R437 frame-515 conjugates. It checks these command/order bounds and expanded
+main/half counts at f=1,2,3,10,64. That is an explicitly **reused-code**
+algebra/control probe, separate from the independent engine and from physical
+payload execution or generated callback/continuation validation.
+
+## 5. The full relative bound that the original implementation would need
+
+Let T(j,w) be a **uniform per-payload-bit upper bound** over the admitted
+descendant programs, complete spectators, compatible stock and generated
+unit-weight contexts. Do not replace it by a measurement at one fixture.
+Write M_f=max_(1<=j<=f) T(j,w); no unproved runtime monotonicity is assumed.
+One family/low packet has v=V_parent/(W N), with N=|GL64(F_q)|.
+
+Let gamma_orig be the total encoded preparation, directory/Count setup,
+source-copy, transfer, and successful cleanup work not already covered by
+the completed primitive contracts. Before absorption, summing all N low
+classes and dividing by their complete total volume N v gives
+
+    C_copy_cover/(N v)
+      <= T(7f,w) + delta_7*49911*M_f
+           + A_q*(f+leaf_toll(w)+1) + gamma_orig/(N v).     (C)
+
+The dropped (1-delta_7) factor in front of T is an upper bound, not a claim
+that bad classes execute a second good child. It is legitimate because T
+is nonnegative. Every bad main and half child remains in the second term.
+The clone and final guarded disposal, intermediate releases, parent checkpoint,
+returned output and source/directory head restoration are in the local bill.
+The outer family's gather/route/scatter costs remain additional as appropriate
+to the containing node. Summing equal complete volumes is essential: using
+the smaller visible seven-coordinate chunk would underprice the spectators.
+
+Under the predecessor's **canonical root-coupling**, encoded preparation,
+complete fixed-tape primitive, and numerical/storage arguments, the last
+terms have the stated source-specific bound A'_q*(n+w^p). C0 has p=1;
+C1 uses p=1-a1 and depends on both reached B1 primitive replacements.
+Then (A),(B) imply the inherited safe envelope
+
+    C_copy_cover/(N v)
+      <= T(7f,w) + 10^-16*(32*64^2)*M_f
+                     + A'_q*(n+w^p).                     (D)
+
+This is the complete rank-seven specialization **relative to those named
+contracts**. It covers arbitrary dirty functions and every high lift;
+it does not create a new low-class error or an extra per-prefix child factor.
+It does not independently prove the required T, gamma absorption or physical
+normal-form/refinement premises. Outside the admitted root/history domain,
+retain (C) with the actual gamma and encoded widths. Reusing the scalar
+endpoint alone would not justify either bound.
+
+## 6. What is established and what is still missing
+
+The written all-parameter specialization proves that the complete dirty R437
+COPY function fits the original **semantic** callee interface, and supplies
+the conditional full cost envelope (C)/(D), including bad-class integration.
+The rank-specific degree/count proof and the independent new fixed-support
+alternative are source-bound results. Full physical supplier admission is
+not established: the repository still labels PR2809's universal refinement
+partially reviewed, and the actual R437 source-code registry and uniform
+coefficient/continuation integration have not been executed or independently
+proved here. Nine reserved family slots are not a complete scratch theorem.
+
+The smallest next physical task is to bind one actual R437 rank-seven
+Specification and generated program to the completed whole-packet
+CALL/RETURN protocol, while preserving exact source identity, all raw prefix
+ordinals, U, encoded width/precision, parent continuation and owned output.
+Verify the actual coefficient and primitive access path at that binding,
+then prove its gamma/time and full visited-cell reserve rather than infer
+them from (D). The alternate conjugacy route has a distinct, explicitly
+larger ordinary bill and does not discharge this remaining implementation.
+
+Credit remains with the original normal-form, weighted child, stock, primitive,
+caller, parking and outer-wrapper authors. In particular no new broad all-size
+caller, new selected tariff or multiplication exponent is claimed.