Bound every coordinate caller of the seven-axis B/C suppliers

research/complementary-caller-capacity-gate/.gitattributes

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research/complementary-caller-capacity-gate/INPUTS.json

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+    "path": "research/native-dual-suppliers/INTERSECTION153-ROUND3-ACTUAL-PROFILE.json",
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+    "path": "research/native-dual-suppliers/HETEROGENEOUS-CONTROLLER.md",
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research/complementary-caller-capacity-gate/PROOF.md

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+# A sign-cut bound for complete complementary callers
+
+Prepared by Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+
+## 1. Contract and claim
+
+Fix a finite set of coordinate axes. On axis a let Q_a be the usual lifted
+Gaussian at width f, and put J_a=Q_a^2. The Q_a commute, J_a^2=I, and f>=3.
+An ordinary operation is complex-linear and has a matrix on field indices whose entries are complex
+scalar combinations of products of the J_a. This grants all such operations
+for free. It includes arbitrary field mixing, COPY, retained observations,
+owned zero fields, CLEAR, and ordinary sector projectors. It does not grant
+an arbitrary address permutation, value conjugation or an odd Gaussian input gauge.
+
+A whole B or C call has 112 main pairs. There are 56 A pairs and 56 exceptional
+pairs of the stated type. In the common coordinate chart its main blocks are
+
+    A = D [[1,0],[Q^-1,1]],
+    B = D [[1,0],[1,Q]],
+    C = D [[Q,0],[1,1]].
+
+Here D is the product on seven distinct transverse axes, and Q is on a
+different central axis. A call can choose its axes, central axis, main fields
+and port placement. Different calls can reuse axes and fields. Inverses,
+transposes, inverse transposes and the source even-power corrections are
+allowed. Each call retains its exact paid profile and all 408 independent
+helper outputs. Helpers are separate from the caller's main computation;
+no helper input or output is a shared main resource. Native internal maps
+are part of the whole-call premise, not extra free caller operations.
+
+An additional query applies a product of coordinate Gaussians or their
+inverses to one field. Its charged rank r is the number of odd axes after
+even powers are removed. All these queries are charged, including idles
+and corrections. There is no upper bound on r. The caller can have any
+finite number of main inputs, fields, steps and calls. It is a linear
+straight-line program; its choices do not depend on the input values.
+
+The productive main outputs are independent full Gaussian jobs on distinct
+original arbitrary fields, on the declared coordinate axes. Their ranks
+r_i are the numbers of odd axes. Invertible ordinary input/output basis
+changes do not affect the proof. A projected component of a field is not
+credited as a full independent Gaussian job.
+
+Let L_B,L_C count the whole calls. Write P_X(t),H_X(t) for the sums of
+rank^t in the exact paid and helper profiles. For every 0<t<1, define
+
+    K(t) = 28 + 140*7^t + 84*8^t,
+    Delta_X(t) = P_X(t) - H_X(t) - K(t).
+
+**Theorem.** The complete target-minus-paid moment satisfies
+
+    F(t) <= -L_B*Delta_B(t) - L_C*Delta_C(t) < 0
+
+if at least one B or C call occurs. Both deltas are strictly positive.
+The result permits any main target ranks and any additional query ranks.
+It holds at each width f>=3. Thus this model has no positive complete
+recursive moment at any saving exponent 0<C<1, where t=1-C.
+It does not claim a universal obstruction for other address alphabets,
+helper sharing, native source changes or nonlinear callers.
+
+## 2. Faithful characters and rank distance
+
+The one-bit Gaussian is
+
+    (1/2) [[1+i,1-i],[1-i,1+i]].
+
+Its two eigenvalues are 1 and i. The f-fold address lift has eigenvalues
+i^k for 0<=k<=f. For f>=3 all four fourth roots occur. In each J_a sector,
+both signs of a chosen square root q_a of J_a therefore occur. Different
+coordinate axes are tensor factors, so all joint sign choices occur in
+every joint even-power sector. This uses tensor products on addresses,
+not tensor powers of the two-field A/B/C matrices.
+
+Fix one even-power sector and a base character q=(q_a). For a set S of
+axes, let q' negate q_a exactly on S. Every ordinary matrix is identical
+at q and q'. For any two field matrices define their rank distance by
+
+    d(M(q),M(q')) = rank(M(q)-M(q')).
+
+The telescoping identity for a product of matrices gives
+
+    d(G_n(q)...G_1(q), G_n(q')...G_1(q'))
+      <= sum_j d(G_j(q),G_j(q')).
+
+Each term is multiplied on its left and right by other matrices, which
+cannot increase its rank. Pad the finite computation to one field bank
+to include copies, zero fields and retained results. Restricting to the
+original input columns and productive output rows cannot increase rank.
+This proves the same inequality with arbitrary interleaving and feedback
+through previously computed linear values. It is not a one-snapshot bound.
+
+For independent diagonal Gaussian targets, the left side is exactly the
+number of target supports that meet S in odd cardinality. Their nonzero
+diagonal values change sign exactly in that case. Invertible ordinary
+boundary matrices preserve this rank. A rank-r additional query has rank
+distance one or zero under the same odd-intersection rule.
+
+## 3. Exact cut charge of a whole B/C call
+
+Let c indicate a central sign flip and e indicate an odd number of flips
+among the seven transverse axes. Then D changes to (-1)^e D. Direct
+subtraction of the two displayed pair matrices gives these ranks:
+
+| c | e | A pair | B pair | C pair |
+|---|---|---|---|---|
+| 0 | 0 | 0 | 0 | 0 |
+| 1 | 0 | 1 | 1 | 1 |
+| 0 | 1 | 2 | 2 | 2 |
+| 1 | 1 | 2 | 1 | 1 |
+
+For example, in the last row the B difference is
+D*[[2,0],[2,0]], while the A difference is 2D*I.
+These calculations hold for every nonzero complex q and D.
+Transposition preserves rank. Also
+
+    M(q)^-1 - M(q')^-1
+      = -M(q)^-1 (M(q)-M(q')) M(q')^-1.
+
+Both outside matrices are invertible, so inverse and inverse-transpose
+calls have exactly the same ranks. The source even-power corrections have
+the same values at both characters and preserve rank. The 56/56 direct
+sum and ordinary port permutations give a whole-call cut charge of 0,
+112,224,168 in the four table rows.
+
+Now flip each global axis independently with probability p, where
+0<p<1/2. Put
+
+    h_p(r) = (1-(1-2p)^r)/2.
+
+This is the probability of an odd number of flips on r distinct axes.
+The central flip and the transverse parity are independent within each
+call. Axis reuse by different calls does not change linearity of
+expectation. The expected whole-call charge is exactly
+
+    224*h_p(7) + 112*p - 168*p*h_p(7)
+      = 140*h_p(7) + 28*h_p(1) + 84*h_p(8).
+
+The verifier also obtains this polynomial by enumerating all 256 masks
+and their exact Bernoulli weights. For any complete caller, the rank
+distance inequality therefore gives
+
+    sum_i h_p(r_i)
+      <= (L_B+L_C)*(140*h_p(7)+28*h_p(1)+84*h_p(8))
+         + sum_extra_queries h_p(r).
+
+No serial topology, fixed number of main owners, disjoint stage axes or
+target/query rank cutoff is used here.
+
+## 4. From random cuts to every positive saving exponent
+
+For 0<t<1, the number
+
+    Z_t = integral_0^infinity (1-exp(-s))*s^(-t-1) ds
+
+is positive and finite. Near zero its integrand has order s^-t; near
+infinity it has order s^(-t-1). The substitution u=rs gives
+
+    r^t = (1/Z_t) integral_0^infinity (1-exp(-r*s))*s^(-t-1) ds.
+
+Set p=(1-exp(-s))/2. Then 1-exp(-r*s)=2*h_p(r).
+Integrate the preceding cut inequality against this positive measure.
+All sums have finitely many nonnegative terms. This gives
+
+    sum_i r_i^t <= (L_B+L_C)*K(t) + sum_extra_queries r^t.
+
+The exact extra-query moment cancels its charged cost. It cannot open a
+gap by using a higher rank or by feeding a later native call. Add the
+separate native helper targets and subtract the exact local paid profiles
+to obtain the asserted upper bound on F(t).
+
+This argument also applies to positive mixtures of the functions
+1-exp(-s*r), with a nonnegative linear term. It does not assert the
+whole-caller capacity inequality for every concave function. The bill
+comparison in the next section does hold for every nondecreasing concave
+g with g(0)=0; these are distinct statements.
+
+## 5. Exact complete bills and strict sign
+
+The frozen PR4703 profiles, audited in PR4712, are:
+
+| Type | Paid rank:count | Independent helper rank:count | Net rank mass |
+|---|---|---|---|
+| B | 1:1956, 2:450, 3:403, 4:127, 5:28, 8:8 | 3:2, 4:26, 6:7, 7:47, 8:326 | 1688 |
+| C | 1:2011, 2:445, 3:427, 4:112, 5:12, 7:8 | 3:26, 4:2, 6:6, 7:48, 8:326 | 1680 |
+
+After subtracting the capacity coefficients 1:28, 7:140, 8:84 and
+cancelling equal ranks, the exact delta coefficients are:
+
+| Type | Positive terms | Negative terms |
+|---|---|---|
+| B | 1:1928, 2:450, 3:401, 4:101, 5:28 | 6:7, 7:187, 8:402 |
+| C | 1:1983, 2:445, 3:401, 4:110, 5:12 | 6:6, 7:180, 8:410 |
+
+Remove 8*g(1) from the positive B terms. The remaining positive and
+negative rank masses are equal: 4567 for B and 4576 for C. Every positive
+rank is at most 5; every negative rank is at least 6. For a nondecreasing
+concave g with g(0)=0, the ratio g(r)/r is nonincreasing. Consequently
+
+    Delta_B(g) >= 8*g(1) + 4567*(g(5)/5-g(6)/6),
+    Delta_C(g) >=           4576*(g(5)/5-g(6)/6).
+
+For g(r)=r^t and 0<t<1, the common ratio gap is strictly positive.
+This proves the sign for the full open interval, without sampling
+exponents. In the linear limit t=1, the deltas are 8 and 0.
+
+As a quantitative example only, at C=9/2500=.0036 the exact interval
+checks give Delta_B>31.58147 and Delta_C>23.98603. These are deficits in
+the raw recursive moment per whole call, not kappa values. The checker
+uses integer powers to enclose each r^(-9/2500); decimal arithmetic only
+proposes the endpoints. Any finite native work or outer reserve that was
+granted free can only worsen the already negative bill in this model.
+
+## 6. What changes this conclusion
+
+PR4848 proves a serial ceiling on the same 224 owners and fresh stage
+axes. This proof removes those premises. Merely adding main owners,
+interleaving paid queries, reusing axes, changing port placement, using
+noncoordinate scalar role gauges, or packing ordinary even-power sectors
+does not evade this theorem while its contract holds. The scalar
+rank-distance and integral facts are classical; the exact application to
+the admitted 56/56 B/C endpoint and complete bill is the new result.
+
+The following changes require a new proof and bill: shared or productive
+helper feedback into the main computation; a joint native call with a
+different endpoint or paid profile; noncommuting address maps between
+calls; a target in another address chart; nonlinear operations; or a
+source resource with an odd input gauge whose preparation has a different
+contract. These are boundaries, not claims of feasible profitable callers.
+
+The eight-transverse-axis A interface has rank capacity 1904 in the linear
+limit, rather than 1680. The checker retains this positive scope control.
+No obstruction for the productive eight-axis P104 family follows. A
+query-only caller also attains equality, at any query rank, including ranks
+above eight. The theorem is not a universal lower bound for multiplication.
+
+## 7. Verification and inherited obligations
+
+The written proof quantifies over all callers in the stated model and all
+f>=3. The standard-library verifier authenticates frozen source bytes,
+checks 864 rational pair/orientation/sign cases, enumerates the cut
+polynomials, and checks 5952 character pairs in twelve mixed finite-field
+circuits. It verifies the bill cancellation, exact power intervals and
+four corruption controls. Those finite checks support the algebra; they
+are not substituted for the all-width or all-caller proof.
+
+The full native word, phases, helper ownership and compiler are inherited
+from the cited source and audit. They were not rerun here. No new native
+caller, formal proof-assistant theorem, complete multiplication algorithm,
+selected-frontier change or independent scientific acceptance is claimed.
+
+Credit Dugongue for PR4703 and the typed-provider work in PR5578/5586;
+Cao Yifan/intqwq for PR4712; Chris R/Clear1nsight for PR4848; Cinna Davis
+for PR4674 and the controller lineage; Ali Romman for PR5626's query-rank
+perspective; DaysSky, zxv, Jacob Alan Sussman and all retained source authors.